Mathematics teachers are encouraged to base their classroom teaching practices upon several components, including providing opportunities for students to solve mathematical tasks in their own original way; listening to students’ mathematical thinking, and using students’ mathematical thinking to make instructional decisions. In this study, seven prospective teachers (PTs) of mathematics were asked to relate on various levels to an authentic case centering on three students’ suggested proofs, whether valid and invalid, to a geometric task. They were further asked to address the brief verbal exchanges that took place in the class between the teacher and the students immediately after the presentation of each suggested proofs, during which the teacher deliberately refrained from commenting on its validly. All the PTs correctly recognized the valid proof, but some of them failed to identify the flaws in the invalid proofs. The PTs suggested raising in their future classes, issues related to the geometric configurations that formed the basis of the task (parallelograms, diagonals, and auxiliary lines). Most PTs voiced dissatisfaction with the sequential presentation of the three proofs, with no indication from the teacher as to the validity of each solution. The article endorses exposing PTs to authentic teaching cases that provide opportunities to discuss subtle issues related to their mathematical knowledge and to obstacles which their future students might encounter when solving such mathematical tasks.

prospective teachers, cases, using errors in learning and teaching mathematics.

Received: February 1, 2021; Accepted: March 17, 2021; Published: May 31, 2021

How to cite this article: Ruthi Barkai, “Do you accept these as geometrical proofs?” using cases as a means of raising prospective teachers’ awareness of the subtle nature of geometrical proofs, Far East Journal of Mathematical Education 21(1) (2021), 93-121. DOI: 10.17654/ME021010093

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] D. L. Ball, With an eye on the mathematical horizon: dilemmas of teaching elementary school mathematics, Elementary School J. 94(4) (1993), 373-397.
[2] C. Barnett, Building a case-based curriculum to enhance the pedagogical content knowledge of mathematics teachers, Journal of Teacher Education 42(4) (1991), 263-272.
[3] J. L. Booth, K. E. Lange, K. R. Koedinger and K. J. Newton, Using example problems to improve student learning in algebra: differentiating between correct and incorrect examples, Learning and Instruction 25 (2013), 24-34.
[4] G. Brousseau, Theory of didactical situations in mathematics: Didactique des Mathematiques 1970-1990, Dordrecht, Kluwer, The Netherlands, 1997.
[5] T. P. Carpenter, E. Fennema, M. L. Franke, S. B. Empson and L. W. Levi, Children’s Mathematics: Cognitively Guided Instruction, Heinemann, Portsmouth, NH, 1999.
[6] T. J. Cooney and K. Krainer, Inservice Mathematics Teacher Education: The Importance of Listening, A. J. Bishop et al., eds., International Handbook of Mathematics Education, Kluwer Academic Publishers, Netherlands, 1996, pp. 1155-1185.
[7] A. Conner, P. S. Wilson and H. J. Kim, Building on mathematical events in the classroom, ZDM - The International Journal on Mathematics Education 43 (2011), 979-992.
[8] T. Dvora, Unjustified assumptions in geometry made by high-school students in Israel, Unpublished Doctoral Dissertation, Tel Aviv University, Israel, 2012.
[9] K. Durkin and B. Rittle-Johnson, The effectiveness of using incorrect examples to support learning about decimal magnitude, Learning and Instruction 22(3) (2012), 206-214.
[10] R. Even and D. Tirosh, Teacher knowledge and understanding of students’ mathematical thinking and knowledge, L. English, ed., Second Handbook of International Research in Mathematics Education, Routledge, NY, 2008, pp. 202-222.
[11] L. Fan, C. Qi, X. Liu, Y. Wang and M. Lin, Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An intervention-based study with two Chinese classrooms, Educational Studies in Mathematics 96 (2017), 229-248.
[12] E. Fennema, T. P. Carpenter and P. L. Peterson, Learning mathematics with understanding: cognitively guided instruction, J. Brophy, ed., Advances in Research on Teaching, Greenwich, JAI Press, CT, 1989, pp. 195-221.
[13] E. Fennema and M. Franke, Teachers’ knowledge and its impact, D. Grouws, ed., Handbook of Research on Mathematics Teaching and Learning, Macmillan Publishing Company, New York, 1992, pp. 147-164.
[14] E. Fennema, T. P. Carpenter, M. L. Franke, L. Levi, V. B. Jacobs and S. B. Empson, A longitudinal study of learning to use children’s thinking in mathematics instruction, Journal for Research in Mathematics Education 27(4) (1996), 403-434.
[15] E. A. Forman, J. Larreamendy-Joerns, M. K. Stein and C. A. Brown, You’re going to want to find out which and prove it: collective argumentation in a mathematics classroom, Learning and Instruction 8(6) (1998), 527-548.
[16] M. L. Franke, T. P. Carpenter, L. Levi and E. Fennema, Capturing teachers’ generative growth: a follow-up study of professional development in mathematics, American Educational Research Journal 38 (2001), 653-689.
