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Effects of Computer Algebra System (CAS)with Metacognitive Training on MathematicalReasoningBracha Kramarski a & Chaya Hirsch aa IsraelPublished online: 04 Jun 2010.

To cite this article: Bracha Kramarski & Chaya Hirsch (2003) Effects of Computer Algebra System (CAS) withMetacognitive Training on Mathematical Reasoning, Educational Media International, 40:3-4, 249-257, DOI:10.1080/0952398032000113167

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Effects of Computer Algebra System (CAS) withMetacognitive Training on Mathematical ReasoningBracha Kramarski Chaya Hirsch, Israel

AbstractsThe main purpose of the present study was to investigate the differential effects of Computer Algebra Systems (CAS) andmetacognitive training on mathematical reasoning. Participants were 83 students (boys and girls) who studied algebra infour eighth-grade classrooms randomly selected to four instructional methods: CAS embedded within metacognitivetraining (CAS + META), metacognitive training without CAS learning (META), CAS learning without metacognitivetraining (CAS) and CONT learning without CAS and metacognitive training (CONT). Results showed that the CAS + METAcondition significantly outperformed the META and the CAS conditions, who in turn significantly outperformed theCONT condition on aspects of mathematical reasoning. No significant differences were found between the META and CASconditions. In addition, the study found that the metacognitive students (CAS + META and META conditions)outperformed their counterparts (CAS and CONT) on their metacognitive knowledge. This paper describes researchwhose main focus is the use of Computer Algebra Systems (CAS) in mathematics classrooms and the didactical possibilitiesbased on the IMPROVE method linked with its use.

Les effets du systeme d’algebre par informatique (CAS) sur une formation metacognitive en raisonnementmathematique.Le but principal de la presente etude etait d’explorer les effets differentiels du CAS et d’une formation metacognitive enraisonnement mathematique. Les participants etaient 83 eleves (garçons et filles) qui etudiaient l’algebre dans les classes(eighth-grade) choisies au hasard avec 4 methodes d’instruction. CAS enchasses a l’interieur d’une formationmetacognitive (CAS + META) metacognitive sans CAS (META), apprentissage CAS sans formation metacognitive (CONT)et apprentissage sans CAS + formation metacognitive (CONT). Les resultats ont montre que la formule CAS + META asurpasse de façon significative les formules META + CAS qui a leur tour ont surpasse la formule CONT quant aux aspectsdu raisonnement mathematique. Il n’y a pas de differences entre les formules META et CAS. De plus l’etude a montre queles eleves soumis aux conditions CAS + META et META ont obtenu de meilleurs resultats que leurs condisciples (CAS etCONT) quant a leurs connaissances metacognitives cet article decrit des recherches dont le but principal est l’utilisationdu CAS dans les classes de mathematiques et les possibilites didactiques fondees sur la methode IMPROVE a son usage.

Wirkungen des Computer Algebra Systems (CAS) mit metacognitivem Training auf mathematisches DenkenDer Hauptzweck der gegenwartigen Studie war, die unterscheidenden Wirkungen des Computer Algebra Systeme (CAS)und der metacognitiven Ausbildung auf mathematische Uberlegung zu untersuchen. Teilnehmer waren 83 Studenten(Jungen und Madchen), die Algebra in vier zufallig ausgewahlten Klassen der achten Stufe in vier Unterrichtsmethodenstudierten: CAS eingebettet in metacognitive Ausbildung (CAS + Meta), metacognitive Ausbildung ohne CAS (Meta), CASLernen ohne metacognitive Ausbildung (CAS) und CONT Lernen ohne CAS und metacognitive Ausbildung (CONT). DieErgebnisse zeigten dass die CAS + Meta-Bedingung die Meta- und die CAS Bedingungen signifikant ubertraf, die wiederumdie CONT Bedingung bedeutend in Bezug auf Aspekte mathematischen Denkens uberflugelten. Keine bedeutsamenUnterschiede wurden Zwischen den Meta und CAS Versionen wurden keine bedeutsamen Unterschiede gefunden.Außerdem ergab die Studie, dass die Studenten der metacognitiv unterrichteten Gruppe (CAS + Meta- undMetabedingungen) ihre Pendants (CAS und CONT) in Bezug auf ihr metacognitives Wissen ubertrafen. Dieses Papierbeschreibt Forschung, deren Hauptfokus auf der Verwendung von Computer Algebra Systemen (CAS) in Mathema-tikklassen ist und die didaktischen Moglichkeiten, die auf der mit ihrer Verwendung verbundenen IMPROVE Methodebasieren.

