Educational Media International, v38 n2-3 p77-82 Jun-Sep 2001
Using Technology to Enhance Mathematical Reasoning: Effects of Feedback and Self-Regulation Learning Bracha Kramarski and Orit Zeichner, Israel
Abstracts Effects of two types of feedback on mathematical reasoning in a computerized environment are compared. Metacognitive feedback (MF) is based on self-regulation learning using metacognitive questions that serve as cues for understanding the problem. Result feedback (RF) means giving cues that pertain only to the nal outcome. A total of 108 sixth-grade Israeli students participated in the study. Results indicated that the MF students signi cantly outperformed the RF students on various measures of mathematical reasoning and mathematical explanations. Theoretical and practical implications of the study are discussed. L’utilisation de la technologie pour développer le raisonnement mathématique : les effets du feed back et de l’apprentissage autorégulé. Ce texte compare les effets de deux types de feed back sur le développement du raisonnement mathématique dans un environnement informatique: a) le feed back métacognitif (MF) qui se base sur un apprentissage autorégulé et utilise des questions de type métacognitif (Mevarech et Kramarski, 1997) qui servent d’indices pour identi er la compréhension du problème ; b) le feed back sur les résultats (RF) qui ne donne des indications que sur les résultats naux. Cent quatre-vingt étudiants israéliens de sixième année ont participé à cette étude. Les résultats indiquent que les étudiants soumis au MF obtiennent de meilleurs résultats que ceux soumis au RF sur la base de mesures du raisonnement et de l’explicitation mathématiques. Der Einsatz von Technologie zur Steigerung mathematischer Logik: Die Wirkung von Feeback und selbstreguliertem Lernen. Im Vergleich stehen zwei Feeback-Arten mathematischer Logik in einem Computer-unterstützten Umfeld : (a) Metakognitives Feedback (MF = metacognitive feedback) basiert auf selbstreguliertem Lernen unter Verwendung von metakognitiven Fragen (Mevarech & Kramarski, 1997) die als Hinweise zum besseren Verständnis des Problems dienen ; (b) Ergebnis-orientiertes Feedback (RF = result feedback), das lediglich Hinweise auf das Endergebnis gibt. 180 israelische Schüler im Alter von ca. 12 Jahren nahmen an dieser Studie teil. Bei der Analyse des Resultats wurde offenbar, dass die MF-unterstützten Schüler wesentlich besser abschnitten, als die RF-Schüler. Theoretische und praktische Auswirkungen dieser Studie werden noch diskutiert.
Among the underlying goals of the NCTM (National Council of Teacher of Mathematics, 1989, 2000) are efforts to enhance students’ mathematical reasoning. The new reforms emphasize the importance of explaining mathematical ideas, not simply isolated answers, but mainly explaining chains of reasoning, using different forms of representations for justifying the answers. Research on the NCTM standards suggests focusing on the conditions under which they can be optimally enhanced in the classroom. Electronic technologies are essential tools for teaching, learning and doing mathematics. When technological tools are available, students can focus on decision making, re ection, reasoning and problem solving (NCTM, 2000, p. 23). But research on the effects of using technology (e.g. Kramarski, 1999) indicated that technology is not a panacea. As with any teaching tools, it can be used well or poorly. Thus, it appears critical that research should not only explore the development of appropriate software to be used, but also explore the role of effective pedagogy that can maximize students’ learning in electronic technologies environments. In recent years, there has been much interest in the role of metacognition in mathematics education (Hacker, Education Media International ISSN 0952-3987 print/ISSN 1469-5790 online © 2001 International Council for Education Media http://www.tandf.co.uk/journals DOI: 10.1080/0952398011004145 8
