According to psychological and cognitive development theories, the preference s of pupils in elementary school toward inductive versus deductive and general types of reasoning when asked to prove or review mathematical claims, changes along the school years. This study examines this hypothesis through a survey in which 267 pupils from the Arabic sector in three different elementary schools in Israel, in grades 4 to 6 participated. The survey, based on the math reasoning tasks by Healy and Hoyles (1998) , is comprised of Algebra and Geometry reasoning tasks. Additionally, 12 of these pupils’ teachers were interviewed in order to explore their attitudes toward mathematical reasoning and math proving tasks. Findings show that : 1 ) There is a difference in students’ preferen ces towards types of reasoning, between grades 4 and 6; 2 ) Sixth graders will be less likely to accept tautologic and inductive reasoning than fourth graders; 3 ) Elementary school pupils tend to prefer empirical arguments (such as inductive and example-based) as their approach in contrast to the arguments that they believe will receive the highest scores from their teachers. However, findings do not support the hypothesis that there will be a difference in teachers’ preferences towards different types of thinking. The research findings and their practical implications are discussed.
Anderson, J. A. (1985). Cognitive Psychology and Its Implications. New York: W.H Freeman.
Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on Teaching Mathematics: The Unsolved Problem of Teachers’ Mathematical Knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (pp. 433-456). New York: Macmillan.
Bartlett, F. A. (1932). A Study in Experimental and Social Psychology. New York: Cambridge University Press.
Bloom, B. S. (1956). Taxonomy of Educational Objectives: The Classification of Educational Goals, Handbook I, Cognitive Domain. New York: Longmans, Green.
Brown, A. (1987). Metacognition, Executive, Control, Self-Regulation and Other More Mysterious Mechanisms. In F. E. Weinet, & R. H. Kluer (Eds.), Metacognition. Motivation and Understanding (pp. 65-116). Hillsdale, NJ: Lawrence Erlbaum.
Dehaene, S., & Cohen, L. (1995). Toward an Anatomical and Functional Model of Number Processing. Mathematical Cognition, 1, 83-120.
Dewey, J. (1933). How We Think a Restatement of the Relation of Reflective Thinking to the Educative Process. Boston, MA: D.C. Heath & Co Publishers.
Duschl, R. (2008). Science Education in Three-Part Harmony: Balancing Conceptual, Epistemic, and Social. Review of Research in Education, 32, 268-291. https://doi.org/10.3102/0091732X07309371
Eshet, Y. (2004). Digital literacy: A Conceptual Framework for Survival Skills in The Digital Era. Journal of Educational Multimedia and Hypermedia, 13, 93-106.
Fischbein, E. (1982). Intuition and Proof. For the Learning of Mathematics, 3, 9-18+24.
Flavel, J. H. (1979). Metacognition and Cognitive Monitoring—A New Era of Cognitive-Developmental Inquiry. American Psychologist, 34, 906-911. https://doi.org/10.1037/0003-066X.34.10.906
Flores, A. (2002). How Do Children Know That What They Learn in Mathematics Is True? Teaching Children Mathematics, 8, 269-274.
Glassner, A. & Schwartz, B. (2001). From Reading Comprehension to Understanding: From Textual Activities to Literacy Activities in Various Contexts. Script, 2, 11-43.
Goetting, M. M. (1995). The College Students’ Understanding of Mathematical Proof. Doctoral Dissertation, College Park, MD: University of Maryland.
Hanna, G. (1989). Proofs That Prove and Proofs That Explain. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the International Group for the Psychology of Mathematics Education Vol. II (pp. 45-51). Paris: PME Books.
Harel, G., & Sowder, L. (1998). Students’ Proof Schemes: Results from Exploratory Studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education III (Vol. 7, pp. 234-283). Providence, RI: AMS, CBMS: Issues in Mathematics Education.
Healy, L., & Hoyles, C. (1998). Justifying and Proving in School Mathematics: Technical Report on the Nationwide Survey. London: Institute of Education, University of London.
Healy, L., & Hoyles, C. (2000). A Study of Proof Conceptions in Algebra. Journal for Research in Mathematics Education, 31, 396-428. https://doi.org/10.2307/749651
Knuth, J. (2002). Secondary School Mathematics Teachers’ Conceptions of Proof. Journal for Research in Mathematics Education, 33, 379-405. https://doi.org/10.2307/4149959
Lewis, A., & Smith, D. (1993). Defining Higher Order Thinking. Theory into Practice, 32, 131-137. https://doi.org/10.1080/00405849309543588
Martin, W. G., & Harel, G. (1989). Proof Frames of Preservice Elementary Teachers. Journal for Research in Mathematics Education, 20, 41-51. https://doi.org/10.2307/749097
Means, M. L., & Voss, J. F. (1996). Who Reasons Well? Two Studies of Informal Reasoning among Children of Different Grade, Ability and Knowledge Levels. Cognition & Instruction, 14, 139-179. https://doi.org/10.1207/s1532690xci1402_1
Newton, P., Driver, R., & Osborne, J. (1999). The Place of Argumentation in the Pedagogy of School Science. International Journal of Science Education, 21, 553-576. https://doi.org/10.1080/095006999290570
Papadakis, S., Kalogiannakis, M., & Zaranis, N. (2016). Improving Mathematics Teaching in Kindergarten with Realistic Mathematical Education. Early Childhood Education Journal, 45, 369-378. https://doi.org/10.1007/s10643-015-0768-4
Passig, D. (2001). A Taxonomy of ICT Mediated Future Thinking Skills. In H. Taylor, & P. Hogenbirk (Eds.), Information and Communication Technologies in Education: The School of the Future (pp. 103-112). Boston: Kluwer Academic Publishers. https://doi.org/10.1007/978-0-387-35403-3_9
Piaget, J. (1965). The Child’s Conception of Number. New York: W.W. Norton & Co.
Sadler, T. D., & Fowler, S. R. (2006). A Threshold Model of Content Knowledge Transfer for Socioscientific Argumentation. Science Education, 90, 986-1004. https://doi.org/10.1002/sce.20165
Salomon, G., & Perkins, D. N. (1989). Rocky Roads to Transfer: Rethinking Mechanisms of a Neglected Phenomenon. Educational Psychologist, 24, 113-142. https://doi.org/10.1207/s15326985ep2402_1
Stylianides, A. J. (2007a). Introducing Young Children to the Role of Assumptions in Proving. Mathematical Thinking and Learning, 9, 361-385. https://doi.org/10.1080/10986060701533805
Stylianides, A. J. (2007b). The Notion of Proof in the Context of Elementary School Mathematics. Educational Studies in Mathematics, 65, 1-20. https://doi.org/10.1007/s10649-006-9038-0
Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the Transition from Empirical Arguments to Proof. Journal for Research in Mathematics Education, 40, 314-352.
Swartz, R. J. (2008). Engineering Learning. Educational leadership, 65, 26-31.
Toulmin, S. E. (1969). The Use of Argument. Cambridge, AT: The University Press.
Van Dormolen, J. (1977). Learning to Understand What Giving a Proof Really Means. Educational Studies in Mathematics, 8, 27-34. https://doi.org/10.1007/BF00302502
Zaranis, N., Kalogiannakis, M., & Papadakis, S. (2013). Using Mobile Devices for Teaching Realistic Mathematics in Kindergarten Education. Creative Education, 4, 1-10. https://doi.org/10.4236/ce.2013.47A1001
Zohar, A. (1996). Learn, Think and Learn to Think. Jerusalem: Branco Weiss Institute for Cultivation of Thinking.