EFFECTS OF NUMERICAL-COGNITION AND EMOTIONAL- FREEDOM TECHNIQUES ON MATHEMATICS ANXIETY AND ACHIEVEMENT AMONG NON-SCIENCE SECONDARY SCHOOL STUDENTS WITH PSEUDO-DYSCALCULIA IN IBADAN Y BY AR R ADEBUKOLA KABIR TAIWO IB Matric No: 87991 L B.Sc, Psychology, University oNf Ibadan M.Ed, Counselling Psychology (ADsseAssment and Testing), University oAf Ibadan. B F I A Thesis in the D epOartment of Guidance and Counselling Submitted to the FaYculty of Education in Partial Fulfillment of the IT requirement for award of S ER The Degree of IV DOCTOR OF PHILOSOPHY N Of the U UNIVERSITY OF IBADAN, IBADAN 1 SEPTEMBER, 2014 ABSTRACT Anxiety and low achievement in Mathematics are critical challenges facing secondary school students in Nigeria, especially non-science. One of the factors responsible for poor performance in Mathematics is phobia. Many children and young adults develop a fear for Mathematics while they are in school, often as a result of inappropriate methods of teaching or lack of interest on the part of the students. Despite previous studies on Mathematics anxiety (such as systematic desensitization; Verbalizing FearYs and Frustrations Techniques; etc.), the problem of Mathematics anxiety and low achievement still persist. Mathematics anxiety and low achievement are both emotional andR cognitive problems hence; there is need for Numerical Cognition (NCT) and EmotAional Freedom Techniques (EFT). This study, therefore, investigated the effects of NCT and EFT on Mathematics anxiety and achievement among selected secondary schoIoBls. R The pretest-posttest, control group, quasi-experimental design waLs adopted. Simple random sampling technique was used to select 120 participants for the study. The participants were randomly assigned to NCT, EFT and control groups. TheN training was conducted for ten weeks. Mathematics Achievement test (r = 0.90), Mathematics Anxiety scale (α = 0.89), Mathematics Efficacy scale (α = 0.86) and Pseudo-dDyscaAlculia scale (α = 0.93) were used for data collection. Fourteen hypotheses were testedA at 0.05 level of significance and data were analysed using Analysis of Covariance. There were significant main effects of treatImBents on Mathematics anxiety (F(2,109) = 173.020, ŋ2 = 0.760) and achievement F(2,109) F= 4 2.161, ŋ2 = 0.432). The treatments accounted for 83.0% variance in the reduction of Mathematics anxiety of the participants while EFT was more effective (x = 33.8) than NCT (x = 45.4) in reducing students‘ Mathematics anxiety. Also, the treatment accounYted f Oor 78.6% variance in Mathematics achievement of the participants while EFT was also more effective (x = 71.7) than NCT (x = 59.3) in enhancing students‘ Mathematics IaTchievement. There were significant main effects of Mathematics efficacy (F(1,109) = S34.973, ŋ 2 = 0.243) on Mathematics anxiety. There were significant interactive effects of treatments and Mathematics efficacy (F(2,109) = 26.394, ŋ = 0.195) on Mathematics anxiety. Also, there were significant main effects of Mathematics efficacy (F(1,109) = 2E1.00R, ŋ 2 = 0.162) on Mathematics achievement. There were significant interactive effects of treatments and Mathematics efficacy (F(2,109) = 6.116, ŋ 2 = 0.053) on Mathematics achievIemVent of the students. There were 3-way interaction effects of treatments, mathematics effNicacy and gender on Mathematics anxiety (F( 2 2,109) = 7.327, ŋ = 0.063). By implication, these two techniques are important in helping students to have positive thought about UMathematics and learn how to adjust their negative thought and believe in their ability to excel in the subject. Numerical-cognition and emotional-freedom techniques were effective in reducing anxiety and enhancing achievement in Mathematics in both male and female. Based on these findings, it is recommended that these techniques could resolve phobia in Mathematics and improve the students‘ performance in the subject. 2 Key words: Numerical cognition, Emotional-freedom, Mathematics anxiety, Mathematics achievement, Pseudo- dyscalculia Word count: 478 DEDICATION This thesis is dedicated to the ALMIGHTY ALLAH, the Alpha, the Omega, the immortal and invisible, Who gave me wisdom, knowledge, emotional and physical health and strength to withstand the storm and stress of the Ph.D programme. I am forever indebted to Him. Y It is also dedicated to my parents Dr. and Mrs. Titilope Taiwo who are tRhe vehicle through which I got to this world, and who are still alive, praying days and nights for my sustenance in this life. I strongly believe that nobody can be compared to Aboth of you. I am forever thankful. LIB R N AD A F I B O SI TY R VE UN I 3 ACKNOWLEDGEMENT With a grateful heart, I give all thanks to the Almighty ALLAH, the Lord of all worlds and Heavens. He is the bestower of wisdom and knowledge to whom He pleases. He is an uncreated creator, unfashioned fashioner, unmoved mover, the first without beginning and last without an end. He is the unshakeable shaker, unchangeable changer, the Alp ha and Omega. His mercy endures over my life and family members. RY I wish to put on record, because human words and expressions may noAt be sufficient enough, my indebtedness and profound gratitude to my indefatigable,R young and ever- dynamic and erudite scholar, supervisor, Prof. Oyesoji Aremu (ICBF, JP). Prof. Oyesoji Aremu is a man with fear of God – the more reason why He Lhas been with him all this while. He is a man endowed with ocean of knowledgeA, anN enigma of hope and of diverse sorts, a man of limitless access and a researcher pDar excellence, a model in academia, an ebullient teacher whose academic prowess hAas propelled students, like me, to greater heights. I am forever grateful to you Bfor your mentoring, thorough supervision, constructive suggestions, invaluable con trIibutions and unparallel and prompt attentions. I have learned so much under yoOur tFutelage and diligent supervision. I have gained so much confidence in this thesis and the wisdom gained would be carried throughout my entire life. Infact you arTe a Ygreat and rare gem worth emulating because I have witnessed so many situations wheIre so many people have celebrated you. I appreciate theR effSorts of the greatest professor in the Department, the academic father of professors Eand doctors in the Department, Prof. Charles Uwakwe. He is my mentor and academIVic grandfather. He encouraged and directed me to the end of tunnel. I would never foNrget the roles he has played in my life to make me who I am today. My good God will Uforever reward you for the good deeds you have shown to me in life. I also wish to specially appreciate another erudite and great scholar, who has in many occasions, shaken the Department and Faculty positively with his ocean of knowledge and wisdom, Prof. David Adeyemo. He is a man who stood by me when the road was 4 rough, he encouraged and lifted me up when I was pessimistic in the course of the study. He showed me the direction when I was completely lost. He, despite his tight schedule, gave me prompt attention, direction and guidelines, most especially on my abstract. He corrected my abstract so many times before it became a whole. The next appreciation goes to other lecturers of the Department: Prof. Ajibola Falaye, Prof. Jonathan Osiki, Prof. Salami, Dr. Ayo Hammed, Dr. Awoyemi, Dr. Animasha un, Dr. Asuzu, Dr. Oluwole, Dr. Jimoh, Dr. Oparah, Dr. Ogundokun, Dr. OwoRdunYni, Dr. Busari, Dr. Adeyemi, Dr. Ofole, Dr. Fehintola and Miss Alade. A I also want to appreciate the efforts of non-academic staff of the RDepartment, with emphasis on Mrs Adeyemi (Mama), who has stood by me fromI Bthe beginning of the study to the end. Mama, saw me through thick and thin; IN am f Lorever grateful to you ma. Others include Mrs Faloye, a good friend to all of us and a dear mother indeed; Mrs Owolaju, who has left the Department to another uniAt; Mrs Akinola, who has also been there for me; Mrs Oladiipo, my dear friend, Mr ODlasehinde and lastly, my wonderful and greatest friend of them, Mr. Atubu UcheB, wAho has ever stood by me, making sure I always do the right thing at the right Ftim e.I I say thank you all. I want to specially appreciate myO academic fathers in the Faculty in persons of Prof. M.K. Akinsola and Prof. M.S. EnYiol a. Both of them have stood by me all the time through the course of my study. PrIoTf. Akinsola has really touched my heart. I can never forget those positive statementsS you always say to me, making sure I do the right thing. He has a slogan that ―finRe boy is not needed to me, what I want is your credence and hard work on your Ph.D‖E; thank you very much, good father. Likewise, Prof. M.S. Eniola has changed my lifIe Vpositively; it was indeed a very beautiful intervention when things were rough. ThNank you very much. UI cannot forget to mention the efforts of other erudite scholars in the Faculty, the like of Prof. Kolawole, former Dean of the Faculty and Prof. Moronkola, the present Dean of the Faculty. I am very grateful to you sirs. I also acknowledge and appreciate the efforts of Dr. Popoola and Dr. Okilagwe, both from the Department of Library, Archival and 5 Information Studies. I also need to mention my senior colleagues from Adult Education, in persons of Dr. Kester and Dr. Olumide who have done wonderfully well in my Ph.D work. I cannot also forget my senior colleague, a brother and friend, Prof. Adegbesan (Skira) from Department of Human Kinetics and Health Education. I have so many others in the Faculty that time may not permit me to mention them: Prof. Abass, Dr. Akinwumi, H.O.D, Educational Management, Dr. Ben Emunemu, Prof. Ajiboye, ASUU Chairm an and so many others. I appreciate you all. RY I would be an ingrate if I didn‘t have a paragraph for an erudite scholar andA my mentor in the Department, Dr. Teslim Ayobami Hammed. I speciallRy reserved the acknowledgement till the end because of his wonderful impact heI Bhas shown to my life. He has touched my life and family positively. God has used himL to make me a viable and worthy person in the society. I pray to the Almighty AllahN tha t whatever that breathes and breathes not, in your property, will never perish. JazaaAkum llahu khaeran. Lastly, I wish to put on record, the wonderful cDontributions of my darling wife, Mrs Titilope Taiwo. I strongly believe that sucIcBess AI have recorded now and always, is simply because I have a good wife like youF. W ithout mincing words, you are the greatest and wonderful wife I have ever seeOn on earth. Your patience, endurance and perseverance that God has endowed you with , is incomparable. You are a wonderful woman who loves a husband to the deepesTt heYart and soul. You have shown to me what another woman has never shown to her huIsband in lifetime. Your enduring pattern of trait has survived me with good inteRrperSsonal relationship, as a husband, making me to see you beyond any other womaEn on earth. Your contributions to my life are peculiar and cannot be repaid with goVld and silver. The great God of heaven and earth will reward you abundantly. You wiNll liIve long to reap the fruit of your labour. I am forever proud of you. You have ever Ugiven me wonderful and lovable seeds on earth in person of Aishat Oyindamola, Azeezat Olabisi and Habeebllah Omogoriola. I can never forget to mention a damsel, who is a gift to the family, Olawunmi Judith Aliyah. Your presence means a lot to me and the family. You will never regret a moment with the family and we will never regret a moment with you, mon sha Allah. I love you all. 6 Taiwo A.K. Y AR LIB R N AD A F I B O SI TY R VE UN I 7 CERTIFICATION I certify that this work was carried out by Adebukola Kabir TAIWO (Matric: 87991) in the Department of Guidance and Counselling, Faculty of Education, University of Ibadan. Y AR BR LI AN BA D I F ……………………… …O………………………………………………. TY Supervisor SI AMOS OYESOJI AREMU (CF, JP) R B.Ed, M.Ed, Ph.D, (Ibadan) E Professor of Counselling and Criminal Justice, IV Department of Guidance and Counselling, N Faculty of Education, University of Ibadan U 8 TABLE OF CONTENT Page Title Page i Abstract ii Dedication iYii Acknowledgment ARiv Certification R vii Table of Content LI B viii CHAPTER ONE Background to the study AN 1 Statement of the problem D 10 Purpose of the study B A 12 Significance of the study F I 12 Scope of the study O 13 Operational Definition of TYerm s 13 CHAPTER TWO IT 2.1 TheoreticalS Background 16 2.1.1 ConEcepRt of Mathematics Anxiety 16 2.1.2 IVConcept of Mathematics Achievement 22 U2.1N.3 Concept of Mathematics Efficacy 26 2.1.4 Concept of Pseudo-dyscalculia 30 2.1.5 Classification of Dyscalculia 31 2.2 Theoretical Framework 43 2.2.1 Information processing theory 43 9 2.2.2 Constructivism theory of cognition 53 2.2.3 Chain reaction or cycle theory 65 2.2.4 Social cognitive theory 65 2.3 Empirical Background 75 2.3.1 Numerical cognition and Mathematics anxiety 75 2.3.2 Numerical cognition and Mathematics achievement R8Y0 2.3.3 Emotional freedom technique and Mathematics anxiety A 87 2.3.4 Emotional freedom technique and Mathematics achievement R 97 2.3.5 Mathematics efficacy and Mathematics anxiety LI B 98 2.3.6 Mathematics efficacy and Mathematics achievemeNnt 99 2.3.7 Gender and Mathematics anxiety A 102 2.3.8 Gender and Mathematics achievement D 103 2.4 Conceptual Model AIB 104 2.4.1 Explanation of conceptual moFdel 105 2.5 Hypotheses O 107 CHAPTER THREE Y METHODOLOGY IT 3.1 ResearcRh DSesign 109 3.2 PopEulation 109 3.3 IVSample and sampling technique 110 U3.4N Eligibility for Participation 111 3.5 Instrumentation 111 3.6 Procedure for the experiment 114 3.7 Objectives of the therapeutic packages 114 3.8 Numerical cognition outline 115 10 3.9 Numerical cognition group 115 3.10 Emotional freedom outline 123 3.11 Emotional freedom technique group 124 3.12 The control group 136 3.13 Data Analysis 136 CHAPTER FOUR Y RESULTS AR138 CHAPTER FIVE 5.1 Discussion BR 154 5.2 Limitations of the study LI 164 5.3 Implication of the study N 165 5.4 Conclusion A 167 5.5 Recommendations AD 167 5.6 Contribution to knowledge IB 168 5.7 Suggestions for further studieFs 169 REFERENCES O 171 Appendix Y 199 I T RS IV E UN 11 LIST OF TABLES Page Table 1.1 Performance in Mathematics at WASSCE from 2001 – 2005 2 Table 2.1 Classification of Calculia by CTLM (1986) 32 Table 4.1 3X2X2 Analysis of Covariance summary table on the treatment 138 Table 4.2 Multiple Classification Analysis on post test Mean Score of RY Mathematics anxiety A 139 Table 4.3 Bonferonni Post hoc Test showing the nature of differencRe in students Mathematics anxiety LI B 141 Table 4.4 Bonferonni Post-Hoc Test showing the natuNre o f difference in Mathematics anxiety with respect to theA interaction between intervention and Mathematics efficaDcy 143 Table 4.5 Bonferonni Post-Hoc Test IshBow Aing the nature of difference on Students‘ MathematicFs an xiety with respect to the interaction Between treatme nOt Mathematics efficacy and gender 145 Table 4.6 3X2X2 AnalYysis of Covariance summary table on Mathematics AchieveImTent 147 Table 4.7 RMulStiple Classification Analysis on Post test Mean score 148 Table 4.8 EBonferonni Post-Hoc Test showing the nature of difference in IV Mathematics achievement 149 UTaNble 4.9 Bonferonni Post-Hoc Test showing the nature of difference in students Mathematics achievement with respect to the interaction between intervention and mathematics efficacy 151 12 CHAPTER ONE INTRODUCTION 1.1 Background to the study The place of Mathematics in the life of any nation is one which is inextricably linked with the place of development in that nation (Okereke, 2002). Indeed no nat ion that wants to develop scientifically and technologically neglects the MathemYatical component of her school curriculum. The increasing attention given to MathemRatics stem from the fact that without Mathematics there is no science, without scienAce there is no modern technology, and without modern technology, there is no modRern society. This therefore suggests that there could be no real technological dIeBvelopment without a corresponding development in Mathematics both as conceiv edLN and as practiced (Salau, 2002 in Anaduaka, 2008). According to Soyemi (2001 in Anaduaka,D 200A8), Mathematics simply put, is the science of structure, order, numbers, space and quantity. It is a relationship which revolves around the elementary practice ofB couAnting, measuring and describing of shapes and objects. To him, it is a way ofF lif e Iand an all embracing body of knowledge that opens up the mind to logical Oreasoning, analytical thinking and the ability to make abstract objects look real or concrete. From social or economic perspective, he saw Mathematics as a key eTlemYent and activity in day to day living that every human being practices in one form oIr the other. The table below represents the trend of performance of students in MatheSmatics in the last five years in six states. The result showed that performancEe ofR students in Mathematics is getting worse and worse. IV UN 13 Table 1.1 Performance in Mathematics at WASSCE from 2001 – 2005 from six states in Nigeria States 2001 2002 2003 2004 2005 % % % % % Lagos 25.5 25.6 30.0 30.2 34.2 Oyo 15.5 14.8 15.6 22.0 22.6 Bauchi 24.5 25.0 46.7 37.0 33.9 Y Akwa-Ibom 19.1 24.2 14.1 30.4 26.1 AR Kano 20.7 16.5 18.8 14.9 17.0 R Nassarawa 4.5 4.9 5.1 5.8 4.7 IB Total L Source: West African Examination Council. N Despite the relative importance of MathemDatiAcs, it is very disappointing to note that students' achievement in the subject hAas remained consistently poor. Statistics abound to show that mass failure in MIatBhematics especially in the Senior Secondary Certificate Examination (SSCE) is rFeal and that the trend of students' performance has been on the decline (Agwagah, 2001; Arnazigo, 2000; Betiku, 2002; Salau 2002; WAEC, 1990, 2000, 2004, 2006, NECO ,O 2010). A lot of researchTes Yhave as a matter of fact been carried out to ascertain the root causes of this andS to Iproffer solutions. Consequently, research efforts geared towards finding reasonsR and possible solutions for the problems have not yielded much positive effect, as thEe result analysis for the 2006, 2009 SSCE and NECO 2010 revealed that only fifteen percent of the candidates qualified for university admission with credits in five suNbjecIts V which includes English and Mathematics (WAEC, 2006, 2009; NECO 2010). UThe poor states in which science and Mathematics are taught in some Nigerian schools have been revealed by many research findings (Betiku, 2002; Oyedele, Eule & Langkuk, 2002). Harbor-Peters (2001) identified several factors as responsible for the poor achievement of students in Mathematics. According to her, some of these factors emanate from sources which are psychological, physiological and environmental. 14 Byrd in Aprebo (2002) asserted that poor achievement in Mathematics emanated from anxiety and fear. Mathematics phobia, he said, has been an academic disease whose symptoms are always expressed on the faces of the learners in the Mathematics classroom. According to Harbor-Peters (2001), it is unfortunate that students who regard Mathematics as hard are compelled to study one form of Mathematics or the other. This is because Mathematics pervades all forms of learning (Harbor-Peters, 2001). Mathematics anxiety is one of the most serious limitations to education.Y Many children and young adults develop a fear for Mathematics while they are Rin school. Mathematics anxiety and low self-efficacy affect many individuals throuAgh feelings of tension, apprehension, or fear that interfere with the manipulation ofR numbers and the solving of Mathematics problems in a wide variety of ordinaIrBy life and academic situations (Ashcraft, 2002). Mathematics is considered as one oLf the toughest subjects by majority of students. There are very few students in theN classroom who really love to learn and explore Mathematical concepts. MathemAatics problems among students especially in Nigeria are very real. It has beAen nDoted that anxiety in Mathematics can cause one to forget and lose one‘s selfB-confidence. Students may acquire it in the classroom from lack of understanding anId self-doubt, which often results in avoidance strategies that further exacerbate theF problem (Ashcraft, 2002). Mathematics anxiety can be viewed as a perpetual cycle oOf knowledge gaps and lack of confidence. It begins when teachers fail to engage TtheiYr students in meaningful and memorable lessons (Jackson & Leffingwell, 1999). StIudents subsequently lay an incomplete foundation for Mathematics knowledge, wRhichS becomes increasingly unstable as new ideas build upon faulty concepts. TEhe eventual collapse of this fragile structure leads to avoidance and anxiety relatedI Vto Mathematics (Turner et al., 2002). In some, this disabling condition could be caNused by teachers assigned to teach Mathematics. When some of these students re-enter Uthe elementary education system as teachers, they restart the treacherous cycle that robbed them of an invaluable knowledge of Mathematics and the balanced education they deserved. Bamidele (2005) stated that in Nigerian schools, students‘ general impression is that Mathematics is a dreadful subject. But ironically, this subject is the basis for 15 scientific and technological advancement of any country. Mathematics anxiety is an intense emotional feeling of anxiety that people have about their ability to understand and do Mathematics. People who suffer from Mathematics anxiety feel that they are incapable of doing activities and participating effectively in classes that involve Mathematics. Some Mathematics anxious people even have a fear of Mathematics called pseudo-dyscalculia, which is described as false belief in Mathematics disability caus ed by lack of, inconsistent, poor, or inappropriate systematic Mathematics instrYuction; inattention, fear, anxiety, or emotion. R Many students often choose their courses in the universities on theA basis of how little Mathematics is required for the degree. Some students may IeBven R experience worse problems when they find out that their alternative degree thLey put in for have some courses that require Mathematics orientation like StatisticNs. B y this, some students find it very difficult to cope with Statistics, some try to reAadjust, while others resign to fate. Mathematics anxiety is an emotional, rather thDan intellectual problem because the problem emanates from inconsistent emotion and therefore interferes with a person's ability to learn Mathematics which later resBultsA in an intellectual problem. Mathematics anxiety can Fbe Ia disabling condition, causing humiliation, resentment, and even panic. StuOdents who experience Mathematics problems have their mind go completely blank and feel they cannot do it. Most of the time they cannot even remember how to solveT theY simplest calculations. Some students often believe that there is always one right anIswer to Mathematics, and if you cannot find it, you have failed. Mathematics eRxamSinations often terrify students. At the time of examination, some students exEperience sweaty palm, breathe too fast, and often they cannot make their eyes focuseIdV on the paper. It is even worse if they look around, because they would see evNerybody else working, and believe that they are the only one who cannot do it. UMathematics anxiety can be extreme; often caused by having a negative attitude due to a previous bad experience. A student who has experienced frequent failure in Mathematics is unlikely to be motivated to improve his or her performance. It is probable that he or she attributes the failure to bad luck, or difficulty of the subject. The student is likely to believe that increased effort and persistence will not make any difference in the outcome, 16 that he or she has no control over success or failure in Mathematics; and to develop a stance of helplessness and passivity (Corral, 1997). Anxiety reactions to Mathematics situations may contribute to failure in Mathematics. In fact, a person who has high Mathematics anxiety may actually be unable to perform well in test, and may be unable to learn in a Mathematics classroom. Mathematics anxiety also directly contributes to avoiding Mathematics. It is commo nly accepted that Mathematics is difficult, obscure, and of interest only to certain peopYle, i.e., the geniuses. The consequence in many English-speaking countries, and espRecially in Nigeria, is that the study of Mathematics carries with it a stigma, and peAople who are talented at Mathematics or profess enjoyment of it are often treated as thRough they are not quite normal. Mathematics anxiety has been related to teachers andI Bthe classroom setting. Mathematics anxious teachers can result in Mathematics a nxLious students (Bamidele, 2005). Mathematics anxious children often show signs oNf nervousness when the teacher comes near, freezing and stopping work or covering it Aup to hide it (Barnes 1984). Self-efficacy is the judgments individualDs make about their potential to learn successfully and the belief in their own capBabiAlities. The choices people make; the efforts they put forth, and how long they persis t Iare influenced by self-efficacy (Bandura, 1997; Schunk, 1996). According to BanduFra, every individual possesses a belief system that exerts control over his/her thOoughts, emotions and actions. Among the various mechanisms of human agYency, none is more central or pervasive than self-efficacy beliefs (Bandura & LoIckTe, 2003; Pajares, 2000). ExpectaRtionSs about doing well in Mathematics (confidence) relates closely to one's beliefs about personal capabilities for successfully performing domain-specific tasks (IsVelf- Eefficacy). Students with higher levels of self-efficacy set higher goals, apply moNre effort, persist longer in the face of difficulty and are more likely to use self-Uregulated learning strategies (Wolters & Rosenthal, 2000). Students make judgments about their Mathematics capabilities based on accumulating knowledge and experience. They tend to see themselves as either mathematically inclined or disinclined. These perceptions of Mathematics efficacy are shaped by an unlimited array of personal, environmental, and behavioural factors. In the academic milieu, learners make judgments 17 about their capabilities based on comparisons of performance with peers (Brown & Inouye, 1978; Schunk, 1987; Schunk & Hanson, 1985; Schunk, Hanson, & Cox, 1987), successful and unsuccessful outcomes on standardized and authentic measures, and feedback from others such as teachers (Bouffard-Bouchard, 1989; Schunk & Rice, 1987), parents, and peers. These sources of information about their capabilities accumulate within individuals to form perceptions of Mathematics competencies. But th ese judgments are fluid in that they are altered along the way according to new experYiences and knowledge. Students whose perceptions of their capabilities are high ofteRn go on to challenge themselves, persevere in the face of difficulties, and expend Agreater effort resulting in more successful experiences. Self-doubters on the oIthBer Rhand often resign early in the face of difficulty (Bandura, 1986; Brown & InouLye, 1978). If students are able to perform a task successfully, then their self-efficacy c an be raised. In contrast, if students are not able to perform a task, then they may beNlieve that they do not have the skills to do the task which, in turn, lowers their selAf-efficacy. Personal goal-setting is influenced by their self-appraisal of capabilities. TDhe stronger the perceived self-efficacy, the higher the goals or challenges peoplBe seAt for themselves and the firmer is their commitment to them (Bandura, 1991). I Competent functioning inO MaFthematics requires self-beliefs of efficacy to perform effectively (Bandura, 1986). Pe ople tend to avoid tasks and situations they believe exceed their capabilities, but TtheYy undertake and perform assuredly activities they judge themselves capable ofI handling (Bandura, 1997). Pajares and Miller (1994) asserted that efficacy in proRbleSm-solving had a causal effect on students‘ performance. Research findings support the view that high achieving Mathematics students have higher and more accuraIteV ef Eficacy beliefs (Pajares & Kranzler, 1995; Bandura, 1997; Pajares & Miller, 19N94). Efficacy beliefs towards a certain task are accurate when they correspond to what Uthe person can actually accomplish. The term cognition refers to a faculty for the processing of information, applying knowledge, and changing preferences. It is also termed as the mental functions, mental processes (thoughts) and states of intelligent entities. It focuses on the specific mental 18 processes such as comprehension, inference, decision-making, planning and learning (Lycan, 1999). Numerical Cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and Mathematics. It deals with how students acquire an understanding of numbers, and how much is inborn. It also has to do with how humans associate linguistic symbols with numerical quantities. T his brings knowledge of how these capacities underlie our ability to perform coYmplex calculations. In each case, it throws light into the neural bases of these abilitiRes, both in humans and non-humans, the metaphorical capacities and processes thaAt allow us to extend our numerical understanding into complex domains such aRs the concept of infinity; the infinitesimal or the concept of the limit in the calculusI. B Numerical Cognition perspective emphasizes that par tiLcipants can become stuck by focusing on their past and current "bad" behavior aNnd failures versus focusing on future solutions. This therapy will try to increaDse Astudent performance by removing obstacles to student learning. Students accompAlish more when they concentrate on their successes and strengths rather than thIeiBr failures and deficits. There are so many advantages for students who know hoFw to constructively solve problems. Students should be looked at as being good andO capable of rational thought but without any influence from teachers or significant ad ults a student will likely focus more on their own negative side. Y Once the therapIiTst or researcher begins to shift to the positives of the good things that are going on iSn a student's life, the students usually will switch to that, open up and talk about it toRo. Students do have the capacity to act on common sense if given the opportIuVnity E to identify common sense problem-solving strategies. Solution-focused prNoblem solving in numerical cognition is based on the theory that small changes in Ubehavior lead to bigger changes in behavior. The therapy would emphasize a role shift for students. Small shifts in role by a student will cause shifts in other places. In this regards, teachers will also be focused to develop an alliance with the student and work together to determine the problem and the cause. Identify the student's strength, and then they can build strengths and foundations which will lead to positive changes. When the plan does 19 not seem to be working and the student seems to be repeating the same pattern or does not have the ability to control compulsive behaviors then the educator has to watch for a pattern and reinforce with positive. This therapy pursues the positive and students are more likely to find a solution to a problem when they concentrate on their successes rather than their failures. Students must realize that they play a huge part in the success of their problem solving process a nd that change will occur. Once the changes begin to happen then the student will Yrealize that their lives can be very different. Then it is time to have the students setR goals and then monitor their progress. The therapist will then try to use comprehensioAn strategies to translate the linguistic and numerical information in the problem andR come up with a solution. For example, the therapist may read the problem moreL thaInB once and may reread parts of the problem as they progress and think through the pro blem. Emotional Freedom Techniques (EFT) is a mNeridian based intervention, a psychotherapeutic tool that is claimed to be DableA to relieve many psychological conditions, including anxiety, low achieveAment and other psychopathology like depression, post traumatic stress disordIerBs, stress, addictions and phobias (Feinstein, 2005). The basic EFT technique inFvol ves holding a disturbing memory or emotion in mental focus and simultaneouslOy using the fingers to tap on a series of twelve specific points on the body (Rowe, 200 4). The theory behind EFT is that negative emotions are caused by disturbances Tin tYhe body's energy field (Swingle, 2000). Human thoughts are constantly creating Ipatterns of electrical energy that cause the release of neurotransmitters, Shormones and other chemicals in the body that people feel as emotions. EStudRent who have Mathematics disabilities have negative emotions which are unhelpIfVul thoughts and beliefs, and are significant factors in the development of deNpression, anxiety, anger, low self-esteem, self-defeating behaviours, difficulty with Ucoping, negative emotion and lack of Mathematics efficacy. When there is a disruption in the body's electrical flow, such as the fight or flight response, humans feel it. If the disruption continues, it can lead to emotional distress and eventually physical problems. When the disruption is removed, the distress stops. 20 Emotional Freedom Techniques (EFT) work for the immediate and permanent elimination of various phobias. Many EFT practitioners have cured all sorts of anxiety and phobias of people who have battled with it for years or all their lives (Perkins & Rouanzoin, 2002). Clinical examination of EFT has proved to solve problems on limiting beliefs about performance, anxieties in general. The client is then asked to think of his/her problem while a desensitization procedure is followed, involving tapping on the body (the client tapping on his or her own body). The tapping appears to disruYpt the previous patterning of cognitive-emotional response, inducing a dissipation oRf distress; the tapping is accompanied by a statement of self-acceptance in relaRtionA to the target problem (which reduces a common tendency to resist the desensitisation). Tapping may, at certain points in the process, be accompanied by eye mLovIemBents, humming and counting, and the tapping is continued until subjective distNress is eliminated. EFT is usually self administered and always eaAsy to learn and can be used to treat any emotional problem ranging from mild to severe, and from short term to chronic. It also helps alleviate chronic physical pain (CalDlahan, 2001). EFT stimulates certain pressure points in the body. This has the eBffecAt of redirecting the body's basic energies. To use EFT tapping therapy, begin by t hIinking of the feeling you want to treat. Perhaps one feels generally stressed, upset, aFnxious (for example Mathematics) or something at the back of the mind that one f eOels negatively about or situation is bothering one, simply focus on the problem (AndrYade & Feinstein, 2003) The practitioneIr Tclosely monitors the client‘s progress from moment to moment, by careful obseRrvatSion and by asking the client to provide ratings of the Subjective Units of Disturbance (SUDs). This feedback is used to guide the process (Ruden, 2005). The methoIdV ma Ey be used by skilled psychological therapists who are able to track the client‘s prNogress through the layers of anxieties, dysfunctional cognitions, and traumatic Umemories. It may also be readily employed by the client as a simple stress-relief and affect-regulation tool. The method does not require the client to relive emotional trauma – nor does it require him or her to talk in detail about the experience (Wells, Polglase, Andrews, Carrington & Baker, 2003). 21 EFT may readily be combined with other psychological methods, including other cognitive-behavioural strategies. In clinical practice the actual tapping procedure is likely to be embedded within much more activity of a conventional verbal cognitive or psychoanalytic (or other) nature. Through the ordinary discourse of psychotherapy, the practitioner will identify the affective, cognitive, and psychodynamic areas to target with EFT (Wells, 2000). The EFT in solving Mathematics anxiety and phobias involve redirecting tYhe old thought patterns or response mechanisms of anxious or phobic people to the Rsubject or teacher they have a phobia or anxiety about and basically creating a newA set of more useful patterns or mechanisms of behaviour to replace the old phobic reRsponse. There has to be acceptance that despite the phobic condition, the person is loIvBed or he or she loves himself or herself. Eventually, the fear or fright will be repl acLed by curiosity and there will be lesser resistance and anxiety (Swingle, 2000). N Reduction of Mathematics anxiety and enhAancing Mathematics achievement through Numerical-Cognition and Emotional FreDedom Techniques appear to be scarce. This study would therefore expand the froBntieArs of knowledge on Numerical Cognition and Emotional Freedom Techniques in Ireducing Mathematics anxiety and enhancing Mathematics achievement among Fnon-science students with fear of Mathematics. Participants in this study wo uOld be trained with Numerical Cognition Strategy and Emotional Freedom TechYniques to reduce anxiety in Mathematics and enhance Mathematics achievemIeTnt. It is believed that when students are trained to reduce their anxiety in the RMatShematics, their achievement will be more enhanced, thereby helping students to Eacknowledge the fact that their problems in the subject have to do with their cognitIioVn and negative emotion and therefore will be prepared to restructure it and build confidence in them. U N1.2 Statement of the problem Anxiety and low achievement in Mathematics pose some serious limitations on non-science students in Nigeria. This is because, more than any subjects offered at the secondary school level, Mathematics seems to be the most dreaded especially for non- 22 science students. It can be thought of as either an aversion or a fear of working with numbers or equations. Many children and young adults develop a fear for Mathematics while they are in school. This often is a result of inappropriate methods of teaching or lack of interest on the part of the students. This usually makes the students to avoid all Mathematics related subjects such as Statistics at higher levels. This could result in many problems such as avoidance, negative emotions, anxiety and low achievement on the subject. A lot of students in Nigeria have been deprived of certain professionYal and personal opportunities when they become graduates simply because they fear oRr perform poorly in Mathematics. These negative experiences could remain throRughAout their adult lives. The fear of, or low achievement in Mathematics is often associated with pain and frustration. For example, some people get frustrated when thLeyI Bhear that the type of career they intend to do or are doing requires some Ma thematics applications like banking, accounting, auditing and so on. Some peopleA evNen find it difficult to play some games that require Mathematical concepts because of their low state of mind in the subject. This leads to questions like: what actualDly causes anxiety in Mathematics and low achievement in the subject? How doB weA solve the problem? What techniques are most appropriate in treating the problem ?I What suggestions do we offer? Various Psychological OTreFatments have been used in the treatment of Mathematics anxiety in the p ast. Such treatments include systematic desensitization (Hembree, 1990); RelaTxatYion Training and Stress Management (Schneider & Nevid, 1993) others include PIersonal Interviews Technique; Verbalizing Fears and Frustrations Techniques; TRransSactional Analysis Model; Anchoring Technique; Journal Writing Technique;E Peer Tutoring; Comprehensive Teaching-Therapeutic Programme. However, the abIovVe studies and other studies have not been able to deal extensively with Numerical CoNgnition and Emotional Freedom Techniques on how they can be used to solve UMathematical problems Although Numerical Cognition Technique has been used by researchers to treat Mathematics Anxiety in the past, yet most of these studies only see Mathematics problems among students as intellectual rather than emotional problems. Therefore, the intellectual problem in Mathematics is a result of negative emotion towards the subject, 23 probably from the teachers that take the subject or lack of motivation by the significant others. In this case, the present study has combined Emotional Freedom Technique together with Numerical Cognition in order to also deal with emotional problems or phobias that students have for Mathematics. 1.3 Purpose of the Study The general purpose of this study is to reduce anxiety in MathematicYs and enhance Mathematics achievement among non-science students with pseudo-dRyscalculia through Numerical Cognition and Emotional Freedom Techniques. It isA believed that these two techniques would serve as an impetus to higher educatioRnal attainment in Nigeria. The specific objectives are to: IB i. assess main effect of numerical cognition and emot ioLnal freedom techniques in reducing anxiety and enhancing achievemenNt in Mathematics among the students. A ii. explore the main and interactive efAfecDt of gender and Mathematics efficacy (moderating variables) on MatheBmatics anxiety of the participants. iii. find out the main and intFerac tIive effect of gender and Mathematics efficacy (moderating variables)O on Mathematics achievement of the participants. 