mathematicians’ perspectives on the transition to formal proof

Lara Alcock

Graduate School of Education, Rutgers University, USA

lalcock@rci.rutgers.edu

1. Introduction and research context

The research reported here addresses the theme of the transition to proof, via the perspectives of mathematicians who teach a course entitled “Introduction to Mathematical Reasoning”. This course is taught in a standard lecture-based format at a large state university in the USA. Classes contain 20-25 students, so that the professor has relatively close contact with individuals and is able to become familiar with their work during the 14-week semester. The five participants in this exploratory research all had extensive experience in teaching the course. Interview data were collected and analyzed according to a cyclic grounded theory-based design (following Glaser, 1992), with questions progressing from the very open-ended to more detailed requests for further explanation and for responses to comments made by other participants.

The participants spoke about many aspects of the course, including issues of ongoing assessment and their own disagreements about whether the course should be more axiomatic or more investigative. This paper focuses first on cognitive issues, citing four modes of thinking used by successful provers, each of which has a distinct associated goal. These emerged from analysis of the participants’ comments on the differences between their own thinking and that of their students. It then notes that in teaching, emphasis on one mode tends to predominate. It concludes by offering comments on the interdependence of the modes and asking whether a more balanced approach might be appropriate.

2. Four modes of thinking

The four modes of thinking are:

1. Instantiation
2. Structural thinking
3. Creative thinking
4. Critical thinking

2.1 Instantiation

The goal of the instantiation mode is to meaningfully understand a mathematical statement by thinking about the objects to which it applies. Professor 1 describes this as a natural response to a new definition.

P1: …what happens is you know that you describe a new definition, you say “let f be a function, let x be a real number, we say that…” and then “some relationship between f and x holds if, blah blah blah.” So then what they have to do, they have to realize that this definition only makes sense in the context of, I have to have a function in mind and I have to have a [number] in mind…

Dahlberg and Housman (1997) also consider this a strong approach to handling a new definition. Professor 1 notes, however, that his students often do not engage in such thinking.

P1: And what they’ll do is typically if you have a sequence, you know, if I have a sequence definition to use in the rest of the problem, and they don’t understand the definition, they’ll just skip that sentence and go on. I will…they will come in for help on a problem, and five or ten minutes into the discussion I’ll realize that, that they never bothered to process this particular definition. They have no idea what this means.

2.2 Structural thinking

The goal of structural thinking is to generate a proof for a statement by using its formal structure. Professor 3 considers this a sensible first approach when trying to construct a proof.

P3: …to try to get as much as possible, you know as many ideas as possible about how one should go about writing a proof, by…just looking at, at the sentence, what it says and what its logical structure is.

Selden and Selden use similar ideas in discussing how an individual validates a proof (Selden & Selden, 2003). Professor 3 notes, however, that students lack the experience that would help them to proceed in this way.

P3: And the main obstacle is, that the students…most students don’t seem to have had any courses in English grammar, in anything else that might teach them how to think in terms of structures, logical structures.

More specifically, students are also often unaware of the mathematical conventions for interpreting connectives and quantifiers. Professor 2 offers this comment (among others) on the errors that can occur as a result of this.

P2: …but students who would say, “we want to prove this, so that” – it was never clear to me whether they meant “so we can conclude that”, or “so we need to verify that”. And there were a lot of proofs which would be correct, if I could only infer that they meant “so we need to verify…so we need to verify…so we need to verify…and then this last one is clearly true so everything stacks up.”

2.3 Creative thinking

Creative thinking is a semantic mode, the goal of which is to examine instantiations of mathematical objects in order to identify a property that can be used as the “key idea” (in the sense of Raman, 2003) in a proof. This might be accomplished directly, by experimenting with a particular example in the hope that one will find an argument or sequence of manipulations that can be generalized. This is often cited as a strategy for finding an induction argument.

P5: See if you can get from 2 to 3, if you can’t get from n to n plus 1.

Alternatively (in the case of a universal statement), it might be accomplished by attempting to construct a counterexample, and attending to what prevents this.

P1: …the way I often think about a proof is that, you know you imagine this as, try to beat this. Meaning, try to find a counterexample. […] If you think about, if you think about the reason why you were failing to find a counterexample, okay, then, that sometimes gives you a clue, to why the thing is true.

There appear to be two opposing obstacles that prevent students from using these strategies successfully. One is that, as noted above, they may not be inclined to think about examples at all. The other is that they may give examples without articulating properties of these, either performing empirical checks or relying on concept images (cf. Harel & Sowder, 1998).

P3: So then I asked her in class, how do you know that 1997 is prime? The idea being that she would say “well I checked”, in which case I was going to answer, I was going to say “okay, you tell me that you have checked. But why should I accept that?” Okay? That was going to be the second part, but then I didn’t even get there because then she said, “It looks like a prime to me”. That’s what she said.

2.4 Critical thinking

The goal in critical thinking is to check the correctness of assertions or deductions. This involves searching for possible counterexamples, and/or checking for preservation of properties or for implied properties that are false. Checking a proof in this way is described by professor 1 as follows.

