# Rigorous proofs vs. illustrative examples

No one would argue against the idea/ observation that proofs are very important in mathematics. Some people are trying to make their notations on a blackboard during a lecture as consistent as possible with their course notes or textbooks; including full reproduction of rigorous and detailed proofs for all theorems, corollaries and so on. On the other hand, some believe that the theory with detailed proofs should be in textbooks, while it is crucially important to explain main ideas via examples. So, my question is:

How do you think, what is more important at a lecture: 1) to give a rigorous proof or 2) only formulate a statement and give an intuitive understanding via illustrative examples?

P.S.: I understand that it would be great to do both. But in real life we have natural time limitations. And feel free to edit this post because of my bad English :).

• I think this is a nice question but too broad as it is. Usually you would mix in different proportions in almost every course. Apr 10, 2014 at 19:24
• @Toscho: I edited the post. Thanks. Apr 10, 2014 at 19:32
• @AndrásBátkai: But i didn't know how to make it more specific. I'm thinking about a general strategy for math courses. Apr 10, 2014 at 19:37
• I would always go for intuitions first, if it is possible. The problem with it is that people think differently and intuitions are not directly transferable. In other words, sometimes formal proof might give students more intution than your explanation. Some reasons are outlined in this answer, you can also find a few concrete examples here. Apr 10, 2014 at 22:38

I like vonbrand's emphasis on "think of what you are trying to accomplish." If the goal is to have the students writing proofs, doing a proof in class is not actually about the theorem, it's about you demonstrating the process of writing a proof.

Often the goal is conceptual understanding, which means that an illustrative example, that hopefully has access so that the students can play with it, is often far superior to a proof. The exception, to me, is a good constructive proof that gets at the mathematical structure. An existence proof by construction, for example.

As the OP points out, time in class is precious! Don't spend it on coverage, spend it on engaged activity.

• +1 for the first paragraph and I'd also give an extra +1 for the last paragraph, if I could: "Don't spend the course time on coverage, spend it on engaged activity." Apr 10, 2014 at 20:36
• Interesting answer. And as for the last sentence, I heard onсe (but I forget where) something like: "don't try to COVER the material, but try to UNCOVER the main idea". Apr 10, 2014 at 20:53

What your class should aim at is understanding. There are some topics (theory of automata, for example) where most proofs of intuitively obvious results are long-winded multiple inductions. In such cases, stick to intuiton on the blackboard, remind students they should at least glance over the proofs, and hopefully follow one of them in detail.

If the result has the happy coincidence of an understandable, short proof, go for it on the blackboard.

Also think of what you are trying to accomplish: A course for non-mathematicians should emphasize intuition and applicability, a course for specialists should dig into proofs. The "understanding" alluded to above can happen at very different levels.

Although, you're aiming at undergraduate, I'd like to point out some aspects of secondary:

Your task as a teacher/prof/lecturer is not to write things at the blackboard, that the students can read in a book. A book is far cheaper than you are. Your task is to make the students learn/understand (a.k.a. teach):

1. foundation of knowledge of your subject
2. basic methodology of your subject (in math usually: finding new results, prooving new results, apply new results + quality of definitions)
3. aims, applications, self-concept of your subject
4. where and how to further their knowledge if they are interested in

So into what aim does a proof fall? Depends on the proof:

• If the proof itself is a classical example showing the fundamental technique, then do it. If it takes long, then do one of this kind and let the students find a similar one for another result. (Then you're teaching 2.)
• If the proof itself is boring/technical/extremely long, but the result is important, then leave the proof out but illustrate the result in its importance/applications/whatever. (Then you're teaching 1. or 3.)
• If neither proof nor result itself are interesting, then simply name the result and a source for it and the proof. The students should read that source, try to understand it and question it. Check that the next course. (Then you're teaching 4.)

My experience in Linear Algebra: while trying to explain how the co-ordinates/matrix change for a different basis I started with a 3d-rotation matrix for a basis consisting of a vector in its axis and two vectors orthogonal to it; then carried out the appropriate computation for similarity change to express it for standard basis. Filling out a board with $a_{ij}, b_{kl}$ etc would have been messy. I generally do more illustrative computations than a formula with lot of notations.

It's not an "either or".

I think there should be two branches of mathematics taught; theoretical and computational. They should have their own chain of prerequisites.

• With an expansion this can be interesting. Apr 11, 2014 at 2:51
• Welcome to the site, Shannon! I agree with Fantini that you could make this into an excellent answer. Let me know by commenting here if you elaborate on the answer, so I can come back and upvote it. Apr 11, 2014 at 3:34