1.4 Significance of tThe YStudy The study serIved as contributions to knowledge in the field of Tests and Measurement, REduScational Psychology and other related specializations. The outcome of this study Eshed light on the efficacy of the two techniques in reducing Mathematics anxietIy Vand enhancing Mathematics achieverment. In the area of Clinical Psychology, the effNicacy of Emotional Freedom Technique was revealed in dealing with all emotional Urelated problems. In the areas of Educational Psychology, the study would alleviate the fear students have for Mathematics and improve their academic life. The study would serve as an eye-opener to researchers who would like to carry out researches on the techniques to treat various psychological issues. The present study would also serve as credible reference tool to psychometricians, clinical psychologists, educational 24 psychologists, teachers, school counselors and other related experts in managing psychopathologies and other students‘ academic problems especially those who have phobia for a particular subject. Students of Mathematics through the findings of this study would begin to have a better understanding of themselves and their capabilities and see that they can all learn Mathematics through whatever their strength intelligence or learning styles. This wo uld help to build their self confidence and get them always prepared for meaningful leYarning in the subject. The results of this study would make clear to teachers the Rfact that a situation where students are left as passive listeners in the classroom tenAd to kill their interest and enthusiasm and so hinder learning. Professional bodiesI,B sch Rool administrators and other interest groups in the educational sector would through the findings of the study, be able to organize more worthwhile seminars an d Lworkshops for students, teachers and school administrators that would yield greateNr result in alleviating anxiety of Mathematics. This study would serve as an empirDicalA basis for future research reference and citations as there are paucity of research evAidence on the use of Numerical Cognition and Emotional Freedom Techniques in rIedBucing anxiety in Mathematics and enhancing Mathematics achievement. F 1.5 Scope of the study O The study investTigaYted the effects of numerical-cognition and emotional freedom techniques on MatheImatics anxiety and achievement among non-science secondary school studentRs wSith pseudo-dyscalculia in Ibadan. The scope of the study was non-science (stEudents who are offering arts subjects) secondary school students in three selecteIdV secondary schools in Ibadan metropolis. SSI students were observed because theNy are just transiting from junior secondary schools. U 1.6 Operational Definition of Terms The following terms are defined as they will be used in the study: Numerical Cognition Technique: Numerical Cognition, is described, within the context of this study, as a technique that is used to treat various Mathematics problems arising 25 from cognitive, developmental and neural bases of numbers and Mathematics that are present within individuals. Emotional Freedom Technique: Emotional Freedom Technique, within the context of this study, is a Emotional-freedom-based intervention that involves tapping some parts of the body in order to energize emotions to be able to correct emotional problems in Mathematics. Mathematics Anxiety: Mathematics anxiety can be viewed in this study as feYar that students have for Mathematics that further leads to lack of self confidence, pooRr problem solving and avoidance of the subject. A Mathematics Efficacy: This can be described as a variablesI uBsed R to moderate the relationship between the techniques (Numerical Cognition and Emotional Freedom) and criterion variables (Mathematics anxiety and achievement). ItL is described as students‘ confidence and positive sense of judgment in theAir aNbilities to solve mathematical problems. Mathematics Achievement: This can be describDed, within the context of this study, as developed and validated Mathematics acBhievAement test or stimulus presented to the student in order to test their performaFnce iIn the subject. Non-Science students: This can be defined as secondary school students who offer arts/humanities subjects. O Pseudo-dyscalculia – TThiYs is described as false belief in Mathematics disability by students, caused by laIck of, inconsistent, poor, or inappropriate systematic Mathematics instruction; inaRttenStion, fear, anxiety, or emotion. VE NIU 26 CHAPTER TWO LITERATURE REVIEW Introduction Designing and identifying sensible intervention strategies and practical ways to alter self-efficacy beliefs and anxiety when they are inaccurate and debilitating to students has been suggested as an important and viable avenue of future resea rch (Bandura, 1997). In this research work, various theories and literature will be revYiewed on numerical cognition and emotional freedom. Empirical review or related sRtudies will be sought to see how these two techniques influence MathematicsA anxiety and Mathematics achievement of students. The subsections of thIiBs c Rhapter is further highlighted below: L Theoretical Background N  Concept of Mathematics Anxiety A  Concept of Mathematics Achievement AD  Concept of Mathematics efficacy B  Concept of pseudo-dyscalculia I  Classification of dyscalcuOlia F Concept of Numerical C ognition  Concept of EmIoTtionYal Freedom Theoretical framework  InformaRtionS processing theory  ConEstructivism theory  IVSocial cognitive theory EmNpirical Findings U  Numerical cognition and Mathematics Anxiety  Numerical cognition and Mathematics Achievement  Emotional freedom techniques and Mathematics Anxiety  Emotional freedom techniques and Mathematics Achievement  Mathematics efficacy and Mathematics Anxiety 27  Mathematics efficacy and Mathematics Achievement  Gender and Mathematics Anxiety  Gender and Mathematics Achievement 2.1 THEORETICAL BACKGROUND 2.1.1 Concept of Mathematics Anxiety According to Fiore (1999), Tobias and Weissbrod (1980) Mathematics anxiety is defined as the panic, helplessness, paralysis, and mental disorganization that Yarises among some people when they are required to solve a Mathematics problem. ItR is both an emotional and cognitive dread of Mathematics. While some measure of aAnxiety can be motivating or even exciting, too much anxiety can cause ―downsBhiftRing‖ in which the brain‘s normal processing mechanisms begin to change by nIarrowing perceptions, inhibiting short term memory and behaving in more prima l Lreactions (McKee, 2002). Pries & Biggs (2001) describe a cycle of MathemaAticsN avoidance: In phase one, the person experiences negative reactions to MathemDatics situations. These may result from past negative experiences with Mathematics,A and lead to a second phase in which a person avoids Mathematics situations. TBhis avoidance leads to phase three, poor Mathematics preparation, which Fbrin gIs them to phase four, poor Mathematics performance. This generates more negative experiences with Mathematics and brings us back to phase one. This cycle cOan repeat so often that the Mathematics anxious person becomes convinced theTy cYannot do Mathematics and the cycle is rarely broken. Arem (2003) equates a lSot oIf Mathematics anxiety with Mathematics test anxiety, which she asserted is threRe-fold: Poor test preparation, poor test-taking strategies and psychological pressures. IVRes Eearch confirms that pressure of timed tests and risk of public embarrassment haNve long been recognized as sources of unproductive tension among many students. UThree practices that are a regular part of the traditional Mathematics classroom and cause great anxiety in many students are imposed authority, public exposure and time deadlines. Although these are a regular part of the traditional Mathematics classroom cause great deal of anxiety. Mathematics anxiety has been defined as feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of 28 Mathematical problems in a wide variety of ordinary life and academic situations. Mathematics anxiety can cause one to forget and lose one‘s self-confidence (Tobias, 1993). Studies have shown that students learn best when they are active rather than passive learners (Spikell, 1993). Everyone is capable of learning, but may learn in different ways. Students‘ prior negative experiences in Mathematics class and at ho me when learning math are often transferred and cause a lack of understandiYng of Mathematics. According to Sheila Tobias, millions of adults are bAlocRked from professional and personal opportunities because they fear or perform poorly in mathematics for many, these negative experiences remain throughoutR their adult lives. This may more particularly true of the study of Mathematics IbBecause "Mathematics offers what is perhaps the clearest and most concentrate dL example" of intelligent learning, "which is to say the formation of conceptualN structures communicated and manipulated by means of symbols" (Skemp, 1971, p.1A6). Anxiety reactions to Mathematical situDations may contribute to failure in Mathematics (Tobias & Weissbrod, 1980)B. In Afact, a person who has high math anxiety may actually be unable to perform w eIll on test, and may be unable to learn in a Mathematics classroom. MathematFics anxiety also directly contributes to avoiding Mathematics (Tobias & We isOsbrod, 1980, p.63). The ways in which avoidance contributes to failure andY anxiety are perhaps a little less clear. Avoidance of Mathematics engendeIrTs failure because a person who has successfully avoided Mathematical sRituaStions for some time may lack the skills and knowledge needed when he or she iEs presented with a situation requiring its use. This is situation in which the individual is very likely to fail. Similarly, the person who has avoided Mathematics and is suNddenIl Vy confronted with a circumstance requiring it is likely to be painfully aware of his Uor her lack of preparation and become anxious about it as a result. Thus the avoidance of Mathematics can lead to failure and/or anxiety with staggering effect. Of course, if one could only continue to avoid Mathematics situations, neither failure nor anxiety would result. 29 The phenomenon of Mathematics anxiety itself is of interest to the education community only because individuals find themselves placed in situation requiring that they either use or learn Mathematics, or both. Without conditions necessitating the use of Mathematics, Mathematics anxiety, however high the individual‘s level would not be of any consequence. An underlying assumption of this model is that Mathematics anxiety is of interest only to those people who have been influenced by it in the past, in car eer choices, for example, those who are influenced by it presently, as in a MathematicsY class, or those who will in influenced by it in the future, as in a required MathematiRcs class or job skill. As long as a person has no need for Mathematics, MathematAics anxiety is unimportant. R Avoidance can occur for many reasons. Sometimes it is a sIimBple as students being extremely gifted in non-Mathematical areas and choosingN to s p Lend their time and energy on the subject in which they are gifted. Individuals mAay also decide that the study or use of Mathematics is not appropriate for them. DThis may be the result of sex-role stereotyping or other beliefs held by their socio-economic group. Tobias asserted that "most peopleI leBave A school as failures at Mathematics" (1978, p.26). Hilton (1980, p. 176) lists the cau ses of failure to be "bad teaching, bad texts, and bad educational instruments (p. O177)F." He went on to include rote calculations, memory dependence, authoritarianism, spurious applications and unmotivated problems as additional factors whichT inYhibit success in many students. Kogelman and Warren (1979) hypothesized that percIeived rigidity of rules and an inordinate emphasis on right answers may drive somRe stSudents, who are intellectually capable of learning Mathematics from success to fEailure. In addition, the cumulative nature of Mathematics may be a source of failureI fVor students who must be absent from school for any length of time. Regardless of hoNw or why individuals fail in Mathematics, they often experience what Tobias (1978) Ucalls "sudden death (p.27)." Whether it was timed tests on multiplication facts, the introduction of operations fractions, multi-stage word problems, or solving equations that caused the difficulties, for many "failure was sudden and very frightening" (Tobias, 1978, p. 44). Presumably, students do not really just suddenly reach a concept or procedure that they cannot learn. In Lazarus‘ (1974) analysis of Mathematics anxiety, he hypothesized a 30 "latency stage" – a period in which the student has been relying on a memorize-what-to- do strategy in learning Mathematics. The transition from Confidence to Anxiety has been hypothesized to be the result of unpleasant experiences associated with learning or doing Mathematics (Byrd, 1982; Kogelman & Warren, 1979; Tobias, 1978). Many people recall their first negative experiences with Mathematics with surprising vividness and clarity. They may remem ber how the teacher looked or dressed and what type of Mathematics task was invYolved. Students recollect Mathematics being taught in an atmosphere of tension creRated by an emphasis on swift computations and correct answers (Tobias, 1978). SomAetimes these negative experiences are not school related, but are associated with aR parent or sibling who acts as tutor (Kogelman & Warren, 1979). In addition to IthBese stresses, tests on mathematics serve as high stress producers. Kogelman & WLarren (1979) found that Mathematics "has long been associated with the pressNures of performing and being evaluated" (p.58), and far too frequently, the associatioAns are not pleasant ones. A student who has experienced frequeAnt fDailure in Mathematics is unlikely to be motivated to improve his or her performanBce. It is probable that he or she attributes the failure to bad luck, or difficulty ofF th eI subject. The student is likely to believe that increased effort and persistence Owill not make any difference in the outcome, that he or she has no control over succe ss or failure in Mathematics; and to develop a stance of helplessness and passiviTty (YCorral, 1997). According to DIodd (1999), the lack of confidence is probably the Mathematics- anxious learnerR‘s gSreatest obstacle. In addition, the loneliness of thinking you‘re the only one with MEathematics anxiety can be debilitating. She asserted that Mathematics anxiety isn‘t aInV inherited tendency, it‘s created. She asserted, ―It can be created when teachers plaNce too much emphasis on memorizing formulas and applying rules‖ and that ―It can Uresult when teachers fail to realize the critical connection between students‘ academic performance and their feelings about themselves and the subject being studied‖ (page 296). Mathematics anxiety can be described as a combination of factors as described by Mitchell (1987) who states that mathematics anxiety is a combination of physical, 31 cognitive and psychobehavioural components. Physical aspects of Mathematics anxiety are biological, consisting of hormonal, chemical and muscular changes in the body which results in a disability to think (Mitchell 1987). A number of different factors have been described as the causes of maths anxiety. Norwood (1994) describes Mathematics anxiety as the results of different factors including the inability to handle frustration, excessive school absences, poor self concept, parental and teacher attitudes towards Mathemat ics and emphasis on learning Mathematics through drill without understanding. A lYack of confidence when working in mathematical situations is described by Stuart (20R00) as the cause of Mathematics anxiety. Hodges (1983) argues that failure oAr success in mathematics may be related to individual learning styles and moIreB spe Rcifically with the coupling of learning styles and the way in which material is preLsented. Dossel (1993) identified several factors leading to the crNeati on of Mathematics anxiety: These are outlined as follows:  Personality factors (the belief that success DcannAot be attributed to effort – feelings associated with lack of control). A  Pressure of perceived authority figIuBres (parents, teachers).  Time pressure (to answer quickly and verbally).  Effect of public failure (askinFg to perform in front of a class).  Right – wrong dichoto mOy (the teacher‘s attention should be directed towards effort rather than acYhievement). The beginnings of SanxIie Tty can often be traced to negative classroom experiences and the teaching of maRthematics (Stodolsky 1985; Williams 1988). It is considered critical to examine claEssroom practice and establish whether the roots of Mathematics anxiety may be in inVstructional methods and in the quality of Mathematics teaching in elementary scNhoolI (Newstead 1998). U Greenwood (1984:663) stated that the principal cause of Mathematics anxiety lies in the teaching methodologies used to convey basic mathematical skills. He asserted that the ―explain – practise – memorize‖ teaching paradigm is the real source of the Mathematics anxiety syndrome. He states that teachers create anxiety by placing too much emphasis on memorising formulae, learning mathematics through drill and 32 practice, applying rote-memorised rules and setting out work in the traditional way. Butterworth (1999) believes that a lack of understanding is the cause of anxiety and avoidance and that understanding based learning is more effective than drill and practice. Another source of mathematics anxiety that has been identified is word problems. Tobias (1978) believes that word problems are the heart of Mathematics anxiety. Learners need higher levels of reasoning and if not taught strategies to solve these problems, learn ers may grow up avoiding Mathematics and science (Tobias 1978). Y The degree of accuracy and ease at which numbers can be manipulatedR has been identified as a cause of Mathematics anxiety. Mathematics anxiety is Athe result of nervousness about the required manipulation of numbers in maRthematics classes including tests, homework or in-class instruction (Ashcraft & FIaBust, 1994). Martinex (1987) identified a significant component of math anxiety to b eL the fear of failure. A long term sense of inadequacy by learners was described asA theN result of the pressures of timed tests, speed drills, and flash cards (Kogelman 1983). Research has shown that a teacher‘s own mDathematics anxiety could be a cause of anxiety for learners. Martinex (1987) stateBs thAat a teacher‘s own Mathematics anxiety is likely to be transmitted to their studeFnts . In a study to determine the underlying anxieties of teacher trainees it was found tOhat many had gaps in their Mathematics knowledge or an awareness of imperfectly lear ned concepts which in turn, can be transmitted to the learners they teach (MTartiYnex, 1987). The effect of having to perform and provide explanations in front oIf teachers or peers has been found to be a source of anxiety. It was found by KogRelmSan (1982) that experiences of learners having been punished or humiliated Eat the blackboard was very damaging. Newstead (1998) concludes from her researIchV that learners learn to do Mathematics before they are able to explain problems anNd communicate about mathematics. To expect learners to provide explanations to UMathematics questions could cause anxiety at the crucial age between the development of skills for doing Mathematics and the development of skills for explaining Mathematics (Newstead 1998). Emotion and anxiety can have a negative effect on the ability of a learner to learn as can be seen from the following research findings. One of the consequences of 33 Mathematics anxiety as stated by Goleman (1996) is that learners who are anxious cannot take in information efficiently or deal with it well, resulting in not being able to learn. Goleman (1996) describes that the ―working memory‖ becomes swamped when excessive emotion is present and the learner is unable to hold in mind all information relevant to the task in hand which results in not being able to think straight. Skemp (1986) similarly states that anxiety becomes debilitating in terms of performance a nd higher mental activities and perceptual processes. Strong emotion blocks reasoninYg and learners under pressure try to remember rather than understand, causing them Rto be hand icapped mathematically (Wells 1994). A It has been noted by Ashcraft and Faust (1994) that highlIy BMa Rthematics anxious learners tend to avoid the distressing Mathematics stimuli. ThisL has far reaching, national consequences as was highlighted by Hembree (1990) whose c oncern was when otherwise capable learners avoid the study of mathematics, theirN options regarding careers are reduced, eroding the country‘s resource base in scienceA and technology. Krantz and Silver (1992) asserted that Dat every level of mathematical skill, Mathematics anxiety had a negative corrBelatAion with interest in scientific careers. As found by research, the speed and accura cIy at which learners complete mathematics tasks is dependent on the anxiety thatO theFy experience. Emotional reactions such as apathy or depression as well as decrea sing motivation can be experienced by learners who consistently experience TfailYure, despite trying to succeed (Gentile & Monaco 1988). This exposure to uncontrolIlable failure experiences is referred to by Gentile and Monaco (1988) as learRnedS helplessness. Skiba (1990) comments that even if a skill is well grounded, tEhe anticipation of possible incompetence may block the ability to carry out the operatIioVn. U2.1N.2 Concept of Mathematics Achievement Student achievement in schools has always been a concern for parents, students and educators. There have been several theories on what help students achieve in a Mathematics class. A number of variables have been identified to be responsible for poor achievement of students especially in mathematics. According to Betiku (2002), these 34 variables are government related, curriculum related, examination body related, teacher related, students related, home related and textbook related. Some specific variables have also been identified by Amazigo (2000), such as poor primary school background in Mathematics, perception that Mathematics is difficult, large classes, psychological fear of the subject, lack of incentives for the teachers, students not interested in the subject, etc. Similarly Salau (2002) listed problems that seem to beset mathematics education in Nigeria, and which have resulted to consistent poor achievement of students in SSCYE as; 1. Acute shortage of qualified professional mathematics teachers. R 2. Undue emphasis on syllabus coverage at the expense of meaningful learnAing. 3. Exhibition of poor knowledge of mathematics content by many teachRers. 4. Over- crowded mathematics classroom. IB 5. Students' negative attitude towards mathematics. L 6. Adherence to odd teaching in-spite of exposure to mAoreN viable alternatives, to mention but a few. It is possible that these factors act singAly oDr in combination for some students in affecting their achievement in mathematicsB. However, Salau (2002) believes that through instructional strategies geared toward sI demystifying the subject in the Nigerian classroom, the poor state of mathemaFtics education in the country can be redressed. There is increasing concern about th e Onumber of learners who drop Mathematics in the latter years of high school. BarneYs (1984) notes that this avoidance is the result of a complex set of interacting factoIrsT, affecting boys and girls differently, but the main cause, for both sexes is the anRxieSty which Mathematics arouses in many students. In an international study involEving forty one countries, the competency of grade seven and grade eight learneIrsV in the fields of mathematics and science was tested. South Africa scored lowest in Nboth Mathematics and science proficiency (Sunday Times, 24 November 1996). In a Ufurther report by Unesco, Unicef and the South African Department of Education, (Sunday Times, 16 July 2000) amongst twelve countries in Africa, a sample of grade four South African learners scored lowest in numeracy skills on the continent. These results will affect the number of learners having the skills required for future careers in the fields of science and technology in South Africa. A decrease in tertiary studies in the fields 35 involving mathematics and science will negatively impact on South Africa‘s technological developments and economy. Because of the importance attached to mathematics at all levels of education, there is a need to study the cognitive and affective implications of mathematics teaching. Teaching methods need to be developed to determine which programmes would be appropriate for those with deficits in Mathematics skills and varying amounts of mathematics anxiety (Kostka 1986). Wu (1999) states that the problem is not in thYe lack of ability of students, but in the teaching method. In most cases, the precision aRnd fluency in the execution of the skills are the requisite vehicles to conveRy thAe conceptual understanding (Wu 1999). Mathematics, by its nature involves both cognition and aIffBective effects. Sutton (1997) argues that the glory of mathematics lies in the Nfact t Lhat Mathematics does not come easily to anyone. It is in the struggle to understand and the manner in which this is met, that one learns life skills. Research is needDed Ain the areas of both Mathematics anxiety, mathematics achievement and instructional techniques for the reduction and prevention of anxiety (Hadfield 1988). BA An increasing number of students arFe e xIperiencing mathematics problems. The national pass rate for mathematics was 49O,5 % in 1996, dropping to 46,3 % in 1997 and to 42,1% in 1998 (Pretoria News 2000 ). More recent statistics with respect to pass rates for mathematics have not bTeenY officially published. Coetzer (2001) notes that progressive schools in the United SItates of America that followed a curriculum similar to curriculum 2005 showed sRimilSar results, especially as reg HemEbree (1990) noted that Mathematics anxiety seriously constrains performance in MIatVhematical tasks and reduction in anxiety is consistently associated with imNprovement in achievement. As such, it is to be expected that highly Mathematics Uanxious individuals will be less fluent in computation, less knowledgeable about mathematics, and less likely to have discovered special strategies and relationships within the mathematics domain (Ashcraft & Faust 1994). In a meta-analysis, Ma (1999) quantified the potential improvement in mathematics achievement when mathematics anxiety is reduced. Ma (1999) found that the relationship between mathematics anxiety 36 and mathematics achievement is significant from grade 4 and that once mathematics anxiety takes shape, its relationship with mathematics achievement is consistent across grade levels. It was also found that the relationship was consistent across gender groups, ethnic groups and instruments used to measure anxiety (Ma 1999). Goleman (1996) describes the relationship between anxiety and performance, including mental performance in terms of an upside-down U. At the peak of the inverted U is the optimal relationship between anxiety and performance. Too little anxiety, the first Yside of the U, brings about apathy or too little motivation to try hard enough to do wellR, while too much anxiety, the other side of the U, sabotages any attempt to do well. A Number manipulation anxiety and test anxiety showIedB si Rgnificant inverse relationships with respect to mathematics achievement. Learners who are anxious about forthcoming tests and number manipulation techniques aNre l ik Lely to perform at a lower level (Wither, 1988). From these it can be noted that the presence of mathematics anxiety has a negative effect on mathematics achievement. ThAe teacher‘s attitude and approach in the classroom has an effect on a learner‘s achieveDment. Dossel (1993) has suggested that the atmosphere in the classroom, includinBg pAerceived warmth, may lower anxiety and improve mathematics performance. Stua rIt (2000) stated that students like to do what they are good at, and to feel good abouFt mathematics, teachers need to build up the self- confidence and refine the skills rOequired to be successful at mathematics. It has been notedT thYat more positive attitudes accompany lower levels of anxiety and are conducive to inIcreased gains in the future (Genshaft 1982). Wells (1994) believes in telling a clasRs beSforehand that the subject matter is difficult, giving learners a truthful picture of mEathematics as something difficult and challenging, but which they can do succesIsVfully, as opposed to leaving learners to draw conclusions from their own failure thaNt mathematics is difficult. Misconceptions about the nature of mathematics have also Ubeen investigated. Gourgey (1992) states that many learners hold misconceptions about what mathematics is, which results in them performing procedures without understanding, often incorrectly distrusting their own intuitions and feeling powerless when they make mistakes. These misconceptions erode a learners‘ confidence and contribute to their learning difficulties (Gourgey 1992). Sutton (1997) states that people 37 misunderstand that in meeting the challenge of the difficulties they experience with mathematics creates an opportunity to learn life skills. There is evidence that different cultural groups have different attitudes to mathematics achievement. Stevenson (1987), in a study showing the difference between American and Asian approaches to mathematics, shows that Americans believe one is either born with a mathematics ability or not. Asians believe mathematics success i s a result of hard work, perseverance and hours of study and believe the virtue of efforYt is the avenue for accomplishment (Stevenson 1987). A learner‘s personality has beeRn cited by some researchers as having an effect on achievement. Tobias (1978) sAtates that the differences in how learners cope with uncertainty, whether theyI Bcan Rtolerate a certain amount of floundering, whether they are willing to take riskLs, what happens to their concentration when an approach fails, and how they feel ab out failure determine how well they will achieve. Attitude and self-image, particularNly during adolescence when the pressures to conform are important, can result in Anegative attitudes that can inhibit intellect and keep one from learning what is withDin one‘s power to understand (Tobias 1978:91). People who trust their intuition,B perAceiving intuition as flashes of insight into the rational mind, rather than emotioFnal , Iirrational thoughts are less mathematics anxious (Tobias 1987). O 2.1.3 Concept of MathemYatics Efficacy Research on MIaTthematics teaching has recently focused on affective variables, which were foundS to play an essential role in influencing behaviour and learning (Bandura, 1E997R). The affective domain is a complex structural system consisting of four main cIoVmponents: emotions, attitudes, beliefs and values (Goldin, 2002). Beliefs can be deNfined as one‘s knowledge, theories and conceptions and include whatever one Uconsiders as true knowledge, although he or she cannot provide convincing evidence to support it (Pehkonen, 2001). Self-beliefs can be described as one‘s beliefs regarding personal characteristics and abilities and include dimensions such as self-concept, self- efficacy and self-esteem. Self-efficacy can be defined as one‘s belief that he/she is able to organize and apply plans in order to achieve a certain task (Bandura, 1997). 38 According to Bandura (1997) every individual possess a belief system that exerts control over his/her thoughts, emotions and actions. Among the various mechanisms of human agency, none is more central or pervasive than self-efficacy beliefs (Bandura & Locke, 2003; Pajares, 2000). Self-efficacy is a task-specific construct and there is a correspondence between self-efficacy beliefs and the criteria task being assessed. In contrast, self-concept is the sense of ability with respect to more global goals (Pajar es, 2000; Bandura, 1986), while self-esteem is a measure of one‘s feeling of pride aYbout a certain trait, in comparison with others (Klassen, 2004). The task-specificity oRf efficacy beliefs implies that related studies are more illuminating when they refer toA certain tasks, such as problem posing; the predictive power of self-efficacy is in thiRs case maximized (Pajares & Schunk, 2002). On the other hand, the level of spIeBcificity could not be unlimited; as Lent and Hackett (1987) have rightly observed, s pLecificity and precision are often purchased at the expense of practical relevance and vNalidity. Research on self-efficacy has recently been acAcumulated providing among other things notable theoretical advances that reinfoArceD the role attributed to this construct in Bandura‘s (1997) social cognitive theoryB. Several works (Pajares & Schunk, 2002; Pajares, 2000; Bandura, 1997) have ind icIated a strong correlation between Mathematics self-efficacy and Mathematics achieFvement (Klassen, 2004). It was further found that Mathematics self-efficacy is a g oOod predictor of Mathematics performance irrespective of the indicators of performaYnce (Bandura, 1986) and regardless of any other variables (Bandura & Locke, 2I0T03). It was found that Mathematics self-efficacy is a better predictor of MatheSmatics performance than Mathematics anxiety, conceptions for the usefulness Eof MRathematics, prior involvement in Mathematics, Mathematics self-concept and prIeVvious Mathematics performance (Klassen, 2004; Pajares & Miller, 1994). It is noNteworthy that self-efficacy beliefs were even found to be a stronger predictor of Uperformance than general mental ability (Pajares & Kranzler, 1995). Self-efficacy beliefs have already been studied in relation to a lot of aspects of Mathematics learning, such as arithmetical operations, problem-solving and problem- posing. Pajares and Miller (1994) asserted that efficacy in problem solving had a causal effect on students‘ performance. Research findings support the view that high 39 achievements in Mathematics students have higher and more accurate efficacy beliefs (Pajares & Kranzler, 1995). Efficacy beliefs towards a certain task are accurate when they correspond to what the person can actually accomplish. Perceptions of self-efficacy come from personal accomplishments, vicarious learning experiences, verbal persuasions, and physiological states (Bandura, 1986; Ingvarson, Meiers, & Beavis 2005; Tanner & Jones, 2003). A self-efficacy impact o n a learner‘s potential to succeed (Bandura, 1977). Self-efficacy is a valuable toYol for Mathematics educators. It is important for educators to know how students fReel, think, and act, about, within, and toward Mathematics (Yates, 2002). The influencAe of attitudes, values and personality characteristics on achievement outcomes and latRer participation in the learning of Mathematics are important considerations for MIBathematics educators (Yates, 2002). L One way to gain insight into how learners feel, thNink, and act, about and toward Mathematics is to examine their psychological doDmaiAns of functioning: the affective, the cognitive, and the conative (Huitt, 1996; Tallon, 1997). It is important to examine each domain as a student may feel efficaciousI wBith Ain the affective domain but less confident within the cognitive domain. Affect iFs a student‘s internal belief system (Fennema, 1989). The affective domain includes Ostudents‘ beliefs about themselves and their capacity to learn Mathematics; their self e steem and their perceived status as learners; their beliefs about the nature of MatheYmatics understanding; and their potential to succeed in the subject (Tanner & JonIeTs, 2003). The cognitive domain considers students‘ awareness of their own MathRemSatics knowledge: their strengths and weaknesses; their abstraction and reification Eof processes; and their development of links between aspects of the subject (TannIerV & Jones, 2000). Cognition refers to the process of coming to know and unNderstand; the process of storing, processing, and retrieving information. The cognitive Ufactor describes thinking processes and the use of knowledge, such as, associating, reasoning, or evaluating. Conation refers to the act of striving, of focusing attention and energy, and purposeful actions. Conation is about staying power, and survival. The conative domain includes students‘ intentions and dispositions to learn, their approach to monitoring their 40 own learning and to self-assessment. Conation includes students‘ dispositions to strive to learn and the strategies they employ in support of their learning. It includes their inclination to plan, monitor, and evaluate their work and their predilection to mindfulness and reflection Tanner & Jones, 2000). Self-efficacy is a domain-specific construct in academics. Many, including Bandura, argue that it is also task-specific, and attempts to measure self-efficacy at the domain level often result in ambiguous or unclear results (Bandura, 1986; PajaYres & Miller, 1994c, 1995). Many of the studies that show self-efficacy to accoAuntR for lesser variance than other personal determinants often stray from Bandura's prescriptions for a microanalytic strategy. Often these studies assess self-efficacy globallRy with just a few scale items; that is, they ask participants to report on their confiIdBence or efficacy with regard to a specific academic domain, and not a specific perf oLrmance task. At this level of self-reporting, it is expected that self-efficacy cannot rNeliably be separated from other personal determinants such as self-concept, anxiety, sAelf-confidence, and background. It thus raises the question of whether one is aActuDally measuring self-efficacy, or more generally measuring attitudes and other coBmmon mechanisms toward a given academic domain. Of course, the latter are impFort anIt in some areas of educational research, but do not always give sufficient evaluOative information for performance on specific, criteria tasks. One possible lens fro m which to view self-efficacy within the context of instructional technologTy iYs to consider one's judgments of personal capabilities to authentically accompliIsh a specific performance objective. Self-efficacy and performance are inextricablyR relSated, and in the domain of Mathematics both are often correlated with gender. E IVIn academic domains, the research on self-efficacy is less extensive. However, it is Nseen as being applied to such diverse academic domains as Mathematics, computer Uliteracy, writing, in service teacher training, choice of academic majors, and so on. Many of these studies are correlational and describe how self-efficacy relates to academic outcomes. Schunk (1997) is one of the more prolific researchers applying self-efficacy as an academic construct. He and his colleagues often used a research paradigm that goes beyond correlational analysis to include instructional interventions designed to raise 41 learners‘ percepts of efficacy and corresponding performance on criteria tasks. Schunk's (1997) treatments to influence self-efficacy include variations on modeling, attributions of success or failure, and goal-setting. Pajares (2000) and colleagues (Pajares & Kranzler, 1995; Pajares & Miller, 1994a; Pajares & Miller, 1994b; Pajares & Miller, 1994c) often used advanced statistical procedures to account for the explanatory and predictive variance of self-efficacy in relation to other personal determinants, such as anxiety, academic backgroundY, self- confidence, and so on (Pajares & Kranzler, 1995; Pajares & Miller, 1994a; RPajares & Miller, 1994b; Pajares & Miller, 1994c; Pajares & Miller, 1995). CoRnsisAtently, Pajares and colleagues find that self-efficacy maintains high explanatoryI Band predictive power for Mathematics performance. L 2.1.4 Concept of Pseudo-dyscalculia N Dyscalculia is described as disabilities expDerieAnced from mathematical activities, which may be as a result of developmental prAocesses of the child, cognitive disabilities, environmental factors, etc. A majority of Bdyscalculia cases, experienced by individuals with average or superior intelligence, Iare exclusively caused by failure to acquire Mathematics fundamentals in schooFl. Worldwide, Mathematics has the highest failure rates, and lowest average grad e Oachievements. Almost all students, regardless of school type or grade, cannot peTrfoYrm in Mathematics on par with their intellectual abilities. This is not surprising becauIse sequential Mathematics instruction requires a perfect command of acquired funRdamSentals. (CTLM 1986) The slightest misunderstanding makes a shaky mathematicEal foundation. IVCohn (1968) explains that having a disability in Mathematics is socially acNceptable. He asserts that math ability is more regarded as a specialized function, rather Uthan a general indication of intelligence. As long as one can read and write, the stigma and ramifications of math failure can be diminished and sufficiently hidden. Sharma, concurs, explaining that in the West, it is common to find people with high IQ's who shamelessly accept incompetency in math. At the same time, they find similar incompetence in spelling, reading, or writing, totally unacceptable. Prevailing social 42 attitudes excuse math failure. Parents routinely communicate to their children that they are "no good at math." (Sharma 1989). Shelia Tobias, in 1978, realized that because only 8% of girls took 4 years of math in high school, 92% of young women were immediately eliminated from careers and study in science, chemistry, physics, statistics, and economics. Half of university majors were closed to them. Tobias states that women avoid math, not because of inability, but because women are "socialized" away from studying math. She advYocates "math therapy" for both sexes to overcome "math anxiety." (Tobias 1978, 12-13R). Sharma asserts that gender differences in math skills are due more to social forces tAhan to gender- specific brain construction and function. He believes that gender dRifferences can be eliminated by equalizing the activities and experiences of both bIoBys and girls at every level of development. The social forces that direct a child's eLxperiences and choice of activities lead to the differences in the neurologicalA soNphistication of boys and girls. (Sharma 1989). For example, most studies show that girls do better than boys in math until the age of 12. Then boys dominate the subjeDct. This difference can be explained by analyzing the gender-specific developmentB of mAath prerequisite, spatial orientation skills. The main reason for this is the method oIlogy of teaching in pre-school and elementary grades, where focus is on fine-motorF skill development. (Sharma 1989). As with all abilities, m Oath aptitude can be inherited or an inborn disposition. Studies of identical twinsY reveal close math scores (Barakat 1951). Research into exceptionally gifted inIdiTviduals shows high levels of math knowledge in early childhood, unexplained bRy exSternal influences. Family histories of mathematically "gifted" and "retarded" individuals, revealed common aptitudes in other family members. (CTLM 1986). EI EVven the most "mathematically gifted" individual can be hindered by inadequate MNathematics education. Likewise, a "mathematically retarded" individual will not attain Ucompetency in math despite intensive systematic training. (CTLM 1986) 2.1.5 Classification of Dyscalculia Quantitative dyscalculia is a deficit in the skills of counting and calculating. Qualitative dyscalculia is the result of difficulties in comprehension of instructions or the 43 failure to master the skills required for an operation. When a student has not mastered the memorization of number facts, he cannot benefit from this stored "verbalizable information about numbers" that is used with prior associations to solve problems involving addition, subtraction, multiplication, division, and square roots. Intermediate dyscalculia involves the inability to operate with symbols, or numbers. (CTLM 1989). Pseudo-Dyscalculia can be described as Mathematics inability caused by lack of, inconsistent, poor, or inappropriate systematic Mathematics instruction: inattentionY, fear, anxiety, illness, absence, or emotion. Psychology is concerned with the disoRrders and disturbances of math abilities. Neurology and psychiatry deal with the distuArbed functions resulting from brain damage. (CTLM 1986) Each profession uses IspBeci Rfic terminology to describe math disabilities. The result is the categorical fragmentation of classes and types of dyscalculia, as seen in the table below: N L Table 2.1: Classification of Dyscalculia byD CAentre for Teaching/Learning of Mathematics ( CTLM, 1986) A Class Name DBefinition Examples I OF Dysfunction in math, in Numerical difficulties individuals with normal with: counting, ITY mental functioning, recognizing numbers, resulting from brain manipulating math RS anomalies inherited or symbols mentally E occurring during and/or in writing, Developmental 1 ClassV A prenatal development. sequential memory for I Dyscalculia N Discrepancy 1-2 numbers and U standard deviations operations, mixing up below the mean, numbers in reading, between mental age and writing, recalling, and math age. Clear auditory processing, retardation in math memory. Much more 44 development. effort is required. 2 Math disability that is the result of brain Post-Lesion Class B damage/ head injury, Dyscalculia cerebral dysfunction, or Minimal Brain Damage. RY Environmentally A Caused Dyscalculia. R Math inability caused B by: lack LIof, Pseudo- inconsistent, pNoor, or 3 Class C Dyscalculia. inappropriatAe systemaDtic math iBnstrAuction; inattention, Ifear, anxiety, illness, F absence, or emotion. Dyscalcu lOia SubtYypes 4 Class A-Type 1 SIo Tccurring with R Normal Mental E Ability IV Dyscalculia coexisting N Secondary with oligophrenia, 5 UClass A- Type 2 Dyscalculia mental retardation, or dementia. 6 Class A-1-a Dyscalculia Total inability to 45 MODERATELY abstract, or consider SEVERE: concepts, numbers, attributes, or qualities apart from specific, tangible examples. Complete inability of 7 Class A-1-b Acalculia Y math functioning. AR A relative decrease of R 8 Class A-1-c Oligocalculia all facets of mathI B ability. L Secondary Dementia N with 9 Class A-2-a Dyscalculia dyscalcuDlia.A Secondary IMBen Atal retardation with 10 Class A-2-b Acalculia dyscalculia. F Seconda ryO Oligophrenia with 11 Class A-2-c OligoYcalculia dyscalculia. STecondary A neurotic aversion to 12 Class A-2-d SI R Paracalculia numbers. VE Math inability caused by: lack of, NI Environmentally inconsistent, poor, or UClass C Caused inappropriate 13 Dyscalculia systematic math instruction; inattention, fear, anxiety, illness, 46 absence, or emotion. 14 Class C-Type 1 Pseudo-acalculia Pseudo- 15 Class C- Type 2 dyscalculia Pseudo- Y 16 Class C- Type 3 oligocalculia AR MOST SEVERE: R Dyscalculia with B 17 Class D Para-calculia I Learned M Lath Avoidance. AN Motor Verbal 18 Class D-1 D Paracalculia BA Dyscalculia + I 19 Class A-1-a-D Learned OMFath Avoidan Y ce ATcalculia + 20 Class A-1-b-D SILearned Math R Avoidance E IV Oligocalculia + 21 NClass A-1-c-D Learned Math U Avoidance Pseudo-acalculia 22 Class C-1-a-D + Learned Math Avoidance 47 Pseudo- dyscalculia + 23 Class C-1-b-D Learned Math Avoidance Pseudo- oligocalculia + 24 Class C-1-c-D Y Learned Math R Avoidance RA IBCannot verbally name L amounts of things, N numbers, terms, A symbols, and DysnomDia for operations. Cannot qBuanAtitative terms, associate numerals to Verbal Ielements and relations. amounts of things. 25 Class A-1-a-I Dyscalculia F Capable of performing Cannot verbally O operations involved. continue counting Y Counting disorders. patterns. May be able IT to read and write S dictated numbers. R Capable of performing VE operations involved I UN May incorrectly write Motor Verbal Cannot read or write 26 Class A-1-a-I-a numbers as they are Dyscalculia dictated numbers. literally pronounced: 48 "Five hundred and 4" as 5004, etc. Brain-damaged. Cannot display a requested number of items Sensory -Verbal Y 27 Class B-1 physically or Dyscalculia pictorially. Cannot read R or write numbers, or A count items. LIB R Cannot manipulate, N add, compare, or Impaired Aability to estimate quantity or manipulDate real or A magnitude of physical pictured items for B or pictured items. May Imathematical purposes. be able to read, write, F Apraxic (Processing Practognostic and imitate written 28 Class A-1-a-II O errors that result in Dyscalcu lia numbers and Y inability to perform operations. Cannot IT purposeful motor compare, comprehend, S actions, especially or describe part-whole R sequences.)-Perceptual relationships, spatial E Dysfunction. details, shapes and IV sizes. UN Cannot use fingers to Inability to recognize assist with math Finger Apraxia 29 Class A-1-a-II-a objects by touching processing, cannot or Gnosia with the fingers. carry numbers or follow computational 49 sequences. Cannot count by heart. Cannot count by heart. Inability to purposeful Cannot use fingers to motor acts, especially a assist with math Apraxic 30 Class A-1-a-II-b sequence of processing, cannot Dyscalculia Y movements. Caused by carry nRumbers or processing errors. foRllowA computational Bsequences. I Performs b elLow intellectual, N developmenAtal, and May transpose (mix acadAemiDc level. up) [21 as 12], DBifficulty with, or interchange similar Iinability to read serial digits[6 and 9], numbers, digits, place inappropriately insert, Numerical F O value, operational signs, or omit digits, words, 31 Class A-1-a-II-c Dyslexia or math symbols, & signs. LTiterYal Dyslexia I fractions, squares, roots, May read without S decimals, and the acknowledging place R language of math. Can value: 5007 as "five E be caused by apatic hundred seven," or 576 IV agnosia, or directional and "five seven six." N confusion. Usually U occurs with other types. Lexical Difficulty with, or May transpose (mix 32 Class A-1-a-III Dyscalculia or inability to read serial up) [21 as 12], Numerical numbers, digits, place interchange similar 50 Dyslexia value, operational signs, digits[6 and 9], math symbols, inappropriately insert, fractions, squares, roots, or omit digits, words, decimals, and the & signs. language of math. Can May read without be caused by apatic acknowledging place agnosia, or directional value: 50R07 Yas "five confusion. Usually hundrAed seven," or 576 occurs with other types. anRd "five seven six." Inability to writeI B numbers because Lof inefficient AmotNor skills or D insufficient coorAdination of visual Numerical 33 Class A-1-a-III-a IpBerception skills and Dysgraphia F fine motor skills. Numerical 34 Class A-1-a-III-b O DyscYalcu lia or Class A-1-a-III- INTumerical 35 (a+b)-c RSDysmbolia VE May be unable to I Disability in writing form/write individual N math symbols. Usually digits, or copy them. 36U Class A-1-a-IV occurs with literal Cannot encode (write) Graphical dysgraphia and literal numbers correctly: [ Dyscalculia or dyslexia. 