[17] M. Franke, E. Kazemi and D. Battey, Mathematics teaching and classroom practice, 2nd ed., F. Lester, ed., Handbook of Research on Mathematics Teaching and Learning, Information Age Publishing, Charlotte, 2007, pp. 225-256.
[18] H. Freudenthal, Revisiting Mathematics Education, Dordrecht, Kluwer, Netherlands, 1991.
[19] K. Gravemeijer, Developing Realistic Mathematics Education, Utrecht, CD-B Press, Netherlands, 1994.
[20] K. Gravemeijer, Local instruction theories as means of support for teachers in reform mathematics education, Mathematical Thinking and Learning 6(2) (2004), 105-128.
[21] A. Halai, Secondary mathematics teaching: should it be all chalk and talk? Mathematics Teaching 161 (1997), 18-19.
[22] P. Herbst and C. Brach, Proving and doing proofs in high school geometry classes: what is it that is going on for students? Cognition and Instruction 24(1) (2006), 73-122.
[23] A. Heinze and K. Reiss, Mistake-handling activities in the mathematics classroom: effects of an in-service teacher training on students’ performance in geometry, J. H. Woo, H. C. Lew, K. S. Park and D. Y. Seo, eds., Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, PME, Seoul, Vol. 3, 2007, pp. 9-16.
[24] M. Henningsen, Getting to know Catherine and David: using a narrative classroom case to promote inquiry and reflection on mathematics, teaching, and learning, Cases in Mathematics Teacher Education: Tools for Developing Knowledge Needed for Teaching, 2008, pp. 47-56.
[25] H. C. Hill, L. D. Ball and S. G. Schilling, Unpacking pedagogical content knowledge: conceptualizing and measuring teachers’ topic-specific knowledge of students, Journal for Research in Mathematics Education 39 (2008), 372-400.
[26] J. C. Hill, M. Blunk, C. Charalambous, J. Lewis, G. Phelps, L. Sleep and D. L. Ball, Mathematical knowledge for teaching and the mathematical quality of instruction: an exploratory study, Cognition and Instruction 26(4) (2008), 430-511.
[27] V. R. Jacobs, L. L. C. Lamb and R. A. Philipp, Professional noticing of children’s mathematical thinking, Journal for Research in Mathematics Education 41 (2010), 169-202.
[28] J. Kilpatrick, J. Swafford and B. Findell, Eds., Adding it Up: Helping Children Learn Mathematics, National Academy Press, Washington, DC, 2001.
[29] M. Lampert, Teaching Problems and the Problems of Teaching, Yale University Press, New Haven, 2001.
[30] C. Lewis and I. Tsuchida, A lesson is like a swiftly flowing river: research lessons and the improvement of Japanese education, American Educator 22(4) (1998), 14-17, 50-52.
[31] P. J. Lin, Using research-based video-cases to help pre-service primary teachers conceptualize a contemporary view of mathematics teaching, International Journal of Science and Mathematics Education 3(3) (2005), 351-377.
[32] M. Markovitz and R. Even, The decimal point situation: a close look at the use of mathematics-classroom-situations in teacher education, Teaching and Teacher Education 15(6) (1999), 653-665.
[33] Z. Markovitz, Is 1¼ the consecutive of ¼? mathematics classroom situations as part of math lessons, Mathematics in School 37(1) (2008), 10-12.
[34] Z. Markovitz and M. Smith, Cases as tools in mathematics teacher education, D. Tirosh and T. Wood, eds., The International Handbook of Mathematics Teacher Education, Sense Publishers, Rotterdam, 2008, pp. 39-64.
[35] National Council of Teachers of Mathematics, Curriculum and Evaluation Standards, Reston, VA, 1989.
[36] National Council of Teachers of Mathematics, Professional Standards for The Teaching of Mathematics, Reston, VA, 1991.
[37] S. F. Ng, Ed., Cases of Mathematics Professional Development in East Asian Countries: Using Videos to Support Grounded Analysis, Springer, Singapore, 2015.
[38] J. Pang, Case-based pedagogy for prospective teachers to learn how to teach elementary mathematics in Korea, ZDM - The International Journal on Mathematics Education 43(6-7) (2011), 777-789.
[39] L. B. Resnick, P. Nesher, F. Leonard, M. Magone, S. Omanson and I. Peled, Conceptual bases of arithmetic errors: the case of decimal fractions, Journal for Research in Mathematics Education 20 (1989), 8-27.
[40] B. Rittle-Johnson and J. R. Star, Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations, Journal of Educational Psychology 99 (2007), 561-574.
[41] S. Rotem and M. Ayalon, Using critical events in preservice training: examining the coherence level between interpretations of students’ mathematical thinking and interpretations of teachers’ responses, E. Bergqvist, M. Osterholm, C. Granberg and L. Sumpter, eds., Proceeding of the 42nd Conference of the International Group for the Psychology of Mathematics Education, PME, Umea, Sweden, Vol. 4, 2018, pp. 51-58.
