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Articles

Characterizing how and when a way of proving develops in a primary mathematics classroom: a commognitive approach

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Pages 3326-3351 | Received 01 Feb 2021, Published online: 01 Jul 2021

References

  • Ahmadpour, F., Reid, D., & Fadaee, M. R. (2019). Students’ ways of understanding a proof. Mathematical Thinking and Learning, 21(2), 85–104. https://doi.org/10.1080/10986065.2019.1570833
  • Balacheff, N. (1987). Processus de preuve et situations de validation [Proving processes and situations for validation]. Educational Studies in Mathematics, 18(2), 147–176. https://doi.org/10.1007/BF00314724
  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). Hodder & Stoughton.
  • Balacheff, N. (2008). The role of the researcher’s epistemology in mathematics education: An essay on the case of proof. ZDM Mathematics Education, 40(3), 501–512. https://doi.org/10.1007/s11858-008-0103-2
  • Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the International Congress of mathematicians: Beijing 2002, August 20–28 (Vol. III, pp. 907–920). Higher Education Press.
  • Bell, A. W. (1976). A study of pupil’s proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1), 23–40. https://doi.org/10.1007/BF00144356
  • Bieda, K. N., & Lepak, J. (2014). Are you convinced? Middle-grade students’ evaluations of mathematical arguments. School Science and Mathematics, 114(4), 166–177. https://doi.org/10.1111/ssm.12066
  • Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183–203. https://doi.org/10.1007/BF00555722
  • Brown, S. (2018). Difficult dialogs about degenerate cases: A proof script study. Journal of Mathematical Behavior, 52, 61–76. https://doi.org/10.1016/j.jmathb.2018.02.002
  • Brunner, E., & Reusser, K. (2019). Types of mathematical proof: Personal preference or adaptive teaching behavior? ZDM Mathematics Education, 51(5), 747–758. https://doi.org/10.1007/s11858-019-01026-y
  • Campbell, T., Boyle, J. D., & King, S. (2019). Proof and argumentation in K-12 mathematics: A review of conceptions, content, and support. International Journal of Mathematics Education in Science and Technology, 51(5), 754–774. https://doi.org/10.1080/0020739X.2019.1626503
  • Campbell, T., King, S., & Zelkowski, J. (2020). Comparing middle grade students’ oral and written arguments. Research in Mathematics Education, 23(1), 21–38. https://doi.org/10.1080/14794802.2020.1722960
  • Common Core State Standards for School Mathematics. (2010). Common core state standards for mathematics. National Governors Association Center for Best Practices and the Council of Chief State School Officers.
  • Department for Education. (2013). Mathematics programmes of study: Key stage 3, national curriculum in England. Department for Education.
  • De Villiers, M. (1990). The role of function of proof in mathematics. Pythagoras, 24, 17–24.
  • Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (Ed.), Theorems in school: From historic, epistemology, and cognition to classroom practice (pp. 163–181). Sence.
  • Flores, A. (2006). How do students know what they learn in middle school mathematics is true? School Science and Mathematics, 106(3), 124–132. https://doi.org/10.1111/j.1949-8594.2006.tb1-8169.x
  • Fukawa-Connelly, T. (2012). Classroom sociomathematical norms for proof presentation in undergraduate in abstract algebra. Journal of Mathematical Behavior, 31(3), 401–416. https://doi.org/10.1016/j.jmathb.2012.04.002
  • Gravemeijer, K., & Prediger, S. (2019). Topic-specific design research: An introduction. In G. Kaiser, & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education, ICME-13 monographs (pp. 33–57). Springer.
  • Güçler, B. (2016). Making implicit metalevel rules of the discourse on function explicit topic of reflection in the classroom to foster student learning. Educational Studies in Mathematics, 91(3), 375–393. https://doi.org/10.1007/s10649-015-9636-9
  • Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. Bishop, M. K. Clements, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 877–908). Kluwer.