5731 as "5000700301" Numerical or omits zeros: 5073 as 51 Dysgraphia "573"] Writing #s in mixed up order, or opposite direction. May be able to write words for numbe rs. Inability to write Y numbers because of R inefficient motor skills A Numerical R 37 Class A-1-a-IV-a or insufficientI Dysgraphia Bcoordination of v isLual perception skiNlls and fine motor sAkills. Numerical D 38 Class A-1-a-IV-b A Dyscalculia IBF Lexical dyscalculia O occurring with Class A-1-a-IV- NumYeric al graphical dyscalculia. 39 (a+b)-c IDT ysmbolia OR Numerical Dyslexia S occurring with R numerical Dysgraphia. IV E Ideognostic Poor mental May be unable to N Dyscalculia or comprehension of calculate the easiest U Asemantic quantitative concepts. A sums mentally, or at an 40 Class A-1-a-V Aphasia or dysfunction of the age/ academically Dysymbolia cognitive function of appropriate level. May TESTS: [a}100- forming or assigning be able to read and 7-7-7-7-7-7- numbers & symbols write numbers but is 52 7.....mentally "notions" or meaning. oblivious to their first/writing 2nd; Inability to do mental meaning. Unable to {b} Series math. Aphasia is the distinguish colors of Completion. inability to express objects, or objects meaningful verbal from a competing identifications (of math background. YC annot symbols). identify a specified numbeAr oRf items. 41 Class A-1-a-V-a Dysymbolia B RI MOST SEVE RLE: Extreme difficNulty in grasping prAinciples and D Is unable to continue logicA of math concepts the sequence of 42 Class A-1-a-V-b Acalculia aBnd reasoning, Gnostic I numbers in the most disturbance is noted F basic of given series. when can do test O mentally but not in Y writing. SI T R Operational Inability to learn and Frequent errors E Dyscalculia apply the rules for include: mixing up IV (Anarithmetie) addition, subtraction, operations like +/-, - N Tests: Note multiplication and 43U Class A-1-a-VI performance division resulting in a or oversimplification strategy. Have disability to of complex operations; subject verbalize successfully perform insisting on written thinking, if math operations. computation over possible. mental calculation, 53 uses fingers to assist mental or written computation. Sensory-verbal 44 Class A-1-a-VI-a (Cannot count out) RY Motor-Verbal 45 Class A-1-a-VI-b A (Cannot name) BR Poor memory Lfor:I Sequential counting seNque nces, 46 Class A-1-a-VI-c Dyscalculia operational Asequences, math Dfacts, time, direcAtion, schedules. Hypo- denotes a IB 47 Hypocalculia lack For deficienc yO in. SI TY Treatment: Appropriately A serious retardation in R motivate student to Oligo- new, developmental math 48 OligocalcEulia learn math, then fill in V recent abilities caused by I the knowledge and N mental retardation. skill gaps with U remedial techniques. Calculasthenia A socially caused 49 (LESS deficit in the level of SERIOUS) math acquisition, skills, 54 and knowledge. 50 Acalculia A- without, not Para- functionally disordered or Y diseased R 51 Paracalculia condition, or A similar to, but R not identical to a IB true condition or L form. N Term from DA Focus on Learning Learning education and 52 A Problems in Disability educational IB Mathematics. psychology. Are fr omO F the fields of Terms containing Y 53 InTeurology and . "calculia" S neuro-R psychology. (CTLIMV 19 E86) U2.2N THEORETICAL FRAMEWORK 2.2.1 Information Processing Theory Information-processing theories of anxiety (Anderson, 1983; Newell and Simon 1972), analyze cognitive performances into complexes of rules, but performances critically depend on interactions among those rules. Each rule can be thought of as a 55 component of the total skill, but the rules are not defined independently of one another. The relevance of this theory to the present study has been explained in four claims and has been further elaborated. The most significant point of this theory is that Mathematics disability is situational and one can improve upon it if proper restructuring is carried out. Again, the theory postulated that knowledge transfer or application of knowledge is very important. The whole purpose of modeling information processing theories of cognitYion on Mathematics anxiety is that people who are mathematical disinclined goeRs through negative information processing in their brain, and this in turn feed Aback negative thoughts within the particular individuals. Unlike earlier behaRviourist theories, information-processing theories do not posit a simple one-to-IonBe mapping between individual rules or knowledge components and individual bi tsL of behaviour. They deny this precisely because continual interaction can be obNserved among components of knowledge and behaviour. Information-processing psAychology has advanced rapidly by developing methods both for identifying the cAomDponents and for studying them in their interactions with their entire contexts. TIhBis is the meaning of the "unified theories of cognition" (Newell, 1991) which haFs gu ided so much of the recent research and theory-building. The information-proces sOing approach tries both to deepen our understanding of the components and to uTndeYrstand the relations among them and with their environments. Examples of these metIhods of componential analysis are the use of think-aloud protocols as data (EricssoRn anSd Simon, 1993) and the use of models that simulate the interactions of perceptual, Ememory, learning and thinking processes over a wide range of cognitive tasks (AndeIrson, 1993; Feigenbaum and Simon, 1984; Newell, 1991). Assessing learning and imNprovi Vng learning methods require careful task analysis at the level of component skills, Uintimately combined with study of the interaction of these skills in the context of broader tasks and environments. It is a well-documented fact of human cognition that large tasks decompose into nearly independent subtasks (Simon, 1981, Card, Moran & Newell, 1983), so that only the context of the appropriate subtask is needed to study its components. For instance, 56 there is no need to teach or assess the ability to perform multi-column addition in the context of calculating income taxes. The process of adding tax deduction items is the same as the process of taking sums in other tasks. And whether one does the sum by hand or by calculator is unlikely to affect the rest of the tax calculation procedures. Thus, the larger procedure is independent of the summing procedure, just as the summing procedure is independent of the larger procedure. In general, situated learning focuses on some well-documented phenomYena in cognitive psychology and ignores many others. While cognition is pAartlyR context-dependent, it is also partly context-independent; while there are dramatic failures of transfer, there are also dramatic successes; while concrete instIructio Rn helps, abstract instruction also helps; while some performances benefit from traininBg in a social context, others do not. The development from behaviourism to cognit ivLism was an awakening to the complexity of human cognition. The analysis offereNd by situated learning seems a regressive move. What is needed to improve learniAng and teaching is to continue to deepen our research into the circumstances thatD determine when narrower or broader contexts are required and when attention Bto nAarrower or broader skills are optimal for effective and efficient learning. I That action situationally OgrouFnded is surely the central claim of situated cognition. It means that the potentialities for action cannot be fully described independently of the specific situation. But thTe cYlaim is sometimes exaggerated to assert that all knowledge is specific to the situationI in which the task is performed, and that more general knowledge cannot and wiRll nSot transfer to real-world situations. Supposed examples of this are Lave's (1988) description of Orange County homemakers who did very well at making supermIaVrke Et best-buy calculations but who did much worse on arithmetically equivalent scNhool-like paper-and-pencil Mathematics problems. Another frequently cited example is UCarraher, Carraher and Schliemann's (1985) account of Brazilian street children who could perform Mathematics when making sales in the street but were unable to answer similar problems presented in a school context. An example in the Nigerian setting is that, some secondary school students who are not mathematically inclined at school may be good when making calculation at sales. This then means that, student can as well 57 perform better in Mathematics at school because their problem of Mathematics is situational. Even if these claims are valid and generalisable beyond the specific anecdotes that have been cited, they demonstrate at most that particular skills practiced in real-life situations do not generalize to school situations. They assuredly do not demonstrate that arithmetic procedures taught in the classroom cannot be applied to enable a shopper to make price comparisons or a street vendor to make change. What such observatioYns call for is closer analyses of the task demands and the use of such analyses to deviseR teachable procedures that will achieve a balance between the advantages of geneArality and the advantages of incorporating enough situational context to make transferR likely. What they also call for is research on the feasibility of increasing the applIicBation and transfer of knowledge by including ability to transfer as a specific goNal in Linstruction – a skill that is given little attention in most current instruction. At one level there is nothing new in this claAim about the contextualization of learning in Mathematics. There have been numDerous demonstrations in experimental psychology that learning in Mathematics BcanA be contextualized (Godden & Baddeley, 1975; Smith, Glenberg, & Bjork, 197 8)I. For instance, Godden and Baddeley (1975) found that divers had difficulty remFembering under water what they learned on land or vice versa. However, it is not t heO case that learning is totally tied to a specific context. In fact, there are many dTemYonstrations of learning that transfer across contexts and of failures to find any cIontext specificity in the learning (Fernandez & Glenberg, 1985; Saufley, OlakaR, & BSaversco, 1985). HowE tightly learning will be bound to context depends on the kind of knowledge being IaVcquired. Sometimes knowledge is necessarily bound to a specific context by the naNture of instruction. In other cases, how contextualized the learning is depends on the Uway the material is studied. If the learner associates the knowledge with material from a specific context, it becomes easier to retrieve the knowledge in that same context (Eich, 1985), but perhaps harder in other contexts. One general result is that knowledge is more contexts bound when it is just taught in a single context (Bjork & Richardson-Klavhen, 1989). 58 Clearly, some skills, like reading, transfer from one context to another. For instance, the very fact that we can engage in a discussion of the context-dependence of knowledge is itself evidence for the context independence of reading and writing competence. Many of the demonstrations of contextual-binding from the situated camp involve Mathematics, but clearly, Mathematics competence is not always contextually bound either. Although the issue has seldom been addressed directly, the psychologi cal research literature is full of cases where Mathematics competence has transferredY from the classroom to all sorts of laboratory situations (sometimes bizarre- the inteRntion was never to show transfer of Mathematics skills (Bassok & Holyoak, 1987A; Elio, 1986; Reder & Ritter, 1992). It is not easy to locate the many published Rdemonstrations of Mathematics competence generalizing to novel contexts; these rIeBsults are not indexed under "context-independence of Mathematics knowledge"N bec a Luse, until recently, this did not seem to be an issue. The literature on situation-specificity ofD leaArning often comes with a value judgment about the merits of knowledge tied to a non-school context relative to school- taught knowledge, and an implied or exBpresAsed claim that school knowledge is not legitimate. Lave (1986, 1988) suggFests Ithat school-taught Mathematics serves only to justify an arbitrary and unfair Oclass structure. The implication is that school-taught competences do not contribute to on-the-career performance. However, numerous studies show modest to large corrYelations between school achievement and work performance (Hunter & Hunter, 198I4T; Brossiere, Knight, & Sabol, 1985) even after partialling out the effects of general aSbility measures (which are sometimes larger). In thEis cRase, the authors of this theory conclude that action is indeed grounded in the sitIuaVtion where it occurs. They (authors) dissent strongly from some of the supposed imNplications that have been attached to this claim by proponents of situated action, and Uthey have shown that their dissent has strong empirical support. Knowledge does not have to be taught in the precise context in which it will be used, and grave inefficiencies in transfer can result from tying knowledge too tightly to specific, narrow contexts. Closer analyses of the task demands to devise teachable procedures that will balance the advantages of generality with the advantages of 59 incorporating enough situational context to make transfer is needed. Individuals also need to study how to increase the application and transfer of knowledge by including ability to transfer as a specific goal in instruction. This second claim, of the failure of knowledge to transfer, can be seen as a corollary of the first. If knowledge is wholly tied to the context of its acquisition, it will not transfer to other contexts. Even without strong contextual dependence, one could s till claim that there is relatively little transfer, beyond nearly identical tasks, to diYfferent physical contexts. For instance, while one might be able to do fractional MathRematics in any context, it might not transfer to learning algebra. A The more recent psychological literature is also full of failures tRo achieve transfer (Gick & Holyoak, 1980; Hayes & Simon, 1977; Reed, Ernst, & BIanBerji, 1974; Weisberg, DiCamillo, & Phillips, 1985), but it is also full of successful dLemonstrations of transfer (Brown, 1990; Brown & Campione, 1993; Kotovsky & FaNllside, 1989; Schoenfeld, 1985; Singley & Anderson, 1989; Smith, 1986). Indeed, inA the same domain, quite different amounts of transfer occur depending on the amAounDt of practice with the target task and on the representation of the transfer task B(Kotovsky & Fallside, 1989). In general, representation and degree of practice a reI critical for determining the transfer from one task to another. F Singley and Anderson (1O989) argued that transfer between tasks is a function of the degree to which the TtaskYs share cognitive elements. This hypothesis had also been put forth very early in theI development of research on transfer (Thorndike & Woodworth, 1901; WoodwRorthS, 1938), but was hard to test experimentally modern capability for identifying Etask components was acquired. Singley and Anderson taught subjects several text edIiVtors, one after another and sought to predict transfer (savings in learning a new edNitor when it was not taught first). They found that subjects learned subsequent text Ueditors more rapidly and that the number of procedural elements shared by two text editors predicted the amount of this transfer. In fact, they obtained large transfer across editors that were very different in surface structure but that had common abstract structures. Singley and Anderson also found that similar principles govern transfer of Mathematics competence across multiple domains, although here they had to consider 60 transfer of declarative as well as procedural knowledge. As a general statement of the research reported by Singley and Anderson, transfer varied from one domain to another as a function of the number of symbolic components that were shared. If anything, Singley and Anderson found empirically slightly more transfer than was predicted by their theory. One of the striking characteristics of such failures of transfer is how relativ ely transient they are. Gick and Holyoak were able to increase transfer greatly jYust by suggesting to subjects that they try to use the problem about the geneAral.R Exposing subjects to two such analogs also greatly increased transfer. The amount of transfer appeared to depend in large part on where the attention of subjects wRas directed during the experiment, which suggests that instruction and training on tIhBe cues that signal the relevance of an available skill might well deserve more emNph as Lis than they now typically receive. As a methodological comment, the authorDs thiAnk that there is a tendency to look for transfer in situations where one is least likely to find it. That is, research tends to look for transfer from little practice in one domBainA to initial performance in another domain. Superficial differences between the two Idomains will have their largest negative effect when the domains are unfamiliar. ThFis does not require that students show the benefit of one day of Calculus on the fir stO day of Physics. Rather, it is expected that they will be better Physics students at yYear's end for having had a year's study of Calculus. In other words, if students werIe Ttaught numerical cognition to restructure their negative thoughts in MathematicsR theSy would perform better. Consequently they would be able to transfer this knowlEedge to university settings – especially when they are exposed to social sciencIe-Vbased or educational statistics. N Representation and degree of practice are critical for determining the transfer Ufrom one task to another, and transfer varies from one domain to another as a function of the number of symbolic components that are shared. The amount of transfer depends on where attention is directed during learning. Training on the cues that signal the relevance of an available skill may deserve much more emphasis than they now typically receive in instruction. 61 Like the second claim, the claim that training by abstraction is of little use is a corollary of the earlier claims. Nonetheless, one might argue for it even if one dismisses the others. The third claim has been extended into an advocacy for apprenticeship training (Brown, Collins, & Duguid, 1989; Collins, Brown, & Newman, 1989). It is argued that, because current performance will be facilitated to the degree that the context closely matches prior experience, the most effective training is an apprenticeship to others in the performance situation. This claim is used more than any other to chaYllenge the legitimacy of school-based instruction. R Abstract instruction can be ineffective if what is taught in the claAssroom is not what is required in the career situation. Often this is an indictment ofR the design of the classroom instruction rather than of the idea of abstract instructIioBn in itself. However, sometimes it is an indictment of the career situation. For in stLance, Los Angeles police after leaving the police academy are frequently told by mNore experienced officers "now forget everything you learned" (Independent CommAission on the Los Angeles Police Department, 1991). The consequence is that pAolicDe officers are produced who, ignoring their classroom training in the face of coBntrary influences during apprenticeship, may violate civil rights and make searchFes wIithout warrants. Clearly, one needs to create a better correspondence between career performance and abstract classroom instruction and sometimes this means changi ngO the nature of the career performance (including the structure of motivationsT anYd rewards) and fighting unwanted and deleterious effects of apprenticeship learninIg. Likewise in this study, students need to be counseled that Mathematics mRay Sbe useful in any career they might find themselves. This is one of the significanceEs of this theory to the present study. Teachers need to stop teaching MatheImVatics in abstract but rather practicalize the usefulness of the subject to the real life sitNuations. U The issue of choosing between abstract and very specific instruction can be viewed in the following way. If abstract training is given, learners must also absorb the money and time costs of obtaining supplemental training for each distinct application. But if very specific training is given, they must completely retrain for each application. Which is to be preferred, and to what extent, do they depend on the balance among (a) 62 the cost of the more general abstract training, (b) the cost of the specific training, (c) the cost of the supplemental training for application of abstract training, and (d) the range of careers over which the learner is likely to have occasion to apply what was learned. Someone who will spend years performing a single set of very specific tasks might be well advised to focus on specific training. But if the cost of supplemental training is not large (i.e., if there is substantial transfer over the range of tasks), or if technological or other changes are likely to alter tasks substantially over the years, or if the range oYf tasks the learner is likely to address over time is substantial, then abstract AtraiRning with supplemental applications training is clearly preferable. It is easy to woRrk out an exercise of this kind by assigning numbers to the various costs and to the variability of the tasks encountered, and thereby to show that there is no solution that is opItBimal for all cases. Most modern information-processing theories arNe "l e Larning-by-doing" theories which imply that learning would occur best with a cAombination of abstract instruction and concrete illustrations of the lessons of this instruction. Numerous experiments show combining abstract instruction with specific concreDte examples (Cheng, Holyoak, Nisbett, & Oliver, 1986; Fong, Krantz, & Nisbett,I 1B986 A; Reed & Actor, 1991) is better than either one alone. One of the most famoFus studies demonstrating this was performed by Scholckow & Judd (described in Judd, 1908; a conceptual replication by Hendrickson & Schroeder, 1941). They had ch ilOdren practice throwing darts at a target underwater. One group of subjects receivTed aYn explanation of refraction of light which causes the apparent location of the targetI to be deceptive. The other group only practiced, receiving no abstract instrucRtionS. Both groups did equally well on the practice task which involved a target 12 inEches under water, but the group with abstract instruction did much better when asked to transfer to a situation where the target was now under only 4 inches of waNter.I V U A variation on the emphasis on apprenticeship training is the emphasis that has been given to using only "authentic" problems (Lesh & Lamon, 1992). What is authentic is typically ill-defined but there seems to be a strong emphasis on having problems be like the problems students might encounter in everyday life. A focus on underlying cognitive process would suggest that this is a superficial requirement. Rather, it is 63 reinstated that the authors would argue as have others (Hiebert, Hearner, Carpenter, Fennema, Fuson, 1994) that the real goal should be to get students motivated and engage in cognitive processes that will transfer. What is important is what cognitive processes problem evokes and not what real-world trappings it might have. The fourth claim that instruction is only effective in a highly social environment. This claim that instruction is best in a highly social environment comes not from tho se advocating situated learning, per se, but from those advocating the advantages Yof co- operative learning (Johnson & Johnson, 1989) as an instructional tool. CoR-operative learning, also known as "communities of practice" and "group learninAg", refers to learning environments where people of equal status work together Rto enhance their individual acquisition of knowledge and skills. This environmenItB or structure is to be contrasted with tutoring (where the tutor and tutee are of une qLual knowledge and status) and team training (where the desired outcome is coNncerned with team or group performance). In a review by the Committee on TeAchniques for the Enhancement of Human Performance (National Research CouncilD, 1994), it was noted that research on cooperative learning has frequently noBt bAeing well controlled (e.g., non-random assignments to treatments, uncontrolled I"teacher" and treatment effects), that relatively few studies "have successfully demoFnstrated advantages for cooperative versus individual learning," and that "a number oOf detrimental effects arising from cooperative learning have been identified –T thYe "free rider," the "sucker," the "status differential," and "ganging up" effects (SIalomon and Globerson, 1989). The autRhor‘Ss point is not to say that cooperative learning cannot be successful or sometimes better than individual learning. But that, it is not a panacea that always providIeVs ou Etcomes superior or even equivalent to those of individual training. It has been seNen that cooperative learning is still part of numerical cognition and could be very Ueffective in teaching students to learning cooperatively. This particular theory also indicates that it is good to familiarize the students with what they could gain with their learning. There is a strong point that students may prepare their minds on the type of career they would like to do, and which would not require Mathematics at all. In this case this would affect their thinking, perception and attitudes towards Mathematics. 64 2.2.2 Constructivism Theory This theory postulates that knowledge is active rather than passive. In other words, learning Mathematics must be an active process. Learning Mathematics requires a change in the learner, which can only be brought about by what the learner does – what he or she attends to, what activities he or she engages in. However, there is a ris ing interpretation of constructivism that rejects the information-processing approach Y(Cobb, 1990). Such views are often espoused by those claiming to practicRe "radical constructivism". Even among radical constructivists, positions vary and sAome theorists seem to be making philosophical claims about the nature of knowRledge rather than empirical claims. Indeed, in the extreme, constructivism denies theI rBelevance of empirical data to educational decisions. However, some of the claims al soL have clear psychological implications that are not always supported. This studyA is Nalso supported by the following claims: That knowledge would be viewed as anD active, constructive process in which students attempt to resolve problems that arise as they participate in the Mathematics practices of the classroom. Such a view emBphAasizes that the learning-teaching process is interactive in nature and involves the i mIplicit and explicit negotiation of Mathematics meanings. In the course of these nFegotiations, the teacher and students elaborate the taken-as-shared Mathematics Oreality that constitutes the basis for their ongoing communication (Cobb, YacYkel, & Wood, 1992). As an exampleI Tof this, Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, & Pertwitz (1991R) deSscribe an effort to teach second graders to count by tens. Rather than telling the Estudents the principle directly, they assigned groups of students the task of countiInVg objects bundled in sets of ten. Invariably, the groups discovered that counting byN tens is more efficient than counting by ones. Building a whole second-grade Ucurriculum around such techniques, they found their students doing as well on traditional skills as students from traditional classrooms, transferring more, and expressing better attitudes about Mathematics. One can readily agree with one part of the constructivist claim: that learning must be an active process. Learning requires a change in the learner, which can only be 65 brought about by what the learner does – what he or she attends to, what activities he or she engages in. The activity of a teacher is relevant to the extent that it causes students to engage in activities they would not otherwise engage in – including, but not limited to, acquiring knowledge provided by the teacher or by books. A teacher may also engage students in tasks, some of which may involve acquisition of skills by working examples. Other tasks include practicing skills to bring them to effective levels, interacting w ith their fellow students and with the teacher, and so on. Y The problem posed to psychology and education is to design a Rseries of experiences for students that will enable them to learn effectively and to moAtivate them to engage in the corresponding activities. On all of these points, it woulRd be hard to find grounds for disagreement between constructivists and other cogniItiBve psychologists. The more difficult problem, and the one that often leads to dLifferent prescriptions, is determining the desirable learning goals and the experienNces that, if incorporated in the instructional design, will best enable students to aDchieAve these goals. Of course, arriving at good designs is not a matter for philosophAical debate; it requires empirical evidence about how people, and children in particIuBlar, actually learn, and what they learn from different educational experiences. One finds frequent referOenceF to Jean Piaget as providing a scientific basis for constructivism. Piaget has had enormous influence on our understanding of cognitive development and indeedT wYas one of the major figures responsible for the emergence of cognitivism from the eIarlier behaviourist era in psychology. It is fair to say that many of his specific claRims Shave been seriously questioned, the general influence of his theoretical perspectiveE remains. Key to constructivism is Piaget's distinction between assimilation and aIccommodation as mechanisms of learning and development. Assimilation is a relNative Vly passive incorporation of experience into a representation already available to Uthe child. However, when the discrepancies between task demands and the child's cognitive structure become too great, the child will reorganize his or her thoughts. This is called accommodation (and often nowadays, "re-representation"). Piaget emphasized how the child internalizes by making changes in mental structure. The constructivists make frequent reference to this analysis, particularly the 66 non-passive accommodation process. (In this respect, constructivism is quite different from situated learning which emphasizes the external bases of cognition.) A more careful understanding of Piaget would have shown that assimilation of knowledge also plays a critical role in setting the stage for accommodation – which the accommodation cannot precede without assimilation. Some constructivists (Cobb, 1990) have mistakenly implied that mod ern information-processing theories deal only with assimilation and do not incorporaYte the more constructive accommodation. Far from this, the learning-by-doing theoriRes that are widely employed in cognitive science are in fact analyses of how cognAitive structure accommodates to experience. The authors briefly describe two such Ranalyses, both to correct the misrepresentation of information-processing theory aIndB to establish a more precise framework for discussing the effects of instruction. L In Anderson's (1993) study, one principal learnNing mechanism is knowledge compilation. When learners come upon problemsD theAy do not know how to solve, they can look at an example of how a similar problem is solved (retrieved either from memory or some external source) and try to solvBe tAhe problem by analogy to this example. Knowledge compilation is the accommo dIation process by which new procedures (rules) are created to produce more dOirectFly the computation that this retrieve-and-analogize process requires. In Feigenbaum aTnd YSimon's (1984) study, learning involves gradually building up discrimination net for Irecognizing objects and taking appropriate actions. Discrimination net is a sequenRce oSf tests that are applied to various features of an object. Gradually, the system develops a complex sensitivity to the situations and stimuli in its environment in a continIuVing Eprocess of re-representation, or accommodation. N These theories provide concrete realizations of what it means for a system to Uconstruct knowledge. As such they provide a basis for examining the constructivist's claim that knowledge cannot be instructed. If passive recording is what is meant by "instruct" these learning mechanisms cannot be instructed. However, it is quite wrong to claim that what is learned is not influenced by explicit instruction. For instance, in Anderson's learning by analogy, instruction serves to determine the representation of the 67 examples from which one "constructs" one's understanding, and Pirolli and Anderson (1985) showed in the domain of recursive programming that what one learns from an example is strongly influenced by the instruction that accompanied the example. In Feigenbaum and Simon's (1984) study, which has had extensive success in modeling human learning in a variety of perceptual and verbal learning tasks (Simon & Feigenbaum, 1964), learning is strongly influenced by the sequence of stimuli and the feedback that tells the system when responses are correct, and when they are wrongY. There is a great deal of research showing that, under some circumstancRes, people are better at remembering information that they create for themselves thaAn information they receive passively (Bobrow & Bower, 1969; Slamecka & Graf, 197R2). However, this does not imply that people do not remember what they are told.I IBndeed, in other cases people remember as well or even better information that is pLrovided than information they create (Slamecka & Katsaiti, 1987; Stern & BransfordN, 1979). When, for whatever reason, students cannAot construct the knowledge for themselves, they need some instruction. The Dargument that knowledge must be constructed is very similar to the earlier arBgumAents that discovery learning is superior to direct instruction. In point of fact, Fthe reI is very little positive evidence for discovery learning and it is often inferior (Charney, Reder & Kusbit, 1990). Discovery learning, even when successful in acquir inOg the desired construct, may take a great deal of valuable time that could have bTeenY spent practicing this construct if it had been instructed. Because most of the lIearning in discovery learning only takes place after the construct has been found, whSen the search is lengthy or unsuccessful, motivation commonly flags. As Ausubel (1R968) wrote, summarizing the findings from the research on discovery learninIgV tw Eenty-five years ago:"actual examination of the research literature allegedly suNpportive of learning by discovery reveals that valid evidence of this nature is virtually Unonexistent. It appears that the various enthusiasts of the discovery method have been supporting each other research-wise by taking in each other's laundry, so to speak, that is, by citing each other's opinions and assertions as evidence and by generalizing wildly from equivocal and even negative findings." (p. 497-498) 68 It is sometimes argued that direct instruction leads to "routinization" of knowledge and drives out understanding: "the more explicit a teacher is about the behaviour he/she wishes his/her students to display, the more likely it is that they will display the behaviour without recourse to the understanding which the behaviour is meant to indicate; that is, the more likely they will take the form for the substance." Brousseau (1984). An extension of this argument is that excessive practice will also driYve out understanding. This criticism of practice (called "drill and kill," as if thRis phrase constituted empirical evaluation) is prominent in constructivist writings. All evidence, from the laboratory and from extensive case studies of professionals, Rindicates that real competence only comes with extensive practice (Hayes, 198I5B; Ericsson, Krampe, Tesche-Romer, 1993). In denying the critical role of pracNtice o Lne is denying children the very thing they need to achieve real competence. ThAe instructional task is not to "kill" motivation by demanding drill, but to find tasks tDhat provide practice while at the same time sustaining interest. Substantial evidence shows that there are a number of ways to do this; learning-from-examples – method, is BoneA such procedure that has been extensively and successfully tested in school situat ioIns. The evidence, then, leads to the following conclusions about the role of studentF and teacher in learning: Learning requires a change in the learner, which can only beO brought about by what the learner does. The activity of a teacher is relevant to the eYxtent that it causes students to engage in activities they would not otherwise engage iIn.T The tasRk is Sto design a series of experiences for students that will enable them to learn effectively and to motivate them to engage in the corresponding activities. The learnin EIgV-by-doing theories that are widely employed in cognitive science are analyses of hoNw cognitive structure accumulates to experience. When students cannot construct the Uknowledge for themselves, they need some instruction. There is very little positive evidence for discovery learning and it is often inferior. In particular, it may be costly in time, and when the search is lengthy or unsuccessful, motivation commonly flags. People are sometimes better at remembering information that they create for themselves than information they receive passively, but in other cases they remember as 69 well or better information that is provided than information they create. Real competence only comes with extensive practice. The instructional task is not to "kill" motivation by demanding drill, but to find tasks that provide practice while at the same time sustaining interest. There are a number of ways to do this, for instance, by "learning-from- examples." The claim of the constructivist school that knowledge cannot be represen ted symbolically is more an epistemological claim in the constructivist's hands Ythan a psychological claim. The claim is that there are subtleties in human understaRnding that defy representation in terms of a set of rules or other symbol structures (CobAb, 1990). The argument is not really about whether knowledge is actually so represenRted in the human head, but whether knowledge, by its very nature, can be repIreBsented symbolically. Searle's well-known attempt to show that, in principle, a Lsymbolic system cannot understand language (the "Chinese Room" metaphor, SAearNle, 1980) is an extension of this claim. Among the misconceptions underlyingA theD claim that knowledge is non-symbolic is the faulty notion that "symbolic" meIanBs "expressed in words and sentences, or in equivalent formal structures." SymboFls a re much more than formal expressions. Any kind of pattern that can be stored andO can refer to some other pattern, say, one in the external world, is a symbol, capable of b eing processed by an information-processing system. A substantial nuTmbYer of symbolic systems have been built that can store symbol structures representingI mental images of external events and can reason about the events pictorially with theS help of these structures (Larkin, 1981). Careful comparison with the behaviour Eof hRuman subjects reasoning about pictures or diagrams shows that these systemIsV capture many of the basic properties and processes of human imagery. Searle's ChNinese Room story fails because the inhabitants of his postulated room, unlike humans Uand other symbolic systems do not have a sensory window on the world – cannot associate a pattern in memory with the external object that can be seen and denoted by that pattern. To know is to represent accurately what is outside the mind; so to understand the possibility and nature of knowledge is to understand the way in which the mind is 70 able to construct such representations (Cobb, Yackel, and Wood, 1992, from Rorty, 1979). The representational view of mind, as practiced in cognitive psychology, certainly makes no claims that the mind represents the world accurately or completely, and no strong claims about the nature of knowledge as a philosophical issue. The true representational position is compatible with a broad range of notions about the relation of the mind to the world, and about the accuracy or inaccuracy and completenYess or incompleteness of our internal representations of the world's features. Its AclaiRm simply: Cognitive competence (in this case Mathematics competence) depends on the availability of symbolic structures (e.g., mental patterns or mental images) thRat are created in response to experience. IB The misinterpretation of the representational view lea dsL to much confusion about external Mathematics representations (e.g., equations, graNphs, rules, Dienes blocks, etc.) versus internal representations (e.g., production DruleAs, discrimination nets, and mental images). Believing that the representational version of learning records these external representations passively and withoIuBt Atransformation into distinct internal representations, constructivists takeF in adequacies of the external representations as inadequacies of the notion of inOternal representation. For instance, if a set of rules in a textbook is inadequate this is taken as an inability of production rules to capture the concepts. However, coTgniYtive theories postulate (and provide evidence for) complex processes for transIforming (assimilating and accommodating) these external representationsR to pSroduce internal structures that are not at all isomorphic to the external representatiEons. IVWhile it is true that education has proceeded for centuries without a theory of intNernal representation, this is no reason to ignore the theories that are now coming from Ucognitive psychology. Let us consider the analogy of medicine: For thousands of years before there was any real knowledge of human physiology, remedies for some pathological conditions were known and used, sometimes effectively, by both doctors and others. But the far more powerful methods of modern medicine were developed concurrently with the development of modern physiology and biochemistry, and are 71 squarely based on the latter developments. To acquire powerful interventions in disease, we had to deepen our understanding of the mechanisms of disease – of what was going on in the diseased body. In the same way, human beings have been learning, and have been teaching their offspring, since the dawn of our species. There is a reasonably powerful "folk medicine," based on lecturing and reading and apprenticeship and tutoring, aided by such technolo gy as paper and the blackboard – a folk medicine that does not demand much knowYledge about what goes on in the human head during learning and that has not changeRd radically since schools first emerged. To go beyond these traditional techniques, thAere is need to follow the example of medicine and build (as we have been doing forR the past thirty or forty years) a theory of the information processes that underlie sIkiBlled performance and skill acquisition: that is to say, we must have a theory of the w aLys in which knowledge is represented internally, and the ways in which such inAternNal representations are acquired. In fact, cognitive psychology has now progressed a long way toward such a theory, and, as we have seen, a great deal is already knownA thaDt can be applied, and is beginning to be applied, to improve learning processes. B In summary, contrary to the cIlaim that knowledge cannot be represented symbolically, the evidence indicatesF the following actual state of affairs: Symbols are much more than formal expres siOons. Any kind of pattern that can be stored and can refer to some other pattern, TsayY, one in the external world, is a symbol, capable of being processed by an infoIrmation-processing system. Cognitive competence (in this case Mathematics coRmpSetence) depends on the availability of symbolic structures (e.g., mental patterns or Emental images) that are created in response to experience. Cognitive theories postulIatVe (and provide evidence for) complex processes for transforming (assimilating anNd accommodating) these external representations to produce internal structures that are Uquite different from the external representations. Today instruction is based in large part on "folk psychology." To go beyond these traditional techniques, there is need to continue to build a theory of the ways in which knowledge is represented internally, and the ways in which such internal representations are acquired. 72 The third claim by construtivists is that knowledge can only be communicated in complex learning situations. Part of the "magical" property of knowledge asserted in the second claim, that there is something in the nature of knowledge that cannot be represented symbolically, is that no simple instructional situation suffices to convey the knowledge, whatever it may be. This assertion is the final consequence of rejecting decontextualization. Thus, constructivists recommend, for example, that children learn all or nearly all of their Mathematics in the context of complex problems (LYesh & Zawojeski, 1992). This recommendation is put forward without any evidencRe as to its educational effectiveness. A There are, of course, reasons sometimes to practice skills inI Bthei Rr complex setting. Some of the reasons are motivational and some reflect the special skills that are unique to the complex situation. The student who wishes to play violin inL an orchestra would have a hard time making progress if all practice were attemptNed in the orchestra context. On the other hand, if the student never practiced as a meAmber of an orchestra, critical skills unique to the orchestra would not be acquiredA. ThDe same arguments can be made in the sports context, and motivational argumentsB can also be made for complex practice in both contexts. A child may not see the Fpoi ntI of isolated exercises, but will when they are embedded in the real-world taskO. Children are motivated to practice sports skills because of the prospect of playing in full-scale games. However, they often spend much more time practicing componTent Yskills than full-scale games. It seems important, but this is not a reason to make this tIhe principal mechanism of learning. While tRhereS may be motivational merit to embedding Mathematics practice in complex siEtuations, Geary (1995) notes that there is a lot of reason to doubt how intrinsIicVally motivating complex Mathematics is to most students in any context. The kinNd of sustained practice required to develop excellence in an advanced domain is not Uinherently motivating to most individuals and requires substantial family and cultural support (Ericsson, Krampe, & Tesch-Romer, 1993). Geary argues, as have others (Bahrick & Hall, 1991; Stevenson & Stigler, 1992), that it is this difference in cultural support that accounts for the large difference in Mathematics achievement between Asian and American children. 