Key Ideas in the Context of a Proof from Collegiate Calculus
Manya Raman
Graduate
In discussions about the nature of mathematical proof, researchers often use dichotomies to characterize different types of proof. There are informal and formal proofs (Schoenfeld, 1991), syntactic and semantic proofs (Weber and Alcock, in press), proofs that explain and proofs that demonstrate (Steiner, 1978; Hanna, 1989). However, as pointed out by Schoenfeld (1991), the classification of proofs in terms of dichotomies masks an important, if not essential, part of mathematics—the connection between the two dichotomous poles. This paper discusses this connection and the role it plays in generating and understanding proofs. A model of proof processes is developed, building on the notion of “key idea” proposed by Raman (2003). This model is applied to describe how mathematics faculty, graduate students, and college freshmen produced and/or understood a proof from collegiate calculus.
Heuristic
ideas, Procedural ideas, and Key ideas
One type of idea used in proof production is called a heuristic idea. This is an idea based on informal understandings, e.g. grounded in empirical data or represented by a picture, which maybe suggestive but does not necessarily lead to a formal proof. A heuristic idea provides a sense that something ought to be true, but by itself does not constitute a formal proof. Heuristic ideas are often used behind the scenes—for instance, as a mathematician tries to develop an intuition for why a claim is true. With few exceptions (such as Polya, 1968) these types of ideas rarely make their way into the final exposition of what most mathematicians (and the textbooks they write and use) would call a proof. Because of this, we say that heuristic ideas constitute a private aspect of proof.
Another type of idea used in proof production is called a
procedural idea. This is an idea based
on logic and formal manipulations that lead to a formal proof. A procedural idea
demonstrates that something is true,
but is not necessarily explanatory or personally meaningful. One might be able to follow, or even produce,
all the steps of a formal proof without being able to understand it. (For
example, after producing a formal proof of a difficult theorem, the Fields Medalist Pierre Deligne said, "I
would be grateful if anyone who has understood this demonstration would explain
it to me." (Alibert and Thomas, 1991)
Procedural ideas, in contrast to heuristic ideas, generate
precisely the type of arguments found in most mathematical textbooks, journal,
and the like, though the level of rigor that one uses
to express a procedural idea may vary from context to context (e.g. one would
need more rigor for a journal article than in a discussion with a
colleague). Because procedural ideas
lead to proofs that are publicly acceptable, we say that they constitute a public
aspect of proof.
So far we have done little more than to introduce more dichotomies in characterizing aspects of proof. Proofs have either a public or private aspect; they are generated by either heuristic or procedural ideas. However, the characterization of proof so far misses what we consider to be the crucial aspect of proof—the key idea. A key idea is a mapping between heuristic idea(s) and procedural idea(s). It links together the public and private domains, and in doing so provides a sense of understanding and conviction. The key idea is the essence of the proof, providing both a sense of why a claim is true and the basis for a formal rigorous argument.
The notion of key idea, while not discussed as such, has precedents in the literature. For example mathematician Bill Thurston, another Fields Medalist, explains how he reads a mathematical paper in a field in which he is conversant:
I might look over several paragraphs or strings of equations and think
to myself, "Oh yeah, they're putting in enough rigamarole to carry
such-and-such idea." When the idea
is clear, the formal setup is usually unnecessary and redundant—I often feel
that I could write it out myself more easily than figuring out what the authors
actually wrote. (Thurston, 1994)
To the mathematician, what is important about a proof is the idea it expresses. The symbols and formalism used to express that idea are just 'rigamarole' for carrying that idea.
The problem, from a pedagogical standpoint, is that students do not view proof this way. For them, the public and private aspects of proof are disconnected (Balacheff, 1988, Raman 2003, Schoenfeld, 1985). This disconnect not only reflects an immature view of proof, but also stands in their way of generating and understanding one.
Below a model is proposed to show the role the key idea can play in generating and understanding a mathematical proof. That is not to say key ideas always play a role in proof production or comprehension. One can produce a correct proof and even have some sense of understanding, without grasping the key idea. The model only purports to illustrate the role key ideas can play. Three different situations are described, illustrated with examples of students and mathematics faculty working on the task: prove that the derivative of an even function is odd.
Before reading on, you might want to prove this claim yourself and see if your approach resonates with the examples given below.
Key
ideas in action
There are at least three ways key ideas can play a role in generating and/or understanding a formal proof, illustrated in the figure below. The first way (Figure 1a) begins with heuristic ideas, perhaps generated by looking at examples or more general exploration. Looking across these patterns, one identifies the key idea that convinces oneself personally that the claim is true. One then tries to rigorize this key idea into a publicly acceptable proof.
