Towards a Comprehensive Conception of Mathematical Proof
Abstract
There is overwhelming evidence that students face serious challenges in learning mathematical proof. Studies have found that students possess a superficial understanding of mathematical proof. With the aim of contributing to efforts intended to develop a comprehensive conception of mathematical proof, literature search was conducted to identify areas where research could be directed in order to increase proof understanding among students. To accomplish this goal, literature on modes of reasoning involved in proof construction, ideas on the classification of activities that constitute a proof path, and categories of proof understanding are exemplified using mathematical content drawn from Real Analysis. These exemplifications were used to illustrate the connections between modes of reasoning and levels of proof understanding. With regard to students’ fragile grasp of mathematical proof this critique of literature has revealed that many previous studies have given prominence to proof validations while there is lack of crucial interplay between structural and inductive modes of reasoning during proving by students. Hence, it is suggested in this paper that current research could also focus on mechanisms that promote an analytic conceptions of mathematical proof that are comprehensive enough to allow students to engage in more robust proof constructions.
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K. Lesseig, “Investigating mathematical knowledge for teaching proof in professional development,” International Journal of Research in Education and Science, vol. 2(2), pp. 253-270, 2016.
T. Cadawallader Olsker, “What do we mean by mathematical proof?”, Journal of Humanistic Mathematics, vol. 1, pp: 1-33, 2011.
A.J. Stylianides, “Towards a comprehensive knowledge package for teaching proof: A focus on the misconception that empirical arguments are proofs,” Pythagoras, vol. 32(1), pp. 1-14, 2011.
H.N. Jahnke, “Proofs and hypotheses,” ZDM: The International Journal on Mathematics Education, vol. 39, pp. 79-88, 2007.
I. Ug ̌urel, S. Morali, K. Yig ̌it and O. Karahan, “Pre-service secondary mathematicsteachers’ behaviors in the proving process,” Eurasian Journal of Mathematics, Science and Technology Education, vol. 12(2), pp. 203-231, 2016.
G. Harel and L. Sowder, “Toward a comprehensive perspective on proof,” In Lester (ed), Second Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics, 2007.
A.H. Schoenfeld, “Mathematical problem solving,” Orlando, Fla: Academic Press, 1985.
Z. Ndemo and D.K. Mtetwa, “Negotiating the transition from secondary to undergraduate mathematics: Reflections by some Zimbabwean students,” Middle Eastern and African Journal of Educational Research, vol. 14, pp. 67-78, 2015.
G. Harel and L Sowder, “Students proof schemes: Results from exploratory studies,” In Schoenfeld, Kaput and Dubinsky (Eds). Research in Collegiate Mathematics Education III, pp. 234-282, 1998.
M. Wilkerson-Jerde and U.J. Wilensky, “How do mathematicians learn math?: resources and acts for constructing and understanding mathematics,” Educational Studies in Mathematics, vol. 78, pp. 21-43, 2011.
J. Duffin and A. Simpson, “A search for understanding,” The Journal of Mathematical Behaviour, vol. 18(4), pp. 415-427, 2000.
K. Charmaz, “Constructing grounded theory: A practical guide through qualitative analysis,” London: Sage, 2006.
K. Pfeiffer, “The role of validation in students’ mathematical learning,” In Joubert ed: Proceedings of the British Society for Research into Learning Mathematics, vol. 29(3), pp. 79-84, 2009.
G. Martin and G. Harel, “Proof frames of pre-service elementary teachers.” Journal for Research in Mathematics Education, 29, pp. 41-51, 1989.
E. Herlina and S. Batusangkar, “Advanced mathematical thinking and the way to enhance it,” Journal of Education and Practice, vol. 6(5), pp. 79-88, 2015.
K. Weber and J. Mejia-Ramos, “Why and how do mathematicians read proofs: an exploratory study,” Educational Studies in Mathematics, vol. 76, pp. 329-344, 2011.
K.N. Bieda, “Enacting proof-related in middle school mathematics: Challenges and opportunities,” Journal for Research in Mathematics Education, vol.41, pp. 351-382, 2010.
J.D. Bostic, “Fostering justification: A case study of preservice teachers,” proof-related tasks, and manipulatives. Journal of Mathematics Education at teachers’ college, vol, 7(1), pp. 35-42, 2015.
N. Balacheff, “Aspects of Proof in pupils' practice of school mathematics,” In D. Pimm ed: Mathematics, teachers and children. London: Hodder & Stoughton, pp. 216-235, 1988.
Z.B. Ersen, “Pre-service mathematics teachers” metaphorical perceptions towards proof and proving,” International Education Studies, vol. 9(7), pp. 88-97, 2016.
D. Stylianou, M.L. Blanton and O. Rotou, “Undergraduate students’ understanding of proof: relationships between proof conceptions, beliefs, and classroom experiences,” International Journal of Research in Mathematics Education, vol. 1, pp. 91-134, 2015.
N. B. Goethe and M. Friend, “Confronting the ideals of proof with the ways of proving of research mathematicians,” an International Journal for Symbolic Logic, vol. 96(2), pp. 273-288, 2010.
K. Weber and J.P. Mejia-Ramos, “On relative and absolute conviction in mathematics,” For the Learning of Mathematics, vol. 35(2), pp. 15-21, 2015.
A. Baki, “Mathematics education from theory to practice (4th Ed.),” Ankra: Harf Education Publishing, 2008.
R. Maya and U. Sumarmo, “Mathematical understanding and proving abilities: Experiment with undergraduate student by using modified Moore learning approach,” IndoMS, Journal of Mathematics Education, vol. 2(2), pp. 231-250, 2011.
DOI: https://doi.org/10.11591/edulearn.v12i4.9557
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