73 Contrary to the contention that knowledge can always be communicated best in complex learning situations, the evidence shows that: A learner who is having difficulty with components can easily be overwhelmed by the processing demands of a complex task. Further, to the extent that many components are well mastered, the student wastes much time repeating these mastered components to get an opportunity to practice the few components that need additional effort. There are reasons sometimes to practice skills in their complex setting. SoYme of the reasons are motivational and some reflect the skills that are unique to thRe complex situation. While it seems important both to motivation and to learning to Apractice skills from time to time in full context, it is important to reiterate thatI tBhis Ris not a reason to make this the principal mechanism of learning. The fourth claim is that it is not possible to apply sta nLdard evaluations to assess learning. The denial of the possibility of objective evaluNation could be the most radical and far-reaching of the constructivist claims. TheyD puAt it last because it is not clear how radically this principle is interpreted by all conAstructivists. Certainly, some constructivists have engaged in rather standard evaluIatBions of constructivist learning interventions (Cobb, Wood, Yackel, Nicholls, WhFeatl ey, Trigaitti, & Perlwitz, 1992). However, others are very uncomfortable with theO idea of evaluation. As Jonassen (1992) writes: "If you believe, as radical constructivi sts do, that no objective reality is uniformly interpretable by all learners, then asTsessYing the acquisition of such a reality is not possible. A less radical view suggests Ithat learners will interpret perspectives differently, so evaluation processes shouRld acScommodate a wider variety of response options." (p. 144). In thEe hands of the most radical constructivists, the fourth claim implies that it is imposIsiVble to evaluate any educational hypothesis empirically because any such test neNcessarily requires a commitment to some arbitrary, culturally-determined, set of values. UIn the hands of the more moderate constructivists, the claim manifests itself in advocacy of focusing evaluation on the process of learning more than the product, in what are considered "authentic" tasks, and by involving multiple perspectives in the evaluation. This milder perspective calls for emphasis on more subjective and less precisely defined instruments of evaluation. While the authors share with most educators their 74 instinctive distaste of four-alternative forced-choice questions and they agree that Mathematics assessment should go beyond merely testing computational skills, they question whether the very open-ended assessment being advocated as the proper alternative will lead to either more accurate or more culture-free assessment. The fundamental problem is a failure to specify precisely the competence being tested for and a reliance on subjective judgment instead. The authors then examined a number of rec ent papers in Wirzup and Streit (1992) addressing this issue. In one of such papers, ReYsnick, Briars, and Lesgold (1992) presented two examples of answers that are oRbjectively equivalent (and receive equal scores in their objective assessment schemAe). However, they are uncomfortable with this equal assessment and feel a subjRective component should be added so one answer would receive a higher score beLcaIusBe it displayed greater "communication proficiency." Although the "better" answNer h ad neater handwriting, one might well judge it as just more long-winded as theA "worse" answer. "Communication proficiency" is very much in the eyes of the beholder. In another paper, Dossey (1992), in explaining the new NAEP open-ended scoring, staDtes that a student will be given 50% (2 points) for the right answer if the justificatBion Afor the answer is "not understandable" but will be given 100% (4 points) for Ithe wrong answer if it "does not reflect misunderstanding of either the probFlem or how to implement the strategy, but rather seems to be a copying error or cOomputational error." While they are sympathetic with the sentiments behind such ideYas, such subjective judgments will open the door to a great deal of cultural bias inI aTssessment (Rist, 1970). Anytime the word "seems" appears in an assessment, it sRhouSld be a red flag that the assessors do not know what they are looking for. The infEormation-processing approach would advocate precisely specifying what one is lookIiVng for in terms of a cognitive model and then precisely testing for that. N Another sign of the constructivist's discomfort with evaluation manifests itself in Uthe motto that the teacher is the novice and the student the expert (Von Glasersfeld, 1991). The idea is that every student gathers equal value from every learning experience. The teacher's task is to come to understand and value what the student has learned. As Confrey (1991) writes: 75 "Seldom are students' responses careless or capricious. We must seek out their systematic qualities which are typically grounded in the conceptions of the student...frequently when students' responses deviate from our expectations, they possess the seeds of alternative approaches which can be compelling, historically supported Y and legitimate if we are willing to challenge our R own assumptions." (p. 122) A R And also as Cobb, Wood, and Yackel (1991) write: IB "The approach holds that students are the Lbest judges of what they find problematiNcal and encourages them to construct solDutioAns that they find acceptable given their current ways of knowing." (p. 158). BA I If the student is supposed to movOe, inF the course of the learning experiences, from a lower to a higher level of compete nce, one wonders why the student's judgments of the acceptability of solutionTs aYre particularly valid. While the teacher who can appreciate children's individualityI, see their insights and motivate them to do their best and to value learning, is alsRo vaSlued, then there must be definite educational goals. More generally, if the "studenEt –as- judge" attitude were to dominate education, it would no longer be clear when IinVstruction had failed and when it had succeeded, when it was moving forward and whNen moving backward. It is one thing to understand why the student, at a particular Ustage in understanding, is doing what he or she is doing. It is quite another matter to help the student understand how to move from processes that are "satisfactory" in a limited range of tasks to processes that are more effective over a wider range. As Resnick (1994) argues, many concepts which children naturally come to (e.g., that motion implies force) are not what the culture expects of education and that in these cases "education must 76 follow a different path: still constructivist in the sense that simple telling will not work, but much less dependent on untutored discovery and exploration (p. 489)." Again, there is an important empirical reason for proceeding in assessment in somewhat different ways from those recommended by constructivists, and particularly, the more radical among them. It is shared by all, an instinctive distaste for four- alternative forced-choice questions, but these are not required to attain validity or reliability in assessment. Accurate and culture-free assessment does requires, hoYwever that the competence being tested for to be specified precisely without undue rReliance on subjective judgment. Subjective judgments open the door to cultural bias inA assessment. It cannot be assumed that students' judgments of the acceptability Rof solutions are particularly valid and as stated earlier, if the "student –as- judge"I Bview were adopted, it would no longer be clear when instruction had failed and wNhen Lit had succeeded 2.2.3 Chain Reaction or Cycle Theory DA Mathematics anxiety has been explained in termAs of a chain reaction or cycle. Spielberger (1972) conceptualised anxiety as a s taIteB, trait and a process. As is described by Spielberger (1972), anxiety is a resFult of a chain reaction that consists of a stressor, a perception of threat, a state reaction, cognitive reappraisal and coping. Mitchell (1987) described a Mathematics aYnxie ty O cycle and stated that maths anxiety experienced in the present has its roots in the past. Anxiety is perpetuated through negative self-talk manifesting in beliSefs Iw Thich cause anxiety. This leads to physical symptoms, an inability to think and avRoidance which, in turn, leads to the inability to perform, causing anxiety and more Enegative self-talk, and the continuation of the Mathematics anxiety cycle (MitchIeVll 1987). This cycle leads to negative educational and societal Mathematics attNitudes which often become a self-fulfilling prophecy, and generally leads to UMathematics avoidance (Williams 1988). 2.2.4 Social Cognitive Theory Social Cognitive Theory is the overarching theoretical framework of the self- efficacy construct (Bandura, 1986). Within this perspective, one's behaviour is constantly 77 under reciprocal influence from cognitive (and other personal factors such as motivation) and environmental influences. Bandura calls this three-way interaction of behavior, cognitive factors, and environmental situations the "triadic reciprocality." Applied to an instructional design perspective, students' academic performances (behavioural factors) are influenced by how learners themselves are affected (cognitive factors) by instructional strategies (environmental factors), which in turn builds on itself in cycli cal fashion. The methods for changing students' percepts of efficacy, according to BYandura (1977, 1986), are categorically subsumed under four sources of efficacy informRation that interact with human nature: (1) enactive attainment, (2) vicarious exAperience, (3) persuasive information, and (4) physiological state. R The reciprocal nature of the determinants of human IfBunctioning in social cognitive theory makes it possible for therapeutic and counse linLg efforts to be directed at personal, environmental, or behavioural factors. StrategieNs for increasing well-being can be aimed at improving emotional, cognitive, or mAotivational processes, increasing behavioural competencies, or altering the sociAal cDonditions under which people live and work. In school, for example, teachers IhBave the challenge of improving the learning achievement and confidence of the sFtude nts in their charge. Using social cognitive theory as a framework, teachers can wOork to improve their students' emotional states and to correct their faulty self-beliefs and habits of thinking (personal factors), improve their academic skills and seTlf-rYegulatory practices (behaviour), and alter the school and classroom structures Ithat may work to undermine student success (environmental factors). S BandurRa's social cognitive theory stands in clear contrast to theories of human functioInVing E that overemphasize the role that environmental factors play in the deNvelopment of human behaviour and learning. Behaviourist theories, for example, show Uscant interest in self-processes because theorists assume that human functioning is caused by external stimuli. Because inner processes are viewed as transmitting rather than causing behaviour, they are dismissed as a redundant factor in the cause and effect process of behaviour and unworthy of psychological inquiry. For Bandura, a psychology without introspection cannot aspire to explain the complexities of human functioning. It 78 is by looking into their own conscious mind that people make sense of their own psychological processes. To predict how human behavior is influenced by environmental outcomes, it is critical to understand how the individual cognitively processes and interprets those outcomes. More than a century ago, William James (1890/1981) argued that "introspective observation is what we have to rely on first and foremost and always" (p. 185). For Bandura (1986), "a theory that denies that thoughts can regulate actio ns does not lend itself readily to the explanation of complex human behaviour" (p. 15)Y. Similarly, social cognitive theory differs from theories of human functiRoning that overemphasize the influence of biological factors in human development aAnd adaptation. Although it acknowledges the influence of evolutionary factors in humRan adaptation and change, it rejects the type of evolutionism that views social behaIviBour as the product of evolved biology but fails to account for the influence tha t Lsocial and technological innovations that create new environmental selection presNsures for adaptiveness have on biological evolution (Bussey & Bandura 1999). AInstead, the theory espouses a bidirectional influence in which evolutionaryA preDssures alter human development such that individuals are able to create increasinBgly complex environmental innovations that, "in turn, create new selection pressurFes f oIr the evolution of specialized biological systems for functional consciousness, thought, language, and symbolic communication" (p. 683). This bidirectional influence r eOsults in the remarkable intercultural and intracultural diversity evident in our planYet. Social cognitivIe Ttheory is rooted in a view of human agency in which individuals agents are proaRctivSely engaged in their own development and can make things happen by their actionEs. Key to this sense of agency is the fact that, among other personal factors, individIuVals possess self-beliefs that enable them to exercise a measure of control over theNir thoughts, feelings, and actions, that "what people think, believe, and feel affects Uhow they behave" (Bandura, 1986). Bandura provided a view of human behaviour in which the beliefs that people have about themselves are critical elements in the exercise of control and personal agency. Thus, individuals are viewed both as products and as producers of their own environments and of their social systems. Because human lives are not lived in isolation, Bandura expanded the conception of human agency to include 79 collective agency. People work together on shared beliefs about their capabilities and common aspirations to better their lives. This conceptual extension makes the theory applicable to human adaptation and change in collectivistically-oriented societies as well as individualistically-oriented ones. Individuals have self-regulatory mechanisms that provide the potential for self- directed changes in their behaviour. The manner and degree to which people self-regul ate their own actions and behaviours involve the accuracy and consistency of theiYr self- observation and self-monitoring, the judgments they make regarding AtheRir actions, choices, and attributions, and, finally, the evaluative and tangible reactions they make to their own behaviour through the self-regulatory process. This lasItB sub Rfunction includes evaluations of one's own self (their self-concept, self-esteem, values) and tangible self- motivators that act as personal incentives to behave in self-d iLrected ways. For Bandura (1986), the capability that is most "distinctly human" (pN. 21) is that of self-reflection, hence it is a prominent feature of social cognitive theoAry. Through self-reflection, people make sense of their experiences, explore their AowDn cognitions and self-beliefs, engage in self-evaluation, and alter their thinking andB behaviour accordingly. Of all the thoughts that affect hu mIan functioning, and standing at the very core of social cognitive theory, are self-efficFacy beliefs, "people's judgments of their capabilities to organize and execute cou rOses of action required attaining designated types of performances" (p. 391T). YSelf-efficacy beliefs provide the foundation for human motivation, well-beingI, and personal accomplishment. This is because unless people believe that their aSctions can produce the outcomes they desire, they have little incentive to act or toE perRsevere in the face of difficulties. Much empirical evidence now supports BanduIraV's contention that self-efficacy beliefs touch virtually every aspect of people's livNes—whether they think productively, self-debilitatingly, pessimistically or Uoptimistically; how well they motivate themselves and persevere in the face of adversities; their vulnerability to stress and depression, and the life choices they make. Self-efficacy is also a critical determinant of self-regulation. Of course, human functioning is influenced by many factors. The success or failure that people experience as they engage the myriad tasks that comprise their life 80 naturally influence the many decisions they must make. Also, the knowledge and skills they possess will certainly play critical roles in what they choose to do and not do. Individuals interpret the results of their attainments, however, just as they make judgments about the quality of the knowledge and skills they posses. Bandura's (1997) key contentions as regards the role of self-efficacy beliefs in human functioning is that "people's level of motivation, affective states, and actions are based more on what they believe than on what is objectively true" (p. 2). For this rYeason, how people behave can often be better predicted by the beliefs they hold aRbout their capabilities than by what they are actually capable of accomplishing, fAor these self- efficacy perceptions help determine what individuals do with the knoRwledge and skills they have. This helps explain why people's behaviours are someLtimIeBs disjoined from their actual capabilities and why their behaviour may differN wid ely even when they have similar knowledge and skills. For example, many taAlented people suffer frequent (and sometimes debilitating) bouts of self-doubt about capabilities they clearly possess, just as many individuals are confident about what theyD can accomplish despite possessing a modest repertoire of skills. Belief and BreaAI lity are seldom perfectly matched, and individuals are typically guided byF th eir beliefs when they engage the world. As a consequence, people's accomplishments are generally better predicted by their self- efficacy beliefs than by their p rOevious attainments, knowledge, or skills. Of course, no amount of confidence or seYlf-appreciation can produce success when requisite skills and knowledge are absent. I T It bearsR noSting that self-efficacy beliefs are themselves critical determinants of how well kEnowledge and skill are acquired in the first place. The contention that self-efficacIyV beliefs are a critical ingredient in human functioning is consistent with the view ofN many theorists and philosophers who have argued that the potent affective, evaluative, Uand episodic nature of beliefs makes them a filter through which new phenomena are interpreted. People's self-efficacy beliefs should not be confused with their judgments of the consequences that their behaviour will produce. Typically, of course, self-efficacy beliefs help determine the outcomes one expect. Confident individuals anticipate successful 81 outcomes. Students confident in their social skills anticipate successful social encounters. Those confident in their academic skills expect high marks in exams and expect the quality of their work to reap personal and professional benefits. The opposite is true of those who lack confidence. Students who doubt their social skills often envision rejection or ridicule even before they establish social contact. Those who lack confidence in their academic skills envision a low grade before they begin an examination or enroll in a course. The expected results of these imagined performances will be diffYerently envisioned: social success or greater career options for the former, social isRolation or curtailed academic possibilities for the latter. A Because the outcomes we expect are themselves the result of Rthe judgments of what we can accomplish, our outcome expectations are unliIkBely to contribute to predictions of behaviour. Moreover, efficacy and outcome jLudgments are sometimes inconsistent. A high sense of efficacy may not result inN behaviour consistent with that belief; however, if the individual also believes thatA the outcome of engaging in that behaviour will have undesired effects. A student Dhighly self-efficacious in her academic capabilities may elect not to apply to a partBicuAlar university whose entrance requirements are such as to discourage all but the hIardiest souls. Low self-efficacy and positive outcome expectations are also possFible. For example, students may realize that strong Mathematics skills are essentia l Ofor a good GRE score and eligibility for graduate school, and this, in turn, may ensurYe a comfortable lifestyle, but poor confidence in Mathematics abilities are likely to kIeeTp them away from certain courses and they may not even bother with the GRE or Sgraduate school. In the social arena, a young man may realize that pleasing soEcialR graces and physical attractiveness will be essential for wooing the young lass wIhVo has caught his eye, which, in turn, may lead to a romantic interlude and even a lasNting relationship. If, however, he has low confidence in his social capabilities and Udoubts his physical appearance, he will likely shy away from making contact and hence miss a potentially promising opportunity. Because individuals operate collectively as well as individually, self-efficacy is both a personal and a social construct. Collective systems develop a sense of collective efficacy—a group‘s shared belief in its capability to attain goals and accomplish desired 82 tasks. For example, schools develop collective beliefs about the capability of their students to learn, of their teachers to teach and otherwise enhance the lives of their students, and of their administrators and policymakers to create environments conducive to these tasks. Organizations with a strong sense of collective efficacy exercise empowering and vitalizing influences on their constituents, and these effects are palpable and evident. Even average-ability students are sometimes known to do poorly in sYpecific subject areas while performing up to standard in others. This phenomAenoRn is often reflected in the domain of Mathematics. The reasons for this phenomenon no doubt reflect the multivariate nature of school learning. We must alsoI Btake R into account the idiosyncratic nature of diverse learners. When capable learners do not perform up to their potential despite positive environmental conditions, we must gLive more attention to the self-regulatory processes within individuals that promoteN or inhibit performance. From the social-cognitive view, self-efficacy is an imporAtant factor that resides within the learner and mediates between cognition and aAffeDct, and results in changes in academic performance (Zimmerman, Bandura, & MIaBrtinez-Pons, 1992). The growth and reduction of self-efficacy is influenced over timFe by social comparison with peers and is therefore more pronounced as one grows older. By the time children re acOh middle school (grades six through eight), the majority of them have made siTgnifYicant judgments regarding their preferences toward certain academic domains. TIhese judgments are no doubt influenced by their perceived capability with regSard to the domains, as a result of social comparison with peers and feedback frEomR teachers. This is particularly true in the domain of Mathematics. At this stage, IcVhildren are already making decisions leading to career directions and choice of claNsses. By high school, these decisions become more solidified. For educators, the Ucritical time to reduce or prevent Mathematics alienation is in middle school, or early in high school. Elementary school children usually have greater confidence in their academic capabilities, and this confidence extends equally across gender to both verbal and Mathematics domains of learning. In later years, however, gender differences regarding 83 Mathematics begin to emerge. Fennema and Sherman (1978) found that there were no significant differences with gender and Mathematics learning, or with gender and motivation for learning, for 1,300 middle school children. There were, however, significant effects on Mathematics confidence and on perceptions of Mathematics as a male domain, with boys reportedly averaging higher on both variables. When these results are compared to previous research by the same authors, using the same design but with high school students (Sherman and Fennema, 1977) the overall results indicaYte that the gender gap on Mathematics confidence and perceptions begins to widenR in middle school and increasingly widens in high school. Although these studieRs didA not measure self-efficacy, per se, the significant variables of confidence and gender stereotyping of a domain are contributing sources of self-efficacy information. IB Bandura (1977), sought to address the relatedN q ue Lstion of what mediates knowledge and action beginning with his seminal worAk on self-efficacy. Bandura (1986) defines the performance component of self-efficacy as people's judgments of their capabilities to organize and execute courses of acDtion required to attain designated types of performances. It is not concerned with BtheA strategies one has but with judgments of what one can do with whatever strateg ieIs one possesses. Students feel self-efficacious when they are able to picture thOemsFelves succeeding in challenging situations, which in turn determines their level of effort toward the task (Paris & Byrnes, 1989; Salomon, 1983; 1984). Y Bandura (BanIduTra 1977, 1986) asserts that self-percepts of efficacy highly influence whether Sstudents believe they have the coping strategies to successfully deal with challeEngiRng situations. One's self-efficacy may also determine whether learners chooseI Vto engage themselves in a given activity and may determine the amount of effort leaNrners invest in a given academic task, provided the source and requisite task is Uperceived as challenging (Salomon, 1983, 1984). Several researchers have since investigated the relationship of self-efficacy to learning and academic achievement, but research in the area of academic performance is still developing (Lent, Brown, & Larkin, 1986; Multon, Brown & Lent, 1991; Schunk, 1994). 84 People make judgments about their capabilities based on enactive experience, vicarious experience (observation), persuasive information, and physiological states. In school, children gain knowledge and experiences through experiential activities. They also gain information based on seeing how peers they judge to be similar to themselves perform at various levels and under given circumstances. They also are told by teache rs, peers, family and others about their expected capabilities. Children give themYselves physiological feedback about their capabilities through symptoms such aAs sRoreness or sweating. These sources of efficacy information are not mutually exclusive, but interact in the overall process of self-evaluation. Bandura, Adams, & Beyer (R1977) advise that enactive experience is a highly influential source of efficacy iInfBormation. Successful experiences raise self-efficacy with regard to the target Nperf o Lrmance while experiences with failure lower it. Another source of efficacy informatioDn Ais vicarious experience through observation. Observing peers, or peer modelAs, especially those with perceived similar capabilities, carry out target performanceIsB which result in evaluative information about one's personal capabilities. Verbal persuasion or convincFing serves as another source of efficacy information. Teachers, for example, can rai sOe or inhibit students' percepts of efficacy by suggesting whether or not they havTe thYe capabilities to succeed in a given task (Bouffard-Bouchard, 1989). Models can alsIo be used to demonstrate to self-doubters that personal capabilities are more often Ra reSsult of effort rather than innate capability. Students often have physical reactions to anticipated events. Many a public speaker testifies to sweaty palms and nervouIsV vo Ecal reactions when performing a speech. These physiological indicators are soNurces of self-efficacy information as well. U Social cognitive theory postulates that the aforementioned sources of self efficacy information are the most influential determinants of performance. An important assumption of Social Cognitive Theory is that personal determinants, such as forethought and self-reflection, do not have to reside unconsciously within individuals. People can consciously change and develop their cognitive functioning. This is important to the 85 proposition that self-efficacy too can be changed, or enhanced. From this perspective, people are capable of influencing their own motivation and performance according to a model of triadic reciprocality in which personal determinants (such as self-efficacy), environmental conditions (such as treatment conditions), and action (such as practice) are mutually interactive influences. Improving performance, therefore, depends on changing some of these influences. Within the model of triadic reciprocality, the ability to influence various peYrsonal determinants is accorded to five basic human capabilities: 1) symbolizing, 2) foRrethought, 3) vicarious, 4) self-regulatory, and 5) self-reflective. People are generally gAifted with the capability of symbolizing. In an academic context, this allows learners tRo process abstract experiences into models that guide their learning and perforImBance. For example, observing someone on computer or videotape vocalize a c omLputational algorithm for calculating may serve as an adequate instructional ArepNresentation of performing that procedure. One can learn how to perform the stratDegy in this manner, and may even gain in self-efficacy by observing a peer model thatA this procedure is within the scope of one's own capabilities. B Forethought, the cognitive rFepr eIsentation of future events, is also a powerful causal influence on one's learnOing. For example, watching a self-efficacious model perform a Mathematics calcula tion using a particular strategy may lead the observer to foresee this within the TscoYpe of his or her own capabilities and consequently expect to perform the procedureI with success. VicarioRus cSapability occurs by observing others and vicariously experiencing what they dEo. According to Bandura (1986), if we had to directly experience everything we leIarVn, we would not have adequate time and opportunity to learn very much. ObNserving a model's thinking through text-based soliloquy, for example, can direct the Uobserver on how to conceptualize a Mathematics calculation or overcome self-doubts about successful performance. Students typically self-regulate by determining what capabilities they have with regard to a given task and in effect compare those capabilities against a set of standards they maintain for themselves. Students who believe that they can achieve a high grade in a Mathematics course may persist in their efforts to achieve 86 the grade. Conversely, low self-efficacy pertaining to a given task may inhibit one's effort and persistence (Bouffard-Bouchard, 1989). People compare their performance with that of their peers in various contexts, especially the classroom. The accuracy of their assessments determines whether they overestimate or underestimate their capabilities. Consequently, accurate self-reflection is critical to the development of self-efficacy. Y 2.3 EMPIRICAL BACKGROUND R 2.3.1 Numerical Cognition and Mathematics Anxiety A There is paucity of literature on numerical cognition and MathemRatics anxiety. So there is no much available literatures on the domain of behaviour. IInB a study by Hopko et al (1999) it was found that Mathematics-anxious individuals hLave a deficient inhibition mechanism whereby working memory resources are Nconsumed by task-irrelevant distracters. A consequence of this deficiency was thatA explicit memory performance was poorer for high-anxious individuals. TheyA aDlso found no relationship between competence and Mathematics anxiety. TIhBere are a great many causes postulated for Mathematics anxiety. In a study of eFigh t adult learners, Zopp (1999) found that unrelated life events, trigger events in education and a lack of support contributed to Mathematics anxiety in her subjects. In add iOtion, parents with Mathematics anxiety pass it along to their children, while teaTchYers with Mathematics anxiety pass it along to their students (Fiore 1999). JacksonI et al (1999) studied 157 students in a senior-level elementary education MathRemSatics class in college by giving them the prompt, ―Describe your worst or most cEhallenging Mathematics classroom experience from kindergarten through collegIe‖V. They were also asked to describe factors that would have made their exNperiences more positive. These subjects were above average in academic achievement, Uhighly motivated with an average age of 26. Mathematics anxiety and performance across several initial studies, have found substantial evidence for performance differences as a function of Mathematics anxiety. These differences typically are not observed on the basic whole-number facts of simple addition or multiplication (e.g., 7 + 9, 6 X 8) but are prominent when somewhat more 87 difficult arithmetic problems are tested. In particular, Ashcraft and Faust (1994; also Faust, Ashcraft, & Fleck, 1996) have shown that high-Mathematics-anxiety participants have particular difficulty on two-column addition problems (e.g., 27 + 18), owing largely to the carry operation. When such problems were answered correctly, the time estimate for the embedded carry operation was nearly three times as long for high-anxiety participants as it was for low-anxiety participants (Faust et al., 1996). Thus, hi gh- Mathematics-anxiety participants showed slower, more effortful processing Yon a procedural aspect of performance, performing the carry operation (for Rsuggestive evidence on Mathematics affect and procedural performance in a numeriAcal estimation task, (LeFevre, Greenham, & Waheed, 1993). Furthermore, their higRher error rates on these problems, often showing classic speed-accuracy tradeoffs IwBhen confronted with relatively difficult arithmetic, indicated a willingness to sNacri fi Lce accuracy on especially difficult trials, either to avoid having to deal with tAhe stimulus problem or merely to speed the experimental session along. An interpretation, of course, is that highD-anxiety participants are simply less competent in Mathematics, unable to perfoBrm tAhe necessary calculations at the same level of accuracy as low-anxiety individuals. IThe literature documents that there is indeed a significant relationship between MaFthematics anxiety and Mathematics competence or achievement in Hembree's (199 0O) meta-analysis. If the correlation holds across all levels of problem difficulty, thYen competence and Mathematics anxiety are completely confounded, and perfoIrTmance differences cannot be uniquely attributed to either factor. Results reporteRd elSsewhere, however, suggest that there is not a complete confounding of MathematicEs anxiety and Mathematics competence. IVFaust et al. (1996), for instance, showed equivalent performance across MNathematics-anxiety groups to simple one- and two-column addition and multiplication Uproblems when those problems were tested in an untimed, pencil-and-paper format. It is important to note that the larger of these problems had shown Mathematics-anxiety effects in laboratory tasks, suggesting strongly that the on-line anxiety reaction had compromised participants' ability to demonstrate their basic competence. Ashcraft and Kirk (1998) examined Mathematics competence and Mathematics anxiety more 88 thoroughly in a study that administered a standardized Mathematics achievement test. Simple whole-number arithmetic problems showed no Mathematics anxiety effects at all, whereas accuracy for the higher Mathematics-anxiety groups did decline more on the later test lines at which more difficult arithmetic (e.g., mixed fractions) and Mathematics problems (e.g., factoring) were tested. Finally, Hembree (1990) noted an interesting outcome in his meta-analysis on Mathematics anxiety. Reports on the most effective treatment interventionYs for Mathematics anxiety, behavioral and cognitive-behavioral approaches, alsoR presented evidence of post treatment increases in Mathematics achievement or compAetence scores to levels "approaching the level of students with low Mathematics Ranxiety" (p. 43). Because the treatments did not involve instruction or practice in MIBathematics, it is quite improbable that treatment itself improved individuals' Mathem Latics competence. Instead, it seems very likely that the low pretreatment achievemeNnt scores of high-Mathematics- anxiety individuals were depressed by Mathematics aAnxiety during the assessment itself and that relief from Mathematics anxiety thenA peDrmitted a more accurate assessment of Mathematics achievement and competence.B On the basis of such eviFden cIe, it would appear that lower Mathematics competence cannot be offered asO a simple, wholesale explanation for all the performance decrements associated with hig h Mathematics anxiety (see Ashcraft & Kirk, 1998, for a full discussion of this arTgumYent). Instead, these performance decrements seem to call for an explanation involviIng numerical cognitive processing. A growing body of evidence attests to the centrSality of working memory to processes such as reading and reading comprehensEionR (Just & Carpenter, 1992), reasoning (Baddeley & Hitch, 1974; Jonides, 1995),I aVnd retrieval from long-term memory (Conway & Engle, 1994; Rosen & Engle, 19N97; Richardson et al., 1996). U The various components of these mental processes are often attributed to one or another of the three major subcomponents—the central executive, the auditory rehearsal loop, or the visuo-spatial sketchpad—in Baddeley's (1986, 1992) well-known model. There is a supportive although not extensive literature on the role of working memory in Mathematics cognition. Since Hitch's (1978) early article on multistep arithmetic problem 89 solving, there have been several reports on the critical role of working memory in Mathematics performance. As an example, Geary and Widaman (1992) demonstrated that working memory capacity was closely related to skill in arithmetic problem solving and, in particular, to the speed of both fact retrieval and execution of the carry operation. In both cases, the higher the capacity of working memory, the faster were the component processes (Ashcraft, 1992, 1995; Lemaire, Abdi, & Fayol, 1996; Widaman, Gea ry, Cormier, & Little, 1989). So, for example, executing the carry operation is thoughYt to be controlled by working memory, thus placing significant demands on the capaRcity of the working memory system (Ashcraft, Copeland, Vavro, & Falk, 1999; HitchA, 1978; Logie, Gilhooly, & Wynn, 1994). Accordingly, they hypothesized that a majorR contributor to the performance deficits found for high-Mathematics-anxiety participIaBnts involves working memory. In particular, such deficits are predicted to stem fr omL that portion of working memory, presumably the central executive that appliNes the various procedures of arithmetic during problem solving (Ashcraft etD al.A, 1999; Butterworth, Cipolotti, & Warrington, 1996; Darke, 1988, and Sorg & Whitney, 1992). More generally, Eysenck and CalvoB (19A92) have proposed an overall model of the anxiety-to-performance relationship in cIognitive tasks, which is called the processing efficiency theory. Their most relevanFt prediction for the present topic is that performance deficits due to generalized anx ieOty will be prominent in exactly those tasks that tap the limited capacity of workinYg memory. In their theory, the intrusive thoughts and worry characteristic of high aInTxiety are thought to compete with the ongoing cognitive task for the limited proRcessSing resources of working memory. The result of such competition is either a slEowing of performance or a decline in accuracy—in other words, lower cognitIivVe efficiency. Because high-anxiety individuals must expend greater cognitive effNort to attain the same level of performance achieved by low-anxiety individuals, Uprocessing efficiency is lower for high-anxiety individuals. Most of Eysenck's work (Eysenck, 1992) is based on results with either generalized anxiety disorder individuals or individuals who exhibit high trait anxiety. Eysenck (1992) discusses a whole range of anxiety-related phenomena, for instance, increased physiological arousal, selective attention, and distractibility. For the 90 tasks under consideration here, however, the consequences of anxiety that affect working memory processes are the most relevant function of Mathematics anxiety, especially on tasks that require intensive processing within working memory. We do not test the specifics of Eysenck and Calvo's (1992) prediction here, which states that it is intrusive thoughts and worry (in this case, about Mathematics) that detract from available working memory capacity. Instead, we assess the more general prediction that Mathemat ics anxiety disrupts working memory processing when the cognitive task involves aritYhmetic or Mathematics-related processes. R Regarding Mathematics intervention, some researchers, SeethalAer and Fuchs (2005) analyze the literature in terms of the efficacy of studies cIoBmp Rleted. In a similar research, Augustyniak, Murphy and Phillips (2005) argue Lthat the research on the definition of a Mathematics disability is lacking with respe ct to identification of core deficits. They identify the core areas needing further eNxplanation as numerical skills, visual/spatial deficits, cognitive skill development (mAemory retrieval, working memory, speed of processing, attention regulation, proDblem solving) and social cognition. Mazzocco (2005) reviewed research reIgBard Aing practices of early identification and intervention for students with MathFem atics difficulties. The commentary discusses the criteria and nature of MathematiOcs difficulties and notes the need for additional research. Butler, Beckingham, an d Lauscher (2005) report on three case studies regarding the support of students TwitYh Mathematics learning challenges. Three 8th grade students were given assistanceI in self-regulating their learning. General strategies found to be successful includeSd: engaging the students in constructive conversation; supporting students in EreflRection on their learning; and, the need for teachers to engage in dynamic, curricuIlum-based forms of assessment. FuNchs, F Vuchs, and Hamlett (2006) report on the validation of an intervention to improve UMathematics problem solving in third grade. The intervention (HotMathematics) involved explicit instructions, self-regulation strategies, and tutoring. Results indicated positive, short-term results for problem-solving skills. 91 2.3.2 Numerical Cognition and Mathematics Achievement Although different theoretical orientations of researchers have often caused differing operational definitions, the common conceptualization of numerical cognition learners is that they are active participants in their own learning (Zimmerman, 1990). The research agrees on at least two major findings with respect to Numerical Cognition and academic achievement: Numerical Cognition is comprised of several components, su ch as cognitive strategies and effort (Miller, Behrens, Greene, & Newman, 19Y93) or metacognition and effort (Pintrich & De Groot; 1990; Yap, 1993), although thRe specific components were not always identical; and students who employ metacRognAition and exert effort perform more successfully (Pintrich & De Groot, 199I0;B Zimmerman, 1986; Zimmerman & Martinez-Pons, 1986, 1988). To make a summLary of the key features in most definitions of Numerical Cognition is the systema tic use of metacognitive, motivational, and/or behavioral strategies. MoreoverA, nuNmerical cognition learners are distinguished by both awareness of the relationship between strategic regulatory processes and learning outcomes, and the usAe oDf these strategies to achieve academic goals (Zimmerman, 1990). B Although there have been num eIrous theoretical and empirical articles about Numerical Cognition (Garcia, 1995F; Garcia & Pintrich, 1991, 1994, 1995; Pintrich & Garcia, 1991; Schunk & Zim mOerman, 1994; Zimmerman, 1994), few have explicitly linked the components Yof Numerical Cognition to academic achievement in Mathematically-giftedI sTtudents and to each other. In those studies that have explicitly investigated thRese Scomponents, the correlational relationships tend to be small (e.g., Pintrich & EDe Groot, 1990, Yap, 1993). In this study, Numerical Cognition conjoins two major IcVonstructs: (a) metacognition, consisting of awareness (consciousness), planning (gNoal setting), self-checking (monitoring), and the cognitive strategies students use to Ulearn, remember, and understand; and (b) management and control of effort. This study additionally investigated the relationship of learning goal orientation, self-efficacy, and worry to high-stakes Mathematics achievement and with each other. Each of the study's variables was discussed in greater detail. 