Educational Studies in Mathematics

, Volume 96, Issue 2, pp 169–186 | Cite as

Enhancing students’ mathematical reasoning in the classroom: teacher actions facilitating generalization and justification

  • Joana Mata-PereiraEmail author
  • João-Pedro da Ponte


A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students’ mathematical reasoning, we conduct design research where whole-class mathematical discussions triggered by exploratory tasks play a key role. We take mathematical reasoning as making justified inferences and we consider generalizing and justifying central reasoning processes. Regarding teacher actions, we consider inviting, informing/suggesting, supporting/guiding and challenging actions can be identified in whole-class discussions. This paper presents design principles for an intervention geared to tackle such reasoning processes and focuses on a whole-class discussion on a grade 7 lesson about linear equations and functions. Data analysis concerns teacher actions in relation to design principles and to the sought mathematical reasoning processes. The conclusions highlight teacher actions that lead students to generalize and justify. Generalizations may arise from a central challenging action or from several guiding actions. Regarding justifications, a main challenging action seems to be essential, while follow-up guiding actions may promote a further development of this reasoning process. Thus, this paper provides a set of design principles and a characterization of teacher actions which enhance students’ mathematical reasoning processes such as generalization and justification.


Mathematical reasoning Teacher actions Generalization Justification Design research 



This work is supported by national funds through FCT – Fundação para a Ciência e Tecnologia by a grant to Joana Mata-Pereira (SFRH/BD/94928/2013).


  1. Becker, J. R., & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th PME conference (vol. 4, pp. 121–128). Melbourne: PME.Google Scholar
  2. Ball, D., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). Reston, VA: NCTM.Google Scholar
  3. Boaler, J. (2010). The road to reasoning. In K. Brodie (Ed.), Teaching mathematical reasoning in secondary school classrooms (pp. v–vii). New York: Springer.Google Scholar
  4. Brodie, K. (2010). Teaching mathematical reasoning in secondary school classrooms. New York: Springer.CrossRefGoogle Scholar
  5. Brousseau, G., & Gibel, P. (2005). Didactical handling of students’ reasoning processes in problem solving situations. Educational Studies in Mathematics. doi: 10.1007/s10649-005-2532-y
  6. Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization. ZDM. doi: 10.1007/s11858-007-0067-7
  7. Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 243–307). Dordrecht: D. Reidel.CrossRefGoogle Scholar
  8. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher. doi: 10.3102/0013189X032001009
  9. Dörfler, W. (1991). Forms and means of generalization in mathematics. In A. Bishop, S. Mellin-Olsen, & J. van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 63–85). Dordrecht: Kluwer.Google Scholar
  10. Francisco, J. M., & Maher, C. A. (2011). Teachers attending to students’ mathematical reasoning: Lessons from an after-school research program. Journal of Mathematics Teacher Education. doi: 10.1007/s10857-010-9144-x
  11. Franke, M. L., Webb, N., Chan, A., Ing, M., Freund, D., & Battey, D. (2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education. doi: 10.1177/0022487109339906
  12. Jahnke, H. N., & Wambach, R. (2013). Understanding what a proof is: A classroom-based approach. ZDM Mathematics Education. doi: 10.1007/s11858-013-0502-x
  13. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Reston: NCTM.Google Scholar
  14. Kosko, K., Rougee, A., & Herbst, P. (2014). What actions do teachers envision when asked to facilitate mathematical argumentation in the classroom? Mathematics Education Research Journal. doi: 10.1007/s13394-013-0116-1
  15. Krussel, L., Edwards, B., & Springer, G. T. (2004). The teacher’s discourse moves: A framework for analyzing discourse in mathematics classrooms. School Science and Mathematics. doi: 10.1111/j.1949-8594.2004.tb18249.x
  16. Lannin, J., Ellis, A. B., & Elliot, R. (2011). Developing essential understanding of mathematics reasoning for teaching mathematics in prekindergarten-grade 8. Reston: NCTM.Google Scholar
  17. Pólya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. I). Princeton: Princeton University Press.Google Scholar
  18. Ponte, J. P. (2005). Gestão curricular em Matemática. In GTI (Ed.), O professor e o desenvolvimento curricular (pp. 11–34). Lisboa: APM.Google Scholar
  19. Ponte, J. P., & Quaresma, M. (2016). Teachers’ professional practice conducting mathematical discussions. Educational Studies in Mathematics. doi: 10.1007/s10649-016-9681-z
  20. Reid, D. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5–29.CrossRefGoogle Scholar
  21. Rivera, F., & Becker, J. (2009). Algebraic reasoning through patterns. Mathematics Teacher in the Middle School, 15(4), 213–221.Google Scholar
  22. Ruthven, K. (1989). An exploratory approach to advanced mathematics. Educational Studies in Mathematics. doi: 10.1007/BF00315610
  23. Sherin, M. (2002). A balancing act: Developing a discourse community in a mathematics classroom. Journal of Mathematics Teacher Education. doi: 10.1023/A:1020134209073
  24. Stein, M. K., Engle, R., Smith, M., & Hughes, E. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning. doi: 10.1080/10986060802229675
  25. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.Google Scholar
  26. Sowder, L., & Harel, G. (1998). Types of students’ justifications. Mathematics Teacher, 91(8), 670–675.Google Scholar
  27. Wood, T. (1999). Creating a context for argument in mathematics class. Journal for Research in Mathematics Education, 30(2), 171–191.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Instituto de EducaçãoUniversidade de Lisboa, Alameda da UniversidadeLisboaPortugal

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