## Abstract

Drawing on prior research on indirect proof, this paper reports on a series of exploratory studies that examine the extent to which findings on students’ ways of reasoning about contradiction and contraposition characterize students’ views of indirect existence proofs. Specifically, Study 1 documents students’ comparative selections and selection rationales when asked to choose the “most convincing” proof, given a constructive and nonconstructive existence proof. Study 2 further examines findings from Study 1 by documenting novices’ levels of conviction and interpretations of a nonconstructive existence proof. Findings show that when presented with a nonconstructive proof, students tended to not only find the proof convincing but also interpreted the proof constructively. Moreover, the data indicate students who exhibit an awareness of the nonconstructive structure were divided in terms of their views of which form – constructive or non-constructive – was the most convincing. The discussion considers students’ reactions to the disjunctive structure of nonconstructive existence proofs and use of pragmatic and theoretical modes of thought.

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## Notes

- 1.
These definitions of constructive and nonconstructive existence proofs are based on those used in tertiary mathematics texts (see Fig. 1), rather than the philosophies of constructivism or intuitionism.

- 2.
Advances in programming have led to debates regarding the production of algorithms and, therefore, the constructive / nonconstructive classifications of some proofs (cf. Gray 1994).

- 3.
Specifically, Hilbert proved that there exists a finite basis for the ring of invariants

*G*= SL*n*(*C*) acting on the ring of polynomials*R*=*S*(*V*), when*V*is a finite dimensional representation of*G* - 4.
It should, however, be noted that Hilbert produced a constructive proof in 1893.

- 5.
It is also important to note that this classification is mathematical rather than philosophical for those who adhere to the philosophy of constructivism or intuitionism would likely reject the mathematical classification.

- 6.
A dilemma proof may involve a false premise, as is the case in Argument B. There are other dilemma forms that do not.

- 7.
Weber and Mejia-Ramos’s definition of

*relative conviction*differs from that offered here and is focused on “the subjective level of probability that one attributes to that claim being true” (p. 16). Their definition is not incompatible with that given above since a higher degree of conviction can be thought of as assigning a higher “subjective level of probability.” - 8.
The absence of rationales may be due to the lack of access to mathematical symbols and notations with an online survey environment.

- 9.
The assumption students would lack familiarity was based on the curricular information available at the university.

- 10.
*Italics*are used to indicate a student’s, as opposed to the researcher’s, emphasis in verbalized remarks.

## References

Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving?

*ZDM – The International Journal on Mathematics Education, 40*, 401–412.Balacheff, N. (1988). Aspects of proof in pupils practice of school mathematics. In D. Pimm (Ed.),

*Mathematics, teachers, and children*(pp. 216–235). Kent: Hodder and Stoughton.Brown, S. (2014). On skepticism and its role in the development of proof in the classroom.

*Educational Studies in Mathematics, 86*(3), 311–335.Chartrand, G., Polimeni, A., & Zhang, P. (2008).

*Mathematical proofs: A transition to advanced mathematics*(3rd ed.). Boston: Pearson.Cunningham, D. (2012).

*A logical introduction to proof*. New York: Springer.de Villiers, M. (1990). The role and function of proof in mathematics.

*Pythagoras, 24*, 17–24.Dawson, J. (1997)

*Logical dilemmas: The life and work of Kurt Gödel.*Wellesley, MA: A. K. Peters, Ltd.Fischbein, E. (1982). Intuition and proof.

*For the Learning of Mathematics, 3*(2), 9–18, 24.Garofalo, J., Triner, C., & Swartz, B. (2015). Engaging with constructive and nonconstructive proof.

*Mathematics Teacher, 108*(6), 422–428.Glaser, B. G., & Strauss, A. L. (1967).

*The discovery of grounded theory: Strategies for qualitative research*. New York: Aldine De Gruyter.Glasser, B. (1965) The constant comparative method of qualitative analysis.

*Social Problems, 12*(4), 436–445.Gray, R. (1994). Georg Cantor and transcendental numbers.

*The American Mathematical Monthly, 101*, 819–832.Hammack, R. (2013).

*Book of proof*, 2nd Edition. Creative Commons. Attribution-No Derivative Works 3.0 License. Richmond: Virginia Commonwealth University.Hanna, G. (1995). Challenges to the importance of proof.

*For the Learning of Mathematics, 15*(3), 42–49.Hardy, G. H. (1967).

*A Mathematician's apology*. London: Cambridge University Press.Harel, G., & Sowder, L. (1998). Students proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.),

*Research on collegiate mathematics education III*(pp. 234–283). Providence: American Mathematical Society.Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra.

*Journal for Research in Mathematics Education, 31*(4), 396–428.Jahnke, H. N. (2010). The conjoint origin of proof and theoretical physics. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.),

*Explanation and proof in mathematics: Philosophical and Educational perspectives*. New York: Springer.Jones, I., & Inglis, M. (2015). The problem of assessing problem solving: Can comparative judgement help?

*Educational Studies in Mathematics, 89*, 337–355.Knuth, E. (2002). Secondary school mathematics teachers conceptions of proof.

*Journal for Research in Mathematics Education, 33*, 379–405.Koichu, B., & Zazkis, R. (2013). Decoding a proof of Fermat’s little theorem via script writing.

*Journal of Mathematical Behavior, 32*, 364–376.Laming, D. (1984). The relativity of Babsolute^ judgements.

*British Journal of Mathematical and Statistical Psychology, 37,*152–183.Leron, U. (1985). A direct approach to indirect proofs.

*Educational Studies in Mathematics, 16*(3), 321–325.Lobato, J. (1996). Transfer reconceived: how “sameness” is produced in mathematical activity. (Doctoral dissertation, University of California, Berkeley, 1996).

*Dissertation Abstracts International,*AAT 9723086.Lobato, J. & Siebert, D. (2002). Quantitivative reasoning in a reconceived view of transfer.

*Journal of Mathematical Behavior, 21*(1), 87–116.Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.),

*Handbook on research of the psychology of mathematics education past present, and future*(pp. 173–204). Rotterdam: Sense Publishers.Mariotti, M. A., Bartolini Bussi, M. G., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorem in contexts: From history and epistemology to cognition. In E. Pekhonen (Ed.),

*Proceedings of the 21*^{st}*PME International conference*,*Vol*(Vol. 1, pp. 180–195).Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics.

*Educational Studies in Mathematics, 79*(1), 3–18.Piaget, J. (1968).

*Six psychological studies*. New York: Random House.Tall, D. (1979). Cognitive aspects of proof, with special reference to the irrationality of √2. In D. Tall (Ed.),

*Proceedings of the third International conference for the psychology of mathematics education*(pp. 206–207). Warwick: Warwick University, Mathematics Education Research Centre.Thurstone, L.L. (1927). A law of comparative judgement.

*Psychological Review, 34,*273–286.Webb, J. C. (1997). Hilbert’s formalism and arithmetization of mathematics.

*Synthese, 110*(1), 1–14.Weber, K., & Mejia-Ramos, J. P. (2015). On relative and absolute conviction in mathematics.

*For the Learning of Mathematics, 35*(2), 15–21.Zaslavsky, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics.

*Educational Studies in Mathematics, 60*, 297–321.

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Brown, S.A. Who’s There? A Study of Students’ Reasoning about a Proof of Existence.
*Int. J. Res. Undergrad. Math. Ed.* **3, **466–495 (2017). https://doi.org/10.1007/s40753-017-0053-6

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### Keywords

- Nonconstructive arguments
- Existence proofs
- Proof by contradiction