On the heterogeneity of mathematical practice with respect to justification and proof

1. On the heterogeneity of mathematical practice with respect to justification and proof Keith Weber Rutgers University

2. Mathematical practice withrespect to proof • What do I think constitutes proof in mathematics? • What types of claims do I make about mathematical practice? • On what evidence do I make these claims? • How should these claims inform mathematics instruction?

3. Mathematical practice withrespect to proof • What do I think constitutes proof in mathematics? Mathematicians are the judge of what proof is, and I treat this as an empirical question. My current thinking is that proof is a polysemous, contextual, and a radial category. • What types of claims do I make about mathematical practice? • On what evidence do I make these claims? • How should these claims inform mathematics instruction?

4. Mathematical practice withrespect to proof • What do I think constitutes proof in mathematics? Mathematicians are the judge of what proof is, and I treat this as an empirical question. My current thinking is that proof is a polysemous, contextual, and a radial category. • What types of claims do I make about mathematical practice? I examine the processes used by mathematicians to construct and understand proofs and (indirectly) the purposes these processes are designed to fulfill. • On what evidence do I make these claims? • How should these claims inform mathematics instruction?

5. Mathematical practice withrespect to proof • What do I think constitutes proof in mathematics? Mathematicians are the judge of what proof is, and I treat this as an empirical question. My current thinking is that proof is a polysemous, contextual, and a radial category. • What types of claims do I make about mathematical practice? I examine the processes used by mathematicians to construct and understand proofs and (indirectly) the purposes these processes are designed to fulfill. • On what evidence do I make these claims? I use both task-based and open-ended interviews to form hypotheses. I use larger scale studies (experiments and surveys) to test these hypotheses. • How should these claims inform mathematics instruction? These findings create desired goals from instruction by illustrating domain-specific competence.

6. The relationship between mathematical practice and instruction • It is widely believed that mathematical practice with respect to proof should inform and constrain instruction. “Since a notion of proof exists in the discipline of mathematics, it might be entitled to exist in classroom activity. And if it were to exist, it would be expected to exist in a form that was accountable to, if not compatible with, how it exists in the discipline” (Herbst & Balacheff, 2009, p. 43). “The goal of instruction must be unambiguous; namely to gradually refine current students’ proof schemes toward the proof scheme shared and practiced by the mathematicians of today” (Harel, 2001, p. 188). “[A proof] employs modes of reasoning that are valid and known to … the classroom community” (Stylianidies, 2007, p. 2910.

7. The relationship between mathematical practice and instruction • It is widely believed that mathematical practice with respect to proof should inform and constrain instruction. “Since a notion of proof exists in the discipline of mathematics, it might be entitled to exist in classroom activity. And if it were to exist, it would be expected to exist in a form that was accountable to, if not compatible with, how it exists in the discipline” (Herbst & Balacheff, 2009, p. 43). “The goal of instruction must be unambiguous; namely to gradually refine current students’ proof schemes toward the proof scheme shared and practiced by the mathematicians of today” (Harel, 2001, p. 188). “[A proof] employs modes of reasoning that are valid and known to … the classroom community” (Stylianidies, 2007, p. 2910.

8. The relationship between mathematical practice and instruction • It is widely believed that mathematical practice with respect to proof should inform and constrain instruction. “Since a notion of proof exists in the discipline of mathematics, it might be entitled to exist in classroom activity. And if it were to exist, it would be expected to exist in a form that was accountable to, if not compatible with, how it exists in the discipline” (Herbst & Balacheff, 2009, p. 43). “The goal of instruction must be unambiguous; namely to gradually refine current students’ proof schemes toward the proof scheme shared and practiced by the mathematicians of today” (Harel, 2001, p. 188). “[A proof] employs modes of reasoning that are valid and known to … the classroom community” (Stylianidies, 2007, p. 2910.

9. The relationship between mathematical practice and instruction • It is widely believed that mathematical practice with respect to proof should inform and constrain instruction. “Since a notion of proof exists in the discipline of mathematics, it might be entitled to exist in classroom activity. And if it were to exist, it would be expected to exist in a form that was accountable to, if not compatible with, how it exists in the discipline” (Herbst & Balacheff, 2009, p. 43). “The goal of instruction must be unambiguous; namely to gradually refine current students’ proof schemes toward the proof scheme shared and practiced by the mathematicians of today” (Harel, 2001, p. 188). “[A proof] employs modes of reasoning that are valid and known to … the classroom community” (Stylianidies, 2007, p. 2910.

10. The relationship between mathematical practice and instruction • Underlying each of these assumptions is that there are core behaviors, beliefs, and activities shared by most mathematicians • To avoid a strawman, no one claims absolute agreement, or a consensus in all areas, but we would expect a consensus on: Validity in the classroom: • What constitutes a proof in a college calculus course Obtaining conviction from non-deductive evidence: • Whether empirical evidence can be convincing • Whether authoritative evidence is convincing

11. Agreement about proofs? • Selden and Selden (2003) conducted a study on students’ evaluations of invalid and valid proofs. • When Weber (2008) and Inglis and Alcock (in press) gave the same proofs to mathematicians, there was not uniformity in their evaluations. • In the two studies, the “real deal” was judged valid by 14 of 20 mathematicians who read it. • The “gap”, a purported invalid proof, was judged invalid by 12 of 20 mathematicians.