P1: …there is a locality principle about proofs, about every proof, and that is that we somehow recognize, that even though you’re proving some very specific thing, that there are portions in the argument. Each portion in the argument is actually doing something more general. […] And therefore I can do a local check on this part of the argument by thinking about that more general situation and doing, examples within that more general situation.

Johnson-Laird & Hasson (2003) report that comparable counterexample checking is a preferentially used way to determine that an inference is invalid. Nevertheless, the professors in this study regularly express exasperation at the fact that students appear not to make these checks and thus make claims that are “obviously wrong”.

P3: For instance, problem: “Express the number 30 as the difference of two squares, or show that it cannot be done.” Answer: “It cannot be done because 30 is divisible by 6 and a number that is divisible by 6 cannot be written as the difference of two squares.” Well, 12 is 16 minus 4. Ah…take any number that’s a difference of two squares, multiply it by 36, you’ll get a number that’s the difference of two squares and is divisible by 6, so I mean…again I could give you loads of examples of the same kind.

3. Teaching emphasis

In this study, it seemed that the majority of the teaching effort is directed toward improving the skills needed for structural thinking. The professors give rules and “templates” for constructing proofs and for writing definitions, and employ a restricted vocabulary in order to help students avoid errors that arise from inappropriate use of connectives and quantifiers. In some cases, this is in a deliberate effort to help students learn to operate in an axiomatic system.

P3: …my own idea as I said is to try to, to tell them…to try to get them to operate in a very systematic way, and systematically tell them that it’s for their own good. For example one of the purposes of having all these rules, these very precise rules, is that you know that a proof is supposed to be an argument that you give, that should convince me.

In other cases, the professor began the course with the intent of structuring it around investigations of interesting questions, but felt compelled to issue guidelines in response to student errors.

P1: …every time the students demonstrate to me that they, that some completely natural logical thing to me, that I couldn’t imagine how anybody could fail – could make this kind of logical error. But they make it. Okay. So what that does is it causes me to institute some…rule that I announce to the class. Not being able to do this, or must do this or something like this. […] Okay…and…well, the problem is the more, the longer that I teach the course, the more of these I end up putting in.

The professors do use tasks that address the other modes of thinking, requesting examples of given constructs, encouraging an “exploration” phase before an attempt at a written proof is begun, and requiring the identification of false statements and provision of counterexamples. However, these tend to be employed less systematically.

4. Interdependence of thinking modes

In my view, the most important thing to note about the modes of thinking described above is their interdependence. An outline of a proof attempt (for a general conditional statement) serves to illustrate the way in which a successful prover may move quickly and flexibly from one to another according to changing goals:

1. Structural: structural analysis to identify statement as conditional.
2. Instantiation: consider or construct one or more simple or typical example objects satisfying premise in order to check whether conclusion holds.
3. Critical: consider more examples, including different “types” or known likely problematic cases to check that conclusion also holds for these.
4. Structural: set up the first line(s) of a proof according to structural conventions, introduce appropriate objects, state that they satisfy the appropriate definitions etc. Follow any “obvious” manipulations or inferences until it is not clear what to do next.
5. Creative: try to construct an object that satisfies the premise but not the conclusion, answering the question “why is that impossible?” with “because of property X”.
6. Critical: check that property X holds for all of the objects that satisfy the premise.
7. Structural: formulate property X in appropriate mathematical language.
8. Creative: check that this is a correct formulation of property X by trying to construct an object that satisfies the original premise but not this new property.
9. Structural: re-work the manipulations in order to arrive at this property, and draw the conclusion.
10. Critical: check each deduction in the proof.
11. Instantiation: instantiate an example and follow through the proof with this example.

A focus on the skills needed for structural thinking does not seem to be misguided, as undoubtedly students need to learn to communicate using the linguistic conventions of mathematics. However, this study raises the question of the degree to which any approach to teaching proof is balanced with respect to the four modes of thinking. In particular, as recommended by Weber (2001), it seems that we might profitably investigate ways to foster the strategic use of different modes according to changing goals.

References

Dahlberg, R.P. & Housman, D.L., (1997), “Facilitating learning events through example generation”, Educational Studies in Mathematics, 33, 283-299.

Glaser, B., (1992), Emergence vs. Forcing: Basics of Grounded Theory Analysis, Sociology Press, Mill Valley, CA.

Harel, G. & Sowder, L., (1998), “Students’ proof schemes: results from exploratory studies”, in A.H. Schoenfeld, J. Kaput & E. Dubinsky (Eds.) CBMS Issues in Mathematics Education, 7, 234-283.

Johnson-Laird, P.N. & Hasson, U., (2003), “Counterexamples in sentential reasoning”, Memory & Cognition, 31(7), 1105-1113.

Raman, M., (2003), “Key ideas: what are they and how can they help us understand how people view proof?”, Educational Studies in Mathematics, 52, 319-325.

Selden, J. & Selden, A., (2003), “Validation of proofs considered as texts: can undergraduates tell whether an argument proves a theorem?”, Journal for Research in Mathematics Education, 34(1), 4-36.

Weber, K., (2001), “Student difficulty in constructing proofs: the need for strategic knowledge”, Educational Studies in Mathematics, 48(1), 101-119.