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1998; Butler and Winne, 1995; Mevarech and Kramarski, 1997). Butler and Winne (1995) describe self regulation learning (SRL) as a style of activities for problem solving that includes evaluating goals, thinking of strategies and choosing the most appropriate strategy for solving the problem. In this model, feedback is a central factor in cognitive–metacognitive processes. In terms of self-regulation, feedback serves as an internal catalyst that oversees the course of activities entailed in performing the tasks, the results and the quality of the cognitive processing of the tasks. Aside from internal feedback there is also external feedback in which students receive cues about the tasks, and the proper process for solving the problem, from an external source such as a teacher, a student or an interactive technological means such as a computer. In general, studies have found that learning with feedback is preferable to learning without feedback (McDaniel and Fisher, 1991; Zellermayer et al., 1991). Studies examining the in uence of learning mathematics in a computerized environment show that computers can serve as an effective aide in learning process and in developing mathematical reasoning because they provide students with immediate feedback about their progress and success in the task (Kramarski and Mevarech, 1997; NCTM, 2000). However, not enough attention has been paid to the comparison between various types of feedback in a computerized environment on mathematical reasoning. The present study examined two types of feedback provided during the process of learning mathematics: metacognitive feedback (MF) and result feedback (RF). Metacognitive feedback is based on SRL, using metacognitive questions (Mevarech and Kramarski, 1997) that serve as cues for understanding the content and structure of the problem as well as ways of solving the problem. Result feedback, in comparison, means giving cues that pertain only to the nal outcome. We hypothesized that providing MF by computer will be more affective than providing RF because the external MF serves as a catalyst for developing an internal feedback. The purpose of the study was twofold: to investigate the effects of two different types of computerized feedback (MF vs. RF on mathematical achievement); and to compare their effects on the ability to explain mathematical reasoning.
Method Participants were 186 eleventh grade students (mean age 17.5), males and females, who studied in eight classrooms selected from four schools. Intact classrooms were randomly assigned to one of two conditions: (a) studying mathematics with computerized MF (n = 102); and (b) studying mathematics with computerized RF (n = 84).
Treatment The researchers dealing with mathematical series developed two computerized units. The units were identical in terms of teaching aims and learning materials but different in the type of feedback provided by the computer. The students learned individually and received feedback from the computer during solving tasks/problems or when they had a mistake. The MF group received an explanation of the importance of using MF for SRL. Metacognitive feedback was based on the IMPROVE method (Mevarech and Kramarski, 1997) that provides each student with the opportunity to be involved in mathematical reasoning via the use of metacognitive questions that focus on: (a) The nature of the problem /task (e.g. What is the problem/ task all about?); (b) The construction of relationships between previous and new knowledge (e.g. What are the similarities/ differences between the problem/ task at hand and the problems/tasks you have solved in the past?); and (c) The use of strategies appropriate for solving the problem/ task (e.g. What are the strategies/tactics/principles that are appropriate for solving the problem/ task and why?). The RF group received an explanation of the importance of using feedback provided by the computer. The feedback included responses such as: ‘think about it, you made a mistake, Try again. Check it once more. Very good! Wonderful job’. In solving problems/tasks, students had three chances to reach the correct answer using the MF or RF and after the third trial, the answer was supplied. The teacher was present in the laboratory but did not interfere in the student-computer interaction unless technical problems arose. Students encountering dif culties were referred back to the lesson and to answering questions provided by the computer.
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Measurement and procedure To assess mathematical reasoning a 27-item test was administered to all students at the beginning and at the end of the study. Students were asked to give an answer and explain their reasoning. The test was based on three measures: general term formula, rule of recursion and word problems. The score ranged from 0 (incorrect answer) to 1 (correct answer). A total score was ranged from 0–27. The scores were translated into percentages. In addition students’ mathematical explanations were analysed using three criteria: verbal arguments, algebraic formula and mixed format (verbal and algebraic formula).
