pp. 3511-3519 | Article Number: iejme.2016.289
Published Online: November 29, 2016
Article Views: 609 | Article Download: 1430
The main goal of the study reported in our paper is to characterize teachers’ choice of examples in and for the mathematics classroom. Our data is based on 54 lesson observations of five different teachers. Altogether 15 groups of students were observed, three seventh grade, six eighth grade, and six ninth grade classes. The classes varied according to their level—seven classes of top level students and six classes of mixed—average and low level students. In addition, pre and post lesson interviews with the teachers were conducted,and their lesson plans were examined. Data analysis was done in an iterative way, and the categories we explored emerged accordingly. We distinguish between pre-planned and spontaneous examples, and examine their manifestations, as well as the different kinds of underlying considerations teachers employ in making their choices, and the kinds of knowledge they need to draw on. We conclude with a dynamic framework accounting for teachers’ choices and generation of examples in geometry class
Keywords: Example, non-example, geometry class
Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from Examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2), 181–214.
Ball, D., Bass, H., Sleep, L., & Thames, M. (2005). A theory of mathematical knowledge for teaching. Paper presented at a Work-Session at ICMI-Study15: The Professional Education and Development of Teachers of Mathematics, Brazil, 15–21 May 2005.
Bills, C., & Bills, L. (2005). Experienced and novice teachers’ choice of examples. In P. Clarkson, A.
Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 146–153). MERGA Inc., Melbourne.
Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 126–154). Prague, Czech Republic.
Denton, C., Bryan, D., Wexler, J., Reed, D. & Vaughn, S. (2007). Effective instruction for middle school students with reading difficulties: The reading teacher’s sourcebook. University of Texas:Austin.
Frayer, D., Frederick, W. C., and Klausmeier, H. J. (1969). A Schema for Testing the Level of Cognitive Mastery, Madison, WI: Wisconsin Center for Education Research
Harel, G. (in press). What is mathematics? A pedagogical answer to a philosophical question. In R. B. Gold, & R. Simons (Eds.), Proof and other dilemmas: Mathematics and philosophy. Mathematical Association of America.
Kennedy, M. M. (2002). Knowledge and teaching . Teachers and teaching: Theory and practice, 8, 355– 370. DOI 10.1080/135406002100000495.
Leikin, R., & Dinur, S. (2007). Teacher flexibility in mathematical discussion. The Journal of Mathematical Behavior, 26(4), 328–347.
Leinhardt, G. (1990). Capturing craft knowledge in teaching. Educational Researcher, 19(2), 18–25.
Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 227–289. DOI 10.1007/BF00312078.
Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38, 135–161. DOI 10.1023/A:1003622804002.
Miles, M. B., & Huberman, A. M. (1987). Qualitative data analysis: a sourcebook of new methods (5th edition). SAGA Publications.
Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also) explain. FOCUS on Learning Problems in mathematics, 19(3), 49–61.
Petty, O. S., & Jansson, L. C. (1987). Sequencing examples and nonexamples to facilitate concept attainment. Journal for Research in Mathematics Education, 18(2), 112–125. DOI 10.2307/749246.
Rissland, E. L. (1991). Example-based reasoning. In J. F. Voss, D. N. Parkins, & J. W. Segal (Eds.), Informal reasoning in education (pp. 187–208). Hillsdale, NJ: Lawrence Erlbaum.
Rowland, T., Thwaites, A., & Huckstep, P. (2003). Novices’ choice of examples in the teaching of elementary mathematics. In A. Rogerson (Ed.), Proceedings of the International Conference on the Decidable and the Undecidable in Mathematics Education (pp. 242–245). Brno, Czech Republic.
Shulman, S. L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4–14.
Shulman, S. L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–22.
Simon, A. M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. DOI 10.2307/749205.
Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth, UK: Penguin Books.
Stake, R. E. (2000). Case studies. In N. K. Denzin, & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 435–454, 2nd ed.). Thousand Oaks, CA: Sage.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.
Strauss, A. L., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, Ca: Sage.
Vinner, S. (1983). Concept Definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14, 293–305. DOI 10.1080/0020739830140305.
Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.
Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111. DOI 10.1207/ s15327833mtl0802_1.
Wiersma, W. (2000). Research methods in education: an introduction (7th ed.). Allyn & Bacon.
Zaslavsky, O., Harel, G., & Manaster, A. (2006). A teacher’s treatment of examples as reflection of her knowledge-base. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 457–464). Prague, Czech Republic.
Zaslavsky, O., & Lavie, O. (2005). Teachers’ use of instructional examples. Paper presented at ICMIStudy15: The Professional Education and Development of Teachers of Mathematics, Brazil, 15–21 May 2005.
Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student–teachers: The case of binary operation. Journal for Research in Mathematics Education, 27(1), 67–78. DOI 10.2307/749198.
Zaslavsky, O., & Zodik, I. (2007). Mathematics teachers’ choices of examples that potentially support or impede learning. Research in Mathematics Education, 9, 143–155. DOI 10.1080/14794800008520176.
Zazkis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195–208.
Zodik, I. & Zaslavsky, O. (2007). Exemplification in the mathematics classroom: What is it like and what does it imply? Proceedings of the 5th Conference of the European Society for Research in Mathematics Education (pp. 2024–2033), Larnaka, Cyprus.