### What should be the object of research with respect to the notion of mathematical proof?

#### Abstract

Despite its central place in the mathematics curriculum the notion mathematical proof has failed to permeate the curriculum at all scholastic levels. While the concept of mathematical proof can serve as a vehicle for mathematical thinking, studies have revealed that students experience serious difficulties with proving that include: not knowing how to begin the proving process, the proclivity to use empirical verifications for tasks that call for axiomatic methods of proving, resorting to rote memorization of uncoordinated fragments of proof facts. While several studies have been conducted with the aim of addressing students’ fragile grasp of mathematical proof the majority of such studies have been based on activities that involve students reflecting and expressing their level of convincement in arguments supplied by the researchers thereby compromising the voice of the informants. Further, research focus has been on the front instead of the back of mathematics. Hence, there is a dearth in research studies into students’ thinking processes around mathematical proof that are grounded in students’ own proof attempts. Therefore current research strides should aim at identifying critical elements of students’ knowledge of the notion of proof that should be informed by students’ actual individual proof construction attempts.

#### Keywords

#### References

M. Duruk and A. Kaplan, “Prospective mathematics teachers’ difficulties in doing proofs and causes of

their struggle with proofs.”, Proceedings of the 1st International Eurasian Educational Research Congress, 2015.

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

Undergraduate Mathematics Preparation of School Teachers: The Journal. Volume 1, pp. 1-13, 2009.

A. Mukuka, and O. Shumba, “Zambian university student teachers’ conceptions of algebraic proofs.”, Journal

of Education and Practice, Vol. 7, Number 32, pp. 157-171, 2017.

D. H. Jonassen and B. Kim, “Arguing to learn and learning to argue: design justifications and guidelines.”, Educational Technology Research and Development, Volume 58 Number 4, pp. 439-457, 2010.

J. Selden and A. Selden, (2009). “Understanding the proof construction process.”, Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education, Volume 2, pp. 196-201, 2009.

Y. Ko and E. Knuth, “Undergraduate mathematics majors’ writing performance producing proofs and

counter examples in continuous functions.”, The Journal of Mathematics Behaviour, Volume. 28, pp. 68-77, 2009.

G. Harel and J.M. Rabin, (2010). Teaching practices associated with the authoritative proof scheme. Journal for Research in Mathematics, Volume 41 Number 1, pp. 14-19, 2010.

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, Volume 96 Number 2, pp. 273-288, 2010.

T. CadawalladerOlsker, “What do we mean by mathematical proof.”, Journal of Humanistic Mathematics, Volume 1, pp. 1-33, 2011.

G. Oflaz., N. Bulut and V. Akcakin, (2016). “Pre-service classroom teachers’ proof schemes in geometry:

A case study of three pre-service teachers.”, Eurasian Journal of Educational Research, Volume 63, pp. 133- 152, 2016.

D. Stylianou., M.L., Blanton and O. Rotou (2015). “Undergraduate students’ understanding of proof:

relationships between proof conceptions, beliefs, and classroom experiences.”, International Journal of Research in Mathematics Education, Volume 1, pp. 91- 134, 2015.

A.J. Stylianides and G.J. Stylianides, “Proof constructions and evaluations.”, Educational Studies in Mathematics, Volume 72 Number 3, pp. 237-253, 2009.

G. Martin and G Harel, “Proof frames of pre-service elementary teachers.”, Journal for Research in Mathematics Education, Volume 29, pp. 41-51, 1989.

G. Harel and L. Sowder, L. “Toward a comprehensive perspective on proof.” , National Council of Teachers of Mathematics, 2007.

A.J. Stylianides, “Towards a comprehensive knowledge package for teaching proof: A focus on the misconception that empirical arguments are proofs.”, Pythagoras, Volume 32, Number 1, p. 1-14, 2011.

K. Weber and J. Mejia-Ramos, “ Why and how do mathematicians read proofs: an exploratory study.”,

Educational Studies in Mathematics, Volume 76, pp. 329-344, 2011.