92 In their review of the research, Alexander, Carr, and Schwanenflugel (1995) found that gifted children possessed greater metacognition than the general cohort. Schwanenflugel, Moore Stevens, and Carr (1997) also found that children who made causal metacognitive comments were likely to be more strategic in their cognitive processing. "Express" [gifted] pupils employed effective retention strategies more frequently than "normal" students (Chang, 198 9). Although metacognition is thought to differ from other cognitive learning strategieYs such as rehearsal, elaboration, and organization, there is mixed evidence aboutA theR extent to which respondents can actually distinguish their use of metacognitive and cognitive strategies. According to Boufard-Bouchard, Parent, and Larivee (1993R), gifted learners monitor comprehension more effectively than non-gifted studentIs.B They also use more strategies in a flexible manner. Some researchers foun dL distinct cognitive and metacognitive factors using exploratory factor analysesN (Pokay & Blumenfeld, 1990; Pintrich & De Groot, 1990). However, the correlatiAons between the scales measuring these factors were high (r = .60 and r = .83) in theDse two studies respectively, and neither correlation was corrected for measurement BerroAr, thus raising concerns about the extent to which students can accurately distinguis hI their use of the various strategies. Further, Yap (1993) found that a composite indeFx of cognitive strategies correlated very high with three commonly used indices o f Ometacognitive strategies, awareness (.97), planning (.95), and self-checking (.96). TheY present study sheds further light on this debate. With respect toI eTffort, both Bandura (1993) and Schunk (1984) see effort as both being directly RinflSuenced by self-efficacy and directly affecting skill or performance. Bandura (1E993) suggested that self-regulatory skills are meaningless if students cannot apply IthVemselves in a persistent manner in the face of difficulties, distractions, and stress, anNd that "self-directed learning requires motivation as well as cognitive and Umetacognitive strategies" (p. 136). Zimmerman (1990) also observed that self-regulated learners display extraordinary effort and persistence during learning and report high self- efficacy, self-attributions, and intrinsic motivation. Additionally, Bandura (1993) posited that "self-efficacy beliefs contribute to motivation in several ways. They determine the 93 goals people set for themselves, how much effort they expend and how long they persevere in the face of failures" (p. 131). There is some debate in the literature concerning the distinction between effort and metacognition. Although conceptually it makes sense to distinguish a generalized motivational disposition (i.e., effort) from more specific metacognitive strategies (e.g., planning, self-checking, awareness), there is some evidence t hat individuals themselves cannot distinguish these strategies through self-report. IYn their correlational study of Numerical Cognition in 7th-graders, Pintrich and De GrRoot (1990) originally intended to treat effort management and Numerical CognitioAn as separate constructs, but a preliminary exploratory factor analysis did not sIupBpor Rt the construction of two separate scales. Along the same lines, Yap (1993) used a confirmatory statistics to examine the effort/metacognition distinction in a diveNrse sa Lmple of 640 12th-grade students and found that self-report scales for Aeffort and metacognition lacked discriminant validity. In contrast, Pokay and BlumDenfeld (1990) report a small zero-order correlation between effort management and metacognitive strategy used early (r = .34) and late (r = .39) in the semester in a samBplAe of 283 high school students. This study posits that Numerical Cognition is com prIised of effort and metacognition, and one of its goals is to further examine the effort/F metacognition distinction. Chang (1989) found tha t Ogifted students expressed greater enjoyment in learning a subject than normal stTudeYnts. The question is whether these students become gifted because they enjoyed lIearning in that particular domain. Determining cause and effect on questions suchR as tShese will have interesting and profound implications for practitioners in this fieldE. According to Dweck (1986, 1990), children who believe in intelligence as a fixed tIraVit or entity tend to orient towards performance goals, whereas those who believe intNelligence is incremental and malleable tend to orient towards learning goals. Her Uresearch indicated that when seeking performance type goals, children based their task choice and pursuit process around ability. With learning goals, however, the choice and pursuit process was focused on progress and mastery through effort. Low performing students believed that ability is a fixed trait, whereas gifted students were more likely to believe that ability to learn can be improved (Schommer & 94 Dunnell, 1997). Students who adopted a learning or mastery orientation increased perceptions of self-confidence (self-efficacy) and success in their courses (Dweck & Leggett, 1988). A number of studies clearly show that students demonstrate high levels of Numerical Cognition when they are oriented toward learning goals (e.g., Meece, 1994; Schunk, 1994). Weiner (1986) found that children with low perceived ability were still mastery-oriented when their goal was to learn rather than to perform. Bandura (199 3) emphasized that learning environments that accept ability as a skill that may be acYquired and de-emphasize competition and social comparison are well suited for buiRlding self- efficacy and promoting academic achievement. Furthermore, Dweck's (1A986) research indicated that students whose focus is based on ability judgments tend Rto withdraw from challenges, "whereas a focus on progress through effort creates a tIenBdency to seek and be energized by challenge" (p. 1041). The adaptive motivationa l Lpattern studied by Dweck (1986) "is characterized by challenge seeking and high, eNffective persistence in the face of obstacles" (p 1040). Dweck contended that childAren with learning goals use these obstacles as a cue to increase their effort or to AanaDlyze and vary their strategies. Based on the assumption that gifted students will bIeB more learning-goal-oriented for this study, it was hypothesized that the results wFill agree with those of Dweck (1986) and Schunk (1994); that is, learning goal Oorientation would be positively related to Numerical Cognition and self-efficacy. Bandura (1986) TdefYined self-efficacy as "people's judgments of their capabilities to organize and execIute courses of action required to attain designated types of performance" (Rp. 3S91). Implicitly, self-efficacy refers to people's specific beliefs about their capabEility to perform certain actions or to bring about intended outcomes in a domaiInV or to otherwise exert control over their lives (Bandura, 1986, 1993; Boekaerts, 19N92; Schunk, 1990). Data on self-efficacy were collected in this study to determine the Urelationship between the proposed factors of Numerical Cognition and their relationship with worry and high-stakes Mathematics achievement for a gifted sample. The focus was test performance. Collins (1984) and Pintrich and Schrauben (1992) noted that more efficacious students monitored their performance and applied more effort than students who were low in self-efficacy. Bandura (1993) said that people with high self-efficacy 95 "heighten and sustain their efforts in the face of failure. And also they attribute failure to insufficient effort or deficient knowledge and skills that are acquirable" (p. 144). An excellent review of self-efficacy research is provided by Pajares (1996b). Research of the gifted (Bogie & Buckholt, 1987; Chan, 1988; Feldhusen & Nimlos-Hippen, 1992; Vallerand, Gagne, Senecal, & Pelletier, 1994; Zimmerman & Martinez-Pons, 1990) examined self-perceptions of competence in gifted students. In general, these stud ies indicated "that gifted students perceive themselves as more competent and areY more intrinsically motivated toward school tasks" (Chan, 1996, p. 184) than their peeRrs. In their path model, Zimmerman, Bandura, and Martinez-Pons (A1992) showed that self-efficacy for Numerical Cognition influenced self-efficaRcy for academic achievement; self-efficacy for academic achievement then inLfluIenBced final grades via student goals for their grades. The combined direct and inNdire ct effect of self-efficacy for academic achievement on final grades was ([BetAa] = .37, p [is less than] .05). Zimmerman and Bandura (1994) found essentially the same results in their study ([Beta] = .38). In their path model, Garcia and Pintrich D(1991) found that intrinsic motivation (comparable to learning goal orientation iBn thAis study) had a substantial effect on self- efficacy ([Beta] = .36), and that both in trIinsic motivation and self-efficacy had moderate effects on Numerical Cognition O([BeFta] = .24 and [Beta] = .26, respectively). This study did not investigate the role of these motivational effects on academic achievement, but did posit that self-efficaTcy wYill be strongly and positively related to Numerical Cognition and Mathematics achieIvement. Most reseaSrch has shown that high worry is associated with low cognitive performance (HRembree, 1988, 1990; Pajares & Urdan, 1996; Seipp, 1991). However, a few stIudVies E showed no relationship (e.g., Wigfield & Meece, 1988). Anxiety, on the other haNnd, may be differentiated into two components: worry (cognitive) and emotionality U(physiological/affective) (Hembree, 1988; Hong, 1998, O'Neil & Fukumura, 1992). In several studies, worry has had a stronger negative correlation with achievement than emotionality; in response, Seipp (1991) recommended that studies predicting academic achievement would be better served by using only the worry component. It was 96 hypothesized that worry would be negatively related to Numerical Cognition, self- efficacy, learning goal orientation, and high-stakes Mathematics achievement. The research on certain types of Mathematics continues to support gender differences in favor of males (Fennema & Carpenter, 1998), although there is evidence indicating "that females' achievement is similar to males in all but the most advanced levels of Mathematics" (National Science Foundation, 1996). According to Seegers a nd Boekaerts (1996), there have been, and continue to be, significant gender differenYces in performance on complex Mathematics tasks (Fennema & Carpenter, 1998). RThis study used the Advanced Placement in calculus as the performance indiAcator. It was hypothesized that males would outpIerBform R females. In summary, self-regulated learners are students who plan and check their work, are aware of their thought processes, use cognitive strategies to ac hLieve their goals, and exert effort. This study investigated Numerical Cognition anNd the effects of self-efficacy, learning goal orientation, and worry on achievement inA a sample of mathematically gifted high school students in an Advanced PlacementD Program course in Mathematics. The study's objectives were to extend the BtheAoretical and empirical research on goal orientation, self-efficacy, and Numerica l CIognition by determining whether learning goal orientation and self-efficacy are reFlated to Numerical Cognition, documenting their relationships to worry and high -Ostakes Mathematics achievement, and controlling for the effects of gender. Y Pintrich and IDTe Groot (1990) found that although self-efficacy facilitated cognitive engaRgemSent, the cognitive engagement variables were more directly tied to performancEe. They also found a negative relationship between worry and self-efficacy but noI sVignificant relationship of test anxiety (worry) with Numerical Cognition. In this stuNdy, self-efficacy was more tied to performance than were Numerical Cognition and its Uconcomitant variables, and that worry had a significant negative relationship with both self-efficacy and Numerical Cognition. Using the Motivated Strategies for Learning Questionnaire (MSLQ), Pintrich and De Groot (1990) found that students with higher self-efficacy, intrinsic value (learning goal orientation), cognitive strategy use, and use of self-regulating strategies (metacognition/effort) had significantly higher grades, better 97 seatwork, and better scores in exams/quizzes and essays/reports. Even though the methodologies used by Pintrich and De Groot (1990) and criterion variables were different, many of their results were comparable; in particular, they both found empirical evidence "for the importance of considering both motivational and Numerical Cognition components in thier models of classroom academic performance" (p. 38). Schunk (1984) determined that self-efficacy had both a direct and indirect (as mediated by persistence) path of influence to cognitive skill development. EmphaYsizing that the goal was to learn to solve problems (rather than simply completing Rthem) can raise self-efficacy for learning and increase Numerical Cognition and persAistence in 4th- grade children (Schunk, 1995). In 1994, Schunk posited that "studeRnts who adopt a learning goal are apt to experience a sense of self-efficacy for IskBill improvement and engage in activities they believe enhance learning (e.g., exLpend effort, persist, use effective strategies)" (p. 89). However, Pintrich and DNe Groot (1990) found a non- significant relationship between learning goal orientaAtion and self-efficacy. All of their other findings, with this one exception, were compDarable to Schunk's (1995). With a group of high school stBudeAnts, Pajares and Kranzler (1995) found significant positive direct paths from s eIlf-efficacy to Mathematics performance and a significant negative path to anxietyF. Pajares and Kranzler found no gender effects for these students, either on self-e ffOicacy or performance. A significant correlation between Mathematics self-efficacy Yand problem-solving performance was indicated in college students (Pajares & MIilTler, 1994, 1995). Pajares and Miller (1994) found a gender effect favoring the MathSematics self-efficacy of male undergraduates but found no gender effect on probleRm-solving performance. IVIn a E study of Mathematics self-efficacy in 8th-grade students, Pajares (1996a) foNund a direct effect of gender on self-efficacy for regular education students but no Udirect effect of gender on performance (boys had higher self-efficacy). For gifted students, there was a direct effect of self-efficacy and gender on performance (girls had higher performance), but no gender effect on self-efficacy. Pysher (1996) also found no significant gender differences in Mathematics test scores, goals, or self-efficacy. Pintrich and De Groot (1990)‘s findings with Mathematically-gifted students generally agree with 98 these authors: A significant direct path was indicated both from self-efficacy to Mathematics performance and from self-efficacy to worry; and whereas no significant gender effects on performance were found, there was a significant effect on self-efficacy. 2.3.3 Emotional Freedom Technique and Mathematics Anxiety Previous research (Salas, 2000; Wells, et al., 2003), theoretical writings (Arens on, 2001, Callahan, 1985, Durlacher, 1994, Flint, 1999, Gallo, 2002, Hover-Kramer,Y 2002, Lake & Wells, 2003, Lambrou & Pratt, 2000, and Rowe, 2003), and many caRse reports have suggested that energy psychology is an effective psychotherapy Atreatment that improves psychological functioning. Research evidence for efficacy oRf EFT and related therapies is only beginning to emerge. Research (Church, 2008) IhBas noted frequent co- occurrence of psychological symptoms such as anxiety and dLepression with addiction. This study examined the psychological conditions of 28 aNdults at an addictions workshop at which participants learned EFT (Emotional FreedoAm Techniques), a widely practiced form of energy psychology. The study employAed Da time-series, within-subjects repeated measures design to evaluate symptoms aItB the start of the workshop, at the end of the workshop, and, to determine long-teFrm effects, 90 days later. A statistically significant decrease in the two global scalesO, the global severity index and positive symptom total, as well as the anxiety, and obsess ive-compulsive symptom scales was observed with gains maintained at follow-upT. ImYprovement in somatization was found at posttest only, while improvement in interIpersonal sensitivity occurred at the 90-day follow-up. These findings suggeRst ESFT may be an effective adjunct to addiction treatment by reducing the severity ofE general psychological distress, and in particular, anxiety and obsessive-compuIlVsive symptoms. Rowe‘s (2005) study was to measure any changes in psNychological functioning that might result from participation in an experiential UEmotional Freedom Techniques (EFT) workshop and to examine the long-term effects. Brattberg (2008) carried out a study to examine if self-administered EFT (Emotional Freedom Techniques) leads to reduced pain perception, increased acceptance, coping ability and health-related quality of life in individuals with fibromyalgia. 86 women, diagnosed with fibromyalgia and on sick leave for at least 3 months, were 99 randomly assigned to a treatment group or a waiting list group. An eight-week EFT treatment program was administered via the Internet. Upon completion of the program, statistically significant improvements were observed in the intervention group (n=26) in comparison with the waiting list group (n=36) for variables such as pain, anxiety, depression, vitality, social function, mental health, performance problems involving work or other activities due to physical as well as emotional reasons, and stress symptoms. P ain catastrophizing measures, such as rumination, magnification and helplessnessY, were significantly reduced, and the activity level was significantly increased. R Wells, Polglase, Andrews, Carrington and Banker (2003) explorAed whether a Emotional-freedom-based procedure, Emotional Freedom Techniques R(EFT), can reduce specific phobias of small animals under laboratory-controlled IcBonditions. Randomly assigned participants were treated individually for 30 mNinut es L with EFT (n = 18) or a comparison condition, Diaphragmatic Breathing (DBA) (n = 17). Findings revealed that EFT produced significantly greater improvement Dthan did DB behaviorally and on three self-report measures, but not on pulse rate.A The greater improvement for EFT was maintained, and possibly enhanced, at 6 -I 9B months follow-up on the behavioral measure. These findings suggest that a singFle t reatment session using EFT to reduce specific phobias can produce valid behavOioral and subjective effects. A research (Swingle, P ulos & Swingle, 2005), studied the effects of EFT on auto accident victims sufferiTng Yfrom post traumatic stress disorder -- an extremely disabling condition that involvIes unreasonable fears and often panic attacks, physiological symptoms of RstreSss, nightmares, flashbacks, and other disabling symptoms. These researchersE found that three months after they had taught EFT (in two sessions) those auto aIcVcident victims who reported continued significant symptom relief also showed sigNnificant positive changes in their brain waves. It was assumed that the clients showing Uthe continued positive benefits were those who continued with home practice of self-administered EFT. Feinsten (2008b) utilizes cognitive operations such as imaginal exposure to traumatic memories or visualization of optimal performance scenarios—combined with physical interventions derived from acupuncture, yoga, and related systems—for 100 inducing psychological change. While a controversial approach, this combination purportedly brings about, with unusual speed and precision, therapeutic shifts in affective, cognitive, and behavioral patterns that underlie a range of psychological concerns. Four tiers of energy psychology interventions include 1) immediate relief/stabilization, 2) extinguishing conditioned responses, 3) overcoming complex psychological problems, and 4) promoting optimal functioning. The first tier is m ost pertinent in psychological first aid immediately following a disaster, with the subsYequent tiers progressively being introduced over time with complex stress reactionAs aRnd chronic disorders. Feinsten (2008) utilizes imaginal and narrative-generated expoRsure, paired with interventions that reduce hyperarousal through acupressure anIdB related techniques. According to practitioners, this leads to treatment outco mLes that are more rapid, powerful, and precise than the strategies used in other expNosure-based treatments such as relaxation or diaphragmatic breathing. The methodD haAs been exceedingly controversial. It relies on unfamiliar procedures adapted fromA non- Western cultures, posits unverified mechanisms of action, and early claims IofB unusual speed and therapeutic power ran far ahead of initial empirical support. TFhe study reviews a hierarchy of evidence regarding the efficacy of energy psychologOy, from anecdotal reports to randomized clinical trials. Benor, Ledger, Toussa int and Zaccaro (2008) explored test anxiety benefits of Wholistic Hybrid, EmoTtionYal Freedom Techniques (EFT), and Cognitive Behavioural Therapy. Participants iIncluding Canadian university students with severe or moderate test anxiety participRatedS. A double-blind, controlled trial was conducted. Standardized anxiety measures inEcluded: the Test Anxiety Inventory (TAI) and Hopkins Symptom Checklist (HSCLI-V21). Despite small sample size, significant reductions on the TAI and HSCL-21 weNre found for WHEE; on the TAI for EFT; and on the HSCL-21 for CBT. There were Uno significant differences between the scores for the three treatments. In only two sessions WHEE and EFT achieved the equivalent benefits to those achieved by CBT in five sessions. Participants reported high satisfaction with all treatments. EFT and WHEE students successfully transferred their self-treatment skills to other stressful areas of their lives. WHEE and EFT show promise as effective treatments for test anxiety. 101 Church (2008a) examined a cross section of 194 healthcare professionals, including physicians, nurses, psychotherapists, chiropractors, psychiatrists, alternative medicine practitioners, and allied professionals. The study examined whether self- intervention with Emotional Freedom Techniques (EFT), a brief exposure therapy that combines a cognitive and a somatic element, had an effect on subjects‘ levels of anxiety, depression, and other psychological symptoms. The study utilizes within-subjects, tim e- series, repeated measures design. Besides measuring the breadth and intensYity of psychological distress, this instrument has nine subscales for specific cRonditions, including anxiety and depression. It was administered to subjects before andA after an EFT demonstration and self-application that lasted about 90 minutes. SRubjects also self- reported physical pain, emotional distress, and cravings on a 10 pIoBint Likert-type scale. Subjects received a single page homework EFT reminder sh eLet, and their frequency of practice was tracked at follow up. EFT self-application reNsulted in statistically significant decreases in pain, emotional distress, and cravingsA, and improvements for all nine subscales. On the two general scales, symptomA seDverity dropped by 34%, and symptom breadth by 40% relative to normal baselineBs (both p<.001). Also pain scores dropped by 68%, the intensity of traumatic memorie s Iby 83%, and cravings by 83% (all p<.001). Callahan (1985) developed a Fcausal diagnostic procedure gleaned in part from the insights and discoveries of Ochiropractor George Goodheart, D.C., who related neuromuscular functionT anYd organ system health to the acupuncture Emotional-freedom system. Callahan (198I5) utilized muscle testing methods found in Goodheart's Applied Kinesiology, and JSohn Diamond's Behavioral Kinesiology (Diamond, 1979) to therapy localize (EidenRtify) which acupuncture Emotional-freedoms were involved in psychoIlVogical issues. Once identified, Callahan has the patient repeatedly tap fingers on a deNsignated treatment point along that acupuncture Emotional-freedom to effect change or Urestore balance in that Emotional-freedom. Frequently, the causal diagnostic methods produce a sequence of acupuncture Emotional-freedom points to be tapped. As an outgrowth of the success of The Callahan Techniques", tapping as treatment on the acupuncture Emotional-freedoms has continued, and has been incorporated into other acupuncture Emotional-freedom based psychotherapies. Callahan (1985) asserted that the 102 tapping provides an external source of energy which, when done correctly, at the right spot, with the mind tuned to the problem being treated, balances the energy in a particular energy system in the body which is suffering from a deficiency or imbalance. A couple of years later Callahan (1992) commented on his practical and theoretical ideas related to tapping. He asserted that the points being tapped are related to the ancient Emotional- freedoms of acupuncture. Tapping the proper point when the person is thinking of the problem is quite effective. He then stressed that these points are transducers of eYnergy; where the physical energy of tapping can be transduced into the approprAiateR (probably electromagnetic) energy of the body so that the person with a problemR can be put into proper balance by a knowledgeable person. Callahan's decision to tap acupoints originated in a proIcBedure introduced by Goodheart in Applied Kinesiology (Callahan, 1985; Gallo, 1L999). In the Five Minute Phobia Cure Callahan wrote asserted that rhythmic tapNping at a specific point on a Emotional-freedom will improve the condition of theA associated vital organ. This, they say, occurs because the "energy flow" withinA thaDt Emotional-freedom is freed to move again." (p. 32) B Walther (1988) described a E mIotional-freedom technique in AK called the "Beginning and Ending Technique" F(B and E) which involves tapping the beginning or ending acupoints of the Yang E mOotional-freedoms. Nearly all the treatment points in The Callahan Techniques aTre aYt or close to the beginning or end points of the involved Emotional-freedoms. WI hile describing the AK Melzack-Wall pain treatment, Walther (1988) stated tRhat Sthe most productive tapping is when there is a bony backup to the tonificationE point. If possible, direct the tapping to obtain a bony backup (Walther, 1988, p.263)I. VAccordingly, it is speculated that tapping may cause a piezoelectric effect due to boNne stimulation at the acupuncture points. The piezoelectric effect occurs when tiny Uamounts of generated electrical current result from stimulating the crystallized calcium in the bone, and thus impacts the Emotional-freedom system (Gallo, 1999). Use of cold lasers, rubbing, imaging of tapping, and pressure holding of the acupuncture points, in Emotional-freedom based psychotherapy, were also reported by Gallo to be "effective at times" when used. Gallo however provided no further explanation about the effective 103 times or related circumstances but opined that, in most instances, percussing appears to more profoundly stimulate the acupoint and produce more rapid results. (Gallo,1999,p.150). Walther (1988), however, writes an interesting hypothesis about when tapping fails to yield results (in pain reduction). He asserts that, another factor that may cause less than adequate results with the Melzack-Wall technique is tapping at an improper frequency. It is often necessary to reduce the tapping rate. Two to four Hz appears to be the most productive (p. 263). Y As Callahan followed Goodheart, Walther and Blaich, other inteArestRed energy therapists now follow Callahan in the continuation of the tapping trReatment to effect change via the acupuncture Emotional-freedoms. However, IthBere is no empirical evidence from experimental studies to establish that it is the tapping that works in the treatment of psychological problems. This author has sNtudi ed L with both Callahan and Gallo, and has exposure to the other similar EmotionAal-freedom based psychotherapies. Like many others who have studied Thought FieDld Therapy, this author has tapped his way to psychotherapeutic success hundreds and hundreds of times. Tapping does work, as evidenced in clinical treatment and thBe mAultitude of anecdotal reports and patient testimonials. While it is true that nothin gI succeeds like success, this author believes that the time has come to empirically vFalidate the tapping approach to treatment, and to explore and evaluate alternate t reOatment approaches. Traditional acupTuncYture Emotional-freedom theory holds that ‗Chi‘ is a form of bodily energy which iIs, in part, generated in internal organs and systems (Tsuei, 1996). Further it is beRlieveSd that Chi enters the body from the outside through breathing and the numerous aEcupuncture points. Chi, often called the Life Force, combines with breath to circulaItVe throughout the body along complex pathways called Emotional-freedoms and veNssels. In essence, breath facilitates the flow of Chi in its most natural state. Imbalance Uof flow or distribution of Chi throughout the body is the blueprint for physical and/or psychological problems. Such imbalances become evident at the acupuncture points through definite changes in electrical activity and possibly, tenderness. The pioneering work of Reinhold Voll (1975) revealed that acupuncture points show a dramatic decrease in electrical resistance on the skin compared to non-acupuncture points 104 on the body. In addition, Voll and his colleagues found that each acupoint seemed to have a standard measurement for individuals in good health, and notable changes when health deteriorated (Voll, 1975). Becker (1990, 1985) reasoned from his research that not only does an electrical current flow along the Emotional-freedoms, but that the acupoints functioned as amplifiers which boost the electrical signals as they move across the body. More recently, the research and theories of William (1997), have shed more li ght on the interplay among mind, body, spirit and subtle energies. His work is particYularly relevant to the applicability of Touch and Breathe for use with Emotional-freeRdom based psychotherapies. Considering the complex array of electrical and elAectromagnetic circuitry in and around the body, Tiller theorizes that the body can be thRought of as a type of transmitting / receiving antenna. (p. 107) IB Tiller cites the autonomic nervous system (ANS) as a sign aLl carrier, waveguide, and signal conductor utilizing both sympathetic and parasympNathetic branches. He describes the acupuncture points as a set of antenna elements Athat "...provide an exquisitely rich array with capabilities exceeding the most advAancDed radar system available today. These sensitive points are coupled to the ANS viBa the fourteen known acupuncture Emotional- freedoms" (p. 117). Walther (1988) als oI reported that Goodheart observed "an antenna effect" regarding the acupoints, whicFh he believed, could be easily demonstrated. From the above it could Obe argued that the body's acupoints have the potential to transmit and receive Chi, Ydepending on the need of the Emotional-freedom system to restore balance. This aIuTthor hypothesizes that insertion of acupuncture needles serve as literal antenna R/ traSnsmitter extensions of the acupoints. When we touch an acupoint we perturb it Eand stimulate ion flow "...which reacts at the etheric level to unclog the EmotiIonVal-freedom flow channel" (Tiller, 1997,p. 121). In maintaining the contact by touNch we extend the antenna / transmitter capacity of the body system with a direct feed Uto the held acupoint. In contrast, while tapping perturbs, it also connects then disconnects the circuits, thus creating an inconsistent and disrupted signal to the body. In TAB, the use of one complete respiration (one easy inhalation and exhalation) is the natural vehicle of Chi circulation, which also creates a piezoelectric effect via vibration and sound (sonic resonance). 105 Tiller (1972) observed and reported that variations in mental alertness caused significant changes in the electrical characteristics of the acupuncture skin points. This author suspects that this reflects the influence and impact of intentional thought attunement which is paramount in TFT and the other Emotional-freedom based psychotherapies. Tiller's experiments from 1977-1979 (several thousand) revealed that mind direction or intentionality is evident and measurable, and was not indicative o f a "classical electromagnetic energy..."(p. 10) Accordingly, this author hypothesizeYs that treatment of therapy localized acupuncture Emotional-freedoms, diagnosed whRile attuned to the specific problem, will be more profound using the TAB approacRh thAan tapping or pressure alone. In addition, there have been therapist reports to this auItBhor that tapping was completely out of the question for some victims of abus eL who refused to tap on themselves. Having said all these, the vast majority ofA paNtients, however, do perform the tapping, as it is a requirement of successful treDatment. Watching patients while they tapped proved most interesting. Often it wAas observed that a full breath or sigh accompanied the tapping procedures. AIdBditionally, when patients were not reminded about the number of taps to do, it wFas observed that they would tap as many times as matched a full respiration beforeO inquiring or looking for guidance. In response to these observations, the author bega n to experiment and develop the Touch and Breathe approach to treatment. MTuYch to this author's surprise, every single patient preferred the TAB approach to theI tapping, and they reported more profound, comfortable, and relaxing effectsR. CoSnsequently this author has exclusively employed TAB over the last 15 months while working within the TFT framework in doing psychotherapy. In addition, this auIthVor Ehas demonstrated and shared the TAB approach over the past year with over a huNndred therapists for use with their patients. Again, the patients were reported to Urespond in similar positive form, as did the therapists when they were treated using TAB. In a study, Swingle (2008) used EFT as a treatment for children diagnosed with epilepsy. The children were administered EFT by their parents every time each day that the parents suspected a seizure might occur. Swingle found significant reductions in seizure frequency among these very young children, as well as extensive clinical 106 improvement in the children's E. E. G. readings after exposure to two weeks of daily in- home EFT treatment In preliminary clinical trials involving more than 29,000 patients from 11 allied treatment centers in South America during a 14-year period, a variety of randomized, double-blind pilot studies were conducted by Feinstein (2008). In one of these, approximately 5,000 patients diagnosed at intake with an anxiety disorder were random ly assigned to an experimental group (tapping) or a control group (cognitive beYhavior therapy /medication). Ratings were given by independent clinicians whAo inRterviewed each patient at the close of therapy, at 1 month, at 3 months, at 6 mRonths, and at 12 months. The raters made a determination of complete remissionI Bof symptoms, partial remission of symptoms, or no clinical response. The raters did not know if the patient received CBT/medication or tapping. They knew only tLhe initial diagnosis, the symptoms, and the severity, as judged by the intake staff. NAt the close of therapy: 63% of the control group was judged as having improved; 90A% of the experimental groups were judged as having improved. 51% of the control groDup was judged as being symptom free; 76% of the experimental group was judgedB as sAymptom free. At one-year follow-up, theF p atIients receiving the tapping treatments were substantially less prone to relapsOe or partial relapse than those with CBT/medication, as indicated by the independent r aters assessments and corroborated by brain imaging and neurotransmitter profileTs. YIn a related pilot study by the same team, the length of treatment was substantIially shorter with energy therapy and associated methods than with CBT/medication. ISf subsequent research corroborates these early findings, it will be a notable devEeloRpment since CBT/medication is currently the established standard of care for anIxiVety disorders and the greater effectiveness of the energy approach suggested by thiNs study would be highly significant. U Despite its odd-seeming procedures and eye-raising claims, evidence is accumulating that energy-based psychotherapy, which involves stimulating acupuncture points or other energy systems while bringing troubling emotions or situations to mind, is more effective in the treatment of anxiety disorders than the current standard of care, which utilizes a combination of medication and cognitive behavior therapy. 107 The clinical trials were conducted for the purpose of internal validation of the procedures as protocols were being developed. When acupoint stimulation methods were introduced to the clinical team, many questions were raised, and a decision was made to conduct clinical trials comparing the new methods with the CBT/medication approach that was already in place for the treatment of anxiety. These were pilot studies, viewed as possible precursors for future research, but were not themselves designed w ith publication in mind. Specifically, not all the variables that need to be controlled in Yrobust research were tracked, not all criteria were defined with rigorous precision, tRhe record- keeping was relatively informal, and source data were not alwaysA maintained. Nonetheless, the studies all used randomized samples, control groups,R and double blind assessment. The findings were so striking that the research teamI dBecided to make them more widely available. L Standard medications for anxiety (benzoAdiazNepines, including diazepam, alprazolam, and clonazepan) were given to 30 patDients with generalized anxiety disorder (the three drugs were randomly assigned to sAubgroups of 10 patients each). Outcomes were compared with 34 generalized anIxBiety disorder patients who received tapping treatment. The medication group haFd 7 0 percent positive responses compared with 78.5 percent for the tapping group. OAbout half the medication patients suffered from side effects and rebounds upon disc ontinuing the medication. There were no side effects in the tapping group, though oTne pYatient had a paradoxical response (increase of anxiety). Specific elemeInts of the treatment were also investigated. The order that the points must beR stimSulated, for instance, was investigated by treating 60 phobic patients with a standard 5-point protocol while varying the order in which the points were stimulIatVed Ewith a second group of 60 phobic patients. Positive clinical responses for the twNo groups were 76.6 percent and 71.6 percent, respectively, showing no significant Udifference for the order in which the points were stimulated. In other studies, varying the number of points that were stimulated, the specific points, and the inclusion of typical auxiliary interventions such as the 9 Gamut Procedure did not result in significant differences between groups, although diagnosis of which energy points were involved in the problem led to treatments that had slightly more favorable outcomes. 108 In a study comparing tapping with acupuncture needles, 40 panic patients received tapping treatments on pre-selected acupuncture points. A group of 38 panic patients received acupuncture stimulation using needles on the same points. Positive responses were found for 78.5 percent from the tapping group, 50 percent from the needle group. The follow-up data on the 29,000 patients coming from the 11 centers in South America included subjective scores after the termination of treatment by independ ent raters. The ratings, based on a scale of 1 to 5, estimated the effectiveness of the Yenergy interventions as contrasted with other methods that might have been used. ThRe numbers indicate that the rater believed that the energy interventions produced: A 1. Much better results than expected with other methods. R 2. Better results than expected with other methods. IB 3. Similar results to those expected with other methods. L 4. Lesser results than expected with other methods (oAnlyN used in conjunction with other therapies). 5. No clinical improvement at all or contraindicateDd. It must be emphasized that the fIoBllow Aing indications and contraindications for energy therapy are tentative guidelines based largely on the initial exploratory research and these informal assessments. OIn aFddition, the outcome studies have not been precisely replicated in other settings, an d the degree to which the findings can be generalized is uncertain. Nonetheless, TbasYed upon the use of tapping techniques with a large and varied clinical population in I11 settings in two countries over a 14-year period, the following impressions caRn sServe as a preliminary guide for selecting which clients are good candidates Efor acupoint tapping. There is also considerable overlap between these tentatiIvVe guidelines and other published reports. U2.3N.4 Emotional Freedom Technique and Mathematics Achievement Daniel, Brenor, Karen and Loren (2005) explored test anxiety benefits of Wholistic Hybrid derived from Emotional Freedom Techniques (EFT), and Cognitive Behavioral Therapy. Participants include Canadian university students with severe or moderate test anxiety. A double-blind, controlled trial of EFT, and CBT was conducted. 109 Standardized anxiety measures included the Test Anxiety Inventory (TAI) and Hopkins Symptom Checklist (HSCL-21). The result of their study showed that Emotional Freedom Technique was better than Cognitive Behavioural Technique. In only two sessions WHEE and EFT achieved the equivalent benefits to those achieved by CBT in five sessions. Participants reported high satisfaction with all treatments. EFT and WHEE students successfully transferred their self-treatment skills to other stressful areas of th eir lives. Y Nilhan and Bahar (2006) investigated the effect on test anxiety wAith REmotional Freedom Techniques (EFT), a brief exposure therapy with somatic and cognitive components. A group of 312 high school students enrolled at aI priv Rate academy was evaluated using the Test Anxiety Inventory (TAI), which containsB subscales for worry and emotionality. Scores for 70 demonstrated high levels of teLst anxiety; these students were randomized into control and experimental grouNps. A statistically significant decrease occurred in the test anxiety scores of both thAe experimental and control groups. The EFT group had a significantly greater decAreaDse than the PMR group. The scores of the EFT group were lower on the emotioInBality and worry subscales. Both groups scored higher on the test examinations afterF tre atment; though the improvement was greater for the EFT group, the difference waOs not statistically significant. 2.3.5 Mathematics EfficaYcy and Mathematics Anxiety In one study oIf T350 college students, Pajares and Miller (1994c) examined the hypothesized medSiational role and predictive power of self-efficacy in Mathematics problem solvinRg. Using previously validated measures, the researchers ran several MatheImVatic Es-related independent variables in relation to Mathematics problem solving. ReNsults show that self-efficacy held greater predictive power for problem solving success Uthan did Mathematics self-concept, background in Mathematics, perceived usefulness of Mathematics, and gender. The effects of background and gender, however, were significantly related to self-efficacy, supporting Bandura's assertion of the mediational role of self-efficacy on performance. Simply put, background and gender are not independently strong predictors of Mathematics performance, but they are influential 110 sources of Mathematics self-efficacy which is highly predictive and plays a strong mediational role on performance. Self-efficacy is a domain-specific construct in academics. Many, including Bandura, argue that it is also task-specific, and attempts to measure self-efficacy at the domain level often result in ambiguous or uninterpretable results (Bandura, 1986; Pajares & Miller, 1994c, 1995). Many of the studies that show self-efficacy to account for les ser variance than other personal determinants often stray from Bandura's prescriptionYs for a microanalytic strategy. Often these studies assess self-efficacy globally with Rjust a few scale items; that is, they ask participants to report on their confidence orA efficacy with regard to a specific academic domain, and not a specific performIance Rtask. At this level of self-reporting, it is expected that self-efficacy cannot reliablLy beB separated from other personal determinants such as self-concept, anxiety, self-con fidence, and background. It thus raises the question of whether one is actually meNasuring self-efficacy, or more generally measuring attitudes and other common mecAhanisms toward a given academic domain. Of course, the latter are important in somDe areas of educational research, but do not always give sufficient evaluative infoBrmaAtion for performance on specific, criterial tasks. One possible lens from which Ito view self-efficacy within the context of instructional technology is to OconFsider one's judgments of personal capabilities to authentically accomplish a spec ific performance objective. Self-efficacy and performance are inextricably related,T andY in the domain of Mathematics both are often correlated with gender. SI 2.3.6 MaEthemRatics Efficacy and Mathematics Achievement IVResearchers have been successful in demonstrating that self-efficacy is positively relNated to, and influences, Mathematics achievement. A meta-analysis published between U1977 and 1988 revealed that self-efficacy was positively related to Mathematics achievement. (Multon, Brown and Lent, 1991). Self-efficacy beliefs were related to academic outcomes and accounted for approximately 14% of the variance. Effects were stronger for high school and college students than for elementary students. How the constructs were operationalised also influenced findings. The strongest effects were 111 obtained where achievement indexes were assessed with basic skills measures or classroom-based indices such as grades than with standardized achievement tests, a finding that supports the context – specific nature of self-efficacy. As with self-concept, researchers have demonstrated that, when self-efficacy beliefs correspond to the academic outcome with which they are compared, prediction is enhanced and the relationship between self-efficacy and academic performance is positive and stro ng (Pajares and Miller, 1994; Pajares and Kranzler, 1995). Y Correlations between self-efficacy and academic performances in invRestigations in which self-efficacy is analysed at the item or task specific levelA and closely corresponds to the criteria task have ranged from .49 to .70; directI eBffec Rts in path analytic studies have ranged from .349 to .545. Results tend to Lbe higher in studies of Mathematics than of other academic areas such as reading o r writing, but even in these areas, relationship are considerably higher than previouNsly obtained if the criteria by which students rate their self-efficacy judgments arAe used as the criteria for scoring essays or assessing reading comprehension (PAajaDres, Miller and Johnson 2001; Pajares and Valiante, 2000). B Zimmerman (1998) and hisF as soIciates have been instrumental to tracing the relationships among self-efficacOy perceptions, academic self-regulatory processes, and Mathematics achievement. T his line of inquiry has demonstrated that self-efficacy influences self-regulatorTy pYrocesses such as goal-setting, self-monitoring, self-evaluation, and strategy use (ZimImerman, 1989, 1990, 1994; Zimmerman and Bandura, 1994; Zimmerman and SMartinez – Pons, 1990). Self-efficacious students embrace more challengingE goRals (Zimmerman, Bonner, and Kovach, 1996). Students with high self-efficacIy also engage in more effective self-regulatory strategies at different levels of abNility, Vand self-efficacy enhances students‘ memory performance by enhancing Upersistence (Bouffard-Bouchard, Parents and Larivee, 2003). In studies of college students who pursue science and engineering courses, high self-efficacy has been demonstrated to influence the academic persistence necessary to maintain high Mathematics achievement (Lent, Brown and Larken 1996; Hacket, 1995). 