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IntroductionAlgebra has emerged as one of the central themes of the Principles and Standards for School Mathematics(NCTM, 2000). ‘To think algebraically, one must be able to understand patterns, relations, and functions;represent and analyze mathematical situations and structures using algebraic symbols; use mathematicalmodels to represent and understand quantitative relationships; and analyze change in various contexts’ (Friel,Rachlin, Doyle, Nygard, Pugalee Ellis and House, 2001, p. 2).

During the last decade the availability of Computer Algebra System (CAS) environments has increaseddramatically. The interaction with the computerized system enabled practicing the algebraic language forsymbolic and numerical computation. It encouraged understanding of algebraic reasoning, findingpatterns, and reflecting on the solution process. But research (e.g., Monaghan, 1994; Lagrange, 1996) onusing Computer Algebra System (CAS) indicated evidence of success but also of difficulties. Lagrange(1996) stated that CAS have a great potential to improve student learning in mathematics’, but easiercalculation did not automatically enhance students’ reflection and understanding. Their conclusion wasthat a great deal of fieldwork is required to establish in what conditions this potential of Computer Algebracan actualize.

In recent years, there has been much interest in the role of metacognition in mathematics education.Metacognition is interpreted as cognition about cognition, which means knowledge of thinking and regulation oflearning processes (Flavell, 1979). Schoenfeld (1987) indicated that in the area of mathematics, metacognitiveknowledge includes knowledge about one’s own thought processes (e.g., ‘How accurate are you in describingyour own thinking?’), control or self-regulation (e.g., ‘How well do you keep track of what you’re doing whenyou’re solving problems/tasks?’), and beliefs and intuitions (e.g., ‘What ideas about mathematics do you bringto your work in mathematics?’

Several previous studies have examined the effects of metacognitive training on mathematics achievement(e.g., Mevarech and Kramarski, 1997; Schoenfeld, 1985; 1987; Garofalo and Lester, 1985). A majorcommon element of these programs is training students to do mathematics by asking and answering aseries of self-addressed metacognitive questions before, during and after attempting a solution. The methodof Mevarech and Kramarski, (1997), called IMPROVE emphasizes the importance of providing eachstudent with the opportunity to construct metacognitive knowledge in mathematical learning via the useof self questioning. This questioning focuses on: (a) comprehending the problem (e.g., ‘What is theproblem all about’?); (b) constructing connections between previous and new knowledge (e.g., ‘Whatare the similarities/differences between the problem at hand and the problems you have solved inthe past? and why?’; (c) using of strategies appropriate for solving the problem (e.g., ‘What are thestrategies/tactics/principles appropriate for solving the problem and why?’; and in some studies also (d)reflecting on the processes and the solution (e.g., ‘What did I do wrong here?’; ‘Does the solution makesense?’).

Evidence shows positive effects of metacognitive training employed in non-computerized environments (e.g.,Mevarech and Kramarski, 1997; Kramarski, Mevarech and Liberman, 2001; Kramarski, Mevarech and Arami,2002) as well as the effects of using such pedagogy in computerized environments (e.g., Teong, Threlfall, andMonaghan, 2001; Kramarski and Zeichner, 2001; Kramarski and Ritkof, 2002). Effects were found onmathematical reasoning, giving mathematical explanations and promoting mathematical discourse. Giventhese studies, there is reason to believe that providing metacognitive training within the CAS environment willimprove students’ mathematical reasoning.

