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**Teachers’ professional learning of teaching proof and proving.**
*(English)*
Zbl 1247.97016

Hanna, Gila (ed.) et al., Proof and proving in mathematics education. The 19th ICMI study. Berlin: Springer (ISBN 978-94-007-2128-9/hbk; 978-94-007-2129-6/ebook). New ICMI Study Series 15, 327-346 (2012).

Summary: This paper reviews studies on teachers’ professional learning of teaching proof and proving. From them we conceptualise three essential components of successful teaching: teachers’ knowledge of proof, proof practices and beliefs about proof. With respect to each component, we examine research studies of primary and secondary teachers. We also discuss the challenges teachers may face in teaching proof and proving, as well as teachers’ professional learning activities. Throughout, we argue that the three components are interrelated in successful teaching of proof and proving. This argument raises a new challenge for further research.

For the entire collection see [Zbl 1234.00015].

For the entire collection see [Zbl 1234.00015].

### MSC:

97E50 | Reasoning and proving in the mathematics classroom |

97B50 | Mathematics teacher education |

97C70 | Teaching-learning processes in mathematics education |

### Keywords:

teaching proof and proving; professional learning of teaching proof; teachers’ knowledge of proof; teachers’ proof practices; teachers’ beliefs about proof
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\textit{F.-L. Lin} et al., New ICMI Stud. Ser. 15, 327--346 (2012; Zbl 1247.97016)

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### References:

[1] | Ball, DL; Bass, H.; Kilpatrick, J.; Martin, WG; Schifter, D., Making mathematics reasonable in school, A research companion to principle and standards for school mathematics, 27-44 (2003), Reston: National Council of Teachers of Mathematics, Reston |

[2] | Ball, DL; Thames, MH; Phelps, G., Content knowledge for teaching: What makes it special?, Journal of Teacher Education, 59, 5, 389-407 (2008) |

[3] | Barkai, R., Tabach, M., Tirosh, D., Tsamir, P., & Dreyfus, T. (2009, February). Modes of argument representation for proving - The case of general proof. Paper presented at the Sixth Conference of European Research in Mathematics Education - CERME 6, Lyon, France. |

[4] | Berg, C. V. (2009). A contextualized approach to proof and proving in mathematics education: Focus on the nature of mathematical tasks (Vol. 1, pp. 100-105). |

[5] | Bieda, K. N. (2009). Enacting proof in middle school mathematics (Vol. 1, pp. 59-64). |

[6] | Bishop, AJ, Mathematics teaching and values education: An intersection in need of research, Zentralblattfuer Didaktik der Mathematik, 31, 1-4 (1999) |

[7] | Biza, I., Nardi, E., & Zachariades, T. (2009). Do images disprove but do not prove? Teachers’ beliefs about visualization (Vol. 1, pp. 59-64). |

[8] | Brown, J., & Stillman, G. (2009). Preservice secondary teachers’ competencies in proof (Vol. 1, pp. 94-99). |

[9] | Chin, C.; Lin, FL, A case study of a mathematics teacher’s pedagogical values: Use of a methodological framework of interpretation and reflection, Proceedings of the National Science Council, Taiwan, Part D: Mathematics, Science, and Technology Education, 10, 2, 90-101 (2000) |

[10] | Cirillo, M. (2009). Challenges to teaching authentic mathematical proof in school mathematics (Vol. 1, pp. 130-135). |

[11] | Cobb, P.; Confrey, J.; DiSessa, A.; Lehrer, R.; Schauble, L., Design experiments in educational research, Educational Researcher, 32, 1, 9-13 (2003) |

[12] | Coe, R.; Ruthven, K., Proof practices and constructs of advanced mathematics students, British Educational Research Journal, 20, 41-53 (1994) |

[13] | Douek, N. (2009). Approaching proof in school: From guided conjecturing and proving to a story of proof construction (Vol. 1, pp. 142-147). |

[14] | Dreyfus, T.; Gagatsis, A., Some views on proofs by teachers and mathematicians, Proceedings of the 2nd Mediterranean Conference on Mathematics Education, 11-25 (2000), Nicosia: The University of Cyprus, Nicosia |

[15] | Engeström, Y.; Engeström, Y.; Miettinen, R.; Punamaki, RL, Activity theory and individual and social transformation, Perspectives on activity theory, 19-38 (1999), Cambridge: Cambridge University Press, Cambridge |

[16] | Fischbein, E., Intuition and proof, For the Learning of Mathematics, 3, 2, 9-18, 24 (1982) |

[17] | Furinghetti, F., & Morselli, F. (2009). Teacher’s beliefs and the teaching of proof (Vol. 1, 166-171). |

[18] | Goetting, M. (1995). The college students’ understanding of mathematical proof. Unpublished doctoral dissertation, University of Maryland, College Park. |

[19] | Goulding, M.; Rowland, T.; Barber, P., Does it matter? Primary teacher trainees’ knowledge in mathematics, British Educational Research Journal, 28, 689-704 (2002) |

