Abstract
This article reports an alternative approach, called the combinatorial model, to learning multiplicative identities, and investigates the effects of implementing results for this alternative approach. Based on realistic mathematics education theory, the new instructional materials or modules of the new approach were developed by the authors. From the combinatorial activities based on the things around daily life, the teaching modules assisted students to establish their concept of the distributive law, and to generalize it via the process of progressive mathematizing. The subjects were two classes of 8th graders. The experimental group (n = 32) received a combinatorial approach to teaching by the first author using a problem-centered with double-cycles instructional model, while the control group (n = 30) received a geometric approach to teaching, from the textbook by another teacher who uses lecturing. Data analyses were both qualitative and quantitative. The findings indicated that the experimental group had a better performance than the control group in cognition, such as for the inner-school achievement test, mid-term examination, symbol manipulation, and unfamiliar problem-solving: also in affection, such as the tendency to engage in the mathematics activities and enjoy mathematical thinking.
Similar content being viewed by others
References
Azevedo, F. (2000). Designing representations of terrain: A study in meta-representational competence. Journal of Mathematical Behavior, 19, 443–480.
Booth, L.R. (1981). Child-methods in secondary mathematics. Educational Studies in Mathematics, 72, 29–41.
Booth, L.R. (1986). Difficulties in algebra. Australian Mathematic Teacher, 42(3), 2–4.
Chaiklin, S. (1989). Cognitive studies of algebra problem solving and learning. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 93–114). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum Associates.
Chang, C.K. (1993). The influence of the spatial ability and the need of cognition to the strategy of problem solving. The reports of the research projects of the NSC 81-0111-S-018-506.
Chang, C.K. (1995a). The discussion of the process and effect and feasibility of the problem-centered teaching in junior high school. Chinese Journal of Science Education, 3(2), 139–165.
Chang, S.H. (1995b). On understanding of multiplicative identities in junior secondary students. Taiwan: National Changhua University of Education.
Chang, C.F. (2003). Inner relation to formal knowledge extended from empirical knowledge -A case of distributive law. Taiwan: National Changhua University of Education.
Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16–30.
Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing. Focus on Learning Problems in Mathematics, 11(2), 77–87.
Davis, R.B., Jockusch, E. & McKnight, C.C. (1978). Cognitive processes in learning algebra. The Journal of Children’s Mathematical Behavior, 2(1), 1–320.
diSessa, A.A., Hammer, D., Sherin, B. & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10, 117–160.
Fang, F.J. (2002). Junior high school students’ learning outcomes on multiplicative identities: With transformation from geometrical representation to algebraic description. Taiwan: National Changhua University of Education.
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3, 413–435.
Freudenthal, H. (1991). Revisiting mathematics education. China Lectures. Dordrecht: Kluwer Academic Publishers.
Gravemeijer, K.P.E. (1994). Developing realistic mathematics education. Utrecht: CD-b Press/ Freudenthal Institute.
Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18, 75–90.
Izsak, A. (2000). Inscribing the winch: Mechanisms by which students develop knowledge structures for representing the physical world with algebra. The Journal of the Learning Sciences, 9(1), 31–74.
Jiang, J.H. (2001). A study of using calculation of areas as analogs to teach the concepts of multiplicative identities for the first grade of junior high school students. Taiwan: National Changhua University of Education.
Kao, F.P. Lin, K.H. & Lin, F.L. (1989). The development of the concept of the symbol of junior high school students. The reports of the research projects of the NSC 77-0111-S004-01A.
Kieran, C. (1992). The learning and teaching of school algebra. In A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390–419) New York: Macmillan.
Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 274–287.
Kirshner, D. (1993). The structural algebra option. Paper presented at the Annual Meeting of the American Educational Research Association. Atlanta, GA, April.
Kuchemann, D. (1981). Algebra. In K.M. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 102–119) London: John Murray.
Laursen, K.W. (1978). Errors in first-year Algebra. Mathematics Teacher, 71(3), 194–195.
Leinhardt, G., Zaslavsky, O. & Stein, M.K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60, 1–64.
Matz, M. (1982). Toward a process model for high school algebra errors. In D. Sleeman & J.S. Brown (Eds.), Intelligent tutoring system. London: Academic Press.
Ministry of Education in Taiwan. (2000). General Guidelines of Grade 1–9 Curriculum of Elementary and Junior High School Education. Taipei, Taiwan.
Petty, R.E. & Cacioppo, J.T. (1986). Communication and persuasion: Central and peripheral routes to attitude change. New York: Springer-Verlag.
Schoenfeld, A.H., Smith, J. & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in instructional psychology 4 (pp. 55–175. )Hillsdale, NJ: Lawrence Erlbaum Associates.
Schwartzman, S. (1977). Helping students understand the distributive property. The Mathematics Teacher, 70, 594–595.
Sherin, B. (2000). How students invent representations of motion: A genetic account. Journal of Mathematical Behavior, 19, 399–441.
Stacey, K. & MacGregor, M. (2000). Learning the algebraic method of solving problems. Journal of Mathematical Behavior, 18, 149–167.
Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht: Kluwer Academic Publishers.
Swafford, J. & Langrall, C. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31, 89–112.
Tai, W.P. (1999). Entering algebra from arithmetic: Learning phenomena and characteristics of the first year secondary school students. Taiwan: National Changhua University of Education.
Yuan, Y. (1993). The concept of the symbol of the 7th grade student. The 4th education academic article collection in Taiwan, 237–262.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tsai, YL., Chang, CK. USING COMBINATORIAL APPROACH TO IMPROVE STUDENTS’ LEARNING OF THE DISTRIBUTIVE LAW AND MULTIPLICATIVE IDENTITIES. Int J of Sci and Math Educ 7, 501–531 (2009). https://doi.org/10.1007/s10763-008-9135-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10763-008-9135-x