There is also evidence showing that students who were exposed to metacognitive training in classroomsimproved their ability to use metacognitive knowledge more than students of the control groups (Masui andDe Corte, 1999; Kramarski, Mevarech and Liberman, 2001). There is reason to suppose that providingmetacognitive training within the CAS environment may affect students’ metacognitive knowledge differently.These students are expected to possess an improved capacity to reflect on using metacognitive knowledge ascompared to students who are not exposed to such learning.

The main purpose of the present study is to investigate the differential effects of Computer Algebra System(CAS) and metacognitive training on mathematical reasoning and metacognitive knowledge. In particular tocompare four instructional methods: Computer Algebra System learning embedded within metacognitivetraining (CAS + META), metacognitive learning in the whole class (META), Computer Algebra Systemlearning with no metacognitive training (CAS), and learning in the whole class with no metacognitivetraining (CONT).

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MethodParticipants were 83 students (boys and girls) who studied in four eighth-grade classrooms randomly selectedfrom four junior high schools.

All classrooms studied Algebra five times a week during a five-month period according to the mathematicscurriculum suggested by the Israeli Ministry of Education.

Classrooms were randomly assigned to one of the following conditions:

CAS + META conditionStudents in this condition were exposed to Computer Algebra System (20 hours in the computer lab, eachweak one hour) embedded within metacognitive training. The CAS software enabled the students to practicethe algebraic language for symbolic and numerical computation. In particular, the students practicedsubstitution of variables, simplifying algebraic expressions and solution of equations. The metacognitivetraining was based on the IMPROVE method. The acronym of IMPROVE represents all the teaching/learning stages of the metacognitive training: introducing the new topics to the whole class; metacognitivequestioning; practicing; reviewing; obtaining mastery on higher and lower cognitive skills; verifying andenrichment. The metacognitive training provides each student with the opportunity to constructmetacognitive knowledge by utilizing self-addressed metacognitive questions: Comprehension questions, strategicquestions, connection questions and reflection questions.

In addressing comprehension questions, students had to read the problem/task, describe the concepts in theirown words, and try to understand what the concepts meant. The strategic questions are designed to promptstudents to consider which strategies are appropriate for solving the given problem/task and for whatreasons.

Connection questions prompt students to focus on similarities and differences between the task at hand and taskthey had already solved.

Reflection questions prompt students to focus on the solution process and to ask themselves ‘what am I doinghere? ‘does it make sense’ ‘what if’. The metacognitive questions were printed in Students’ Booklets, TeacherGuide, and on the hand held index cards that students used in problem solving. Students used themetacognitive questions orally in their small group/individualized activities, and in writing when they usedtheir booklets.

META conditionIn this condition, the metacognitive training was exactly the same as in the above condition, except that themetacognitive training was implemented only in the classroom.

CAS conditionUnder this condition, students studied as in the CAS + META condition, but they were not exposed to themetacognitive training (20 hours in the computer lab, each weak one hour).

Control conditionUnder this condition students were exposed to neither CAS environment nor to metacognitive training. Thisgroup served as a control group.

MeasurementsThree measures were used in the present study to assess students’ mathematical reasoning and metacognitiveknowledge: (a) a pre-test that focused on students’ mathematical knowledge prior to the beginning of thestudy; (b) a post-test that assessed students’ mathematics reasoning on five criteria: algebraic techniques,algebraic reasoning, patterns and change; and (c) a metacognitive questionnaire that assessed students’metacognitive knowledge regarding their thoughts about using problem solving strategies before, during andafter the solution process.

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Mathematics: prior knowledgeA 21-item pre-test was administered to all students at the beginning of the school year. The test coveredoperations with positive and negative numbers, order of operations, the basic laws of mathematics operations,algebraic expressions, and open-ended computation problems.

