Elementary school teachers’ noticing of essential mathematical reasoning forms: justification and generalization

  • Kathleen MelhuishEmail author
  • Eva Thanheiser
  • Layla Guyot


Justifying and generalizing are essential forms of mathematical reasoning, yet, teachers struggle both to produce and identify justifications and generalizations. In comparing elementary school teachers’ self-reported levels of noticing justifying and generalizing in their own classrooms and the levels researchers observed in two consecutive lessons in those classrooms, we found significant discrepancies. In applying a framework we developed to characterize the teachers’ noticing in terms of mathematical content and reasoning form, we found that teachers rarely attended to justifying and generalizing in a manner consistent with the mathematics education community’s view and that their lenses for noticing these activities may account for discrepancies between the teachers’ reports and the researchers’ observations. We conclude by reflecting on the complexity of asking teachers to attend to justifying and generalizing and how these results may affect teacher professional development.


Justifying Generalizing Noticing Professional development 



The research study and preparation of this manuscript was supported by a grant from the National Science Foundation (NSF) (DRL-1223074).


  1. Belton, R. (1996). Art history: A preliminary handbook. Vancouver: University of British Columbia.Google Scholar
  2. Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers’ algebra eyes and ears. Teaching Children Mathematics, 10(2), 70–78.Google Scholar
  3. Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446.Google Scholar
  4. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. Scholar
  5. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 37(2), 87–115.Google Scholar
  6. Cirillo, M., Kosko, K. W., Newton, J., Staples, M., & Weber, K. (2015). Conceptions and consequences of what we call argumentation, justification, and proof. In T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American chapter of the psychology of mathematics education (pp. 1343–1351). East Lansing: Michigan State University.Google Scholar
  7. Cohen, D. K. (1990). A revolution in one classroom: the case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12(3), 311–329. Scholar
  8. Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.Google Scholar
  9. Foreman, L. C. (2013). Best practices in teaching mathematics: How math teaching matters. West Linn, OR: Teachers Development Group.Google Scholar
  10. Hill, H. C. (2014). Mathematical quality of instruction (MQI) [Coding tool]. Cambridge, MA: Harvard Graduate School of Education. Retrieved from Accessed 1 Jne 2014.
  11. Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.Google Scholar
  12. Jacobs, V. R., & Spangler, D. A. (2017). Research on core practices in K-12 mathematics teaching. In J. Cai (Ed.), Compendium for research in mathematics education. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  13. Kirwan, J. V. (2015). Preservice secondary mathematics teachers’ knowledge of generalization and justification on geometric-numerical patterning tasks. Normal: Illinois State University, ProQuest Dissertations.Google Scholar
  14. Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405. Scholar
  15. Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. Scholar
  16. Lo, J. J., & McCrory, R. (2010). Teaching teachers through justifying activities. Teaching Children Mathematics, 17(3), 149–155.Google Scholar
  17. Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51. Scholar
  18. Mason, J. (2002). Researching your own practice: The discipline of noticing. London: Routledge.Google Scholar
  19. Melhuish, K. M., & Thanhesier, E. (2017). Using formative evaluation to support teachers in increasing student reasoning. In L. West & M. Boston (Eds.), Annual perspectives in mathematics education 2017: Reflective and collaborative processes to improve mathematics teaching (pp. 183–199). National Council of Teachers of Mathematics.Google Scholar
  20. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.Google Scholar
  21. National Governors Association. (2010). Standards for mathematical practice. Retrieved from Accessed 1 May 2018.
  22. National Research Council and Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.Google Scholar
  23. Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3–31. Scholar
  24. Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447–462. Scholar
  25. Star, J. R., & Strickland, S. K. (2008). Learning to observe: Using video to improve preservice mathematics teachers’ ability to notice. Journal of Mathematics Teacher Education, 11(2), 107–125. Scholar
  26. Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321.Google Scholar
  27. Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.Google Scholar
  28. Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352.Google Scholar
  29. Szydlik, J. E., & Seaman, C. E. (2012). Prospective elementary teachers’ evolving meanings for generalizing, doing mathematics and justifying. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 15th annual conference on research in undergraduate mathematics education (pp. 1–32). Portland, OR: RUME.Google Scholar
  30. Teachers Development Group. (2010). About the mathematics studio program: Transforming a school’s culture of mathematics professional learning. Teachers Development Group. Retrieved From Accessed 29 Aug 2016.
  31. van Es, E. A. (2011). A framework for learning to notice student thinking. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes. New York: Routledge.Google Scholar
  32. van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24(2), 244-276.CrossRefGoogle Scholar
  33. van Es, E. A., & Sherin, M. G. (2010). The influence of video clubs on teachers’ thinking and practice. Journal of Mathematics Teacher Education, 13(2), 155–176. Scholar
  34. Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402. Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentTexas State UniversitySan MarcosUSA
  2. 2.Fariborz Maseeh Department of Mathematics + StatisticsPortland State UniversityPortlandUSA

Personalised recommendations