As an undergraduate, it was easy for me to assume that as my professors conducted mathematical research, beautiful, complete proofs came to them in moments of epiphany. Their work was mysterious to me, and I believed that somehow their superior intelligence and vast mathematical knowledge gave them immediate access to all things abstract. Had I been asked then, I likely would have said that mathematicians didn’t need to think about examples in their own research – surely they had outgrown the need for concrete examples.
This perspective may be attributable to the fact that throughout my math classes to that point, it had been ingrained in me early and often that showing a statement is true for a few examples is not a valid proof of a universally quantified true statement. The belief that several examples do suffice as a proof has been called the empirical proof scheme (Harel & Sowder, 1998), and a good amount of literature on students’ reasoning on proof has focused on this perspective as a limitation (e.g., Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Porteous, 1990; Stylianides & Stylianides, 2009; Zaslavsky, Nickerson, Stylianides, Kidron, & Winicki, 2012). In light of this, teachers expend a considerable amount of effort in making sure students do not incorrectly cultivate this notion, and rightly so, given that we do not want students to wrongly believe that examples are valid substitutes for proofs.
However, my colleagues and I wonder whether the repeated caution against empirical-based arguments has led students to undervalue the role that examples can play in proof. Mathematicians certainly use examples in their development of conjectures and proofs. As Epstein and Levy (1995) contend, “Most mathematicians spend a lot of time thinking about and analyzing particular examples…. It is probably the case that most significant advances in mathematics have arisen from experimentation with examples (p. 6).” Therefore, while we acknowledge the danger of developing incorrect notions of examples as proof, we worry that the emphasis on such limitations, perhaps coupled with assumptions that mathematicians do not use examples, may preclude postsecondary students from engaging meaningfully with concrete examples as they prove.
In graduate school, I had an excellent professor who demonstrated an infectious curiosity. He was always willing to explore problems in front of students, not as someone who had prepared notes or who was simply recalling information, but as a true problem solver who was deeply engaged in the task. As he solved problems or proved theorems with us, he would get his hands dirty with some concrete examples, and all of a sudden the problem would become more real. We realized that he was doing the kinds of things we could be doing ourselves – carefully writing down a handful of concrete examples, searching for patterns, using examples to determine whether a conjecture might be true, and looking at generic examples to see how a proof might be developed. This experience was incredibly illuminating for me, and it helped me to formulate a more accurate view of mathematicians’ activity.
I want to encourage students to be more aware of and open to the valuable roles that examples can play in proof-related activity. The goal is not to encourage an overreliance on examples in the context of proof, or to deride the warnings against an empirical proof scheme. Rather, the point I want to raise here is that there is a key aspect of many mathematicians’ proof-related activity that I am not sure students consider: Examples can play an integral role in developing conjectures and formulating proofs.
There has been a considerable amount of research conducted recently on examples, and more particularly on examples in proof (e.g., Bills and Watson, 2008; Sandefur, et al., 2013; Weber, 2008). In our studies (reported elsewhere in Ellis, et al., 2012; Lockwood, et al., 2012; Lockwood et al., 2013), my colleagues and I found that mathematicians in a variety of fields regularly draw on examples as a part of their proving activity. Indeed, in all of the 250 survey responses and 19 interviews we gathered, no mathematician indicated that they do not use examples. Below are just three of the responses from this data, which reveal instructive insights into the nature of mathematical exploration and proof.
M1: “I explore examples to find out what statements mean. For instance, yesterday I was trying to understand the meaning of “If E is an elliptic curve/Q, then there is associated a representation \(Gal(\overline Q/Q) \to GL_2(Z/3)\). So I chose an elliptic curve, specifically the one of equation \(y^2=x^3+x+1\), and tried to find the points of order 3. It took a while, but after I was through I knew what the statement meant. Generally, the difficulty in dealing with a new mathematical concept is to form a mental image of it. Examples help develop such mental images.”
M2:“I start with the simplest conceivable example, then I try to come up with slightly more complicated examples. In parallel to this procedure, I also try to guess counterexamples. This guessing typically fails, and if it does, I try to find specific properties of my guess examples that prevent them from doing what I want them to do. Sometimes this allows a slow “building up” of properties that can eventually say something useful about the conjecture. Other times, it is clear what the counterexamples should be, but it is still unclear how to prove the conjecture.”