  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Issues in mathematics education: Vol. 7. Research in collegiate mathematics education III (pp. 234–282). American Mathematical Society.
  • Healy, L., & Hoyles, C. (2000). A study of proof conception in algebra. Journal for Research in Mathematics Education, 31(4), 396–428. https://doi.org/10.2307/749651
  • Hemmi, K. (2008). Students’ encounter with proof: The condition of transparency. ZDM Mathematics Education, 40(3), 413–426. https://doi.org/10.1007/s11858-008-0089-9
  • Herbst, P., & Chazan, D. (2003). Exploring the practical rationality of mathematics teaching though conversations about proving. For the Learning of Mathematics, 23(1), 2–14.
  • Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1–16. https://doi.org/10.1007/s10649-017-9761-8
  • Kieran, C. (2019). Task design frameworks in mathematics education research: An example of a domain-specific frame for algebra learning with technological tools. In G. Kaiser, & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education, ICME-13 monographs (pp. 265–287). Springer.
  • Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. The Journal of Mathematical Behavior, 29(1), 1–10. https://doi.org/10.1016/j.jmathb.2010.01.003
  • Komatsu, K. (2016). A framework for proofs and refutations in school mathematics: Increasing content by deductive guessing. Educational Studies in Mathematics, 92(2), 147–162. https://doi.org/10.1007/s10649-015-9677-0
  • Kosko, K. W. (2016). Making use of what’s given: Children’s detailing in mathematical argumentative writing. The Journal of Mathematical Behavior, 41, 68–86. https://doi.org/10.1016/j.jmathb.2015.11.002
  • Maher, C., & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214. https://doi.org/10.2307/749600
  • Mariotti, M. A. (2012). Proof and proving in the classroom: Dynamic geometry systems as tools of semiotic mediation. Research in Mathematics Education, 14(2), 163–185. https://doi.org/10.1080/14794802.2012.694282
  • Mariotti, M. A., Durand-Guerrier, V., & Stylianides, G. J. (2018). Argumentation and proof. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education (pp. 75–89). Routledge.
  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–289. https://doi.org/10.1007/BF00312078
  • Miyazaki, M., & Fujita, T. (2015). Proving as an explorative activity in mathematics education: New trends in Japanese research into proof. In B. Sriraman (Ed.), First Sourcebook on Asian Research in Mathematics Education: China, Korea, Singapore, Japan, Malaysia and India (pp. 1375–1407). Information Age Publishing.
  • Mueller, M., Yankelewitz, D., & Maher, C. (2012). A framework for analyzing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics, 80(3), 369–387. https://doi.org/10.1007/s10649-011-9354-x
  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66(1), 23–41. https://doi.org/10.1007/s10649-006-9057-x
  • Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5–29. https://doi.org/10.2307/749867
  • Reid, D. A., & Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching. Sense Publishers.
  • Reid, D. A., & Zack, V. (2009). Aspects of teaching proving in upper elementary school. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 133–146). Routledge.
  • Sasa, H., Hiwaki, M., Yamamoto, S., & Fujita, T. (2012). Students’ understanding of the generality of algebraic proofs and operative proof in secondary school mathematics education. Pre-proceedings of the 12th International Congress on Mathematical Education (pp. 2166–2174), 8th–15th July 2012, Seoul, Korea.
  • Selden, A., & Selden, J. (2013). Proof and problem solving at university level. Montana Mathematics Enthusiast, 10, 303–344. https://scholarworks.umt.edu/tme/vol10/iss1/14
  • Semadeni, Z. (1984). Action proofs in primary mathematics teaching and in teacher training. For the Learning of Mathematics, 4(1), 32–34.
  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715
  • Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint. Journal of the Learning Sciences, 16(4), 567–615. https://doi.org/10.1080/10508400701525253
  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.
  • Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70(2), 191–200. https://doi.org/10.1007/s10649-008-9161-1
  • Sfard, A. (2012). Introduction: Developing mathematical discourse—some insights from communicational research. International Journal of Educational Research, 51/52, 1–9. https://doi.org/10.1016/j.ijer.2011.12.013
  • Sfard, A. (2020). Commognition. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 95–101). Springer.