Results The rst research question refers to the differences between the two feedback groups on mathematical achievement. Table 1 presents the mean scores, adjusted mean scores and standard deviations on mathematical achievement by time and treatment. A one way MANCOVA was carried out on the various measures of the posttest scores with the corresponding pre-test scores used as a covariant. Adjusted mean scores indicated signi cant differences between the two feedback groups. The MF group outperformed the RF group on the total score and on all the measures: general term formula, rule of recursion and verbal problems. The term ‘adjusted mean scores’, refers to the post-treatment scores adjusted for the differences in the pre-treatment scores. The second research question refers to the differences between the two feedback groups on mathematical explanations. Results indicated a similar pattern of differences between groups as described above on frequencies of using mathematical explanations (Chi Sq = 6.07; df = 2; p < 0.05). Figure 1 presents students’ frequencies of using mathematical explanations. It was found that students who were exposed to MF explained their reasoning with a richer format. They used more often verbal arguments (30.2%) than students who were exposed to RF (20%) and they also used a mixed format (algebraic rules and verbal arguments) more often (63.5%) than students who were exposed to RF (31.6%).
Discussion and conclusions Electronic technologies are essential tools for teaching, learning, and doing mathematics (NCTM, 2000). The aim of the study was to explore how technology can be used to support mathematical reasoning. In particular to nd what was the effect of feedback and self regulation in a computerized environment. Results indicated that using technology with MF was more affective then using technology with RF. The ndings of the present study raise the question: Why did the MF students outperform the RF students? In the present study both conditions were designed in a computerized environment to activate mathematical processes. Results indicated that explicit MF can enhance mathematical achievement (general term formula, rule of recursion and word problems) and mathematical explanations (verbal arguments, algebraic formula and mixed format) in a computerized environment. It seems that focusing on the similarities and differences between previous and new tasks, as well as on comprehending the problem before attempting a solution and re ecting on the use of strategies that are appropriate for solving the problem enhanced the ability to focus on decision making, re ection which in turn enhanced mathematical reasoning. It seems that the external MF of the computer served as an internal catalyst that oversee the course of activities entailed in performing the tasks. These results con rm ndings of previous studies that focused primarily on the effects of metacognitive instruction on mathematical reasoning in non-computerized environment (Clements, 1990; Schoenfeld, 1985, Mevarech and Kramarski, 1997) as well as in a computerized environment (Kramarski and Mevarech, 1997; Teong et al., 2000). It supports the constructivist theories, that information is retained and understood through elaboration and construction of connections between prior knowledge and new knowledge (Wittrock, 1986). The likelihood of constructing networks of knowledge under the MF group was greater than that of the compared RF group. Furthermore, there is reason to suppose that the MF facilitated metacognitive knowledge, which in turn affects mathematical reasoning. Masui and De Corte (1999) found that students who were exposed to metacognitive instruction in a non-computerized environment had more knowledge about orienting and selfregulating themselves than students in the control groups. In that study, the meta-knowledge was positively related to academic performance and to a transfer task. More work is needed to test whether the superiority of MF found in a computerized environment study would remain the same on a transfer task as well as found on the achievement test.
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Table 1 Mean scores, adjusted mean scores and standard deviations on mathematical achievement by time and treatment
Pre-test Mean SD Post-test Mean Adjusted mean SD
Metacognitive feedback
Result feedback
F
19.03 16.00
24.80 10.70
8.13**
88.10 88.15 14.43
67.15 66.80 18.25
75.98***
Metacognitive feedback
Result feedback
F
32.73 24.56
44.10 23.54
10.34**
88.91 88.87 17.29
81.18 80.22 21.07
7.54**
Metacognitive feedback
Result feedback
General term formula
Pre-test Mean SD Post-test Mean Adjusted mean SD Rule of recursion
Pre-test Mean SD Post-test Mean Adjusted mean SD
F
6.52 14.74
2.24 8.02
5.90*
90.20 89.76 20.23
60.86 61.30 30.57
60.48***
Metacognitive feedback
Result feedback
F
10.30 18.23
8.98 15.70
82.81 84.37 21.08
62.00 59.49 28.00
Verbal problems
Pre-test Mean SD Post-test Mean Adjusted mean SD
2.78 35.46***
*p < 0. 05, **p < 0.01, ***p < 0.001.