[20] | Grant, TJ; Lo, J.; Clarke, B.; Millman, R.; Grevholm, B., Reflecting on the process of task adaptation and extension: The case of computational starters, Effective tasks in primary mathematics teacher education, 25-36 (2008), New York: Springer Science + Business Media, LLC, New York |

[21] | Hanna, G., Challenges to the importance of proof, For the Learning of Mathematics, 15, 3, 42-49 (1995) |

[22] | Hanna, G., Proof, explanation and exploration: An overview, Educational Studies in Mathematics, 44, 5-23 (2000) |

[23] | Harel, G., Two dual assertions: The first on learning and the second on teaching (or vice versa), The American Mathematical Monthly, 105, 497-507 (1998) · Zbl 0942.68652 |

[24] | Harel, G.; Campbell, S.; Zazkis, R., The development of mathematical induction as a proof scheme: A model for DNR-based instruction, Learning and teaching number theory (2001), Norwood: Ablex Publishing Corporation, Norwood |

[25] | Harel, G.; Sowder, L.; Schoenfeld, AH; Kaput, J.; Dubinsky, E., Students’ proof schemes: Results from exploratory studies, Research in collegiate mathematics education III, 234-283 (1998), Providence: American Mathematical Society, Providence |

[26] | Harel, G.; Sowder, L.; Lester, FK, Toward comprehensive perspectives on the learning and teaching of proof, Second handbook of research on mathematics teaching and learning, 805-842 (2007), Greenwich: Information Age, Greenwich |

[27] | Healy, L.; Hoyles, C., Justifying and proving in school mathematics (1998), London: Institute of Education, University of London, London |

[28] | Healy, L.; Hoyles, C., A study of proof conception in algebra, Journal for Research in Mathematics Education, 31, 396-428 (2000) |

[29] | Healy, L., Jahn, A. P., & Frant, J. B. (2009). Developing cultures of proof practices amongst Brazilian mathematics teachers (Vol. 1, pp. 196-201). |

[30] | Herbst, P., Engaging students in proving: A double bind on the teacher, Journal for Research in Mathematics Education, 33, 3, 176-203 (2002) |

[31] | Herbst, P. (2009). Testing a model for the situation of “doing proofs” using animations of classroom scenarios (Vol. 1, pp. 190-195). |

[32] | Hsieh, CJ, On misconceptions of mathematical induction, Letter of the History and Pedagogy of Mathematics in Taiwan (HPM), 8, 2-3, 14-21 (2005) |

[33] | Kaiser, G.; Schwarz, B.; Krackowitz, S., The role of beliefs on future teachers’ professional knowledge, The Montana Mathematics Enthusiast, Monograph, 3, 99-116 (2007) |

[34] | Knuth, EJ, Teachers’ conceptions of proof in the context of secondary school mathematics, Journal of Mathematics Teacher Education, 5, 61-88 (2002) |

[35] | Kunimune, S., Fujita, T., & Jones, K. (2009). “Why do we have to prove this?” Fostering students’ understanding of ‘proof’ in geometry in lower secondary school (Vol. 1, pp. 256-261). |

[36] | Lampert, M., Teaching problems and the problems of teaching (2001), New Haven: Yale University Press, New Haven |

[37] | Lesh, R.; Hamilton, E.; Kaput, J.; Lesh, R.; Hamilton, E.; Kaput, J., Directions for future research, Foundations for the future in mathematics education, 449-453 (2007), Mahweh: Lawrence Erlbaum Associates, Mahweh |

[38] | Lin, FL; Pinto, MMF; Kawasaki, TF, Mathematical tasks designing for different learning settings, Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, 83-96 (2010), Belo Horizonte: PME, Belo Horizonte |

[39] | Lo, J., & McCrory, R. (2009). Proof and proving in mathematics for prospective elementary teachers (Vol. 2, pp. 41-46). |

[40] | Lo, J.; Grant, T.; Flowers, J., Challenges in deepening prospective teachers’ understanding of multiplication through justification, Journal of Mathematics Teacher Education, 11, 5-22 (2008) |

[41] | Martin, WG; Harel, G., Proof frames of preservice elementary teachers, Journal for Research in Mathematics Education, 20, 41-51 (1989) |

[42] | Mason, J.; Burton, L.; Stacey, K., Thinking mathematically (1982), London: Addison-Wesley, London |

[43] | Mingus, T.; Grassl, R., Preservice teacher beliefs about proofs, School Science and Mathematics, 99, 8, 438-444 (1999) |

[44] | National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Virginia: National Council of Teachers of Mathematics. |

[45] | National Council of Teachers of Mathematics [NCTM], Principles and standards for school mathematics (2000), Reston: National Council of Teachers of Mathematics, Reston |

[46] | Philip, R.; Lester, FK, Mathematics teachers’ beliefs and affect, Second handbook of research on mathematics teaching and learning, 257-318 (2007), Charlotte: Information Age, Charlotte |