ScoringFor each item, students received a score of either 1 (correct answer) or 0 (incorrect answer), and a total scoreranging from 0 to 21. The Kuder Richardson reliability coefficient was � = .87.

Mathematical ReasoningThe post-test included 17 items that are based on the following four criteria:

Algebraic techniques (e.g., solution of equations, operations with algebraic expressions); algebraic reasoning(e.g., what can you say about r if: r = s + t and r + s + t = 30); patterns (e.g., the following design is composed fromwhite and black squares, if you know the number of the black squares can you find the number of the whitesquares? If the number of the black squares is K find the pattern for the number of the white squares); andchanges (e.g., which changes faster: 2n or n + 2, explain).

ScoringFor each item, students received a score of either 1 (correct answer) or 0 (incorrect answer), and a total scoreranging from 0 to 17. The Kuder Richardson reliability coefficient was � = .87.

Metacognitive knowledgeThe metacognitive questionnaire, adapted from the study of Montague and Bos (1990) and Kramarski,Mevarech and Liberman (2001) assessed students’ metacognitive knowledge regarding their thoughts aboutlearning processes. The questionnaire includes 24 items: Eight items referred to strategies used before thesolution process (e.g., ‘Before I solve a problem in mathematics, I try to say it in my own words’), eight itemsreferred to strategies used during the solution process. (e.g., ‘When I solve a problem in mathematics, Iorganize the data in a table’), eight items referred to strategies used at the end of the solution process (e.g.,‘After I solve a problem, I check whether the answer is logical’).

ScoringEach item was constructed on a 5-point Likert type scale ranging from 1 (never) to 5 (always), and a totalscore ranging from 24 to 120. Cronbach’s alpha reliability coefficient was. 83.

ResultsMathematical ReasoningThe first purpose of the present study was to investigate students‘ mathematical reasoning under the fourconditions described above. Table 1 presents the mean scores, adjusted mean scores, and standard deviationson mathematical reasoning by time and treatment.

One-way MANCOVA on the five topics of the mathematical reasoning post-test scores as dependent variables(algebraic techniques, algebraic reasoning, patterns and change) and the pre-test as a covariate indicatedsignificant differences at the end of the study between the four conditions (F(3, 78) = 7.19, p < .0001). Furtheranalysis (ANCOVA) on each dependent variable indicated significant differences between the learninggroups on two criteria of mathematical reasoning: algebraic reasoning (F(3, 78) = 9.43, p < .001) and patterns(F(3,78) = 10.7, p < .001).

Figure 1 presents the differences on mean scores of mathematical reasoning by treatment.

It was found that the CAS + META condition significantly outperformed the META and the CAS conditions,which in turn significantly outperformed the CONT condition on algebraic reasoning and discovering patterns.But no significant differences were found between the META and CAS conditions on mathematicalreasoning.

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Metacognitive KnowledgeThe secondary purpose of the present study was to compare the differences in metacognitive knowledgeamong the two learning groups.

Table 2 presents the mean scores, adjusted means and standard deviations on metacognitive knowledge bytreatment.

Results indicated that prior to the beginning of the study no significant differences were found between thefour groups on metacognitive knowledge (F(3, 78) = 0.79, p > .05). MANCOVA with the pre-scores as a

Figure 1 Mean scores on mathematical reasoning test by time and treatment

Table 1 Mean scores and standard deviations on mathematical reasoning test by time and treatment

