Descriptive Thinking vs. Formal Writing

Question

Sometimes I come across some exam answers which describe a proof sketch or a counterexample very well but are not written formally. Such proofs show that a particular student understands the general picture of the question completely. Furthermore in the case of counterexamples sometimes these descriptive proofs are really interesting and one can find essentially new examples to refute a particular conjecture (or a possible theorem without one of its essential assumptions) using such descriptions.

In the usual process of the course, I encourage my students to understand the general picture of a question/phenomenon and describe it sketchy on their own. I appreciate this approach too much. But I don't know what I should do when I receive such creative descriptive proofs of my best students in an exam, especially when some of the other students write the usual formal proof correctly.

I am worried about "forcing" my students to write their true ideas in exams too formally because I think it could affect their motivation for being creative. On the other hand, formal proofs are necessary parts of formal mathematics and they should adopt themselves with this culture.

Question 1. What should I do with inexact true descriptive proofs in exams? Should I give them a full mark (which I think such proofs deserve it) or considering a full mark could be harmful to encourage them to be more formal?

Question 2. How can I encourage my students to write their ideas as formal as possible but think about them descriptively?

Answer 1

I have come to view mathematical proofs as legal briefs, intended to convince a certain audience that the theorem is true. The degree of formality very much depends on the target audience. To echo Jim, the argument should be "coherent and convincing" to that audience.

One phrase that encapsulates this view is: mathematical proofs are social constructs. I first encountered this viewpoint in a paper published by three theoretical computer scientists that caused quite a stir at the time (1979):

De Millo, Richard A., Richard J. Lipton, and Alan J. Perlis. "Social processes and proofs of theorems and programs." Communications of the ACM 22.5 (1979): 271-280. (PDF download link). "We believe that, in the end, it is a social process that determines whether mathematicians feel confident about a theorem."

Discussions of this topic continue to this day. Here's a 2012 blog entry addressing the issue:

Cathy O'Neil. What is a Proof?: A proof is "a convincing argument of why you think something is true....It is the responsibility of the person proving something to convince his or her audience that it's true."

Answer 2

This depends to a large extent on the background of your students, which varies a lot from country to country.

In the United States, many students who take calculus have not seen any formal proofs before, while others had some limited exposure to two-column proofs in geometry class. In this context, it is nearly impossible to require students to consistently write formal mathematical proofs during calculus. At this stage in their mathematical development, students are only starting to become aware that proofs are required in mathematics, and you should be happy when students write arguments that are informal, but convincing.

Part of the reason for this is that calculus is a course that's primarily about calculus, not about formal proofs. Typically calculus textbooks are relatively low on formal proofs, and do not convey the sense that the entire subject takes place within a formal framework. Indeed, most calculus textbooks include a large number of relatively informal arguments designed to help the students' intuition for the subject.

In other countries, it is much more common for calculus students to have experience with formal proofs in their previous math classes, in which case it may be reasonable to expect student proofs to be both convincing and formal. The same may be true in the United States for students in an honors calculus class -- even if such students don't have much experience in proofs, it is possible for very talented students to learn how to write formal proofs within the context of calculus, especially if this pedagogical goal is supported by both the professor and the textbook.

But for most students in the United States, writing formal proofs is a skill that will need to wait for later math classes. Many colleges have a proof-writing class for sophomores for precisely this purpose, while others emphasize proof-writing in linear algebra and subsequent classes.

In any case, my recommendation would be to accept as correct any argument that is coherent and convincing. Think of it as more like an essay question -- it's fine to deduct for poor writing, but any mathematically correct argument that's written in an understandable way should be given full credit. Students need to get used to making arguments like these before formality can be added into the mix.

Answer 3

If I had to simplify this problem to one question, then what I would ask myself is:

                                        To what extent the student convinced me that he
                                          knows and understands the required material?

For a class that required formal proofs (e.g. it was stated in the task) writing just descriptions wouldn't convince me at all. On the other hand, if we were studying something else, and the description was accurate enough (e.g. there were no ambiguous expressions, no guessing and the mental structure was clear), then the student would receive a perfect score (for this part).

In fact, if there is a proof with a clear intention, then this is all well; even if the student would make some typos, I would fix it for him. However, for a correct formal derivation that contains a lot of "noise" (unnecessary formulas, inadequate diagram, etc.), I would ask the student to explain and only after that I would acknowledge his solution (or not, if he does not understand his own work). Of course, there might not be enough time to do this in large courses.

Your second question comes, in my opinion, from an entirely different area. Many mathematicians consider something done once they know the proof, there are even jokes and anecdotes about it. To make students want to write a formal proof, you need to make it worth it. The downside is that requiring a formal proof decreases the incentive to think first in intuitive terms. Moreover, formal proofs need much more work than descriptions and there might be not enough time during an exam (or you might have to give easier problems, but this is undesirable). Perhaps you could make it into two assignments graded separately, and the formal derivation could be done at home.

I hope this helps $\ddot\smile$

Answer 4

It depends on how you have dealt with these proofs during the course. You have written, that you appreciate the approach of describing proofs in sketchy way. In that case, you should accept any coherent proof in exams, regardless of its formality.

If formality was part of your course, you should give part of the points to coherent, informal proofs.

Catching up Jim Belks evaluation:

In Germany, formality is mostly dealt with by the Algebra profs along Algebra itself whereas Calculus profs have to deal with the conceptual difficulties of inequalities and infinitesimal math outside of pure limits.

Answer 5

In an exam (unless specifically about formal proofs) I'd want a reasonably coherent, even if rather informal, argument. In an exam I want to check general understanding (there is the time limit, and you want to cover a range of subject matter). Homework is another kettle of fish, there I expect orderly, reasonably formal arguments. No time pressure, can check other sources (citing/acknowleding them!), can redo the work as needed.