Figure 1a Figure 1b Figure 1c
Example: Professor A starts by thinking of examples of canonical even functions like y=x2 and y=cosx. From these examples he deduces the key idea of the proof, namely that the slope at a point x is the opposite of the slope at –x. He illustrates this property by sketching a picture of a generic even function with tangent lines of opposite slope:
Prof. A: Let's see, an even function. There is only one thing about it, and that is its graph is reflected across the axis. Yeah, and you can be quite convinced that it is true by looking at the picture. If you said enough words about the picture, you'd have a proof.
He goes on to prove the claim applying the chain rule to f(x)=f(-x), the procedural idea which gives him the proof.
The second way key ideas can be involved in proving (Figure 1b), is to begin by producing a formal proof, then understanding the key idea, and finally trying to get a deeper sense of the underlying ideas by looking at examples and/or diagrams.
Example: Professor B is able to produce a proof using the definition of derivative without recourse to specific functions. However, he knows that the formal definition could connect to examples through the key idea. When he looks at the graphical argument he sees that it is essentially the same argument as his proof using the definition of derivative.
Prof. B: If I were going to use that picture, I would take it and turn it into a proof. Although if you do that, it comes down to pretty much this (#3).
Later in the interview he is asked which proof argument (empirical, graphical, definition of derivative, or chain rule) demonstrates the best understanding. He responds:
Prof.
B: Depends on understanding of
what. Understanding of why even
functions have odd derivatives, in some sense the picture does that best.
Even though Prof. B is able to generate a proof without diagrams, he sees that picture (which presumably he could produce himself) provides a deeper sense of understanding, and that it is connected to the formal proof through the key idea.
The third role of key ideas (Figure 1c), is to connect the heuristic and procedural ideas. One could start with either informal explorations or formalizations. It is not immediately clear how any of the heuristic ideas lead to proof, nor how to proceed with the formal proof. One goes back and forth between the informal and formal approaches, and if fortunate, one finds the key idea that connects them. This is how most of the undergraduates in the study proceeded. However, few of the students succeeded in identifying the key idea and they fall short of producing a formal proof.
Examples: Student C starts by taking derivatives of polynomial functions which gives him a strong sense that the claim ought to be true, however he realizes that his examples do not constitute a formal proof. He then tries to prove the claim using the definition of derivative, but he soon gets stuck. Later he compares his two approaches.
Student C: So my understanding of derivative is that you subtract the power by one. Right, so if you have an even function, the power is even, so it always comes out to be odd. That's my… my intuitive understanding of the problem. And then… I don't know… I tried to get somewhere, but I really couldn't, so I just write down the formula for the… I guess the definition for what the derivative is. So, that's what I have. And I couldn't go anywhere from there.
This is a case where a student tries to connect the heuristic and procedural ideas, but fails to do so. In the next example, a graduate student also tries to see the connection between his (successful) proof using the definition of derivative and a graphical argument. We see below his construction of the graphical argument as well as his realization that this argument is in essence the same as the formal proof.
Grad. D: Any simple geometric reasons for as to why
this happens? I'm trying to think… […] I
mean maybe, if you look at a parabola, then it is sort of a flipped version… I
guess as you take this limit then what is happening to your slope lines is that they too are becoming inverted. They are just getting flipped around in a
sense and that makes them negative. […] It is a fine argument and it can be rigorized very easily.
In fact it is quite rigorous. It
is in essence nothing more than this (points to his formal proof).
Once Grad B notices that symmetry of the parabola results in tangent lines of opposite slopes—the key idea—he sees the connection between the graphical argument and the rigorous proof.
Conclusion
Key ideas are what connect the private and public aspects of proof, that is to say the informal, heuristic ideas that provide a sense that a claim ought to be true, and the procedural ideas that can help one produce a formal, rigorous proof. Key ideas can play an important role in producing and generating proofs, as illustrated above. If one does not know immediately how to produce a proof, one can start by looking at examples, then identifying the key idea, and then converting that idea into a formal proof. And even if one can produce a proof using procedural ideas, such as writing out definitions and manipulating symbols, one gets a deeper understanding of the proof by seeing how it connects to examples, diagrams, and/or other informal representations.
Often when one is trying to produce a proof or gain a deeper understanding of a proof, one goes back and forth between the informal, heuristic ideas and the formal, procedural ones. The goal of this ping-ponging should be to identify the key idea that links the informal and formal realms. However, it turns out that finding a key idea is difficult for many students (and even distinguished mathematicians like Deligne!) So it is incumbent upon us as teachers to help students not only to identify key ideas, but also to see that key ideas are in fact the grail that we seek.
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