CAS + META N = 20 META N = 20 CAS N = 23 CONT N = 20

Pretest M: 72.4 73.2 73.6 71.1S: 14.6 19.0 11.6 10.5

Posttest M: 76.1 64.5 62.6 53.2Adjusted M: 76.3 64.9 61.7 54.5

S: 17.0 22.8 18.8 18.7

Algebraic Techniques M: 11.9 12.6 11.5 11.4Adjusted M: 11.7 12.5 11.4 11.3

S: 2.4 2.4 2.2 2.0

Algebraic Reasoning M: 10.3 7.6 8.1 6.9Adjusted M: 10.2 7.5 8.2 6.8

S: 1.7 2.4 2.1 2.1

Patterns M: 5.3 4.2 4.4 3.2Adjusted M: 5.2 4.3 4.3 3.1

S: 0.9 1.5 1.2 0.9

Changes M: 9.9 9.7 9.2 8.5Adjusted M: 9.8 9.8 9.3 8.4

S: 1.8 2.1 1.9 1.5

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covariant indicated significant differences between the four groups on the post-test (F(3,78) = 6.87; p <0.001). Further analysis (ANCOVA) on each dependent variable indicated significant differences between thelearning groups on two measures of metacognitive knowledge: Using problem solving strategies before thesolution process (F(3, 78) = 9.48, p < 0.01) and using problem solving strategies during the solution process(F(3, 78) = 8.28, p < 0.01).

Figure 2, 3 and 4 present the differences on mean scores of using metacognitive knowledge by time andtreatment.

It was found that the CAS + META condition significantly outperformed the META condition, which in turnsignificantly outperformed the CAS condition and the CAS condition outperformed the CONT condition onthe criteria: using problem solving strategies before the solution process. On the criteria using problem solvingstrategies during the solution process it was found that the two conditions that were exposed to metacognitivetraining (META + CAS and META) significantly outperformed the CAS condition, which in turn significantlyoutperformed the CONT condition.

No significant differences were found between the four conditions on the measure of using problem solvingstrategies at the end of the solution process (F(3, 78) = 1.18, p > 0.05).

DiscussionThe study raises the question: How can computer technology be used to enhance cognitive development inmathematics?

Figure 2 Mean scores on using metacognitive knowledge before solving the task/problem by time and treatment

Table 2 Mean scores and standard deviations on metacognitive knowledge test by time and treatment

CAS + META N = 20 META N = 20 CAS N = 23 CONT N = 20

Pretest M: 85.4 81.1 84.8 85.1S: 10.3 12.8 8.2 9.1

Posttest M: 90.9 85.3 83.9 79.2Adjusted M: 90.4 86.7 83.5 78.6

S: 9.1 9.8 6.6 11.7

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Our study investigates the differential effects of CAS environment and metacognitive training on algebraicthinking and metacognitive knowledge. Although all environments focus on promoting algebraic thinking,the study found that students who were exposed to CAS embedded within metacognitive training(CAS + META) showed more positive effects on different aspects of algebraic thinking than students who wereexposed only to CAS or metacognitive training. The effects were observed on algebraic reasoning anddiscovering patterns. The study found no differences between the groups on manipulating algorithms andanalyzing changes. In addition, the study found that the CAS + META students outperformed theircounterparts on metacognitive knowledge.

Figure 3 Mean scores on using metacognitive knowledge during solving the task/problem by time and treatment

Figure 4 Mean scores on using metacognitive knowledge after solving the task/problem by time and treatment

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Two factors may explain the findings of our study. First, students who are trained to reflect on their problemsolving processing probably focused on the information provided in the problem. The self-addressedcomprehension questions (e.g., ‘What is the task/problem all about?’) probably guided students to look for allthe relevant information. Also the connection questions (e.g., ‘How is this problem/task different/similar fromwhat you have already solved?’) might lead students to pay attention to all the information and the structureof the task. Therefore, there is reason to believe that students who use the self-addressed questions succeededin discovering patterns.

These conclusions support other research (Garofalo and Lester 1985; Schoenfeld 1987; Kramarski, Mevarechand Liberman, 2001) which indicates that students’ problem-solving failures are often due to the ineffectiveuse of what they know, and not due to a lack of mathematical knowledge. Successful problem solvers becomeaware of what they are doing and frequently monitor, or self-assess, their progress or adjust their strategies asthey encounter and solve problems. Such metacognitive knowledge is much more likely to be developed ina classroom environment that supports it.