Further research based on systematic observations in the MF and the RF groups may identify measures of self regulation behaviours, so better evidence can be provided regarding the relationship between MF in a computerized environment, metacognitive knowledge, self regulation and mathematical reasoning. These ndings have important educational implications in learning and teaching mathematics. Thus, it indicates ways of learning/teaching that direct SLR by constructing an explicit example model of MF during the learning
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70 63.5 60
50
40 31.6
30.2
30
Results Feedback
20
20
Metacognitive Feedback
10.5 10
0
6.3
Algebraic formula
Verbal arguments
Mixed format
Figure 1 Frequencies of students’ using mathematical explanations by type of feedback
process. This form of learning brings the message to the student that their learning processes can be regulated by themselves and are their responsibility. In summary, relating the ndings of this research to issues of how technology can be used in the wider eld of mathematics learning leads to the conclusion that it serves for effective learning and that further research and evaluation on the conditions under which effective pedagogy work best in this area must be a priority.
References Butler, DL and Winne, PH (1995) Feedback and self-regulated learning: a theoretical synthesis, Review of Educational Research, 65(3), 245–281. Clements, DH (1990) Metacomponential development in a logo programming environment, Journal of Educational Psychology, 82(1), 141–149. Hacker, D (1998) De nitions and empirical foundations in metacognition. In D Hacker, J Dunlosky and A Graesser (eds) Educational Theory and Practice, LEA, London, pp. 1–23. Kramarski, B (1999) Is easier better? The study of graphs by computers, Educational Media International, 36(3), 203–209. Kramarski, B and Mevarech, ZR (1997) Cognitive-metacognitive training within a problem solving based Logo environment, British Journal of Educational Psychology, 67, 425–445. Masui, C and De Corte, E (1999) Enhancing learning and problem solving skills: Orienting and self-judging, two powerful and training learning tools, Learning and Instruction, 9(6), 517–542. McDaniel, MA and Fisher, RP (1991) Tests and tests feedback as learning sources, Contemporary Educational Psychology, 16, 192–201. Mevarech, ZR and Kramarski, B (1997) IMPROVE: a multidimensional method for teaching mathematics in heterogeneous classrooms, American Educational Research Journal, 34, 365–394. National Council of Teachers of Mathematics (1989) Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA.
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National Council of Teachers of Mathematics (2000) Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA. Schoenfeld, AH (1985) Mathematical Problem Solving, Academic Press, San Diego, CA. Teong, SK, Threlfall, J and Monagham, J (2000) The effects of metacognitive training in mathematical word problem solving in a computer environment. In Proceedings of the Twenty Fourth Annual Conference of International Group for the Psychology Mathematical Education, Hiroshima, Japan, vol. 4, pp. 193–200. Wittrock, MC (1986) Students’ thought processes. In MC Wittrock (ed.) Handbook of research on teaching, 3rd edn, Macmillan, New York. Zellermayer, M, Salomon, G, Globerson, T and Givon, H (1991) Enhancing writing-related metacognitions through a computerized writing partner, American Educational Research Journal, 28, 373–391.
Biographical notes Dr Bracha Kramarski serves as Head of the Teacher Training of the School of Education as well as Director of Teaching Mathematics in the Institute for the Promotion of Social Integration at the Bar-Ilan University. She specializes in Teaching of Mathematics, as well as use of New Technologies, Cognition and Metacognition. Miss Orit Zeichner graduated (MA) in the Department of Mathematics and School of Education at the Bar-Ilan University and is at present a doctoral student and an assistant at the same university. She specializes in ICT applications in education.
Address for correspondence Bracha Kramarski, School of Education, Bar-llan University, Israel; e-mail:
[email protected]