112 Self-efficacy is also related to self-regulated learning variables and use of learning strategies (Feather 2002; Fincham and Cain, 1986; Paris and Oka, 1986; Pokay and Blumenfeld, 2003; Schunk, 1985; Zimmerman and Martinez-Pons, 1990). Students who believe they are capable of performing tasks use more cognitive and meta-cognitive strategies and persist longer at those tasks than those who do not. Academic self-efficacy influences cognitive strategy and use self-regulation through the use of meta-cognit ive strategies, and it is correlated with in-class seat work, and homework, examinYations, quizzes, essays and reports. Pintrich and DeGroot (1990) suggested that seRlf-efficacy plays a facilitative role in the process of cognitive engagement, that raisingA self-efficacy beliefs might lead to increased use of cognitive strategies and,R thereby, higher achievement. In addition, students need to have both the wiIllB and the skill to be successful in classrooms. Students with similar previous achie vLement and cognitive skills may differ in subsequent achievement as a result of difNfering self-efficacy perceptions because these perceptions mediate between prioAr attainments and Mathematics achievement. D As a consequence, such performaBnceAs are generally better predicted by self- efficacy than by prior attainments. CFoll inIs (1982) identified children of low, middle and high Mathematics ability who had, within each ability level, either high or low Mathematics self-efficacy. Af teOr instruction, the children were given new problems to solve and an opportunitTy toY rework those they missed. Collins reported that ability was related to performancIe, but that regardless of ability level, children with high self- efficacy completedS more problems correctly and reworked more of the ones they missed. When reseEarcRhers tested the joint contribution to Mathematics performance of MatheImVatics self-efficacy and general mental ability (the variable typically acNknowledged as the most powerful predictor of academic performances), they found Uthat, despite the influence of mental ability, self-efficacy made a powerful and independent contribution to the prediction of performance (Pajares and Valiante, 2000). Clearly, it is not a matter of how capable one is, but how capable one believed oneself to be. Schunk (1989, 1991) has suggested that variables such as perceived control, outcome 113 expectations, perceived value outcomes, attributions, goals and self-concept may provide a type of cue used by individuals to access their efficacy beliefs. 2.3.7 Gender and Mathematics Anxiety In 1992, researchers at the University of Florida circulated a questionnaire to 9,093 students and found that 25.9% had a moderate to high need of help w ith Mathematics anxiety (Jones 2001). According to Burns (1998) 2/3 of Americans feYar and loathe Mathematics (see Furner page 68). According to Zaslavsky (1994), peRople of all races and economic backgrounds fear Mathematics, but women and minorAities are most hindered by it. She reported research which points out that around the sReventh grade girls begin to doubt their ability to do Mathematics. Since self-confidIeBnce and Mathematics performance are so closely related, it plays a major rolNe in Lgirls‘ choices to continue Mathematics into high school. Preis & Biggs (2001) cite research that findAs that women, in particular older women, often experience more Mathematics anxDiety. However, in a recent CBS News story (May 23, 2003), some people are coInBcern Aed that boys are not performing as well as girls: ―Girls are being told, ―Go for Fit, y ou can do it. Go for it, you can do it.‖ They are getting an immense amount of sOupport,‖ (Dr. Michael Thompson, a noted author on the subject) says. ―Boys hear that t he way to shine is athletically. And boys get a lot of mixed messages about what it TmeYans to be masculine and what it means to be a student. Does being a good student mIake you a real man? I don‘t think so…It is not cool.‖ Whether this perceived lower pSerformance is attributable to anxiety is questionable, but must be determined. R IVRac Ee may also play a role in Mathematics anxiety: African American, Hispanic, AsNian and Native American males had high Mathematics anxiety while African UAmerican and Hispanic females rated high as well. In the Bernstein, Reilly and Cote-Bonanno study cited in Preis & Biggs (2001), Asian and Native American male college students had higher Mathematics anxiety than did the females, while Caucasian females scored higher than Caucasian males. However, Lusser (1996) failed to find a significant relationship between gender and Mathematics anxiety, stating that Mathematics 114 background had to be considered. Developmental Mathematics students were found to be more Mathematics anxious than other college students, while nontraditional students have more anxiety about college in general, including Mathematics (Preis & Biggs 2001). However, the correlation between Mathematics anxiety and verbal achievement or aptitude is -0.06 and -.17 with IQ (Ashcraft & Kirk 2001). It seems that all learners have some degree of Mathematics anxiety, which is related to gender, ethnic background, age, attitude towards Mathematics & previous MatheYmatics experience. Apparently we all fear Mathematics to some degree. AR 2.3.8 Gender and Mathematic Achievement R There is a potential gender effect in Mathematics learning and MaIthBematics self-efficacy. Fennema and Sherman (1977) and Sherman and Fennema (19 7L8) found that Mathematics confidence and gender stereotyping are significantN predictors of Mathematics performance for middle and high school studentsD. SAtudies with college students show that gender influences self efficacy in MatheAmatics-related actions, such as academic major and career decisions (Hackett, 1985I; BLent, Lopez, & Beischke, 1991; Matsui, Ikeda & Ohnishi, 1989; Matsui, Matsui & Ohn ishi, 1990). Other studies found that gender is an influential source of efficacy informFation in modeling (for example, Schunk, Hanson & Cox, 1987; Schunk, 1987). In pOersonalization studies, Murphy and Ross (1990) found gender to be an influential Yfactor in determining Mathematics success for eighth graders. Other researchers (LoIpTez, 1989; Lopez & Sullivan, 1992) found that personalization significantly bReneSfited seventh-grade Hispanic boys in performing Mathematics calculationsE. Together, these lineages of research suggest that gender maintains a signifiIcVant influence on Mathematics self-efficacy. UN 115 2.4 CONCEPTUAL MODEL FOR THE STUDY st nd1 Order 2 Order intervening intervening Independent Dependent variables variables variables Variables Y * Lack of basic * Peer Influence R cognitive skills A Numerical * Students‘ home * Continuous background R Cognition Failure B* Poor parental I * Lack of interest in influence Mathematics L * School * Emotional N instability backgroun DA d * Personality * ALack of skilled Mathematics Emotional * Life style IBTeachers freedom * Attitude F * Inadequate curriculum * Lack of motivation O * IntelliYgence * GIeTnder. S* Poor study habit ER * Physical and V health factors I U N S O R 116 * Reduction in Mathematics Anxiety * Enhancement of Mathematics Achievement 2.4.1 Explanation of Conceptual Model for the study A model is a framework applied to the field of study to aid the understanding of how the target behaviour is to be managed. The model for this study consists of the independent variables (i.e. numerical cognition and emotional freedom) to be manipulated by the researcher to see their effectiveness on the dependent variables which are Mathematics anxiety and Mathematics achievement. In-between the independent and dependent variables are the intervening varYiables. An intervening variable is one that surfaces between the time the independenRt variables operate to influence the dependent variables. There is thus a time dimAension to the intervening variables. The intervening variables are those factors wRhich account for internal and unobservable psychological processes that, in turn, mIaBy affect the outcome of the treatment. They are divided into organismic and enviro nLmental factors. (a) Organismic factors N These are the first order intervening variaDblesA inherent in the individual. They, among others, include basic cognitive skills, physical and health factors, lack of interest in school work or poor study habit, coInBtinu Aous failure, intelligence, age, emotional instability, gender, lack of motivation, e tc. (b) Environmental factors F These are second order iOntervening variables inherent in the environment. They include peer influenceT, stYudent‘ home background, poor parental influence, school background, lack of sIkilled Mathematics teachers, inadequate curriculum, poor study habit, poor morRal dSevelopment, etc. DepEendent variables are measurable behavioural outcomes that are expected to be broughItV about by the manipulation of the independent variables. The dependent variables in Nthis study are Mathematics anxiety and achievement. U The behavioural equation S-O-R represents the total interaction of various variables in the study (Kanfer & Phillips, 1970). S - Stimulus (i.e. the independent variables) O - Organism (i.e. the intervening variables inherent in the organism) 117 R - Response (i.e. the dependent variables which are the resultant effects of independent variables). Many things determine the Mathematics anxiety and achievement of an individual among which are: student‘s home background, factor resident in the child, school background could contribute both negatively and positively to the achievement in Mathematics of an individual, the society and government policy all contribute to achievement in Mathematics of students whether positively or negatively. There areY some factors which could be used to explain this. These factors are: nature of home Rdiscipline, the way an individual is being trained matter a lot. It is associated with theA socialization and eventual achievement of adolescence in school; discipline in thRe home could be authoritarian permissive or democratic. Harsh authoritarian disciIpBline produces a child that is insecure, uninquisitive and lacking in initiative. TNhese t Lraits affect negatively the school performance and mathematics achievement. Students who receive democratic type of discipline are better adjusted to their school woDrk bAecause they are emotionally stable, law abiding, more motivated to pursue school Awork without duress and therefore achieve better academically. Other factors undIeBr family or home background are: family relationship, level of cognitive stimFula tion, lack of role model, available financial and medical facilities, etc. The factOors resident in the child include: basic cognitive skills, physical and health factors, ps ycho-emotional factor, lack of interest in school work or poor study habit, continTuouYs failure, etc. Factors resident in the school include: deficient school environment inI terms of school location, school building, school topography, quality of teachRingS staff, existence or non- existence of teachers, evaluation of learning, poor condiEtion of service, inadequate curriculum. Causal resident in the society are: ethnicI aVnd inter-tribal wars, class differences, inadequate medical facilities and so on. Mathematics achievement is a significant decision that often resolves the identity UprNoblem and the much perceived pertinent question ―who am I‖, which is a reflection of self worth and a practical understanding of the self. A combination of interest and concentration in study will enable one to achieve integrated learning that will operate most effectively in the solution of problems. Aremu (2004, 2002 and 2000) is acknowledged for the above factors. Thus, if all the above intervening factors were 118 properly controlled, the end result would be reduction of anxiety and enhancement of Mathematics achievement. 2.5 Hypotheses The following hypotheses will be tested for the purpose of this study at 0.01 level of significance. Y 1. There is no significant main effect of treatment on mathematics anxietyA of Rsecondary school students. R 2. There is no significant main effect of mathematics efficacy oInB mathematics anxiety test score of secondary school students. L 3. There is no significant main effect of gender onA maNthematics anxiety test score of secondary school students. D 4. There is no significant interactive effeBct Aof treatment and mathematics efficacy on mathematics anxiety test score of second aIry school students. 5. There is no significant interaFctive effect of treatment and gender efficacy on mathematics anxiety test score oOf secondary school students 6. There is no signiIfiTcanYt interactive effect of mathematics efficacy and gender on mathematics anxieSty test score of secondary school students 7. There Eis nRo significant interactive effect of treatment, mathematics efficacy and gender oVn mathematics anxiety test score of secondary school students. 8. NTheIre is no significant main effect of treatment on Mathematics achievement of Usecondary school students. 9. There is no significant main effect of mathematics efficacy on mathematics achievement test score of secondary school students. 119 10. There is no significant main effect of gender on mathematics achievement test score of secondary school students 11. There is no significant interactive effect of treatment and mathematics efficacy on mathematics achievement test score of secondary school students. 12. There is no significant interactive effect of treatment and gender on mathematics achievement test score of secondary school students. Y 13. There is no significant interactive effect of mathematics efficacy aAnd Rgender on mathematics achievement test score of secondary school students R 14. There is no significant interactive effect of treatment, mathemaItiBcs efficacy and gender on mathematics achievement test score of secondary sc hLool students DA N IB A O F ITY RS IV E UN 120 CHAPTER THREE METHODOLOGY This chapter presented the methodology on which the study leans. This involves the design that was adopted for the study, population, sample and sampling technique, instruments, validity of the instrument, procedure for data collection and method of data analysis. 3.1 Research Design Y The study adopted a pre-test post-test control group design using a R3 x 2 x 2 experiment. The factorial method consists of the two experimental gRrouAps (Numerical cognition, Emotional-freedom) and control group; gender of the participants varying at two levels: male and female and Mathematics efficacy of the partIicBipants varying at two levels: high and low. L N Table 3.1: A 3 x 2 x 2 Factorial experiment for the pAsychDological treatment of Mathematics anxiety and achievement B Treatment Gender and MatFhem Iatics efficacy Male Female High Y L Oow High Low Numerical n = T8 n = 1 2 n = 7 n = 1 3 Cognition I Emotional RSn = 9 n = 11 n = 6 n = 14 Freedom ControIlV E n = 9 n = 11 n = 10 n = 10 U3.2N Population The target population for this study consisted of the senior secondary students (SS1) in some selected public secondary schools in Ibadan who had consistent records of low achievement in Mathematics. These records were obtained from the schools of the study‘s participants with the permission and cooperation of the school authorities. These 121 records were obtained from at least three sessions of academic records of the identified participants. 3.3 Sample and sampling techniques The sample for this study comprised one hundred and twenty (120) Senior Secondary One (SS I) students drawn from three selected public secondary schools in Ibadan through random sampling technique. Initial selection of the participants waYs done through stratified random sampling. Stratified random sampling was done by Rstratifying the population into strata according to gender, age and class of the RpartAicipants. Two secondary schools were selected in each local government for the experiment. Final selection of participants was preceded by a preliminary investigatioInB for selecting the low achievers in Mathematics in the selected secondary schoo lsL that were used for the experiment. N All the schools operate a common poDlicyA of conducting two continuous assessment tests each term of the session totaling six continuous assessment scores. All schools also conduct the end-of-term examBinaAtion making up three examinations. The two continuous assessment scores haFve aI weight of 40%, while the terminal examination has a weight of 60% totaling 100% for each candidate in every subject offered. At the end of the session, the average oOf the three terms scores are computed for each student to determine suitability for prYomotion. This practice of cumulating and averaging offers a consistent picture of eaIcTh student‘s academic status as to whether he/she is high, average or low in the subjecSt. AccEordRingly, subject grade in all the schools is based on the number of percentage passedI iVn the subject thus: N(a) 75 and above = A1. U (b) 70 – 74 = B2 (c) 65 – 69 = B3 (d) 60 – 64 = C4 (e) 55 – 59 = C5 (f) 50 – 54 = C6 122 (g) 45 – 49 = P7 (h) 40 – 44 = P8 (i) 39 and below = F9 Participants for this study would be selected from categories G to I 3.14 Inclusion/Exclusion Criteria Only participants who met the following criteria were enlisted for participatioYn: a. Participants were SS1 with consistent records of low achievement in MathRematics. b. Participants were willing and interested in participating. A c. School authorities consent was sought and presentatioIn Bof R letter from the department was approved. d. Parental consent was sought after sending parental cons eLnt form to parents through participants. N A 3.15 Instrumentation D The four assessment scales that weBre uAI sed in this study are Mathematics Anxiety Scale (MAS), Mathematics EfficacyF Sc ale (MES), Mathematics Achievement Test and Pseudo-dyscalculia Scale (PDS). a. Mathematics Anxiety SOcale Mathematics AnTxieYty Scale was a paper and pencil psychological instrument developed by Betz (19I78) with coefficient alpha of 0.90 to assess students‘ Mathematics Anxiety level. R ThSe items include statements on the factors to which students attribute their predicEaments in Mathematics. It was a 14-item scale with response format ranging from sItrVongly agree = 5 to strongly disagree = 1. Participants made to tick the option that beNst describes their Mathematics Anxiety level. The items were added to give a total Uscore of 70 and a minimum of 14. A score of 42 and above reveals that the participants‘ Mathematics Anxiety is high, while a score below the norm reveals low anxiety. The items were coded because there were both positive and negative statements which were reversed. The psychometric property of the instrument was established through a pilot study on a sample of 30 students. The Cronbach Coefficient observed was 0.89 and 123 internal consistency of the instrument ranges between 0.46 and 0.75. The scale was adapted to suit the culture of the participants by ensuring its face and content validity by the researcher‘s supervisor. b. Mathematics Achievement Scale This measuring instrument was developed by the researcher. This has to dYo with some items that test Mathematical skill. This is a 30-item instrument developeRd from the Mathematics curriculum by Mathematics teachers. The psychometric prAoperty of the instrument was established through a pilot study on a sample of 20R students. Kuder Richardson 20 (KR 20), which measures achievement test, wasI uBsed to determine the overall coefficient of the instrument. Item analysis was a lsLo used to carry out the difficulty index and discriminatory power of the test. ThiNs was done between the higher achievers and lower achievers in Mathematics. DurinAg the process of item analysis, ten (10) items were removed as too easy or too dAiffiDcult while twenty two (20) items were retained. However the coefficient got fIroBm KR 20 was 0.90, which means that the achievement scale was reliable. F c. Mathematics Efficacy SOcale The MathematicTs EYfficacy Scale was adapted from Betz and Hackett (1983) with coefficient alpha of 0.I88. The version adapted by the researcher was tailored for use in the Nigerian SRecoSndary Schools students. It spans such areas as: problem-solving; performance accomplishment; verbal persuasion and other areas of mathematical abilitieIsV. T Ehe scale comprises 16 items spread over two sections A and B with A coNntaining personal information and B containing items on Mathematics Efficacy. The Uitems were scored as follows: Strongly Agree = 5, Agree = 4, Not sure = 3, Disagree = 2 and Strongly Disagree = 1. The points that were scored on all items were summed up to give participant‘s score on the scale. The items were also coded because there were both negative and positive statements which was be reversed. Scores on the scale ranged between 16 and 80. A score above 48 indicated high Mathematics-efficacy and score 124 below the norm indicated low Mathematics-efficacy. The psychometric property of the instrument was established through a pilot study on a sample of 30 students. The Cronbach Coefficient Alpha of 0.86 was observed with internal consistency ranging between 0.46 and 0.69. d. Pseudo-dyscalculia Scale This measuring instrument was also developed by the researcher. This hasY to do with 40-item instrument that measure individual‘s false belief in mathematical sRkills. This scale was used to identify participants with pseudo-dyscalculia or faAlse belief in Mathematics. A norm was established to indicate participants who hRave high and low pseudo-dyscalculia. Participants with high pseudo-dyscalculia wIoBuld be used for the study while those with low pseudo dyscalculia would beL dropped. Psychometric properties of the scale were established on a sample of 30N students. Reliability techniques that were adopted were Cronbach Coefficient AlpDha, aAnd Split half reliability. Method of validity involved content, convergence constrAuct, discriminant and concurrent validity. The Factor analysis was used to establisBh the factor structure of the scale. Principal component analysis was used with V aIrimax Rotation, iteration method and Kaiser Normalization. Eigen value wOas dFetermined and overall variance of the scale was observed. The initial coefficien t alpha was 0.67. The analysis removed 14 items that were not internally consistent Y(items with less than 0.3) and this increased the overall coefficient to 0.88. AIftTer this procedure, some items (8 items) still reported negative correlation andR remSoved. This later increased the coefficient to 0.93. The Correlation between foErms from the split half method was 0.79, Equal length Spearman Brown was 0.84, UIVnequal length was also 0.86, Guttman Split-half was 0.88, Alpha for part 1 was 0.7N9, alpha for part 2 was 0.80. Factor analysis was carried out using extraction method U(Principal Component Analysis). The overall total variance explained was 61.76% meaning that the scale describes the domain of behaviour. The initial eigen-values were between 0.54 and 4.763. Three components were extracted initially using extraction method of principal component analysis. The factors were later rotated using varimax with Kaiser Normalization. The rotation converged in 8 iterations and all factors: False 125 beliefs, environmental factors, inappropriate concentration, fear or anxiety, negative emotion, avoidance, inability to follow systematic instruction and low self efficacy were loaded in factor 1 (Pseudo dyscalculia). The correlations between the initial and extraction were between 0.51 and 0.68 meaning that the items are really describing the domain of behaviour. 3.6 Procedure for the Experiment Y The research experiment spanned through a period of ten weeks durRing which time there was researcher-participants‘ interactions. There were four Amain phases: randomization, pretest, treatment and post-treatment evaluations. PermRission was sought and obtained to involve both staff and students in the study. ALccIorBdingly, the principals assigned members of staff –the vice-principal or the counsel lor (s) who could assist the researcher. A pseudo-dyscalculia scale was used to idNentify people with anxiety in Mathematics. Mathematics efficacy scale was useDd toA identify those students with high and low mathematics efficacy. The researcher and his assistants met and agreed on the day of the week and time for the therapeIutBic s Aessions in each of the schools where there were sessions. The control group meFt on ly twice i.e. during pretest and post-test. 3.7 Objectives of the Ther aOpeutic Packages The objectives of thYe treatment sessions are:  To enable particIipTants identify what actually lead to their anxiety in Mathematics and help RthemS to reduce such.  To enhance Mathematics-efficacy of participants and thus reduce their state of lIeaVrne Ed helplessness.  NTo equip participants with learning skills in order to improve their academic U performance in the subject.  To reduce negative emotions of the students in Mathematics.  To enable the students to restructure their bad cognition and emotion with positive self statements.  To help the participants learn self-respect and self-esteem 126  To enable them overcome feelings of helplessness  To promote a generalized sense of capability  To equip students with the right problem solving skills  To promote resiliency and solution-focused problem solving  To help them shift away from emphasizing problems  To help them discover considerable power and possibilities they have in themselves. RY 3.8 Numerical Cognition Outline A Session I : Recruitment and Introduction R Session II : General orientation and administration ofI Binstrument to obtain baseline information (pre-treatNmen t Lmeasures) from the participants. Session III : Introduction of the conDceptA of numerical cognition with illustrations Session IV : Explaining cognitiveB disAtortions in Mathematics. Session V : Identifying paFrtic ipIants‘ problems in Mathematics and replacing their cognOitive distortions through numerical cognition technique. Session VI : Evaluati on of the participants. Session VII : GTeneYralization to other subject/setting and allowing higher level of Icritical thinking to occur. Conclusion and Summary delivery. Session VIII R : SPost-treatment and refreshment. 3.9 IVNum Eerical Cognition Group SeNssion I : Recruitment and Introduction U With the help of members of school staff of the school, the researcher specified the category of participants desired for the experiment – low achievers in Mathematics. This category were students with Mathematics anxiety or poor Mathematics achievement. Accordingly, the assistants helped to identify this category of students with their previous academic records in Mathematics. A pseudo-dyscalculia scale was administered to the 127 participants to identify those with Mathematics disabilities. Thereafter, a mathematics efficacy questionnaire was administered to the participants to dichotomize them into high and low mathematics efficacy. An average score on mathematics efficacy indicated the norm or reference point for classifying them into high or low mathematics efficacy i.e participants who fell above average point were considered high while those below were considered low on mathematics efficacy. The participants included a mixture of stude nts who were promoted on trial to SS I and those asked to repeat SS I class. On meetYing the students, there was exchange of pleasantries followed by writing of names Ato aRvoid ghost participants. The day and time of meeting were agreed upon. The researcher and his assistants dwelled much on cooperation for the period of the experimentR. Session II: General Orientation and Administration of InstrumIeBnt to obtain baseline information (pretreatment measures). L  The research assistants assisted the researcher toN assemble participants at the provided venue and at the time earlier agreedD upoAn.  The researcher then explained the nature and objectives of the programme emphasizing that most of the things thBey Awere to learn were new to them.  The researcher also stressed the im pIortance of confidentiality.  Participants were acquaintOed wFith the benefits derivable from the programme – it created in them a new th inking cap that changed their erstwhile lukewarm attitude to studies in MathTemYatics and made them act right to have improved academic performance in thIe subject.  Regular aRttenSdance was emphasised. The need to adhere to all instructions and compElete the sessions was emphasized. Participants were encouraged to feel free aInVd ask questions for clarifications.  NThe researcher then distributed the scales on mathematics efficacy to assess their U level of mathematics efficacy, Mathematics anxiety scale to assess their level of anxiety in the subject before the treatment, Mathematics achievement scale to assess their level of achievement in Mathematics before the introduction of the treatment packages. The researchers also appealed for objectivity and 128 independence of completion as well as ensure that all participants complete the scales.  Thereafter, the researcher enlisted the cooperation of participants to the effect that proceedings of subsequent sessions were video-recorded by the cameramen standing in front of them. The researcher also elicitd their cooperation and seriousness in terms of attendance and attentiveness. He would also explain that the sessions would be cumulative, any gap resulting from absenteeism disYrupted outcome and put the absentee at a disadvantage – entrapping himself/herRself in the old habit of Mathematical situation. Participants were promiseRd toAkens for full attendance.  A take-home assignment was given to them. They were aLskeIdB to state some reasons why many students perform poorly in MathematicNs a nd suggest things to do to overcome the problems. The reason is also to get some baseline information on their deficiencies in Mathematics and addresDs sucAh. Session III: Introduction of the concept of numerical cognition with illustrations (This and subsequent sessions were video-rBecoArded).  The researcher welcomed the paFrtic ipIants and thanked them for making themselves available to improve their Mathematic deficiencies.  The researcher then allowe dO the participants to identify various problems they have been having in MathTemYatics. This reveals if the participants were aware of the factors for poor MathematIics performance and appropriate remedies.  The researcRher Sthen introduced the term ―numerical cognition‖ explaining that it is a treatmeEnt approach that is based on the concept that the way we think affects how we resIpVond/act. It deals with how students acquire an understanding of numbers, and Nhow much is inborn. It also has to do with how students associate linguistic symbols U with numerical quantities. This brings the knowledge of how these capacities underlie their ability to perform complex calculations. The researcher then added that students could interpret the same life event differently leading to many and varied emotional and behavioural consequences. Such consequences could lead to wrong 129 interpretations, lack of self confidence, both of which could lead to poor Mathematics performance. Session IV: Explaining Cognitive Distortions  The researcher thereafter asked the participants how well they have understood the concept of numerical cognition and in what ways they have analysed the illustration.  The researcher further shed more light on what the participants came up with.Y  The researcher further added that it is not actually the event or situation thRat directly impacts on how we feel and behave but rather the thoughts about thAe event. If a student failed a class test in Mathematics for example, it wasI nBot t Rhe poor mark that would make him fail further but what he thought about the poor mark. The student could feel that for one reason or the other, the teacher d iLsliked him. He could feel he was not up to the task (lack of self-efficacyA). NHe could feel his sadness was responsible (negative emotion). He could blame the enemy/devil for being at work. He could feel no matter what he wrote in tDhe subject, he would still fail (lack of self-efficacy). BA  The researcher then explained Fthat tIhese negative thoughts were unhelpful thoughts and beliefs, which were Osignificant factors in the development of depression, anxiety, anger, low self- esteem, self-defeating behaviours, difficulty with coping, negative emotion aTnd Ylack of Mathematics efficacy.  Based on the forIegoing, participants were asked to make a list of their cognitive distortions anSd unhelpful beliefs. Session VE: IdRentifying participants‘ problems in Mathematics and replacing their cIoVgnitive distortions through numerical cognition technique. NThe researcher welcomed the participants to the session and demand for the U assignment given. This would be discussed.  The researcher then tried to identify participants‘ problems in Mathematics. This was enhanced by the previous administration of Mathematics Anxiety and Mathematics Efficacy Scales. Numerical cognition skills were used to train the participants as follows: 130  Discover the sign: The researcher trained the participants to discover Mathematical signs and operations. The step includes the participants: - Scanning the problem; - Circling and saying name of operation sign; and - Saying what the sign meant  Decoding figures and translating them into expression: Here, the researc her explained to the participants the importance of transforming the figuYres to expression. This was to help the participants verbalize MathematicaAl opReration in the way they could understand it.  Read the problem: The research trained the participants to be abRle to: - Read the whole problem; IB - Say the problem aloud as they read it; and L - Deliver feedback swiftly and systematicaNlly. Here the participants would be trained to ask the followAing questions: - What is the problem? AD- What are some plans? B - How is my plan workin gI? - How did I do? F  The researcher was s uOpportive and highlighted shining moments, encourage participants to seTlf-eYvaluate strengths.  The researcherI then trained the participants to become self-aware and self- observaRnt (eS.g., “how do you feel about…? you must be proud that …”)  TheE researcher also trained the participants to develop a new, positive behaviour IVto replace negative ones. N Logical procedures to answer or draw and check: The participants were trained to: U - Answer problem, if they know how to solve it, whenever it arises; - Or draw pictures to solve it; - State the problem; - Develop some plans; - Explore the plans; 131 - Ask themselves if the plan is working; and - See if they are successful. Problem Solving Plans in Numerical Cognition were in four Steps: 1. Clues:  Read the problem carefully.  Underline clue words.  Ask yourself if you've seen a problem similar to this one. If so, what is simiYlar about it? R  What did you need to do? A  What facts are you given? R  What do you need to find out? IB 2. Game Plan: L  Define your game plan. N  Have you seen a problem like this before? A  Identify what you did. D  Define your strategies to solve this pBrobAlem.  Try out your strategies. (UsinFg fo rImulas, simplifying, use sketches, guess and check, look for a pattern, etc.)  If your strategy doesn't wOork, it may lead you to an 'aha' moment and to a strategy that does work. Y 3. Solve: IT  Use youRr stSrategies to solve the problem. 4. Reflect:  EIVThis part is critical. Look over your solution. N Does it seem probable? U  Did you answer the question?  Are you sure?  Did you answer using the language in the question?  Same units? Clue Words: 132 When deciding on methods or procedures to use to solve problems, the first thing you will do is look for a clue which is one of the most important skills in solving problems in Mathematics. If you begin to solve problems by looking for clue words, you will find that these 'words' often indicate an operation. For instance: Clue Words for Addition  sum Y  total R  in all A  perimeter R IB Clue Words for Subtraction L  difference N  how much more A  exceed AD Clue Words for Multiplication B  product I  total F  area Y O  times Clue Words for DivisIioTn  share  distribuRSte  quoEtient N I Vaverage U The participants were also trained to familiarize themselves with the problem situation. This enabled them to be able to collect the appropriate information, identify strategies and utilize the strategies appropriately. The package helped the participants to:  Learn self-respect and self-esteem; 133  Overcome feelings of helplessness;  Promote a generalised sense of capability;  Equip students with the right problem solving skills;  Promote resilience and solution-focused problem solving;  Shift away from emphasizing problems;  Discover considerable power and possibilities they have in themselves;  Set goals; Y  Monitor their progress; R  Use comprehension strategies to translate the linguistic and numericAal information in the problem and come up with a solution; and R  Identify the important information and even underline partsI oBf the problem. Session VI: Evaluation of the participants L Numerical Cognition perspective emphasizes AthatN participants can become stuck by focusing on their past and current "bad" behaviours and failures versus focusing on future solutions. This therapy tried to increasDe student performance by removing obstacles to student learning. Students acBcomAplished more when they concentrated on their successes and strengths rather tha nI their failures and deficits. There are so many advantages for students who knoOw hoFw to constructively solve problems. Students should be looked at as being good an d capable of rational thought but without any influence from teachers or significTantY adults a student will likely focus more on their own negative side. I Once thRe thSerapist or researcher begins to shift to the positives of the good things that are goiEng on in a student's life, the students usually will switch to that, open up and talk abIoVut it too. Students do have the capacity to act on common sense if given the opNportunity to identify common sense problem-solving strategies. Solution-focused Uproblem solving in numerical cognition is based on the theory that small changes in behavior lead to bigger changes in behavior. The therapy would emphasize a role shift for students. Small shifts in role by a student will cause shifts in other places. In this regards, teachers will also be focused to develop an alliance with the student and work together to determine the problem and the cause. Identify the student's strength, and then they can 134 build strengths and foundations which will lead to positive changes. When the plan does not seem to be working and the student seems to be repeating the same pattern or does not have the ability to control compulsive behaviors then the educator has to watch for a pattern and reinforce with positive. This therapy pursues the positive and students are more likely to find a solution to a problem when they concentrate on their successes rather than their failures. Stude nts must realize that they play a huge part in the success of their problem solving proceYss and that change will occur. Once the changes begin to happen then the student wRill realize that their lives can be very different. Then it is time to have the studRentsA set goals and then monitors their progress. The therapist then tries to use comprehension strategies to translate the linguistic and numerical information in the problemI Band come up with a solution. For example, the therapist may read the problem mor eL than once and may reread parts of the problem as they progress and think through thNe problem. Session VII: Generalization to other subject/setting aAnd allowing higher level of critical thinking to occur. Conclusion and Summary Ddelivery. Session VIII: Post-test BA  The researcher tried to summaFriz e Iall that had transpired during the therapeutic sessions. - Participants were giv enO the two instruments – Mathematics Anxiety and Mathematics AchievemeTnt YScales – to complete again as post-test.  The researcher theIn expressed gratitude and prayed for the participants. - RefreshRmenSt was served. - GiftEs were distributed - IVThe researcher gave the participants his phone number for further follow-up. U3.1N0 Emotional-freedom-based intervention Outline Session I : Recruitment and Introduction Session II : General orientation and administration of instrument to obtain baseline information (pre-treatment measures) from the participants 135 Session III : Introduction of the concept of Emotional Freedom Techniques with illustrations Session IV : Explaining negative emotions in Mathematics Session V : Identifying participants‘ problems in Mathematics and replacing their negative emotions through Emotional Freedom Technique Session VI : Evaluation of the participants Session VII : Conclusion and Summary delivery Y Session VIII : Post-treatment and refreshment AR 3.11 Emotional Freedom Technique Group R Session I : Recruitment and Introduction IB With the help of members of school staff of the scho oLl, the researcher specified the category of participants desired for the experiment –N low achievers in Mathematics. These were junior students with Mathematics anxietyA. A pseudo-dyscalculia scale was administered to identify the participants with MatDhematics disabilities. Accordingly, the assistants also helped identify this categoIryB of Astudents based on their previous academic records in Mathematics. A MathemaFtics efficacy questionnaire was then administered to the participants to dichotomize thOem into high and low Mathematics efficacy. An average score on Mathematics efficacy scale indicated the norm or reference point to classify them into high or low MTathYematics efficacy i.e participants who fell above average point were considered high Iwhile those below were considered low on Mathematics efficacy scale. The partRicipSants included a mixture of students who were promoted on trial to SS I and those aEsked to repeat SS I class. On meeting the students, there was exchange of pleasaInVtries followed by writing of names to avoid ghost participants. The day and time ofN meeting were agreed upon. The researcher and his assistants dwelled much on Ucooperation for the period of the experiment. Session II: General Orientation and Administration of Instrument to obtain baseline information (pre-treatment measures).  The research assistants assisted to assemble participants at a venue and at the time earlier agreed upon. 136  The researcher then explained the nature and objectives of the programme emphasizing that most of the things they were to learn would be new to them.  The researcher then stressed the importance of confidentiality.  Participants were acquainted with the benefits derivable from the programme – it will create in them a new thinking cap that would change their erstwhile lukewarm attitude to studies in Mathematics and make them act right to have improv ed academic performance in the subject. Y  Regular attendance was emphasized. The need to adhere to all instruRctions and complete the sessions were also emphasized. Participants were encoAuraged to feel free and ask questions for clarifications. R  The researcher then distributed the Mathematics efficacy scIalBe to assess their level of Mathematics efficacy, Mathematics Anxiety Scale to aLssess their level of anxiety in the subject before the treatment, Mathematics EffNicacy Scale to assess their level of efficacy in Mathematics before the introdDuctiAon of the treatment packages. The researcher appealed for objectivity andA independence of completion as well as ensured that all participants completeBd the three scales.  Thereafter the researcher enlisFted thIe cooperation of participants to the effect that proceedings of subsequent sessions were video-recorded by the cameramen standing in front of them. O  The researcher elicTiteYd their cooperation and seriousness in terms of attendance and attentiveness. HIe also explained that the sessions would be cumulative, any gap resulting RfromS absenteeism would disrupt outcome and put the absentee at a disadvEantage – entrapping him/her in the old habit of Mathematical situation. PIaVrticipants were promised tokens for full attendance.  