  • Shinno, Y. (2018). Reification in the learning of square roots in a ninth grade classroom: Combining semiotic and discursive approaches. International Journal of Science and Mathematics Education, 16(2), 295–314. https://doi.org/10.1007/s10763-016-9765-3
  • Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1–20. https://doi.org/10.1007/s10649-006-9038-0
  • Stylianides, A. J. (2016). Proving in the elementary mathematics classroom. Oxford University Press.
  • Stylianides, A. J., & Al-Murani, T. (2010). Can a proof and a counterexample coexist? Students’ conceptions about the relationship between proof and refutation. Research in Mathematics Education, 12(1), 21–36. https://doi.org/10.1080/14794800903569774
  • Stylianides, G. J., & Stylianides, A. J. (2008). Proof in school mathematics: Insights from psychological research into students' ability for deductive reasoning. Mathematical Thinking and Learning, 10(2), 103–133. https://doi.org/10.1080/10986060701854425
  • Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In Á Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315–351). Sense Publishers.
  • Stylianides, G. (2008). Analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
  • Stylianides, G. J., & Stylianides, A. J. (2017). Research-based interventions in the area of proof: The past, the present, and the future. Educational Studies in Mathematics, 96(2), 119–127. https://doi.org/10.1007/s10649-017-9782-3
  • Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). National Council of Teachers of Mathematics.
  • Tsujiyama, Y., Sakuma, J., & Kakinouchi, M. (2019). Mondaisettei ni okeru shomei no setsumeisei no kenzaika: Chugakkou ni okeru jissen wo toshite [Making explanatory power of proving explicit in problem-posing: Through the teaching practice in lower secondary schools]. Proceedings of the 52nd fall conference of Japan Society of Mathematical Education (pp. 423–426), 16th–17th November 2019, Tokyo, Japan. (In Japanese)
  • Wang, S., & Kinzel, M. (2014). How do they know it is a parallelogram? Analysing geometric discourse at van Hiele level 3. Research in Mathematics Education, 16(3), 288–305. https://doi.org/10.1080/14794802.2014.933711
  • Weber, K., & Alcock, L. (2004). Semantic and syntactic of proof productions. Educational Studies in Mathematics, 56(2–3), 209–234. https://doi.org/10.1023/B:EDUC.0000040410.57253.a1
  • Whitenack, J. W., & Knipping, N. (2002). Argumentation, instructional design theory and students’ mathematical learning: A case for coordinating interpretive lenses. The Journal of Mathematical Behavior, 21(4), 441–457. https://doi.org/10.1016/S0732-3123(02)00144-X
  • Widjaja, W., Vale, C., Herbert, S., Loong, E. Y.-K., & Bragg, L. A. (2020). Linking comparing and contrasting, generalising and justifying: A case study of primary students’ levels of justifying. Mathematics Education Research Journal, https://doi.org/10.1007/s13394-019-00306-w
  • Wittmann, E. C. (1996). Operative proofs in primary mathematics [Paper presentation]. 8th International Congress of Mathematics Education, 14th–21st July, Seville, Spain.
  • Wittmann, E. C. (2009). Operative proof in elementary mathematics. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. De Villiers (Eds.), Proceedings of the ICMI study 19 conference: Proof and proving in mathematics education (Vol. 2) (pp. 251–256). Department of Mathematics National Taiwan Normal University.
  • Wittmann, E. C. (2019). Understanding and organizing mathematics education as a design science – Origins and new developments. Hiroshima Journal of Mathematics Education, 12, 13–32. https://www.jasme.jp/hjme/download/02_Erich%20Ch.%20Wittmann.pdf
  • Wittmann, E. C., & Müller, G. (1990). When is a proof a proof? Bulletin de la Société Mathématique Belge Ser. A., 42(1), 15–42.