[42] R. Santagata and J. Guarino, Using video to teach future teachers to learn from teaching, ZDM - The International Journal on Mathematics Education 43(1) (2011), 133-145.
[43] K. Schenke and L. E. Richland, Preservice teachers’ use of contrasting cases in mathematics instruction, Instructional Science 45 (2017), 311-329.
[44] A. H. Schoenfeld and J. Kilpatrick, Toward a theory of proficiency in teaching mathematics, D. Tirosh and T. Wood, eds., International Handbook of Mathematics Teacher Education, Tools and Processes in Mathematics Teacher Education, Rotterdam, Netherlands, Sense Publishers, Vol. 2, 2008, pp. 321-354.
[45] A. Schoenfeld, Reframing teacher knowledge: a research and development agenda, ZDM - The International Journal on Mathematics Education 52 (2020), 359-376.
[46] S. L. Senk, How well do students write geometry proofs? The Mathematics Teacher 78(6) (1985), 448-456.
[47] D. Shifter, Learning to see the invisible: what skills and knowledge are needed to engage with students’ mathematical ideas? T. Wood, B. S. Nelson and J. Warfield, eds., Beyond Classical Pedagogy: Teaching Elementary School Mathematics, Erlbaum, Mahwah, NJ, 2001, pp. 109-134.
[48] L. S. Shulman, Those who understand: knowledge growth in teaching, Educational Researcher 15(2) (1986), 4-14.
[49] E. A. Silver, L. M. Clark, D. L. Gosen and V. Mills, Using narrative cases in mathematics teacher professional development: strategic selection and facilitation issues, Cases in Mathematics Teacher Education: Tools for Developing Knowledge Needed for Teaching, 2008, pp. 89-102.
[50] M. S. Smith and S. Friel, Eds., Cases in mathematics teacher education: tools for developing knowledge needed for teaching, Fourth Monograph of the Association of Mathematics Teacher Educators, Association of Mathematics Teacher Educators, San Diego, CA, 2008.
[51] M. Smith and M. K. Stein, Five practices for orchestrating productive mathematical discourse, National Council of Teachers of Mathematics, Reston, VA, 2011.
[52] M. K. Stein, R. A. Engle, M. S. Smith and E. K. Hughes, Orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell, Mathematical Thinking and Learning 10 (2008), 313-340.
[53] J. W. Stigler and J. Hiebert, The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom, The Free Press, New York, 1999.
[54] C. Suurtamm and N. Vezina, Transforming pedagogical practice in mathematics: moving from telling to listening, International Journal for Mathematics Teaching and Learning 31 (2010), 1-19.
[55] D. Tirosh, P. Tsamir, E. Levenson and R. Barkai, Using theories and research to analyze a case: learning about example use, Journal of Mathematics Teacher Education 22 (2019), 205-225.
[56] J. Visnovska, P. Cobb and C. Dean, Mathematics teachers as instructional designers: what does it take? G. Gueudet, B. Pepin and L. Trouche, eds., Mathematics Curriculum Materials and Teacher Development: From Text to ‘lived’ Resources, Springer, New York, 2011, pp. 323-341.
[57] A. E. Van Es, J. Tunnet, T. L. Goldsmith and N. Seago, A framework for the facilitation of teachers’ analysis of video, Journal of Teacher Education 65(4) (2014), 340-356.
[58] S. B. Walen and S. R. Williams, Validating classroom issues: case method in support of teacher change, Journal of Mathematics Teacher Education 3(1) (2000), 3-26.
[59] P. H. Wilson, G. F. Mojica and J. Confrey, Learning trajectories in teacher education: supporting teachers understandings of students mathematical thinking, Journal of Mathematical Behavior 32 (2013), 103-121.
[60] X. Yao, The geometric transformation and auxiliary lines in squares, Essential Readings for Junior Secondary School Students 1 (2010), 34-36.
[61] A. You, The difficulties in geometry and some solutions, Secondary School Mathematics 12 (2009), 8-9.
[62] R. Zazkis, Lesson play tasks as a creative venture for teachers and teacher educators, ZDM - The International Journal on Mathematics Education 49 (2017), 95-105.
[63] R. Zazkis and P. Herbst, Eds., Scripting Approaches in Mathematics Education: Mathematical Dialogues in Research and Practice, Springer, New York, NY, 2018.
[64] R. Zazkis and P. Liljedahl, Teaching Mathematics as Storytelling, Sense Publishers, The Netherlands, 2009.
[65] R. Zazkis, N. Sinclair and P. Liljedahl, Lesson Play in Mathematics Education: A Tool for Research and Professional Development, Springer, New York, NY, 2013.
[66] R. Zazkis and O. Marmur, Scripting tasks as a springboard for extending prospective teachers’ example spaces: a case of generating functions, Canadian J. of Science, Mathematics and Technology Education 18(4) (2018), 291-312.