NA take-home assignment was given to them. They were asked to state some reasons U why many students perform poorly in Mathematics and also suggested what things to do to overcome the problems. The reason was to get some baseline information on their deficiencies in Mathematics and address such. Session III: Introduction of the concept of emotional freedom with illustrations (This and subsequent sessions would be video-recorded). 137  The researcher welcomed the participants and thanked them for making themselves available to improve their Mathematic deficiencies.  The assignment given was discussed. This was to reveal if the participants were aware of the factors for poor Mathematics performance and appropriate remedies. The researcher then introduced the term ―emotional freedom‖ explaining that the cause of all negative emotions is a disruption in the body's energy system. Negati ve emotions come about because you are tuned into certain thoughts or circumsYtances which, in turn, cause your energy system to disrupt. Otherwise, you function noRrmally.  The researcher added that students could interpret the same life eveAnt differently leading to many and varied emotional and behavioural consRequences. Such consequences could lead to wrong interpretations, lack of seIlBf confidence, both of which could lead to poor Mathematics performance. L Session IV: Explaining negative emotion N  The researcher asked the participants how well tAhey have understood the concept of emotional freedom and in what ways theyA haDve analysed the illustration.  The researcher further shed more ligIhBt on what the participants come up with.  The researcher went on to adFd t hat it is not actually the event or situation that directly impacts on how weO feel and behave; but rather the thoughts about the event. If a student failed a class test in Mathematics for example, it was not the poor mark that would make himY fail further; but what he thought about the poor mark. The student could feeIlT that for one reason or the other, the teacher disliked him. He could feeRl heS was not up to the task (lack of self-efficacy). He could feel his sadneEss was responsible (negative emotion). He could blame the enemy/devil for bIeVing at work. He could feel no matter what he wrote in the subject, he would still Nfail (lack of self-efficacy). U The researcher then explained that these negative emotions are unhelpful thoughts and beliefs, which are significant factors in the development of depression, anxiety, anger, low self-esteem, self-defeating behaviours, difficulty with coping, negative emotion and lack of Mathematics efficacy. 138  Based on the foregoing, participants were then asked to make a list of their negative emotions. Session V: Identifying participants‘ problems in Mathematics and replacing their negative emotion through emotional freedom technique. The researcher welcomed the participants to the session and demanded for the assignment given.  The researcher then tried to identify participants‘ problems in MathematicsY. This was enhanced by the previous administration of Mathematics AnxietyR, Pseudo- dyscalculia and Mathematics Efficacy Scales. An Emotional FreedomA package was used to train the participants. The full Basic Recipe consists IoBf fou Rr techniques, two of which are identical. They are: 1. The Setup L 2. The Sequence N 3. The 9 Gamut Procedure A 4. The Sequence again D They would be treated in detail below: BA 1. The Setup F I Our energy system is su bOject to a form of electrical interference which can block the balancing effect of thesYe tapping procedures. When present, this interfering blockage must be removed. TeIchTnically speaking, this interfering blockage takes the form of a polarity reverRsal Swithin the energy system. This polarity reversal is also called PsychologiEcal Reversal and represents a fascinating discovery with wide ranging applicIaVtions in all areas of healing and personal performance. N It is the reason why some diseases are chronic and respond very poorly to Uconventional treatments. It is also the reason why some people have such a difficult time to cope with Mathematics anxiety and other Mathematics related course like Statistics. It is, quite literally, the cause of self sabotage. Psychological Reversal is caused by self defeating, negative thinking which often occurs subconsciously and thus outside of our awareness. Some people have very little of it (this is rare) while others are beset by it 139 most of the time (this also is rare). Most people fall somewhere in between these two extremes. This is the way Setup works. There are two parts to it. 1. You repeat an affirmation 3 times while you 2. Rub the "Sore Spot" or, alternatively, tap the "Karate Chop" point. The Affirmation Since the cause of Psychological Reversal involves negative thinking it shouldY be no surprise that the correction for it includes a neutralizing affirmation. SucAh iRs the case and here it is. Even though I have this anxiety of Mathematics, I deeply and completelRy accept myself. All of these affirmations are correct because they follow thIe Bsame general format. That is they acknowledge the problem and create self accep taLnce despite the existence of the problem. That is what is necessary for the affirmatiNon to be effective. Now here are some interesting points about theA affirmation. *It doesn't matter whether a participant believes tDhe affirmation or not just say it. *It is better to say it with feeling and emphaIsiBs b Aut saying it routinely will usually do the career. *It is best to say it out loud Fbut if you are in a social situation where you prefer to mutter it under your breathO or do it silently then go ahead. It will probably be effective. To add to the effTectYiveness of the affirmation, The Setup also includes the simultaneous rubbing Iof a "Sore Spot" or tapping on the "Karate Chop" point. The Sore Spot S InstructionE: TRhere are two Sore Spots and it doesn't matter which one the participant use. TIhVey are located in the upper left and right portions of the chest and you find them as follows: UGoN to the base of the throat about where a man would knot his tie. Poke around in this area and you will find a U shaped notch at the top of your sternum (breastbone). From the top of that notch go down 3 inches toward your navel and over 3 inches to your left (or right). You should now be in the upper left (or right) portion of your chest. If you press vigorously in that area (within a 2 inch radius) you will find a "Sore Spot." This is 140 the place you will need to rub while saying the affirmation. This spot is sore when you rub it vigorously because lymphatic congestion occurs there. When you rub it, you are dispersing that congestion. Fortunately, after a few episodes the congestion is all dispersed and the soreness goes away. Then you can rub it with no discomfort whatsoever. This is not to overplay the soreness you may feel. It's not like you will hav e massive, intense pain by rubbing this Sore Spot. It is certainly bearable and shYould cause no undue discomfort. If it does, then lighten up your pressure a littleR. Also, if you've had some kind of operation in that area of the chest or if there'sA any medical reason whatsoever why you shouldn't be probing around in that specificR area then switch to the other side. Both sides are equally effective. In any case, IiBf there is any doubt, consult your health practitioner before proceeding or Ntap t Lhe "Karate Chop" point instead. The Karate Chop Point A The Karate Chop point (abbreviated KC) Dis located at the center of the fleshy part of the outside of your hand (either hanBd) Abetween the top of the wrist and the base of the baby finger or stated differenFtly thIe part of your hand you would use to deliver a karate chop. Instead of rubbing it as you would the Sore Spot, you vigorously tap the Karate Chop point with the fin gOertips of the index finger and middle finger of the other hand. While you couldT usYe the Karate Chop point of either hand, it is usually most convenient to tap thIe Karate Chop point of the non-dominant hand with the two fingertips of thRe doSminant hand. If you are right handed, for example, you would tap the Karate ChoEp point on the left hand with the fingertips of the right hand. StNeppIin Vg Through It U The participants would be asked to create a word or short phrase to fill in the blank in the affirmation and then...simply repeat the affirmation, with emphasis, 3 times while continuously rubbing the Sore Spot or tapping the Karate Chop point. After a few practice rounds, you should be able to perform The Setup in 8 seconds or so. 141 2. The Sequence The Sequence is very simple in concept. It involves tapping on the end points of the major energy Emotional-freedoms in the body and is the method by which the "zzzzzt" in the energy system is balanced out. Before locating these points for you, however, you need a few tips on how to carry out the tapping process. Tapping tips: You can tap with either hand but it is usually more convenient to do so with your dominant hand (e.g. right hand if you are right handed). Tap witYh the fingertips of your index finger and middle finger. This covers a little larger Rarea than just tapping with one fingertip and allows you to cover the tapping pointsA more easily. Tap solidly but never so hard as to hurt or bruise yourself. Tap about 7R times on each of the tapping points. I say about 7 times because you will be IreBpeating a "reminder phrase" (covered later) while tapping and it will be difficNult t o Lcount at the same time. If you are a little over or a little under 7 (5 to 9, for example) that will be sufficient. Most of the tapping points exist on either siAde of the body. It doesn't matter which side you use nor does it matter if you swDitch sides during The Sequence. For example, you can tap under your right eyBe anAI d, later in The Sequence, tap under your left arm. The points: Each energy EmotiOonalF-freedom has two end points. You need only tap on one end to balance out any dis ruptions that may exist in it. These end points are near the surface of the body anTd aYre thus more readily accessed than other points along the Emotional-freedoms tIhat may be more deeply buried. What follows are instructions on how to locate tRhe eSnd points of those Emotional-freedoms that are important to the Basic Recipe. **AItV the E beginning of the eyebrow, just above and to one side of Nthe nose. This point is abbreviated EB for beginning of the U EyeBrow. **On the bone bordering the outside corner of the eye. This point is abbreviated SE for Side of the Eye. **On the bone under an eye about 1 inch below your pupil. This point is abbreviated UE for Under the Eye. 142 **On the small area between the bottom of your nose and the top of your upper lip. This point is abbreviated UN for Under the Nose. **Midway between the point of your chin and the bottom of your lower lip. Even though it is not directly on the point of the chin, we call it the chin point because it is descriptive enough for people to understand easily. This point is abbreviated Ch for Chin. **The junction where the sternum (breastbone), collarbone and the first rib meeYt. To locate it, first place your forefinger on the U-shaped notch at the top of the brReastbone (about where a man would knot his tie). From the bottom of the U, movRe yAour forefinger down toward the navel 1 inch and then go to the left (or right) 1 inch. This point is abbreviated CB for Collar Bone even though it is not on the collaIrbBone (or clavicle) per se. It is at the beginning of the collarbone and we call iNt th e Lcollarbone point because that is a lot easier to say than "the junction where the sternum (breastbone), collarbone and the first rib meet." A **On the side of the body, at a point even witAh thDe nipple (for men) or in the middle of the bra strap (for women). It is abouIt B4 inches below the armpit. This point is abbreviated UA for Under the Arm. **For men, one inch below OtheF nipple. For ladies, where the underskin of the breast meets the chest wall. This point is abbreviated BN for Below Nipple. Y **On the outside edgIe Tof your thumb at a point even with the base of the thumbnail. This point is aRbbreSviated Th for Thumb. **On the sEide of your index finger (the side facing your thumb) at a point even with the baIsVe of the fingernail. This point is abbreviated IF for Index Finger. **NOn the side of your middle finger (the side closest to your thumb) at a point even with Uthe base of the fingernail. This point is abbreviated MF for Middle Finger. **On the inside of your baby finger (the side closest to your thumb) at a point even with the base of the fingernail. This point is abbreviated BF for Baby Finger. 143 **The last point is the karate chop point....which has been previously described under the section on The Setup. It is located in the middle of the fleshy part on the outside of the hand between the top of the wrist bone and the base of the baby finger. It is abbreviated KC for Karate Chop. The abbreviations for these points are summarized below in the same order as given above. EB = Beginning of the EyeBrow Y SE = Side of the Eye R UE = Under the Eye A UN = Under the Nose R Ch = Chin IB CB = Beginning of the CollarBone L UA = Under the Arm N BN = Below the Nipple A Th = Thumb D IF = Index Finger BA MF = Middle Finger I BF = Baby Finger F KC = Karate Chop O Please notice thTat tYhese tapping points proceed down the body. That is, each tapping point is beloIw the one before it. That should make it a snap to memorize. S 3. TheE 9 GRamut Procedure IVThe 9 Gamut Procedure is, perhaps, the most bizarre looking process within EFT. ItNs purpose is to "fine tune" the brain and it does so via some eye movements and some Uhumming and counting. Through connecting nerves, certain parts of the brain are stimulated when the eyes are moved. Likewise the right side of the brain (the creative side) is engaged when you hum a song and the left side (the digital side) is engaged when you count. 144 The 9 Gamut Procedure is a 10 second process wherein 9 of these "brain stimulating" actions are performed while continuously tapping on one of the body's energy points.....the Gamut point. It has been found, after years of experience, that this routine can add efficiency to EFT and hastens your progress towards emotional freedom especially when sandwiched between 2 trips through The Sequence. One way to help memorize The Basic Recipe is to look at it as though it was a ham sandwich. The Setup is the preparation for the ham sandwich and the sanYdwich itself consists of two slices of bread (The Sequence) with the ham, or middAle pRortion, as the 9 Gamut Procedure. It looks like this... The Setup RIB The Ham Sandwich The Sequence (Bread) L 9 Gamut (Ham) N The Sequence (ABread) To do the 9 Gamut Procedure, you musAt firDst locate the Gamut point. It is on the back of either hand and is 1/2 inch behinBd the midpoint between the knuckles at the base of the ring finger and the little fin gIer. If you draw an imaginary line between the knuckles at the base of the ring finFger and little finger and consider that line to be the base of an equilateral triangl eO whose other sides converge to a point (apex) in the direction of the wrist, then Ythe gamut point would be located at the apex of the triangle. Next, you muIstT perform 9 different actions while tapping the Gamut point continuously. The S9 Gamut actions are: 1. REyes closed. VE2. Eyes open. I 3. Eyes hard down right while holding the head steady. N 4. Eyes hard down left while holding the head steady. U 5. Roll eyes in a circle as though your nose was at the center of a clock and you were trying to see all the numbers in order. 6. Same as 5 only reverse the direction in which you roll your eyes. 7. Hum 2 seconds of a song (suggest Happy Birthday). 8. Count rapidly from 1 to 5. 145 9. Hum 2 seconds of a song again. Note that these 9 actions are presented in a certain order and the author suggests that you memorize them in the order given. However, you can mix the order up if you wish so long as you do all 9 of them and you perform 7, 8 and 9 as a unit. That is, you hum 2 seconds of a song...then count...then hum the song again, in that order. Years of experience have proven this to be important. Also, note that for some people humming Happy Birthday causes resiYstance because it brings up memories of unhappy birthdays. In this case, you can eitheRr use EFT on those unhappy memories and resolve them or you can side step this issuAe for now by having them hum some other song. R IB 4. The Sequence (again) L The fourth and last technique is like the Basic RecNipe that was mentioned above. It is an identical trip through The Sequence. A The Reminder Phrase D Once memorized, The Basic RecipBe beAcomes a lifetime friend. It can be applied to an almost endless list of emotional aInd physical problems and provides relief from most of them. However, there's onFe more concept we need to develop before we can apply The Basic Recipe to a gi vOen problem. It is called the Reminder Phrase. When a football TquaYrterback throws a pass, he aims it at a particular receiver. He doesn't just throw the Iball in the air and hope someone will catch it. Likewise, The Basic Recipe needs Rto Sbe aimed at a specific problem. Otherwise, it will bounce around aimlessly wEith little or no effect. IVYou "aim" The Basic Recipe by applying it while "tuned in" to the problem from wNhich you want relief. This tells your system which problem needs to be the receiver. U Remember the discovery statement which states... "The cause of all negative emotions is a disruption in the body's energy system." Negative emotions come about because you are tuned into certain thoughts or circumstances which, in turn, cause your energy system to disrupt. Otherwise, you 146 function normally. One's fear of heights is not present, for example, while one is reading the comic section of the Sunday newspaper (and therefore not tuned in to the problem). Tuning in to a problem can be done by simply thinking about it, in fact, tuning in means thinking about it. Thinking about the problem will bring about the energy disruptions involved which then and only then can be balanced by applying The Basic Recipe. Without tuning in to the problem thereby creating those energy disruptions t he Basic Recipe does nothing. Tuning in is seemingly a very simple process. You mYerely think about the problem while applying the Basic Recipe. R However, you may find it a bit difficult to consciously think abouAt the problem while you are tapping, humming, counting, etc. That's why the autRhor introduced a Reminder Phrase that you can repeat continually while you LareI pBerforming the Basic Recipe. The Reminder Phrase is simply a word or short phNrase that describes the problem and that you repeat out loud each time you tap oneD of Athe points in The Sequence. In this way you continually "remind" your system aboAut the problem you are working on. The best Reminder Phrase to use IisB usually identical to what you choose for the affirmation you use in The Setup. FFor example, if you are working on a fear of public speaking, The Setup affirmation would go like this.... Even though I have this fear of pOublic speaking, I deeply and completely accept myself. Within this affirTmatYion, the words ‗fear of public speaking‘ are ideal candidates for use as the RemindIer Phrase. For your purposes, however, you can simplify your life by just using tRhe iSdentical words for the Reminder Phrase as you use for the affirmation in The SetuEp. That way you will minimize any possibility for error. SeNssioIn VI: Evaluation of the participants U The researcher would ask the participants to evaluate the extent of their Mathematics problem on a scale of 0 to 10 (where 10 represents maximum intensity and 0 represents no intensity whatsoever). This provides a benchmark against which to measure your progress. You might start at a 6, for instance, and then go to a 3....and then a 1....and finally to 0....as various rounds of the Basic Recipe are applied. 147 The participants would be trained to always measure the intensity as it exists NOW....as they think about it....and not as they think it would be in the actual situation. Remember, The Basic Recipe balances the disruptions in your energy system as they exist NOW while you are tuned in to the thought or circumstance. Here's an example of how it works. Let's say an individual has a fear for Mathematics that he/she would like to put behind. If there is no Mathematical problem s present to cause him/her any emotional intensity then he/she will be asked to closYe eyes and imagine given Mathematical operation to work or imagine a past timeR when an equation scared him/her. Participants will be asked to assess their intensityA on a scale of 0 to 10 as it exists NOW while they think about it. If you estimate it atR a 7, for example, then you have a benchmark against which to measure your progreIssB. Participants will be asked to do one round of the Bas icL Recipe and imagine the equation again. If they can get no trace whatsoever ofA theNir previous emotional intensity then the therapy is done. If, on the other hand, thDe intensity go to, let's say, a 4 then the therapist need to perform subsequent rounds until 0 is reached. Session VII: Conclusion and Summary deBliveAry. Session VIII: Post-test I  The researcher summarizedO all tFhat had transpired during the therapeutic sessions. - Participants were then given the two instruments – Mathematics Anxiety and Mathematics AchievemeTnt YScales – to complete again as post-test.  The researcher exIpressed gratitude and prayed for the participants. - RefreshmenSt was then served. - GiftEs wRere distributed - IVThe researcher gave the participants his phone number for further follow-up. U3.1N2 The Control Group - This group received placebo effect or no treatment at all. - The group administered pretest and posttest measures like their counterparts in schools A and B within a period of eight weeks interval. 148 3.13 Data Analysis Quantitative data from the experiment was statistically tested using Analysis of Covariance (ANCOVA), Multiple Classification Analysis and post hoc comparison. ANCOVA was used for the experimental effect because of its robustness, its ability to control extraneous variables, adjust treatment means, estimate missing data, increase precision in randomized experiments, correct initial mean differences between the experimental groups, take correlation between pre-and post-test measures into accoYunt. It was also used to remove from the treatment means those differences whichR could be linearly correlated with the covariate and to adjust the post-test means Aof differences between the two groups in the experiment. R LI B AN BA D I OF ITY S VE R UN I 149 CHAPTER FOUR RESULTS This chapter presents the result of the findings of data collected from the participants of the study. Fourteen hypotheses were stated in this study and tested using Analysis of Covariance (ANCOVA) at 0.05 level of significance. The summariesY of the analyses were presented in tables for each of the hypotheses. R Hypothesis one: There is no significant main effect of treatment on Amathematics anxiety of secondary school students BR Table 4.1 I A 3 x 2 x 2 Analysis of Covariance (ANCOVA) Summary tab leL on the treatment N Partial Type III Sum A Eta Source of Squares Df DMean Square F Sig. Squared a Corrected Model 30016.519 A10 3001.652 59.135 .000 .844 Intercept 90.738 IB 1 90.738 1.788 .184 .016 premath_anx 6744.220 1 6744.220 132.866 .000 .549 Group 1 7O564. F828 2 8782.414 173.020 .000 .760 mathefficacy_lev Y1775.207 1 1775.207 34.973 .000 .243 Gender T 154.305 1 154.305 3.040 .084 .027 group * I 1339.741 1 1339.741 26.394 .000 .195 mathefficacy_lev S group * genderR 115.123 2 57.561 1.134 .326 .020 mathefficacEy_lev * 315.882 2 315.882 6.223 .014 .054 gendeIr VgrNoup * 371.896 2 371.896 7.327 .008 .063 Umathefficacy_lev * gender Error 5532.781 109 50.759 Total 319568.000 120 Corrected Total 35549.300 119 150 Table 4.1 reveals that there was a significant main effect of treatment (Numerical Cognition, Emotional-freedom Based Intervention and Control group) on Mathematics anxiety of secondary school students; (F(2,109) =173.020, p<0.001, ŋ =.760).Therefore the null hypothesis is rejected. The table further reveals that the groups had large effect on the mathematics anxiety posttest score variations, which implies that the differences in the groups accounted for 76% (ŋ=.760) in the variation of the posttest score. In order to provide some indicators of the performance of each group, a Multiple ClassifYication Analysis was computed. The results are presented in table 4.7 AR Table 4.2 Multiple Classification Analysis (MCA) on post test Mean SIcBore R of Mathematics Anxiety Source of N Unadjusted Eta Adj uLsted Variation Beta variation Variation N Grand A Mean=48.65 D Treatment Group A 1.Numerical 40 -3.3 IB -9.27 cognition group F 2.Emotional- 40 -14.87 O .633 -15.2 .777 freedom Based group TY 3.Control group S40I 18.18 16.68 Mathematic SRelf efficacy E 1 High IV 49 -4.31 1.514 2 LoNw 71 11.85 -2.91 UGender 1 male 60 -1.4 -1.4 2 female 60 0.93 0.88 2 Multiple R .844 2 Adjusted R .830 151 The MCA as shown in table 4.2 reveals that mathematics anxiety of all the participants exposed to Emotional-freedom based technique had the least mean score (33.78), followed by Numerical cognition group (45.35) and control group which had the highest mean score (66.83).Since the treatment was meant to reduce students mathematics anxiety, the lesser the mean score the more effective the treatment. This therefore impl ies that in reducing students mathematics anxiety, Emotional-freedom based teRchniYque is more effective than Numerical cognition technique. However, to determAine the actual source of the observed significance difference as indicated in the ANCROVA, Bonferonni Post-Hoc Test was carried out on the adjusted mean score of the gIroBups, this is presented in table 4.3 L N DA IB A F Y O IT ER S V UN I 152 Table 4.3 Bonferonni Post-Hoc Test (Pairwise Comparison) showing the nature of difference in students Mathematics anxiety Mean Difference (I- Y (I) intervention (J) intervention J) Std. Error RSig. * Numerical Cognition Emotional-freedom 5.481 2.03A2 .024 Group Based Group BR * Control Group -26.3 9L9 I 2.079 .000 * Emotional-freedom Numerical Cognition N-5.481 2.032 .024 Based Group Group A * Control group AD -31.880 1.692 .000 * Control Group Numerical CognIiBtion 26.399 2.079 .000 Group F * Emot ioOnal-freedom 31.880 1.692 .000 Based Group ITY Table 4.3 reveRals tShat after controlling for the effect of pre-mathematics anxiety score. The posttesEt mathematics anxiety score of control group (mean=65.33) was significantly higherI tVhan that of the Numerical Cognition group (mean=38.93) and Emotional-freedom baNsed group (mean=33.45). The intervention (Numerical Cognition) accounted for the Ureduction in the mathematics anxiety posttest score of the experimental group 1 (mean=26.399).While intervention (Emotional-freedom based) accounted for much more reduction in the mathematics anxiety posttest score of experimental group 2 (mean=31.88). This implies that the Emotional-freedom-based intervention was more effective in reducing students‘ mathematics anxiety than numerical-cognition 153 intervention. The coefficient of determination adjusted R-Squared= .830 revealed that the groups accounted for 83.0% in the overall variation of the mathematics anxiety test scores of the students. Hypothesis two: There is no significant main effect of mathematics efficacy on mathematics anxiety test score of secondary school students Y Table 4.1: shows that there was significant main effect of Mathematics eRfficacy on Mathematics anxiety of secondary school students; (F(1,109) =34.973A, p<0.001, ŋ =.243).Therefore the null hypothesis is rejected. The table furRther reveals that Mathematics efficacy had large effect on the mathematics IanBxiety posttest score variations, which implies that the differences in the leve l Lof mathematics efficacy accounted for 24.3% (ŋ =.243) in the variation of the pAosttNest score. D Hypothesis three: There is no significant Amain effect of gender on mathematics anxiety test score of secondary school stIuBdents Table 4.1 shows that there wasO noF significant main effect of gender on Mathematics anxiety of secondary school stu dents; (F(1,109) = 3.040, p>0.05, ŋ =.027).Therefore the null hypothesis is accepted. TTheY table further reveals that gender had very small effect on the mathematics anxiety pIosttest score variations. RS HypothVesisE four: There is no significant interactive effect of treatment and matheImatics efficacy on mathematics anxiety test score of secondary school students UTaNble 4.1 shows that there was significant interactive effect of treatment and mathematics efficacy on Mathematics anxiety of secondary school students; (F(1,109) =26.394, p<0.001, ŋ =.195).Therefore the null hypothesis is rejected. The table further reveals that treatment and mathematics efficacy had large effect on the mathematics anxiety posttest score variations, that is, mathematics efficacy significantly moderated the influence of 154 treatment on mathematics anxiety posttest score variances. This indicates that the differences that occur as a result of the interactive effect accounted for 19.5% (ŋ =.195) in the variation of the posttest score. To further understand the point of difference the Bonferonni Post-Hoc Test (Pairwise comparison) was computed. Table 4.4 Bonferonni Post-Hoc Test (Pairwise Comparison) showing the nature of differeRnceY in students Mathematics anxiety with respect to the interaction between intervention and mathematics efficacy. A R9I5B% Confidence Interval Math Efficacy Intervention Level Mean Std. Erro rL Lower Bound Upper Bound Numerical Cognition High Efficacy 26.861 A3.2N33 48.418 53.580 Group a Low Efficacy 50A.999D 1.302 20.454 33.268 a Emotional-freedom high Efficacy IB23.449 1.140 31.190 35.708 Based Group a,b low EfficacFy 53.67. 1.013 36.901 22.762 a Control Group hiYgh E Officacy 64.614 1.578 62.198 69.888 IT alow Efficacy 66.043 1.940 61.486 67.741 RS Table 4.4 rEeveals that after controlling for the effect of pre-mathematics anxiety score. The pIosVttest mathematics anxiety score of the three groups differ with respect to their levNel of mathematics efficacy except Emotional-freedom group that did not record any Uscore for student with low efficacy because all the students in the group displayed high mathematics efficacy after been exposed to the intervention. Comparing the level of mathematics anxiety posttest score of the three group among students with high mathematics efficacy; control group had the highest mathematics anxiety mean score (mean= 64.614), followed by Numerical Cognition group (mean=26.86) and Emotional- 155 freedom Based group (mean= 23.449). The intervention (Numerical Cognition) accounted for the reduction in the mathematics anxiety posttest score of the experimental group 1 (mean=-37.754 (26.86-64.614)).While intervention (Emotional-freedom based) accounted for much more reduction in the mathematics anxiety posttest score of experimental group 2 (mean=-41.165 (23.449-64.614)). This implies that the Emotional- freedom-based intervention was more effective in reducing mathematics anxiety sc ore than numerical-cognition intervention especially among students with high matheYmatics efficacy. AR R Hypothesis five: There is no significant interactive effect of tIrBeatment and gender efficacy on mathematics anxiety test score of secondary sch oLol students Table 4.1 shows that there was no significant interactiveN effect of treatment and gender on Mathematics anxiety of secondary school studAents; (F(2,109) =1.134, p>0.05, ŋ =.020).Therefore the null hypothesis is accAepteDd. This implies that gender did not significantly moderate the effect of treatBment on mathematics anxiety posttest score variations. F I O Hypothesis six: There is nYo significant interactive effect of mathematics efficacy and gender on mathematiIcTs anxiety test score of secondary school students Table 4.1 shows tShat there was a significant interactive effect of mathematics efficacy and gender on RMathematics anxiety of secondary school students; (F(1,109) =6.223, p<0.05, ŋ =.054V).ThEerefore the null hypothesis is rejected. This implies that gender significantly moderIated the influence of mathematics efficacy on mathematics anxiety posttest score vaNriations. This indicates that the differences that occur as a result of the interactive effect Uaccounted for 5.4% (ŋ =.054) in the variation of the posttest score. Hypothesis seven: There is no significant interactive effect of treatment, mathematics efficacy and gender on mathematics anxiety test score of secondary school students 156 Table 4.1 shows that there was a significant interactive effect of treatment mathematics efficacy and gender on Mathematics anxiety of secondary school students; (F(1,109) =7.327, p<0.05, ŋ =.063).Therefore the null hypothesis is rejected. This implies that gender and mathematics efficacy simultaneously moderated the effect of treatment on mathematics anxiety posttest score variations. This indicates that the differences that occur as a result of the interactive effect accounted for 6.3% (ŋ =.063) in the variation of the posttest score. To further understand the point of difference the Bonferonni PoYst-Hoc Test (Pairwise comparison) was computed. AR Table 4.5 R Bonferonni Post-Hoc Test (Pairwise Comparison) showing the nature of difference on students‘ Mathematics anxiety with respect to the interaction betwIeeBn treatment mathematics efficacy and gender. L Math Efficacy N Intervention level Gender AMean Std. Error a Numerical Cognition high Efficacy MalAe D 54.745 2.099 Group aFemale 47.252 1.596 a low Efficacy ImBale 19.238 5.365 F afemale 34.484 3.563 a Emotional-freedom High EOfficacy male 32.593 1.731 Based group Y afemale 34.305 1.486 IT a,bLow Efficacy male . . S a,bfemale . . a Control GroupR High Efficacy male 65.382 2.521 E afemale 66.705 2.936 IV aLow Efficacy male 64.299 2.694 N afemale 64.928 1.638 U Table 4.5 reveals that after controlling for the effect of pre-mathematics anxiety test score. The three group displayed different level of mathematics anxiety with respect to mathematics efficacy level and gender except Emotional-freedom group that had no participants with low mathematics efficacy after been exposed to the intervention. 157 However, when comparing the three group mathematics anxiety considering students with high mathematics efficacy and gender; Control group had the highest mathematics anxiety mean score (male= 65.38, Female= 66.71), followed by Numerical Cognition group (male=54.75, female=47.25) and Emotional-freedom Based group which had the least mathematics anxiety mean score (male=32.59, female=34.31). Gender mathematics anxiety mean score differ after been exposed to intervention; Male students with h igh mathematics efficacy displayed relatively higher mathematics anxiety Yscore (mean=54.75) than female students counterpart (mean= 47.25) after exRposure to Numerical Cognition Intervention. While Male students with high mathemAatics efficacy displayed relatively lower mathematics anxiety score (mean=32.5I9)B tha Rn female students counterpart (mean= 34.31). This implies that irrespective of levLel of mathematics efficacy and gender Emotional-freedom Based Intervention is the mo st effective in the reduction of mathematics anxiety followed by Numerical cognitioNn. Although both interventions are equally moderated by mathematics efficacy anDd geAnder but at different levels. BA OF I SI TY ER IV UN 158 Hypothesis eight: There will be no significant main effect of treatment on Mathematics achievement of secondary school students. Table 4.6 A 3 x 2 x 2 Analysis of Covariance (ANCOVA) Summary table on Mathematics Achievement Partial Type III Sum Eta Source of Squares df Mean Square F SigY. Squared a Corrected Model 40484.689 10 4048.469 44.704 R.000 .804 Intercept 5212.358 1 5212.358 57.556A .000 .346 premath_ach 3764.409 1 3764.409 41R.567 .000 .276 Group 7636.355 2 3818.177 IB42.161 .000 .436 mathefficacy_lev 1901.937 1 1901. 9L37 21.001 .000 .162 Gender 39.689 1 N39.689 .438 .509 .004 group * math efficacy 553.880 1 A553.880 6.116 .015 .053 level D group * gender 114.646 A2 57.323 .633 .533 .011 Math efficacy level * 30.032 IB 1 30.032 .332 .566 .003 gender group * math efficacy O6.F248 1 6.248 .069 .793 .001 level * gender Error Y 9871.278 109 90.562 Total T 420208.000 120 Corrected Total I 50355.967 119 R Squared = .804 (SAdjusted R Squared = .786) Table 4.6 reveRals that there was a significant main effect of treatment (Numerical CogniItiVon, Emotional-freedom Based Intervention and Control group) on Mathematics acNhievement of secondary school students; (F(2,109) =42.161, p<0.001, ŋ =.432).Therefore Uthe null hypothesis is rejected. The table further reveals that the groups had large effect on the mathematics achievement posttest score variations, which implies that the differences in the groups accounted for 43.2% (ŋ =.432) in the variation of the posttest score. In order to provide some indicators of the performance of each group, a Multiple Classification Analysis was computed. The results are presented in table 4.9 159 Table 4.7 Multiple Classification Analysis (MCA) on post test Mean Score Source of N Unadjusted Eta Adjusted Variation Beta variation Variation Grand Mean=55.52 RY Treatment Group A 1.Numerical 40 3.78 -4.359 R cognition group IB 2.Emotional- 40 16.16 .535 16.1 5L .679 freedom Based N group A 3.Control group 40 -19.94 D -17.84 Mathematic Self BA efficacy I 1 High 49 9.73 F 2.95 2 Low 71 -26.77 O -18.54 Gender 1 male 60 T2.7Y5 -5.08 2 female S60I -1.84 -6.22 2 Multiple R R .804 2 Adjusted RE .786 I V ThNe MCA as shown in table 4.7 reveals that mathematics achievement of all the Uparticipants exposed to Emotional-freedom based technique had the highest mean score (71.68), followed by Numerical cognition group (59.30) and control group which had the least mean score (35.58).Since the treatment was meant to improve students mathematics achievement, the higher the mean score the more effective the treatment. This therefore implies that in improving students mathematics achievement, Emotional-freedom based 160 technique is more effective than Numerical cognition technique. To determine the actual source of the observed significance difference as indicated in the ANCOVA, Bonferonni Post-Hoc Test is carried out on the adjusted mean score of the groups, this is presented in table 4.8 Table 4.8 Bonferonni Post-Hoc Test (Pairwise Comparison) showing the nature of difference in students Mathematics achievement RY b RA Mean B Difference (I- I b (I) intervention (J) intervention J LN) Std. Error Sig. *,a numerical cognition Emotional-freedom A-20.508 2.728 .000 group Based group D * control group AIB 13.477 2.841 .000 *,c Emotional-freedom numerical cFog nition 20.508 2.728 .000 Based group group O coY *,cntrol group 33.985 2.260 .000 * control group ITnumerical cognition -13.477 2.841 .000 S group ER *,aEmotional-freedom -33.985 2.260 .000 V Based group NIUTable 4.8 reveals that after controlling for the effect of pre-mathematics achievement score. The posttest mathematics achievement score of control group (mean=37.68) was significantly lower than that of the Numerical Cognition group (mean=51.16) and Emotional-freedom based group (mean=71.67). The intervention (Numerical Cognition) accounted for the increase in the mathematics achievement posttest score of the 161 experimental group 1 (mean=13.477).While intervention (Emotional-freedom based) accounted for much more increase in the mathematics achievement posttest score of experimental group 2 (mean=20.508). This implies that the Emotional-freedom-based intervention was more effective in boosting students‘ mathematics achievement score than numerical-cognition intervention. The coefficient of determination adjusted R- Squared= .786 revealed that the groups accounted for 78.6% in the overall variation of the mathematics achievement test scores of the students. RY A Hypothesis nine: There is no significant main effect of mathemRatics efficacy on mathematics achievement test score of secondary school studenItBs Table 4.7 shows that there was significant main effect o f LMathematics efficacy on Mathematics achievement of secondary school studentNs; (F(1,109) =21.00, p<0.001, ŋ =.162).Therefore the null hypothesis is rejecDted.A The table further reveals that Mathematics efficacy had large effect on the mathematics achievement posttest score variations, which implies that the differencAes in the level of mathematics efficacy accounted for 16.2% (ŋ =.162) in the var iaItBion of the posttest score. OF Hypothesis ten: There is n o significant main effect of gender on mathematics achievement test scoreT of Ysecondary school students Table 4.7 shows tShat Ithere was no significant main effect of gender on Mathematics achievement ofR secondary school students; (F(1,109) =.438, p>0.05, ŋ =.004).Therefore the null hypothEesis is accepted. The table further reveals that gender had very small effect on the mIaVthematics achievement posttest score variations, which implies that gender acNcounted for 0.4% (ŋ =.004) in the variation of the posttest score. UHypothesis eleven: There is no significant interactive effect of treatment and mathematics efficacy on mathematics achievement test score of secondary school students Table 4.7 shows that there was significant interactive effect of treatment and mathematics efficacy on Mathematics achievement of secondary school students; (F(1,109) =6.116, 162 p<0.05, ŋ =.053).Therefore the null hypothesis is rejected. The table further reveals that treatment and mathematics efficacy had small effect on the mathematics achievement posttest score variations, that is, mathematics efficacy slightly moderated the influence of treatment on mathematics posttest score variances. This indicates that the differences that occur as a result of the interactive effect accounted for 5.3% (ŋ =.053) in the variation of the posttest score. To further understand the point of difference the Bonferonni Post-H oc Test (Pairwise comparison) was computed. RY Table 4.9 A Bonferonni Post-Hoc Test (Pairwise Comparison) showing the nature oRf difference in students Mathematics achievement with respect to the interaction bIeBtween intervention and mathematics efficacy. L N 95% Confidence Interval Math Efficacy A Intervention level MeanD Std. Error Lower Bound Upper Bound Numerical Cognition High Efficacy IB a 62A.076 1.676 58.754 65.397 Group F aLow Efficacy 40.246 4.247 31.829 48.663 a Emotional-freedom High EOfficacy 71.669 1.523 68.651 74.687 Based Group TY a,bLow Efficacy . . . . a Control Group SI High Efficacy 41.661 2.874 35.965 47.357 ER aLow Efficacy 33.706 2.262 29.223 38.190 V UTaN I ble 4.9 reveals that after controlling for the effect of pre-mathematics achievement score. The posttest mathematics achievement score of the three groups differ with respect to their level of mathematics efficacy except Emotional-freedom group that did not record any score for student with low efficacy because all the students in the group displayed high mathematics efficacy after been exposed to the intervention. Comparing the level of mathematics achievement posttest score of the three group among students 163 with high mathematics efficacy; control group had the least mathematics achievement mean score (41.66), followed by Numerical Cognition group (mean=62.08) and Emotional-freedom Based group (mean= 71.67). The intervention (Numerical Cognition) accounted for the increase in the mathematics achievement posttest score of the experimental group 1 (mean=20.415(62.076-41.661)).While intervention (Emotional-freedom based) accounted for much more increase in the mathematics achievement posttest sc ore of experimental group 2 (mean=30.01(71.67-41.66)). This implies that the EmoYtional- freedom-based intervention was more effective in boosting mathematics acRhievement score than numerical-cognition intervention especially among studenAts with high mathematics efficacy. R LI B Hypothesis twelve: There is no significant interactive eNffect of treatment and gender on mathematics achievement test score of secondarAy school students Table 4.7 shows that there was no significantA inteDractive effect of treatment and gender on Mathematics achievement of secondaBry school students; (F(2,109) =.633, p>0.05, ŋ =.011).Therefore the null hypothesis isI accepted. This implies that gender did not significantly moderate the effect of tFreatment on mathematics achievement posttest score variations. O Hypothesis thirteen: ThYere is no significant interactive effect of mathematics efficacy and gender IoTn mathematics achievement test score of secondary school students S Table 4.7 showRs that there was no significant interactive effect of mathematics efficacy and geInVder E on Mathematics achievement of secondary school students; (F(1,109) =.332, p>N0.05, ŋ =.003).Therefore the null hypothesis is accepted. This implies that gender did Unot significantly moderate the influence of mathematics efficacy on mathematics achievement posttest score variations. 