12. Agreement about proofs? • 109 mathematicians were shown a proof that by demonstrating that purportedly was submitted for publication in the Mathematical Gazette.

13. Agreement about proofs? • 109 mathematicians were shown a proof that by demonstrating that purportedly was submitted for publication in the Mathematical Gazette. • 27% said the proof was valid, 73% invalid.

14. Agreement about proofs? • 109 mathematicians were shown a proof that by demonstrating that purportedly was submitted for publication in the Mathematical Gazette. • 27% said the proof was valid, 73% invalid. • Pure mathematicians: 17% valid. Applied mathematicians: 50%.

15. Agreement about proofs? • 109 mathematicians were shown a proof that by demonstrating that purportedly was submitted for publication in the Mathematical Gazette. • 27% said the proof was valid, 73% invalid. • Pure mathematicians: 17% valid. Applied mathematicians: 50%. • When asked about commuting the limit and integral, 83% said it was a legitimate complaint about the proof (9% unsure). • When asked if this was sufficient to invalidate the proof, 65% of the valid responses agreed. 3% of the invalid responses agreed

16. Empirical evidence and conviction • The majority of mathematicians take it as a priori obvious that mathematicians are not convinced by empirical evidence sans proof. • Yet many claim mathematicians aresometimes convinced by empirical evidence. • Goldbach’s conjecture (e.g., Ecceveria, 1996; Paseau, 2011) • Reimann hypotheses (e.g., Conrey, 2003) • Experimental mathematics (e.g., Borwein, 2008). “Contrary to the belief common amongst many mathematics teachers that only proof provides certainty for the mathematician, mathematicians are often convinced by the truth of their results (usually on the basis of quasi-empirical evidence) long before they have proofs” (de Villiers, 2004, p. 402).

17. Empirical evidence and conviction I observed eight mathematicians as they determined if eight arguments were valid proofs (Weber, 2008).

18. Empirical evidence and conviction I observed eight mathematicians as they determined if eight arguments were valid proofs (Weber, 2008). • Reading the claim n≅3 (mod 4) implies n is not a perfect square. • “I’m using examples to see where the proof is coming from. 52 is 25 and that’s 1 mod 4. 62 is 0 mod 4. 72 is 1 mod 4. 82 is 0 mod 4. I’m thinking, 24 times 24 that’s 0 mod 4. So a perfect square has to be 1 mod 4, doesn’t it? Alright” • Reading the claim 32n+1≅2 (mod 8). • “ Is that true? 32 + 1 equals 10. Yeah that seems fine”.

19. Empirical evidence and conviction • 118 research-active mathematicians were shown the statement: “When I read a proof in a respected journal and I am not immediately sure that a statement in the proof is true, it is not uncommon for me to gain a sufficiently high level of confidence in the statement by checking it with one or more carefully chosen examples to assume the claim is correct and continue reading the proof”. 55% agreed, 9% disagreed • 55 who had experience refereeing, when asked the same about refereeing, had 42% agree, 27% disagree. (Mejia-Ramos & Weber, submitted)

20. Empirical evidence and conviction • In another study, 97 research-active mathematicians were convinced that a claim was correct with empirical evidence but no proof. 26 said yes (Weber, submitted).

21. Authority and conviction I interviewed nine mathematicians, where one question was why they read published proofs in journals: “One reason is to find out whether I should believe it’s true… knowing it’s true frees me to use it. If I don’t follow their proof then I would be psychologically disabled from using it. Even if somebody I respect immensely believes that it’s true.” “Now notice what I did not say. I do not try and determine if a proof is correct. If it’s in a journal, I assume it is. I’m much more interested in the ideas of the proof”

22. Authority and conviction 118 mathematicians completed the following survey items: It is not uncommon for me to believe that a proof is correct because it is published in an academic journal. 75% agree, 11% disagree When reading a proof in a reputable journal, it is not uncommon for me to be very confident that the proof is correct because it was written by an authoritative source that I trust. 82% agree, 5% disagree When refereeing a proof, … written by an authoritative source that I trust. 38% agree, 40% disagree

23. Implications for research • If we believe that mathematicians are heterogeneous in their practice, then research on mathematical practice should: • Questions of “do mathematicians do/believe/think/engage in X?” (e.g., “do mathematicians consider the authority of the author?”) might not have yes or no answers. • More caution should be given when drawing inferences from case studies, quotes from mathematicians (Thurston, 1994) , and our own experience. We should notassume mathematicians do not regularly engage in this behavior from this evidence. • In general, as a researcher, I implicitly acted with a “homogeneity default” perspectives, seeking out regularities in mathematical behavior and perhaps ignoring markers for diversity.

24. Connecting mathematical practice to instruction? • If we believe that if mathematicians do/believe/participate in X, then students should be lead to do/believe/participate in X, then what should we do if some mathematicians do X and some do not X? • Perhaps we need to be more fine-grained about asking whether mathematicians engage in behavior/activity X. It might be more worthwhile to discuss all types of behaviors, highlighting their strengths and weaknesses with the different purposes they may serve. • Appeals to authority are good for estimating the truth of a claim but useless for generating understanding.