The Korea Society Of Educational Studies In Mathematics
[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.135-152
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020

Exploring Mathematical Reasoning of Elementary Preservice Teachers

GwiSoo Na* ; Dong-Won Kim*,
*Professor, Cheongju National University of Education, South Korea

Correspondence to: Email:,

Please cite this article as: Na, G. & Kim, D. Exploring mathematical reasoning of elementary preservice teachers.


The purpose of this study is to understand the characteristics of mathematical reasoning of elementary preservice teachers (EPTs). For this purpose, 68 EPTs were presented with two tasks related to mathematical reasoning, and their written responses were analyzed. The EPTs’ mathematical reasonings were investigated on the one hand as a whole and on the other hand between the three groups -the full successful, partial successful, and unsuccessful provers-, with focusing on the affordances from and strategies for using examples proposed in the CAPS framework. As the results of this study, it was revealed that the EPTs had insufficient competencies in terms of exploring with examples, finding structural features, building generalizations, developing conjectures, and producing proofs. In contrast, the EPTs had great strength with regard to representing their chosen examples in formal expression, which was a decisive contributor on the one hand but an impediment on the other in producing valid justifications to the given conjectures. Based on the results, the implications for elementary preservice teacher education were suggested.


Mathematical reasoning, Preservice elementary teacher, Example, Conjecture, Structure, Generalization, Justification, Proof


  • Artzt, A. F. (1999). Mathematical reasoning during small-group problem solving. In L.V. Stiff (Ed.), Developing mathematical reasoning in grades K-12 (1999 Yearbook, pp. 115–127). Reston, VA: NCTM.
  • Balacheff, N. (1988). Aspects of proof in pupil’s practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children. London: The Open University, 216-235.
  • Bieda, K., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71–80. []
  • Bills, L., & Rowland, T. (1999). Examples, generalization and proof. In L. Brown (Ed.), Making meaning in mathematics. Advances in mathematics education (pp. 103–116). York, UK: QED. []
  • CCSSI (2010). Common core state standards for mathematics. Common Core State Standards Initiative,
  • DFE (2013). The national curriculum in England: Key stages 3 and 4 framework. Department for Education: Retrieved from, .
  • Ellis, A., Bieda, K., & Knuth, E. (2012). Developing essential understanding of proof and proving for teaching mathematics in grades 9–12. Reston, VA: National Council of Teachers of Mathematics.
  • Ellis, A., Ozgur, Z., Vinsonhaler, R., Dogan, M. F., Carolan, T., Lockwood, T., Lynch, A., Sabouri, P., Knuth, E., & Zaslavsky, O. (2017). Student thinking with examples: The criteria-affordances-purposes-strategies framework. Journal of Mathematical Behavior []
  • Furinghetti, F., & Morselli, F. (2009). Teachers’ beliefs and the teaching of proof. In Fou-Lai Lin, Feng-Jui Hsieh, Gila Hanna, & Michael de Villiers (Eds.), Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education. Taipei, Taiwan: National Taiwan Normal University, 166-171
  • Goetting, M. (1995). The college students’ understanding of mathematical proof (Doctoral dissertation, University of Maryland). Dissertations Abstracts International, 56, 3016A.
  • Hanna, G. (2000). Proof: explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23. []
  • Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 185–212). Norwood, NJ: Ablex Publishing Corporation.
  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Washington, DC: Mathematical Association of America. []
  • Herberta, S., Valeb, C., Braggb, L. A., Loongb, E., & Widjajab, W. (2015). A framework for primary teachers’ perceptions of mathematical reasoning. International Journal of Educational Research, 74, 26-37. []
  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428. []
  • Inhelder, B. & Piaget, J. (1958). The growth of logical thinking: from childhood to adolescence. London: Routledge & Kegan Paul Ltd. []
  • Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96, 1–16. []
  • KMOE (2015). 2015 revised Korea mathematics curriculum. Korea Ministry of Education.
  • Knuth, E. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405. []
  • Knuth, E. (2002b). Teachers’ conceptions of proof in the context of secondary school Mathematics. Journal of Mathematics Teacher Education, 5, 61-88.
  • Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K–16 perspective (pp. 153–170). New York, NY: Routledge. []
  • Knuth, E., Zaslavsky, O., & Ellis, A. (2017). The role and use of examples in learning to prove. Journal of Mathematical Behavior []
  • Lannin, J., Ellis A., Elliot, R., & Zbiek, R. M. (2011). Developing essential understanding of mathematical reasoning for teaching mathematics in prekindergarten-grade 8. Reston, VA: National Council of Teachers of Mathematics.
  • Lesseig, K. (2016). Investigating mathematical knowledge for teaching proof in professional development. International Journal of Research in Education and Science, 2(2), 253-270. []
  • Lesseig, K., Hine, G., Na, G., & Boardman, K. (2019). Perceptions on proof and the teaching of proof: a comparison across preservice secondary teachers in Australia, USA and Korea. Mathematics Education Research Journal, 31, 393–418. []
  • Martin, W. & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51. []
  • Melhuish, K., Thanheiser, E., & Guyot, L. (2020). Elementary school teachers’ noticing of essential mathematical reasoning forms: justification and generalization. Journal of Mathematics Teacher Education, 23, 35-67. []
  • MES (2013). Secondary mathematics syllabuses. Ministry of Education Singapore: Retrieved from, .
  • Na, G. (2014) Mathematics teachers’ conceptions of proof and proof-instruction. Communications of Mathematical Education, 28, 513-528. []
  • Na, G. (2009). Teaching geometry proof with focus on the analysis-method. Journal of Educational Research in Mathematics, 19, 185-206.
  • NCTM (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  • Peretz, D. (2006). Enhancing reasoning attitudes of prospective elementary school mathematics teachers. Journal of Mathematics Teacher Education, 9, 381-400. []
  • Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3-31. []
  • Stylianides, A. J. (2016). Proving in the elementary mathematics classroom. Oxford University Press. []
  • Stylianides, A. J., & Stylianides, G. J. (2006). Content knowledge for mathematics teaching: the case of reasoning end proving. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 5, 201-208. Prague: PME.
  • Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
  • Stylianides, G. J., & Silver, E. A. (2009). Reasoning-and-proving in school mathematics. In Stylianou, D. A., Blanton, M. L., & Knuth, J. (Eds.), Teaching and Learning Proof Across the Grades: A K-16 Perspective (pp. 235-249). New York, NY: Routledge. []
  • Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10(3),145-166. []
  • Stylianides, G. J., Stylianides, A. J., & Shilling-Traina, L. N. (2013). Prospective teachers’ challenges in teaching reasoning-and-proving. International Journal of Science and Mathematics Education, 11, 1463–1490. []
  • Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Shifter (Eds.), A research companion to the Principles and standards for school mathematics (pp. 227-236). Reston, VA: National Council of Teachers of Mathematics.