A. Azrou, “ Proof writing at undergraduate.”, Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education, pp. 79-85.

S. G. Stavrou, “Common errors and misconceptions in mathematical proving by education undergraduates.”, Issues In The Undergraduate Mathematics Preparation of School Teachers, Volume 1, pp. 1-8, 2014.

L. Manilla and S. Wallin, “Promoting students’ justification skills using structured derivations.”, Proceedings of ICMI study 19 conference on proof and proving in Mathematics Education, Taiwan, 2009.

I. Ug ̌urel, S. Morali, K. Yig ̌it and O. Karahan, “ Pre-service secondary mathematics teachers’ behaviors in the proving process.”, Eurasian Journal of Mathematics, Science and Technology Education, Volume 12 Number 2, pp. 203-231, 2016.

J.A. Maxwell and K. Mittapalli, (2007). “The value of critical realism in qualitative research.”, Annual

conference of the International Association for Critical Realism, Philadelphia, 2007.

M. Hennink, I. Hutter and A. Bailey, “Qualitative research methods.”, London: Sage, 2013.

S.K. Bleiler, D.R. Thompson and M. Krajc ̌evski, “Providing feedback on students’ mathematical arguments: Validations of prospective secondary mathematics teachers.”, Journal of Mathematics Teacher Education, Vol. 17, pp. 105-127, 2014.

A.M.Recio and J. Godino, “Institutional and personal meanings of mathematical proof.”, Educational Studies in Mathematics, Volume 48, pp. 83-99.

O ̈zdemir, E., & O ̈vez, “An investigation into logical thinking skills and proof writing levels of prospective mathematics teachers.”, Journal of Education and Training Studies, Volume 5 Number 2, pp. 10-20, 2017.

A. Cusi and N. Malara , “Proofs problems in elementary number theory: Analysis of trainee teachers’ productions.”, Proceedings of the Fifth Conference of the European Society for Research in Mathematics Education, pp. 591-600, 2007.

T.A. Iskenderoglu and A. Baki, “Quantitative analysis of pre-service elementary teachers’ opinions about doing mathematical proof.”, Educational Sciences: Theory and Practice, Volume 11Number 4, pp. 2285-2295, 2011.

M. Cirillo and P.G. Herbst, “ Moving towards authentic proof and practices in mathematics.”, The Mathematics Educator, Volume 21 Number 2, pp. 11-33, 2012.

P. Boero, “Argumentation and mathematical proof: A complex, productive, unavoidable relationship in

mathematics and mathematics education.”, International Newsletter on the Teaching and Learning of Mathematical Proof, Volume 7 Number 8, 1999.

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, Volume 2, Number. 2, pp. 231-250, 2011

I. Kidron and T. Dreyfus, “Proof image.”, Educational Studies in Mathematics, Volume 87, pp. 297-321, 2014.

M.A. Mariotti, “Proof and proving in mathematics education.”, Handbook of research on the Psychology of Mathematics Education- Past, present and future, pp:173-204. Rotterdan, Netherlands: Sense Publishers, 2006.

A. Selden and J. Selden, “Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem?”, Journal for Research in Mathematics Education, Volume 34 Number, pp. 4–36, 2003.

Y. Imamog ̌lu and A.Y. Tog ̌rol, “Proof construction and evaluation practices of prospective

mathematics educators.”, European Journal of Science and Mathematics Education, Volume 3 Number 2, pp. 130-144, 2015.

K. Weber, “How mathematicians determine if an argument is a valid proof.”, Journal For Research

in Mathematics Education, Volume 39 Number 4, pp. 431-459, 2008.

DOI: http://dx.doi.org/10.11591/edulearn.v13i1.10000

### Refbacks

- There are currently no refbacks.

Copyright (c) 2018 Universitas Ahmad Dahlan

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

**Journal of Education and Learning (EduLearn)**

ISSN: 2089-9823, e-ISSN 2302-9277

Published by: Universitas Ahmad Dahlan (UAD) *in collaboration with* Institute of Advanced Engineering and Science (IAES)