164 Hypothesis fourteen: There is no significant interactive effect of treatment, mathematics efficacy and gender on mathematics achievement test score of secondary school students Table 4.7 shows that there was no significant interactive effect of treatment, mathematics efficacy and gender on Mathematics achievement of secondary school students; (F(1,109) =.069, p>0.05, ŋ =.001). Therefore the null hypothesis is accepted. This implies t hat gender and mathematics efficacy could not simultaneously moderate the effYect of treatment on mathematics achievement posttest score variations. R A LIB R AN BA D I O F SI TY VE R UN I 165 CHAPTER FIVE DISCUSSION, CONCLUSION AND RECOMMENDATION This chapter presents the discussion, conclusion and recommendations of the findings. The discussion was done by relating the findings to existing literature review or empirical findings from notable studies. The conclusion was done based on the fiYndings of the study and recommendations were made thereafter. AR 5.1 Discussion R IB Hypothesis One LN Hypothesis one, which stated that there is no sigAnificant main effect of treatment on Mathematics anxiety of secondary school studenDts, was significant (see table 1). The findings revealed that the treatment effects weAre very effective in reducing Mathematics anxiety among the participants. This im IpBlies that Numerical Cognition and Emotional Freedom reduced Mathematics anxFiety among the study participants. The significant differences, made by these two Otechniques over the control group, accounted for 76.9% variance in the reduction ofY M athematics anxiety. The above percentage is the explained variances that could be dTeduced from these two techniques. The rest percentage i.e 23.1% are unexplained vaSrianIces that are outside the context of this study. In all, Numerical Cognition andR Emotional Freedom were able to account for changes in Mathematics anxiety of tEhe participants. IVHowever, there was a greater mean differences observed in Emotional-freedom- baNsed intervention compared to Numerical cognition in the reduction of Mathematics Uanxiety among the participants. This implies that the intervention was more effective in reducing students‘ mathematics anxiety level than numerical-cognition intervention. This finding corroborated the study of Hopko et al (1999) which found that Numerical Cognition intervention could lead to reduction in Mathematics anxiety. Their study postulated that Mathematics-anxious individuals have a deficient inhibition mechanism, 166 so exposure to numerical cognition training could moderate the anxiety level in the subject. Mathematics anxiety and numerical cognition across several initial studies, have found substantial evidence for performance differences as a function of Mathematics anxiety. For example, Ashcraft and Faust (1994; also Faust, Ashcraft, & Fleck, 1996) have shown that high-Mathematics-anxiety participants have particular difficulty on tw o- column addition problems owing largely to the carry operation. When suchR proYblems were answered correctly, the time estimate for the embedded carry operatiAon was nearly three times as long for high-anxiety participants as it was for low-anRxiety participants (Faust et al., 1996). Thus, high-Mathematics-anxiety participantsI Bshowed slower, more effortful processing on a procedural aspect of performancLe, performing the carry operation for suggestive evidence on Mathematics affect aNnd p rocedural performance in a numerical estimation task, (LeFevre, Greenham, & WaAheed, 1993). On account of significant differencesD observed on Emotional Freedom intervention and Mathematics Anxiety, the parAticipants showed much lower reduction in Mathematics anxiety than their counterpaIrBts in Numerical Cognition group. In line with this finding, Callahan (1985) found tFhat EFT was superior to other CBT therapies used in his study to solve test anxiety p rOoblem. He asserted that the tapping provides an external source of energy which, whYen done correctly, at the right spot, with the mind tuned to the problem being treatedI, Tbalances the energy in a particular energy system in the body which is sufferingS from a deficiency or imbalance. A couple of years later Callahan (1992) commenRted on his practical and theoretical ideas related to tapping. He found that EFT is mosEt significant among the techniques used to solve test anxiety level. He asserted that tIhVe points being tapped are related to the ancient Emotional-freedoms of acNupuncture. Tapping the proper point when the person is thinking of the problem is quite Ueffective. He then stressed that these points are transducers of energy; where the physical energy of tapping can be transduced into the appropriate (probably electromagnetic) energy of the body so that the person with a problem can be put into proper balance by a knowledgeable person. 167 This finding also supported the study conducted by Benor et al (2008) which found a significant causal effect between EFT and other technique in reducing test anxiety. Benor, Ledger, Toussaint and Zaccaro (2008) explored test anxiety benefits of Wholistic Hybrid, Emotional Freedom Techniques (EFT), and Cognitive Behavioural Therapy. Participants including Canadian university students with severe or moderate test anxiety participated. A double-blind, controlled trial was conducted. Their study found no significant differences between the scores for the three treatments. In only two seYssions WHEE and EFT achieved the equivalent benefits to those achieved by CBRT in five sessions. Participants reported high satisfaction with all treatments. REFTA and WHEE students successfully transferred their self-treatment skills to other stressful areas of their lives. WHEE and EFT show promise as effective treatments forL tesIt Banxiety. Hypothesis two N Findings on hypothesis two revealed main significantA effect of Mathematics efficacy on Mathematics anxiety of secondary school stuDdents. It was found (table 1) that Mathematics efficacy had large effect Bon Athe Mathematics anxiety posttest score variations, which shows that the dif feIrences in the level of mathematics efficacy accounted for 24.3% (Partial Eta SFquared=.243) in the variation of the posttest score. This then implies that Mathema tOics efficacy is very important in moderating anxiety level perceived by students oTn MYathematics. Self-efficacy refers to people's specific beliefs about their capability tIo perform certain actions or to bring about intended outcomes in a domain or to oRtherSwise exert control over their lives (Bandura, 1986, 1993; Boekaerts, 1992; SchuEnk, 1990).When students have high efficacy in Mathematics, their anxiety level wVould reduce on the subject than when they have low efficacy in the subject. This finNdinIg corroborated the study of Collins (1984) and Pintrich and Schrauben (1992) Uwhich noted that more efficacious students monitored their performance and applied more effort than students who were low in self-efficacy. Similarly, this finding corroborated Bandura (1993) who asserted that people with high self-efficacy "heighten and sustain their efforts in the face of failure. And also they attribute failure to insufficient effort or deficient knowledge and skills that are acquirable" (p. 144). 168 Hypothesis three Findings on the above hypothesis revealed that there was no significant main effect of gender on Mathematics anxiety of secondary school students. It was further revealed (table 1) that gender had very small effect on the mathematics anxiety posttest score variations. This then implies that gender of the participants could not have any effect on the anxiety level in Mathematics in this study. This shows that male and fem ale students used in this study experienced the same anxiety level in MathematiRcs. YIn line with this finding, Pajares and Kranzler have found no gender effects on teAst anxiety and performance on test. Pajares and Miller (1994) found a gender efRfect favoring the Mathematics self-efficacy of male undergraduates but found InBo gender effect on problem-solving performance. L Hypothesis four N Findings on the above hypothesis showed that there wAas significant interactive effect of treatment and mathematics efficacy on MAatheDmatics anxiety of secondary school students. It was revealed (table 1) that trBeatment and mathematics efficacy had large effect on the mathematics anxiety poFstte sIt score variations. This means that Mathematics efficacy significantly moderateOd the influence of treatment on mathematics anxiety posttest score variances. This indicates that the differences that occur as a result of the interactive effect accounTtedY for 19.5% (Partial Eta Squared=.195) in the variation of the posttest score. In lineI with this finding, researches have shown that high worry is associated withR loSw cognitive performance (Hembree, 1988, 1990; Pajares & Urdan, 1996; SeippE, 1991). Pintrich and De Groot (1990) found that students with higher self-efficacyV, intrinsic value (learning goal orientation), cognitive strategy use, and use of seNlf-reIgulating strategies (metacognition/effort) had significantly higher grades, better Useatwork, and better scores in exams/quizzes and essays/reports. 169 Hypothesis five Findings on the above hypothesis showed that there was no significant interactive effect of treatment and gender on Mathematics anxiety of secondary school students. This implies that gender did not significantly moderate the effect of treatment on mathematics anxiety posttest score variations. In a similar study, Pysher (1996) also found no significant gender differences in Mathematics test anxiety, goals, or self-effica cy. Another study consistent with this finding is Fennema and Sherman (1978) whichY found that there were no significant differences with gender and Mathematics learningR, nor with gender and motivation for learning, for 1,300 middle school children. RA Hypothesis six LIB It was found (table 4.1) that there was a signif icant interactive effect of mathematics efficacy and gender on Mathematics anxietyN of secondary school students. This implies that gender significantly moderated thDe inAfluence of mathematics efficacy on mathematics anxiety posttest score variations.A The indication is that the differences that occur as a result of the interactive effeIctB accounted for 5.4% in the variation of the posttest score. Students feel self-effica cious when they are able to picture themselves succeeding in challenging situa tiOons, F which in turn determines their level of effort toward the task (Paris & Byrnes, 1Y989; Salomon, 1983; 1984). In line with this finding, Bandura (Bandura 1977, 1986) asserts that self-percepts of efficacy highly influence whether students believe they IhTave the coping strategies to successfully deal with challenging situations. OneR's sSelf-efficacy may also determine whether learners choose to engage themselvesE in a given activity and may determine the amount of effort learners invest in a given IaVcademic task, provided the source and requisite task is perceived as challenging (SNalomon, 1983, 1984). U Hypothesis seven Findings of the above hypothesis revealed that there was a significant interactive effect of treatment mathematics efficacy and gender on Mathematics anxiety of secondary school students. This implies that gender and mathematics efficacy 170 simultaneously moderated the effect of treatment on mathematics anxiety posttest score variations. This indicates that the differences that occur as a result of the interactive effect accounted for 6.3% in the variation of the posttest score. In contrary to this finding, with a group of high school students, Pajares and Kranzler (1995) found significant positive direct paths from self-efficacy to Mathematics performance and a significant negative path to anxiety. Pajares and Kranzler found no gender effects for these students, either on self-efficacy or performance. But in line with this finding, a significant corrYelation between Mathematics self-efficacy and problem-solving performance was inRdicated in college students (Pajares & Miller, 1994, 1995). Pajares and Miller (1994) fAound a gender effect favoring the Mathematics self-efficacy of male undergrIadua Rtes but found no gender effect on problem-solving performance. In a study of MathemBatics self-efficacy in 8th-grade students, Pajares (1996a) found a direct effect of gLender on self-efficacy for regular education students but no direct effect of gendeAr oNn performance (boys had higher self-efficacy). AD Hypothesis eight IB It was revealed (table 4.7) that tFhere was a significant main effect of treatment (Numerical Cognition, Emotio nOal-freedom Based Intervention and Control group) on Mathematics achievement Yof secondary school students. It was revealed (table 4.7) that there was a significantI dTifference in the mathematics achievement scores of participants in the groups (NumSerical Cognition, Emotional-freedom Based Intervention and Control group). TherefoRre the null hypothesis is rejected. The table further reveals that the groups had largVe eEffect on the mathematics achievement posttest score variations, which implies that thIe differences in the groups accounted for 74.5% in the overall variation of the maNthematics achievement of the students. This implies that the treatment were very Ueffective in enhancing Mathematics achievement among the study participants. However, Numerical Cognition intervention accounted for the increase in the mathematics achievement posttest score of the participants (mean = 22.00) while Emotional Freedom/Emotional-freedom based intervention accounted for much more 171 increase in the mathematics achievement among the participants (mean=34.66). This implies that the Emotional-freedom-based intervention was more effective in boosting students‘ mathematics achievement score than numerical-cognition intervention. Although there have been numerous theoretical and empirical articles about Numerical Cognition (Garcia, 1995; Garcia & Pintrich, 1991, 1994, 1995; Pintrich & Garcia, 1991; Schunk & Zimmerman, 1994; Zimmerman, 1994), few have explic itly linked the components of Numerical Cognition to academic achievRemeYnt in Mathematically-gifted and non-gifted students. A Researchers agree on at least two major findings with respect to NumerRical Cognition and academic achievement: Numerical Cognition is comprised of sevIeBral components, such as cognitive strategies and effort (Miller, Behrens, Green eL, & Newman, 1993) or metacognition and effort (Pintrich & De Groot; 1990A; YNap, 1993), although the specific components were not always identical; and students who employ metacognition and exert effort perform more successfully (Pintrich & De Groot, 1990; Zimmerman, 1986; Zimmerman & Martinez-Pons, 1986, 198I8)B. A A number of studies, in lineF wi th this finding, have clearly shown that students demonstrate high levels of NumOerical Cognition when they are oriented toward learning goals (e.g., Meece, 1994; Sch unk, 1994). Weiner (1986) found that children with low perceived ability wereI sTtillY mastery-oriented when their goal was to learn rather than to perform. S On EaccoRunt of significant differences that exists between Emotional Freedom and MathemVatics achievement, the findings revealed that participants in this group showed better Iperformance in Mathematics than their counterpart in Numerical Cognition group. UInN line with this finding, Daniel, Brenor, Karen and Loren (2005) explored test anxiety benefits of Wholistic Hybrid derived from Emotional Freedom Techniques (EFT), and Cognitive Behavioral Therapy. Participants include Canadian university students with severe or moderate test anxiety. A double-blind, controlled trial of EFT, and CBT was 172 conducted. The result of their study showed that Emotional Freedom Technique was better than Cognitive Behavioural Technique. Hypothesis nine Findings on the above hypothesis revealed that there was significant main effect of Mathematics efficacy on Mathematics achievement of secondary school students. It w as revealed (table 4.7) that Mathematics efficacy had large effect on the matheYmatics achievement posttest score variations, which implies that the differences in thRe level of mathematics efficacy accounted for 16.2% in the variation of the postteAst score. self- efficacy beliefs help determine the outcomes one expects. ConfRident individuals anticipate successful outcomes. Students confident in their sIoBcial skills anticipate successful social encounters. Those confident in their academ iLc skills expect high marks on exams and expect the quality of their work to reap perNsonal and professional benefits. The opposite is true of those who lack confidenceD. StuAdents who doubt their social skills often envision rejection or ridicule even before they establish social contact. Those who lack confidence in their academic skills envAision a low grade before they begin an examination or enroll in a course. In lin eI wBith this finding, Pajares and Kranzler (1995) found significant positive direct pathFs from self-efficacy to Mathematics performance and a significant negative path to Oanxiety. Similarly, Pajares and Kranzler (1995) found significant positive direct Ypaths from self-efficacy to Mathematics performance and a significant negative paIthT to anxiety. Hypothesis tenR S Finding onE the above hypothesis revealed no significant main effect of gender on MatheImVatics achievement of secondary school students. It was further revealed (table 4.7N) that gender had very small effect on the mathematics achievement posttest score Uvariations, which implies that gender accounted for 0.4% in the variation of the posttest score. This finding also supported the study of Lusser (1996) which found no significant difference between gender and Mathematics achievement. His study pointed to the fact no variation existed in the Mathematics performance of male and female participants. 173 Hypothesis eleven Findings on the above hypothesis revealed a significant interactive effect of treatment and mathematics efficacy on Mathematics achievement of secondary school students. It was further revealed (table 4.7) that treatment and mathematics efficacy had small effect on the mathematics achievement posttest score variations, that is, mathematics efficacy slightly moderated the influence of treatment on mathematics posttest score varianc es. This indicates that the differences that occur as a result of the interactive effect accYounted for 5.3% in the variation of the posttest score. This then implies that MRathematics efficacy is a significant and important factor that could enhanceR acAhievement in Mathematics. Self-efficacy beliefs provide the foundation for humBan motivation, well- being, and personal accomplishment. This is because unlessL peIople believe that their actions can produce the outcomes they desire, they have l ittle incentive to act or to persevere in the face of difficulties. Much empirical eviNdence now supports Bandura's contention that self-efficacy beliefs touch virtuDally Aevery aspect of people's lives— whether they think productively, self-debilitatiAngly, pessimistically or optimistically; how well they motivate themselves and perse vIeBre in the face of adversities; their vulnerability to stress and depression, and the life choices they make. This finding is in line with Pajares and Kranzler (1995) wOho Ffound significant positive direct paths from self- efficacy to Mathematics performance. In a similar study, a significant correlation between Mathematics self-Yefficacy and problem-solving performance was indicated in college students (PSajarIes T & Miller, 1994, 1995). R HypothVesisE twelve FindinIgs on the above hypothesis revealed no significant interactive effect of treatment UanNd gender on Mathematics achievement of secondary school students. This implies that gender did not significantly moderate the effect of treatment on mathematics achievement posttest score variations. This finding also supported the study of Lusser (1996) which found no significant difference between gender and Mathematics achievement. His study pointed to the fact no variation existed in the Mathematics performance of male and 174 female participants. This finding has been supported by previous literatures. In a study of Mathematics self-efficacy in 8th-grade students, Pajares (1996a) found a direct effect of gender on self-efficacy for regular education students but no direct effect of gender on performance (boys had higher self-efficacy). Hypothesis thirteen Y Findings on the above hypothesis showed that there was no significant interacRtive effect of mathematics efficacy and gender on Mathematics achievement of secAondary school students This implies that gender did not significantly moderate Rthe influence of mathematics efficacy on mathematics achievement posttest scorLe vIaBriations. One explanation that could be given to this is that, gender and Mat hematics efficacy could not moderate the treatment when it comes to Mathematics aNchievement. The reason is that gender was a stumbling block when taken with DMatAhematics efficacy to moderate the treatment in enhancing Mathematics achievement. This finding is not in line with Murphy and Ross (1990) who found gendBer Ato be an influential factor in determining Mathematics success for eighth graders. TIhe finding is not also in line with Fennema and Sherman (1977) who found that OgendFer has a correlation with Mathematics performance. Y Hypothesis fourteen IT The hypothesiRs wShich stated that there will be no significant interactive effect of treatment, gender and Mathematics efficacy on Mathematics achievement of secondary school VstudEents was not significant. It was revealed (table 4.7) that there was no sigNnifiIcant interactive effect of treatment, gender and mathematics efficacy on Umathematics anxiety posttest scores of participants. Therefore the null hypothesis is accepted. This implies that gender and mathematics efficacy level could significantly moderate the influence of interventions on group post mathematics anxiety scores. The table further reveals that there is a significant main effect of group variances (Numerical cognition, Emotional-freedom Based and Control Group) on mathematics anxiety posttest 175 scores, which implies that group variances had large effect on the differences that exist in their mathematics anxiety posttest scores. The explanation to this finding is similar to hypothesis five, that gender was not a good match to moderate the treatment with Mathematics efficacy. This finding has supported previous literatures. In a study of Mathematics self-efficacy in 8th-grade students, Pajares (1996a) found a direct effect of gender on self-efficacy for regu lar education students but no direct effect of gender on performance (boys had hRigheYr self- efficacy). For gifted students, there was a direct effect of self-efficacy aAnd gender on performance (girls had higher performance), but no gender effect on selRf-efficacy. Pysher (1996) also found no significant gender differences in MathematiIcsB test scores, goals, or self-efficacy. Pintrich and De Groot (1990)‘s findings wLith Mathematically-gifted students generally agree with these authors: A significantN dir ect path was indicated both from self-efficacy to Mathematics performance andA from self-efficacy to worry; and whereas no significant gender effects on performaDnce were found, there was a significant effect on self-efficacy. BA Similarly, with a group of hig h Ischool students, Pajares and Kranzler (1995) found significant positive direct OpathFs from self-efficacy to Mathematics performance and a significant negative path to a nxiety. Pajares and Kranzler found no gender effects for these students, either onT seYlf-efficacy or performance. A significant correlation between Mathematics self-efficIacy and problem-solving performance was indicated in college students (PajarRes &S Miller, 1994, 1995). E 5.2 IVLimitations of the study UN The study, in its unique contribution to knowledge, has used two therapeutic techniques in reducing Mathematics anxiety and enhancing Mathematics achievement of the students. However, the study encountered a lot of limitations in the course of the study. A limited number of students (120) may not be enough to make generalizations on the population of the study. This limited number of participants was used due to 176 administrative, logistics, time and financial constraints to achieve the objectives of the study; a greater number could have achieved a better result. Another similar limitation observed was that the study was carried out in Ibadan metropolis, this may also affect the generalization of the findings. The researcher has used gender and Mathematics efficacy as the moderating variables in the study. Although, Mathematics efficacy had significant interactive eff ect with the two therapeutic techniques; but this interactive effect was not significaRnt wYhen it comes to 3-way interaction with gender. This is because gender was haAving negative effect on academic efficacy and no significant effect on MathemRatics anxiety and achievement, thus limiting the findings of this study. IB This research work was limited because it was carrie dL out on public secondary school students in urban areas in Ibadan metropolis. So Nthe study could not establish a comparative analysis between public and private schAools, and between urban and rural secondary schools within the setting of the studAy. WDhereas, the initial observation that the researcher made was that background of tBhe study students could affect their academic performance. So, students from pFriva teI school have solid background than public secondary schools. This could have made this research work a robust one if the research had also examined the students fOrom private schools. Moreso, one impTortYant limitation observed in the course of this study was paucity of literatures. The reIsearch could not get enough literature to back his findings on numerical cognRitioSn and emotional freedom in relation to the Mathematics anxiety and MathematicEs achievement. 5.3 VN IImplications of the study U The importance of Mathematics on the social, economic, political and educational life of Nigerians cannot be over emphasized with changing time and advancement in science and technology. Mathematics phobia if not curbed and overcome will become more sophisticated and eventually be accepted as a normal way of life among Nigerian students. Therefore, this study demands for an intensive experimental study to find ways 177 on how this phobia could be reduced among the students. Towards this end, counselors and all other helping professionals must be alert and sensitive to effective techniques like those used on this study to help this emotional problem related to Mathematics. The implication of this is that all concerned: teachers, parents, counselling psychologists, school administrators, government etc should embark on intensive studies to identify the different types of Mathematics related problems and find solution to such. It is also of importance to make a clear distinction between cogRnitivYe and emotional distortions resulting in Mathematics failure, which is limiting oAur students to offer courses related to Mathematics like Statistics in the future. TRhat is why some students in Education and Social Sciences continue to run away frIoBm Statistics till today. The factors underlying these behaviours will not be the sameL in all students, and thus, interventions will be needed towards solving anxiety in MNathe matics The present study has proved that numericalA cognition and emotional freedom techniques were very effective in reducing DMathematics anxiety and enhancing Mathematics achievement among the stuBdy Aparticipants. The study has exposed the participants to how to reduce their fFalse Ibelief in Mathematics disability caused by lack of, inconsistent, poor, or inapprOopriate systematic Mathematics instruction; inattention, fear, anxiety, or emotion thereb y improving their academic self-efficacy to achieve in the subject. It is believed tThat Ythese two techniques will help the students to have positive thought about MathemIatics and learn how to adjust their negative thought and believe in their ability to RexcSel in the subject. Students will know how to plan, control and direct their mentaEl processes toward the achievement of Mathematics in other areas they find themselVves. They will understand how to control the time and effort to be used on tasks, anNd hIow to create and structure favourable learning environments, such as finding a Usuitable place to study, and seeking help from teachers and classmates when they have difficulties in the subject. 178 5.4 Conclusion Identifying and designing effective intervention strategies and practical ways to solve anxiety in Mathematics and enhance Mathematics achievement taking cognizance of efficacy in the subject has been suggested as an important and viable avenue of future research. This study, therefore, sought to investigate the effectiveness of two strategic techniques (Numerical Cognition and Emotional Freedom) in reducing Mathemat ics anxiety and enhancing Mathematics achievement of participants in Ibadan, ROyoY State with moderating effect of Mathematics efficacy. Findings of this study has Aclearly shown that numerical cognition and emotional freedom have significant eRffect in reducing anxiety and enhancing achievement in the subject. The two treatImBents were superior to control group; and emotional freedom technique was superior Lto numerical cognition in reducing anxiety and enhancing achievement in MathemaNtics. The results clearly showed that Mathematics efficacy of adolescents moderatedA the relationship between the two treatments in their effect on anxiety and achievemDent in Mathematics. It could be averred that low Mathematics efficacy could increBaseA anxiety in the subject and decrease their performance in the subject. Students wIith higher levels of Mathematics efficacy set higher goals, apply more effort, persFist longer in the face of difficulty in the subject and are more likely to cope better. O It was observedT frYom this study that Mathematics anxiety was an emotional, rather than intellectualI problem because the problem emanates from inconsistent emotion and therefore inRterfSeres with a person's ability to learn Mathematics which later results in an intellectEual problem. Therefore, competent functioning in Mathematics requires self-beliefs oVf efficacy to perform effectively. I 5.5N Recommendations U Based on the findings of this study, the following recommendations were made: 1. The treatment strategies reviewed in this study are recommended for use by educational psychologists, guidance counsellors, teachers and principals of 179 secondary schools. The strategies will provide these personnel with requisite educational diagnosis aimed at improving the educational system in Nigeria. 2. The study is recommended for policy makers in education to serve as an input on educational issues relating to the improvement of learning. 3. The study has revealed that Mathematics is not only an intellectual problem but emanating from emotional problems. Rather than concentrate only on cognit ive distortions in the subject, the researchers recommended that these techYniques could resolve emotional crisis in Mathematics and improve AtheR students‘ performance in the subject. 4. There is the need to re-orientate teachers with contents of the traRining packages to enable them impart same to their students on a regularL baIsBis. This would make students help themselves. 5. The training packages can be used by counsellorNs in schools to give students a new orientation to enhance positive thinkDingA pattern and a new belief in their capability. 6. Counselling psychologists and teaBcheArs should help students in building their efficacy in Mathematics becaFuse MI athematics efficacy has found to moderate the techniques used to solve OMathematics anxiety. 5.6 Contribution tIoT KnYowledge TheS study has added to the existing scanty literature on the effect of Numerical ECogRnition and Emotional Freedom on Mathematics anxiety and achievement. This stuVdy is novel in Nigeria and Africa in general because to the best of the researcher‘s knNowlIedge, nothing of such has been done here in Nigeria. This makes the study very Upeculiar and a contribution to knowledge. So, this study has served as frontier of knowledge in reducing anxiety and enhancing achievement in Mathematics through these two techniques. One major achievement of this study is that it has been able to demonstrate that students‘ anxiety and achievement in Mathematics as the core and significant subject 180 when it comes to academic success can be improved upon. To this extent, the research has focused on low-achieving students in Mathematics by working on the need to improve their performance. Students with dyscalculia could be assisted using these two techniques in Nigeria and Africa. The study is also unique in terms of the treatment strategies employed. The therapeutic techniques – Numerical Cognition and Emotional Freedom (to the best of the researcher‘s knowledge) - have not been used in Nigeria for the same purposeR forY which they have been used in this study. The treatment techniques have beeAn effective in improving Mathematics anxiety and achievement of students with pRseudo-dyscalculia. This makes the study peculiar and thus contributing significantIlyB to knowledge. The study will most likely enable learners ‗look inward‘ as well asL make them come up with an academic plan for action to make them not to be lockedN up again in the old habit. DA 5.7 Suggestions for further studies BA This study is recommended foFr r epIlication because of the following reasons: A limited number of s tuOdents (120) were used for this study. It will be quite revealing if interventions established could actually reveal vivid information on findings relationship between aInTxieYty in mathematics and factors that cause. So, survey studies could be carried oSut so as to further widen researchers‘ perspectives and increase the numbers of parRticipants in order to solve the problem. VTheE study was carried out in Ibadan metropolis, which may affect the generaIlization of the findings. So similar researches could be expanded throughout UNiNgeria. The researcher has used gender and Mathematics efficacy as the moderating variables in the study. Although, Mathematics efficacy had significant interactive effect with the two therapeutic techniques; but this interactive effect was not significant when it comes to 3-way interaction with gender. Therefore, other researchers may look for other 181 moderating variables apart from gender to complement Mathematics efficacy in solving the problem. This research work carried out on public secondary school students in urban areas in Ibadan metropolis. So the study could not establish a comparative analysis between public and private schools, and between urban and rural secondary schools within the setting of the study. So, other researchers may expand their purpose by looking for comparative studies between urban and rural students, private and public schRoolsY. This will help to see how background of the students affect their anxiety in MaAthematics and other related subjects. R The research could not get enough literature to back Lhis IfBindings on numerical cognition and emotional freedom in relation to the Mathemati cs anxiety and Mathematics achievement. So other researches should look for AextNensive literature to back their findings. 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Nature 417:138–139. 210 UNIVERSITY OF IBADAN FACULTY OF EDUCATION DEPARTMENT OF GUIDANCE AND COUNSELLING Dear Respondent, This questionnaire is designed basically for a research purpose. It seeks to know how you would react to these statements. All information provided would be trea ted confidentially. Please be honest as much as possible. Y SECTION A R Background Information. A R INSTRUCTION: Below are some statements that relate to yoBu. Put a tick on the statement that relate to you. I L 1) Age: ……………………... N 2) Gender: Male ( ) Female ( ) A 3) Religion: Christian ( ) Muslim ( D ) Others (specify)………………….. 4) Class in school…………… BA SE CITION B PSEUDO-FDYSCALCULIA SCALE Kindly respond by marking the rOesponse as it occurs to you using the format below: Strongly Agree = SA, AgreYe = A, Disagree = D and Strongly Disagree = SD No ITEMS SA A U D SD 1. I find MathematiIcsT interesting. 2. Mind goes blSank and I am unable to think clearly when doingE myR Mathematics test. 3. I wVorry about my ability to solve Mathematics problems. 4. NI become physically agitated when I have to go to U Mathematics class. 5. I am always worried about being called on in Mathematics class. 6. Mathematics makes me feel confused 7. I find Mathematics challenging 211 8. I enjoy learning with Mathematics 9. I would like to take more Mathematics classes 10. It's clear to me in Mathematics class, but when I go home it's like I was never there. 11. Mathematics makes me feel nervous 12. I worry about my ability to solve Mathematics problem 13. I worry that I will get poor marks in mathematics R Y 14. I tend to lose my concentration in Mathematics class. A 15. I don't know how to study for Mathematics tests. R 16. I am always worried about being called on in Mathemati cLs I B class. 17. I'm afraid I won't be able to keep up with the rest of Nthe class A 18 I often worry that it will be difficult for mAe inD Mathematics classes. IB FSECTION C MATHEMATICS EFFICACY SCALE Kindly respond by markingY the r Oesponse as it occurs to you using the format below: Strongly Agree = SA, ATgree = A, Disagree = D and Strongly Disagree = SD No I ITEMS SA A U D SD 1. When MatheSmatics exercises are hard I don‘t give up or study onlRy the easy parts. 2. I wVorkE on Mathematics exercises even when I don't have tIo do so. 3. NI work hard to earn a good grade in Mathematics even U when I do not like the subject. 4. I can plan my Mathematics school work. 5. I finish Mathematics assignments by deadlines. 6. I can study Mathematics when there are other interesting things to do. 212 7. I can organize my school work. 8. I would feel confident to solve Mathematics problems. 9. I can boldly discuss Mathematics problem with teacher 10. I can willingly come out of the class to express myself on Mathematics matters 11. I am confident enough to talk on Mathematics issue 12. I think I can still compete with my mates on Mathematics R Y 13. If I am called many times, I will be willing to solve A Mathematics problems even if I did not get the right answer. BR 14. Even though I had Mathematics problem, I am not ready I to submit to fate L 15. I won‘t be part of the students who always say AN Mathematics is difficult BA D SECTION D MATHEMAT ICIS ANXIETY SCALE Kindly respond by marking the respoFnse as it occurs to you using the format below: Strongly Agree = SA, Agree = AO, Disagree = D and Strongly Disagree = SD No Y ITEMS SA A U D SD 1. I cannot explain IwThat actually accounts for my difficulty in MathematiScs 2. I don‘t beRlieve in my ability to do well in Mathematics 3. I oVnlyE try to manage myself with somebody who knows bIetter in class to pass Mathematics 4. NI hate to see Mathematics teacher in my class. U5. I don‘t know how to follow Mathematics syllabus 6. I lose concentration in Mathematics class 7. I don‘t know how to focus on Mathematics problems 8. I am always overwhelmed with fear when asked to solve 213 Mathematics 9. I feel jittery in Mathematics tests or exams. 10. My mind is not compatible with anything called Mathematics 11. I feel emotionally down when it comes to solving of Mathematical problems 12. I avoid Mathematics and any other subject related to it Y 13. I always feel like skipping Mathematics classes. A R 14. I don‘t know how to follow laid down rules in Mathematics BR 15. I don‘t usually understand Mathematics instruction LI 16. I am not capable of solving Mathematics N 17. I feel inferior finding myself among students whAo understand Mathematics better than me. D 18. I feel inferior with other subjects relatBed tAo Mathematics. OF I SECTION E MATHEM ATICS ACHIEVEMENT SCALE 2 3 Y 1. Simplify 10 X 1T0 2 (a) 20 I 5 (b) 100 S 5 (c) E10 R2V(d) 10 I(e) 310 N 7 3U 2. Simplify 22n / 2n 4(a) 11n 4 (b) 24n 4(c) 11n 10(d) 11n 10 (e) 24n 0 3. Simplify 19 214 (a) 19 (b) 0 (c) 1 (d) 9 0(e) 19 -2 4. Solve 5 (a) 10 (b) -10 Y (c) 1/25 R 2(d) 10 A (e) 25 R 5. 3 -1Solve 2 X (1/6) IB (a) 48 L 2 (b) 2 3-1(c) 12 N 1/6(d) 8 2 (e) 8 AD A 0 -56. r X r X r IB4(a) 1/r 5 (b) 1/r F (c) 4r 5 O (d) r (e) -41/r Y IT 7. ApproximaSte 0.0567 to 2 decimal places (a) 0.05 (b) 0.05R7 IV(c) E0.06 (d) 0.5 N (e) 0.0567 U 8. Find the difference between LCM and HCF of 18 and 30 (a) 84 (b) 90 (c) 6 (d) 80 (e) 18 215 9. Find he sum o 11012 and 1112 (a) 11002 (b) 100002 (c) 100102 (d) 101002 (e) 101102 Y 10. Find the value of angle x R A R LI B 0 (a) 205 N 0(b) 195 A 0 (c) 175 D (d) 0172 A 0(e) 160 IB 11. Find the simple interest oOn NF36,000 for 2 years (a) 3% per annum (b) N7200 Y (c) N216 (d) N3600 IT (e) N2160 S 12. DetEermRine angle m IV UN 0 (a) 65 (b) 0130 0 (c) 50 216 0 (d) 25 0 (e) 30 13. Solve the simultaneous equations: 5x – 2y = 14; 2x + 2y = 14 (a) x = 4, y = 3 (b) x = 3, y = 6 (c) x = 2, y = 5 (d) x = 4, y = 6 (e) x = 4, y = 28 Y 14. Using the diagram below, find the value of Sin θ AR R IB L (a) 5/2 (b) 5/13 N (c) 12/13 A (d) 13/12 D (e) 12/5 BA 15. A triangle in which only two ang leIs are equal is said to be ____________ (a) Equilateral F (b) Equiangular O (c) Isosceles (d) Scalene Y (e) Right-angleIdT 16. CalculaRte thSe area of the shape IV E UN 2 (a) 28m 2(b) 15m 2 (c) 35m 2(d) 21m 2 (e) 10m 217 17. Convert 37ten to binary number (a) 1010012 (b) 1011002 (c) 1001102 (d) 1001012 (e) 100112 18. Ade, Bola and Jide share N400 in the ratio 4:7:9 respectively. How mucRh diYd Bola receive? (a) N200 A (b) N180 (c) N140 R (d) N120 IB (e) N80 L 19. What is the square root of 144 N (a) 15 A (b) 14 AD(c) 10 (d) 11 B (e) 12 F I 20. Solve 4x – 30 = 2x + 52 O (a) 38 (b) 41 (c) 39 ITY (d) 42 (e) 49 RS 21I. VExp Eress 2.0495 correct to 4 significant figure. (a) 2.102 N (b) 2.045 U (c) 2.050 (d) 2.06 (e) 2.10 22. What is the one-twentieth of N10,000? (a) N1,000 (b) N5,000 218 (c) N500 (d) N100 (e) N1,500 23. Calculate the circumference of a circle of diameter14m (take π = 22/7) 2 (a) 22cm 2 (b) 44cm 2(c) 140cm Y 2 (d) 154cm R 2 (e) 20cm A R 24. Which of these marked angles are alternate angles? LIB (a) s and m (b) r and s N (c) b and m A (d) b and p D (e) b, r and s BA 25. What is the mean score of theF fol loIwing 21, 40, 32 (a) 33 (b) 31 O(c) 93 (d) 61 Y (e) 3 IT 26. The sumR ofS ages of two persons is 98, and the difference of their ages is 16. How old is each person? IV(a) E41, 57 (b) 32, 66 (c) 42, 56 UN (d) 28, 70 (e) 45, 56 27. Fig. 1 is a pie chart showing the distribution of types of secondary schools in a country Gra2m1m9 ar schools Technical college Teacher’s colleges Fig. 1 Y What is the ratio of Teachers‘ Colleges to Grammar Schools? R (a) 1:5 A (b) 1:6 (c) 3:8 R (d) 2:9 IB (e) 3:10 L 28. Table 1 shows the shoe sizes of 20 children N A Shoe size 36 37 AD38 39 40 Number of 1 3 IB 8 5 3 children Table 1 F O Which size is the mode? (a) 36 Y (b) 37 IT (c) 38 S (d) 39 R (e) E40 I N29. VIn table 1, which is the median shoe size? (a) 36 U (b) 37 (c) 38 (d) 39 (e) 40 220 30. The average speed, in km/h, of a car which travels 72 km in 45 mins is (a) 45 (b) 54 (c) 64 (d) 88 (e) 96 31. The height of a closed cylinder is equal to its radius r. Express the total surface area of the cylinder in terms of π and r. Y 2 (a) Π r R 2 (b) 2 Π r 2 A (c) 3 Π r (d) 24 Π r R (e) 26 Π r LI B 32. A conical cup of base radius r and height h is filled with sand. How many cups of sand will be needed to fill a cylindrical container oNf base radius r and height 2h? (a) 2 DA(b) 3 (c) 4 A (d) 6 B (e) 9 F I 33. A cone and a cylinder haOve equal heights and volumes, r is the radius of the cylinder. What the rYadiu s of the cone in terms of r? (a) r √3 (b) 3r IT (c) r S (d) 1/3r (e) E1/√3R 34I. VXYZ is a straight line such that XY = YZ = 3cm. A point P moves in the place of N XYZ so that PY ≤ XY. Which of the following describes the locus of P? U (a) Line through X perpendicular to XZ (b) Line through Y perpendicular to XZ (c) Line through Z perpendicular to XZ (d) Circular disc, centre Y, radius 3cm (e) Circular disc, centre X, radius 3cm 221 0 35. If 0 ≤ θ ≤ 180 and sin θ = 0.225, then θ = 0 (a) 13 only (b) 0 013 or 167 0 (c) 77 only 0 0(d) 77 or 103 0 (e) 103 36. The slant height of a cone is twice its base radius r. express the total surface are a of the cone in terms of π and r Y 2(a) 2/3π r R 2 (b) π r 2 A (c) 1 1/3 π r 2 (d) 2π r R 2 (e) 3π r IB L 37. A circle, centre 0, radius 5cm is drawn on a sheet of paper. A point P moves on the paper so that it is always 2cm from the circle. TNhe locus of P is (a) A circle, centre 0, radius 3cm A (b) Two circles, centre 0, radius 6cm D (c) Two circles, centre 0, radii 3cmB andA 7cm (d) Two circles, centre 0, radii 4cm and 6cm (e) A circle, centre 0, radius 3F.5c mI 0 0 38. Express 468 as an angleO between 0 and 360 (a) 072 0 (b) 108 Y (c) 0234 T 0 I (d) 252 0 (e) 288R S 39. VIf AEI = {3, 5, 6, 8, 9} and B = {2, 3, 4, 5} write down the sets A ᴗ B and Aᴖ B. show A and b on a Venn diagram N (a) A ᴗ B = {2, 3, 4, 5, 6, 8, 9}, Aᴖ B = {3, 5} U (b) A ᴗ B = {2, 4, 5, 6, 8, 9}, Aᴖ B = {3, 5, 6} (c) A ᴗ B = {2, 3, 4, 8, 9}, Aᴖ B = {3, 5} (d) A ᴗ B = {2, 3, 4, 5, 6,}, Aᴖ B = {3, 5} (e) A ᴗ B = {2, 3, 4, 5, 6, 9}, Aᴖ B = {3, 5} 222 40. If C = {grapefruit, orange, pear} and D = {grapefruit, pear, apple, pawpaw} write the sets C ᴗ D and C ᴖ D (a) C ᴗ D {grapefruit, orange, pear, apple, pawpaw}, C ᴖ D = {grapefruit, pear} (b) C ᴗ D { orange, pear, apple, pawpaw}, C ᴖ D = {grapefruit, pear} (c) C ᴗ D {grapefruit, pear, apple, pawpaw}, C ᴖ D = {grapefruit, pear} (d) C ᴗ D {grapefruit, orange, pear, pawpaw}, C ᴖ D = {grapefruit, pear} (e) C ᴗ D {grapefruit, orange, pear, apple}, C ᴖ D = {grapefruit, pear} 2 41. Evaluate 2a bc/2b – c when a = 3, b = - 4, c = - 5 Y (a) -120 R (b) 120 A (c) -110 (d) 110 R (e) -122 IB L 42. In a class of p students, the average mark is x and in another class of n students the average mark is y. what is the average mark forN both classes? (a) Px + ny / p + n A (b) Px + ny / pn D (c) Px + ny / p – n A (d) Px + py / p + n B (e) nx + py / p + n I F 43. A car travels d km at an aOverage speed of u km/h. How long does it take? (a) d/u hr (b) u/d hr Y (c) u hr IT (d) d hr (e) d/duR hr S 44I. VSolv Ee 4 – 4x = 9 – 12x (a) x = 8/5 N (b) x = 5/8 U (c) x = 5 (d) x = 8 (e) x = 2 223 45. Solve 3(4c – 7) – 4(4c – 1) = 0 (a) -4 ¼ (b) -2 ½ (c) -4 ½ (d) 4 ¼ (e) 2 ¼ 46. Solve the equation 3x + 2 / 6 – 2x – 7 / 9 = 0 Y (a) x – 4 R (b) x = - 4 A (c) x = 4 (d) x = -4x R (e) x = 4x LI B 2 47. If 3 is a root of the equation x – kx +42 = 0, find the value of k and the other root of the equation. N (a) K = 17 A (b) K = 16 D (c) K = 7 A (d) K = 12 B (e) K = -17 F I 248. If 5 is one of the roots ofO the quadratic equation x + 4x – 45 = 0, what is the other root? (a) x = -9 Y (b) x = 9 IT (c) x = -3 (d) x = 3 S (e) VE x = R-9x 49I. A ladder 20m long rests against a vertical wall so that the foot of the ladder is 9m N from the wall. Find, correct to the nearest degree, the angel that the ladder makes U with the wall 0(a) 27 (b) 032 0 (c) 24 0 (d) 28 0 (e) 30 224 50. A point T is on the same horizontal level as the foot of a tower. If the distance of T from the foot of the tower is 80m and the height of the tower is 60m, find the angle of depression of T from the top of the tower. Give your answer to the nearest degree 0 (a) 37 0 (b) 45 0 (c) 36 0 (d) 40 0 (e) 32 Y AR LIB R AN AD F I B O ITY ER S NI V U 225