Of particular interest here are the findings that students who were exposed to metacognitive training(CAS + META and META) were more effective in correctly using mathematical reasoning. There is reason tobelieve that metacognitive knowledge affects algebraic thinking. Perhaps the poor performance of the CASand CONTROL students may have stemmed from the limited metacognitive knowledge employed in solvingword problems. These conclusions support other research. Schoenfeld (1987) also reported that unsuccessfulproblem solvers, who exhibited limited metacognitive knowledge, mathematical problem solving is often seenas a system of taking one step at a time, without understanding the general principle of the problem.

Masui and De Corte (1999) and Kramarski, Mevarech and Liberman (2001) found that students who wereexposed to metacognitive training had more metacognitive knowledge about orienting and self-judgingthemselves than students in the control groups. In these studies, both metacognitive knowledge and transferbehavior were positively related to academic performance. Further research based on systematic observationsmay identify other measures of metacognitive knowledge, so more empirical evidence can be collectedregarding the relation between metacognitive knowledge and mathematical reasoning.

Conclusions and practical implicationsOur findings indicate that effectively using technology to enhance students’ learning opportunities inmathematics classrooms depends on the didactic possibilities linked with its use. The study shows that featuresof self-questioning such as giving reasons before, during and the end of problem solving must be practiced andreinforced (Webb. 1991; Webb and Farivar, 1994; Kramarski, Mevarech and Liberman, 2001). Students’exposure to metacognitive training enhances metacognitive knowledge that in turn affects algebraicthinking.

These findings call for the design of additional computer learning environments based on similarcomponents. Future research based on interviews and observations may explore the development of generaland specific metacognitive knowledge under different conditions.

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Kramarski, B, Mevarech, ZR and Liberman, A (2001) The Effects of Multilevel- versus Unilevel- MetacognitiveTraining on Mathematical Reasoning, The Journal of Educational Research, 94, 5, 292–300.

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Monagham, J (1994) On the successful use of DERIVE. International DERIVE Journal, 1, 1, 57–70.Montague, M and Bos, CS (1990). Cognitive and metacognitive characteristics of eighth-grade students’

mathematical problem solving, Learning and Individual Differences, 2, 371–388.National Council of Teacher of Mathematics (2000) Principles and Standards for School Mathematics, Reston, VA,

Author.Schoenfeld, AH (1985) Mathematical Problem Solving, San Diego, CA: Academic Press.Schoenfeld, AH (1987) What’s all the fuss about metacognition? In Schoenfeld, AH (ed.). Cognitive Science and

Mathematics Education, (pp. 19–216). Hillsdale, NJ: Erlbaum.Teong, SK, Threlfall, J and Monaghan, J (2000) The effects of metacognitive training in mathematical word problem

solving in a computer environment. In Tadao, Nakahara and Masataka, Koyama (eds). Proceedings of the 24thConference of the International Group for the Psychology of Mathematics Education, HiroshimaUniversity: Hiroshima, Japan, 4, 193–200.

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Biographical notesBracha Kramarski has a doctorate and specialized in mathematics and education in different learningenvironments. Her main interests are Teaching of Mathematics, Computers and Education, Cognition andMetacognition, Different Learning Environments and Teacher’s Training. She also develops programs for theteaching of mathematics by computers, the development and management of a project IMPROVE: Methodfor teaching mathematics in heterogeneous classrooms. She is currently Deputy Director of the School ofEducation and Head of the Teachers Training and Inservice Education.

Chaya Hirsh is a senior teacher in mathematics and computing. She holds a B.A. in Education and an M.A.in Technology and Communications in Education. She is currently working on a thesis whose subject is ‘Theeffects of a computerized algebra system (CAS), using a meta-cognitive approach, on problem solving andmathematical discourse’.

Addresses for CorrespondenceBracha Kramarski, School of Education, Bar-Ilan University, Ramat-Gan 52900, Israel; e-mail:kramab@mail.biu.ac.il

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