[47] | Raymond, AM, Inconsistency between a beginning elementary school teacher’s mathematics beliefs and teaching practice, Journal for Research in Mathematics Education, 28, 550-576 (1997) |

[48] | Schoenfeld, AH; Green, JL; Camilli, G.; Elmore, PB; Skukauskaite, A.; Grace, E., Design experiments, Handbook of complementary methods in education research, 193-205 (2006), Washington, DC: American Educational Research Association, Washington, DC |

[49] | Schwarz, B., & Kaiser, G. (2009). Professional competence of future mathematics teachers on argumentation and proof and how to evaluate it (Vol. 2, pp. 190-195). |

[50] | Shulman, LS, Those who understand: Knowledge growth in teaching, Educational Researcher, 15, 2, 4-14 (1986) |

[51] | Simon, MA; Blume, GW, Justification in the mathematics classroom: A study of prospective elementary teachers, The Journal of Mathematical Behavior, 15, 3-31 (1996) |

[52] | Stylianides, AJ, Proof and proving in school mathematics, Journal for Research in Mathematics Education, 38, 289-321 (2007) |

[53] | Stylianides, AJ, The notion of proof in the context of elementary school mathematics, Educational Studies in Mathematics, 65, 1-20 (2007) |

[54] | Stylianides, A.; Ball, D., Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving, Journal of Mathematics Teacher Education, 11, 307-332 (2008) |

[55] | Stylianides, AJ; Stylianides, GJ, Proof constructions and evaluations, Educational Studies in Mathematics, 72, 2, 237-253 (2009) |

[56] | Stylianides, GJ; Stylianides, AJ, Facilitating the transition from empirical arguments to proof, Journal for Research in Mathematics Education, 40, 314-352 (2009) |

[57] | Stylianides, AJ; Stylianides, GJ; Philippou, GN, Undergraduate students’ understanding of the contraposition equivalence rule in symbolic and verbal contexts, Educational Studies in Mathematics, 55, 133-163 (2004) |

[58] | Stylianides, GJ; Stylianides, AJ; Philippou, GN, Preservice teachers’ knowledge of proof by mathematical induction, Journal of Mathematics Teacher Education, 10, 145-166 (2007) |

[59] | Sun, X. (2009). Renew the proving experiences: An experiment for enhancement trapezoid area formula proof constructions of student teachers by “one Problem Multiple Solutions” (Vol. 2, pp. 178-183). |

[60] | Sun, X., & Chan, K. (2009). Regenerate the proving experiences: An attempt for improvement original theorem proof constructions of student teachers by using spiral variation curriculum (Vol. 2, pp. 172-177). |

[61] | Tabach, M., Levenson, E., Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2009). Teachers’ knowledge of students’ correct and incorrect proof constructions (Vol. 2, pp. 214-219). |

[62] | Tall, D., Inconsistencies in the learning of calculus and analysis, Focus on Learning Problems in Mathematics, 12, 3-4, 49-63 (1990) |

[63] | Tirosh, D., Inconsistencies in students’ mathematical constructs, Focus on Learning Problems in Mathematics, 12, 3-4, 111-129 (1990) |

[64] | Tirosh, D.; Stavy, R.; Cohen, S., Cognitive conflict and intuitive rules, International Journal of Science Education, 20, 10, 1257-1269 (1998) |

[65] | Tsamir, P., Tirosh, D., Dreyfus, T., Barkai, R., & Tabach, M. (2008). Inservice teachers’ judgment of proofs in ENT. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sépulveda (Eds.), Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 345-352). Morélia: PME. |

[66] | Tsamir, P.; Tirosh, D.; Dreyfus, T.; Barkai, R.; Tabach, M., Should proof be minimal? Ms T’s evaluation of secondary school students’ proofs, Journal of Mathematics Behavior, 28, 58-67 (2009) |

[67] | Tsamir, P., Tirosh, D., Dreyfus, T., Tabach, M., & Barkai, R. (2009b). Is this verbal justification a proof? (Vol. 2, pp. 208-213). |

[68] | Vinner, S., Inconsistencies: Their causes and function in learning mathematics, Focus on Learning Problems in Mathematics, 12, 3-4, 85-98 (1990) |

[69] | Wittmann, E. (2009). Operative proof in elementary mathematics (Vol. 2, pp. 251-256). |

[70] | Wood, T., Creating a context for argument in mathematics class, Journal for Research in Mathematics Education, 30, 171-191 (1999) |

[71] | Yackel, E., What we can learn from analyzing the teacher’s role in collective argumentation, The Journal of Mathematical Behavior, 21, 423-440 (2002) |

[72] | Yackel, E.; Cobb, P., Sociomathematical norms, argumentation, and autonomy in mathematics, Journal for Research in Mathematics Education, 27, 458-477 (1996) |

[73] | Zaslavsky, O., Seizing the opportunity to create uncertainty in learning mathematics, Educational Studies in Mathematics, 60, 297-321 (2005) |

[74] | Zaslavsky, O.; Peled, I., 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 (1996) |

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