Student teachers’ conceptualizations of mathematical problem solving and the nature of their warrants

Zakaria Ndemo(1*), David K. J. Mtetwa(2)

(1) Bindura University of Science Education
(2) University of Zimbabwe
(*) Corresponding Author


Absence of inquiry about meaning of mathematical objects learners deal with has permeated the school mathematics curriculum. Deep learning through questioning situations can be achieved if learners are helped to embrace the argumentation discourse in their problem solving efforts.  This paper reports on a study that investigated linkages between the kinds of mathematical arguments constructed by students and the students’ grasp of the concept of problem solving.  With the overall goal of teasing out such linkages a questionnaire that elicited likert-style and textual data was administered to 30 undergraduate and 5 postgraduate mathematics education students. The two data sources were triangulated with reflective interviews guided by students’ written responses and their validations on likert items. Descriptive statistics were applied to likert data and directed content analysis was used to analyze and interpret the qualitative data. The study concluded that students lacked appropriate conceptions of the notion of problem solving and in particular their thinking as reflected in their arguments that contradicted current understandings on the construct. The fragile grasp of the idea of mathematical problem solving uncovered by this study has the potential to inform mathematics instruction


Mathematical problem solving, mathematical argumentation, student conceptualizations, mathematical warrants



Bleiler, S.K.,Thompson, D.R., & Krajc ̌evski, M. (2014). Providing feedback on students’

mathematical arguments: Validations of prospective secondary mathematics teachers. Journal of Mathematics Teacher Education, 17, 105-127.

Carlson, M.P., Bloom, I. (2005). The cyclic nature of problem solving: an emergent multidimensional

problem-solving-framework. Educational Studies in Mathematics, 58, 58-75.

Charmaz, K. (2014). Constructing grounded theory: A practical guide through qualitative analysis.

London: Sage.

Creswell, J.W. (2014). Research design: Qualitative, quantitative and mixed methods approaches.

London: Sage.

Dede, A.T. (2019). Arguments constructed within the mathematical modelling cycle. International

Journal of Mathematics Education in Science and Technology, 50(2), 292-314.

Driver, R., Newton, P., & Osborne, J. (2000). Establishingthe norms of scientific argumentation in

classrooms. Science Education, 84, 287-312.

Furingheti, F., & Morselli, F. (2011). Beliefs and beyond: hows and whys in the teaching of proof.

Mathematics Education, 43, 587-599.

Greiff, S.,Holt, V., & Funke, J. (2013). Perspective on problem solving in education: Analytical,

interactive and collaboration problem solving. The Journal of Problem Solving, 5(2), 71-91.

Greiger, V., & Galbraith, P. (1998). Developing a diagnostic framework for evaluating students’

approaches to applied mathematics. International Journal of Mathematics Education, Science and Technology, 29, 533-559.

Housman, D. & Porter, M. (2003). Proof schemes and learning strategies of above-average

mathematics students. Educational Studies in Mathematics, 53, 139-158.

Jonassen, D.H, & Kim, B. (2010). Arguing to learn and learning to argue: Design justifications and

guidelines. Educational Technology Research and Development, 58 (4), 439-457.

Kuhn, D. (1991). The skills argument. Cambridge: Cambridge University Press.

Lewis, J. (2009). Redefining qualitative methods: Believability of the fifth moment. International

Journal of Qualitative Methods, 8(2), 1-13.

Lawson, T. (2009). Cambridge sociology: an interview with Tony Lawson. Erasmus Journal of

Philosophy and Economics, 2(1), 100-122.

Liljedahl, P. , Santos-Trigo, M., Malaspina, U., & Bruder, R. (2016). Problem solving in mathematics

education. ICME-13 Topical surveys. Hambury: Hamburg University.

Lovett, M.C. (2002). Memory and cognitive processes. In D. Medin (Ed), Stevens handbook of

experimental psychology, 317-362.

Mamona-Downs, J., & Downs, M. (2013). Problem solving and its elements in forming proof. The

Mathematics Enthusiast, 1(1), 136-160.

Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Harlow: Pearson Prentice Hall.

Maxwell, J.A., & Mittapalli, K. (2010). Realism as a stance for mixed methods research. In A.

Tashakkori and T. Teddlie (Eds), Sage handbook of mixed methods in social behavioural research (pp.145-166). Washington DC: Sage Publications.

Maya, R., & Sumarmo, U. (2011). Mathematical understanding and proving abilities: Experiment

with undergraduate student by using modified Moore learning approach. IndoMS, Journal of Mathematics Education, 2(2), 231-250.

Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27,


Perkins, D.N., Farady, M., & Bushey, B. (1991). Everyday reasoning and the roots of intelligence. In

J.F. Voss, D.N.Perkins, & J.W. Segal (Eds.), Informal reasoning and education (pp. 83-106), Hillsdale, NJ: Erlbaum.

Punch, K.F.(2005). Introduction to social science research: Quantitative and qualitative approaches.

London: Sage.

Puri, B.K. (1996). Statistics in practice: An illustrated guide to SPSS. New York: Oxford University


Schoenfeld, A.H.(1992). Learning to think mathematically: Problem solving, metacognitionand sense-

making in mathematics. In D.A. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning, Macmillan Publishing Company, New York, pp.334-370.

Schoenfeld, A.H. (1982). Some thoughts on problem solving research and mathematics education. In

F.K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in Research. Franklin Institute Press, Philademia, pp. 27-37.

Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell

whether an argument proves a theorem? Journal for Research in Mathematics Education, 34, 4–36.

Stylianides, A.J., & Stylianides, G.J. (2009). Proof constructions and evaluations. Educational

Studies in Mathematics, 72(3), 237-253.

Toulmin, S. E. (2003). The uses of argument. Cambridge, UK: Cambridge University Press.

Ubuz, B., Dincer, S., & Bu ̈bu ̈l, A. (2013). Argumentation in undergraduate math courses: A study on

definition construction. In A.M. Lindmeier, & A. Heinze (Eds.), Proceedings of the 37th Conference of International Group for the Psychology of Mathematics Education: Vol. 4. (pp. 313-320). Kiel University: PME.

van Eemeren, F., Grootendorst, R., & Henkemans, F.S. (1996). Fundamentals of argumentation

theory: A handbook of historical backgrounds and contemporary developments. Mahwah. NJ: Erlbaum.

Varghese, T. (2009). Secondary-level teachers’ conceptions of mathematical proof. Issues in the

Undergraduate Mathematics Preparation of School Teachers: The Journal, 1, 1-13.

Vinner, S. (1997). The pseudo-conceptual and pseudo-analytical thought processes in mathematics

learning. Educational Studies in Mathematics, 34, 97-129.

Walton, D.N. (1996). Argumentation schemes for presumptive reasoning. Mahwah NJ: Erlbaum


Weber, K. (2001). Student difficulty in constructing proofs. The need for strategic knowledge.

Educational Studies in Mathematics, 48, 101-119.

Weber, K., & Mejia-Ramos, J.P. (2015). On relative and absolute conviction in mathematics. For the

Learning of Mathematics, 35(2), 3-18.

Article Metrics

Abstract view(s): 518 time(s)


  • There are currently no refbacks.