Effects of Computer Algebra System (CAS) with Metacognitive Training on Mathematical Reasoning

The main purpose of the present study was to investigate the differential effects of Computer Algebra Systems (CAS) and metacognitive training on mathematical reasoning. Participants were 83 students (boys and girls) who studied algebra in four eighth-grade classrooms randomly selected to four instructional methods: CAS embedded within metacognitive training (CAS + META), metacognitive training without CAS learning (META), CAS learning without metacognitive training (CAS) and CONT learning without CAS and metacognitive training (CONT). Results showed that the CAS + META condition significantly outperformed the META and the CAS conditions, who in turn significantly outperformed the CONT condition on aspects of mathematical reasoning. No significant differences were found between the META and CAS conditions. 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 research whose main focus is the use of Computer Algebra Systems (CAS) in mathematics classrooms and the didactical possibilities based on the IMPROVE method linked with its use. Les effets du système d’algèbre par informatique (CAS) sur une formation métacognitive en raisonnement mathématique. Le but principal de la présente étude était d’explorer les effets différentiels du CAS et d’une formation métacognitive en raisonnement mathématique. Les participants étaient 83 élèves (garçons et filles) qui étudiaient l’algèbre dans les classes (eighth-grade) choisies au hasard avec 4 méthodes d’instruction. CAS enchassés à l’intérieur d’une formation métacognitive (CAS + META) métacognitive sans CAS (META), apprentissage CAS sans formation métacognitive (CONT) et apprentissage sans CAS + formation métacognitive (CONT). Les résultats ont montré que la formule CAS + META a surpassé de façon significative les formules META + CAS qui a leur tour ont surpassé la formule CONT quant aux aspects du raisonnement mathématique. Il n’y a pas de différences entre les formules META et CAS. De plus l’étude a montré que les élèves soumis aux conditions CAS + META et META ont obtenu de meilleurs résultats que leurs condisciples (CAS et CONT) quant à leurs connaissances métacognitives cet article décrit des recherches dont le but principal est l’utilisation du CAS dans les classes de mathématiques et les possibilités didactiques fondées sur la méthode IMPROVE à son usage. Wirkungen des Computer Algebra Systems (CAS) mit metacognitivem Training auf mathematisches Denken Der Hauptzweck der gegenwärtigen Studie war, die unterscheidenden Wirkungen des Computer Algebra Systeme (CAS) und der metacognitiven Ausbildung auf mathematische Überlegung zu untersuchen. Teilnehmer waren 83 Studenten (Jungen und Mädchen), die Algebra in vier zufällig ausgewählten Klassen der achten Stufe in vier Unterrichtsmethoden studierten: CAS eingebettet in metacognitive Ausbildung (CAS + Meta), metacognitive Ausbildung ohne CAS (Meta), CAS Lernen ohne metacognitive Ausbildung (CAS) und CONT Lernen ohne CAS und metacognitive Ausbildung (CONT). Die Ergebnisse zeigten dass die CAS + Meta-Bedingung die Metaund die CAS Bedingungen signifikant übertraf, die wiederum die CONT Bedingung bedeutend in Bezug auf Aspekte mathematischen Denkens überflügelten. Keine bedeutsamen Unterschiede wurden Zwischen den Meta und CAS Versionen wurden keine bedeutsamen Unterschiede gefunden. Außerdem ergab die Studie, dass die Studenten der metacognitiv unterrichteten Gruppe (CAS + Metaund Metabedingungen) ihre Pendants (CAS und CONT) in Bezug auf ihr metacognitives Wissen übertrafen. Dieses Papier beschreibt Forschung, deren Hauptfokus auf der Verwendung von Computer Algebra Systemen (CAS) in Mathematikklassen ist und die didaktischen Möglichkeiten, die auf der mit ihrer Verwendung verbundenen IMPROVE Methode basieren. D ow nl oa de d by [ B ar -I la n U ni ve rs ity ] at 0 0: 39 1 5 Ja nu ar y 20 12 250 EMI 40:3/4 – ICEM-CIME ANNUAL CONFERENCE, GRANADA Introduction Algebra 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 mathematical models 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 increased dramatically. The interaction with the computerized system enabled practicing the algebraic language for symbolic and numerical computation. It encouraged understanding of algebraic reasoning, finding patterns, and reflecting on the solution process. But research (e.g., Monaghan, 1994; Lagrange, 1996) on using 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 easier calculation did not automatically enhance students’ reflection and understanding. Their conclusion was that a great deal of fieldwork is required to establish in what conditions this potential of Computer Algebra can 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 of learning processes (Flavell, 1979). Schoenfeld (1987) indicated that in the area of mathematics, metacognitive knowledge includes knowledge about one’s own thought processes (e.g., ‘How accurate are you in describing your own thinking?’), control or self-regulation (e.g., ‘How well do you keep track of what you’re doing when you’re solving problems/tasks?’), and beliefs and intuitions (e.g., ‘What ideas about mathematics do you bring to 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 major common element of these programs is training students to do mathematics by asking and answering a series of self-addressed metacognitive questions before, during and after attempting a solution. The method of Mevarech and Kramarski, (1997), called IMPROVE emphasizes the importance of providing each student with the opportunity to construct metacognitive knowledge in mathematical learning via the use of self questioning. This questioning focuses on: (a) comprehending the problem (e.g., ‘What is the problem all about’?); (b) constructing connections between previous and new knowledge (e.g., ‘What are the similarities/differences between the problem at hand and the problems you have solved in the past? and why?’; (c) using of strategies appropriate for solving the problem (e.g., ‘What are the strategies/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 make sense?’). 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, and Monaghan, 2001; Kramarski and Zeichner, 2001; Kramarski and Ritkof, 2002). Effects were found on mathematical reasoning, giving mathematical explanations and promoting mathematical discourse. Given these studies, there is reason to believe that providing metacognitive training within the CAS environment will improve students’ mathematical reasoning. There is also evidence showing that students who were exposed to metacognitive training in classrooms improved their ability to use metacognitive knowledge more than students of the control groups (Masui and De Corte, 1999; Kramarski, Mevarech and Liberman, 2001). There is reason to suppose that providing metacognitive 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 as compared 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 to compare four instructional methods: Computer Algebra System learning embedded within metacognitive training (CAS + META), metacognitive learning in the whole class (META), Computer Algebra System learning with no metacognitive training (CAS), and learning in the whole class with no metacognitive training (CONT). D ow nl oa de d by [ B ar -I la n U ni ve rs ity ] at 0 0: 39 1 5 Ja nu ar y 20 12 Effects of Computer Algelora System 251 Method Participants were 83 students (boys and girls) who studied in four eighth-grade classrooms randomly selected from four junior high schools. All classrooms studied Algebra five times a week during a five-month period according to the mathematics curriculum suggested by the Israeli Ministry of Education. Classrooms were randomly assigned to one of the following conditions: CAS + META condition Students in this condition were exposed to Computer Algebra System (20 hours in the computer lab, each weak one hour) embedded within metacognitive training. The CAS software enabled the students to practice the algebraic language for symbolic and numerical computation. In particular, the students practiced substitution of variables, simplifying algebraic expressions and solution of equations. The metacognitive training 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; metacognitive questioning; practicing; reviewing; obtaining mastery on higher and lower cognitive skills; verifying and enrichment. The metacognitive training provides each student with the opportunity to construct metacognitive knowledge by utilizing self-addressed metacognitive questions: Comprehension questions, strategic questions, connection questions and reflection questions. In addressing comprehension questions, students had to read the problem/task, describe the concepts in their own words, and try to understand what the concepts meant. The strategic questions are designed to prompt students to consider which strategies are appropriate for solving the given problem/task and for what reasons. Connection questions prompt students to focus on similarities and differences between the task at hand and task they had already solved. Reflection questions prompt students to focus on the solution process and to ask themselves ‘what am I doing here? ‘does it make sense’ ‘what if’. The metacognitive questions were printed in Students’ Booklets, Teacher Guide, and on the hand held index cards that students used in problem solving. Students used the metacognitive questions orally in their small group/individualized activities, and in writing when they used their booklets.