M3: “First test the easiest cases. (E.g., for integers, test 2, 3, 4, 5, 10) Then test something that is qualitatively different from the easiest cases. If it still works, make a first attempt at a proof. If you can’t prove it, try to cook up counter-examples that exploit the holes in your “proof”. If you can’t make counter-examples, use what you learned from the failed counter-examples to fix the holes in the proof. Go back and forth between proof and disproof, using the failures of each side of the argument to build up your attempt on the other side.”
For those readers who are currently undergraduate or graduate students, when you go to prove a theorem, what do you do first? Do you launch into the proof, trying to recall certain techniques? Do you read back through the book looking for similar proofs that you can mimic? Or, do you first play around with some concrete examples, using them to make sense of the statement of the problem? Our research suggests that this kind of experimentation with examples can be a useful first step in understanding a conjecture and ultimately coming up with a solid proof. In fact, as we spoke with mathematicians, we found that they often use concrete examples to make sense of conjectures (M1’s response), or to try to convince themselves whether a conjecture might be true (M2’s response), and even to provide concrete insights into how they might go about proving a conjecture (M3’s response). I would also encourage students to reflect on the metacognitive aspect of these mathematicians’ responses. They are clearly being intentional about how they are selecting and using examples. This kind of flexibility with examples is something that students should develop in their mathematical activity. The takeaway for students is this: There is no substitute for getting your hands dirty with specific examples in mathematics – whether you are solving problems, developing conjectures, or proving or disproving conjectures.
For those of us who teach mathematics, I suggest that we should give explicit attention to the role that examples can play in conjecturing and proving. Students may benefit from being encouraged to work with examples and from seeing how specific examples can actually play a crucial role in proof. This can be modeled for them and also reinforced through tasks and homework problems that develop this activity.
Because mathematicians use examples so regularly and in a variety of ways, students should similarly incorporate example-related activity as a fundamental aspect of their work. Students may greatly benefit from grounding their proof writing and conjecturing in concrete examples that can serve a variety of purposes.
I would like to thank my colleagues Eric Knuth and Amy B. Ellis, whose collaboration led to many of the ideas in the post. The research described here is supported in part by the National Science Foundation under grants DRL-0814710 (Eric Knuth, Amy Ellis, & Charles Kalish, principal investigators) and DRL-1220623 (Eric Knuth, Amy Ellis, & Orit Zaslavsky, principal investigators). The opinions expressed herein are those of the author and do not necessarily reflect the views of the National Science Foundation.
Ellis, A. E., Lockwood, E., Knuth, E., Dogan, M. F., & Williams, C. C. W. (2013). Choosing and using examples: How example activity can support proof insight. In A. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Annual Meeting of the International Group of the Psychology of Mathematics Education. Kiel, Germany.
Ellis, A. E., Lockwood, E., Williams, C. C. W., Dogan, M. F., & Knuth, E. (2012). Middle school students’ example use in conjecture exploration and justification. In L.R. Van Zoest, J.J. Lo, & J.L. Kratky (Eds.), Proceedings of the 34th Annual Meeting of the North American Chapter of the Psychology of Mathematics Education (Kalamazoo, MI).
Epstein, D., & Levy, S. (1995), Experimentation and proof in mathematics. Notice of the AMS, 42(6), 670–674.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Issues in Mathematics Education, 7, 234-283.
Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K–16 perspective (pp. 153–170). New York, NY: Routledge.
Lockwood, E., Ellis, A.B., Dogan, M.F., Williams, C., & Knuth, E. (2012). A framework for mathematicians’ example-related activity when exploring and proving mathematical conjectures.
Lockwood, E., Ellis, A., & Knuth, E. (2013). Mathematicians’ example-related activity when proving conjectures. In S. Brown, G. Karakok, K. H. Roh, & M. Oehrtman (Eds.), Electronic Proceedings for the Sixteenth Special Interest Group of the MAA on Research on Undergraduate Mathematics Education. Denver, CO: Northern Colorado University.
Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589–598.
Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics. Doi: 10.1007/s10649-012-9459-x.
Stylianides, G. & Stylianides, J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352.
Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Winicki, G. (2012). The need for proof and proving: mathematical and pedagogical perspectives. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI Study (New ICMI Study Series, Vol. 15). New York: Springer.
Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431-459.