# Teaching the use of pictures in proofs

Today I realized that many of my students in an upper-division undergraduate (projective) geometry class don't really understand how to use pictures in proofs. Some draw no pictures at all, making their proofs nearly impossible to read; some oppositely draw a picture but don't explain it at all; and others try to explain their picture but get confused about things like which elements of the picture are givens and which are constructed.

Now that I've noticed this problem, it seems obvious and that I should have expected it. Probably no one has ever taught them how to use pictures in proofs! I can't remember ever seeing this topic discussed in a logic textbook or an introduction-to-proofs class. And the correct use of pictures in proofs is actually a very subtle thing, e.g. how a picture should be "generic" so as to not give false intuitions.

Can anyone point to any good resources (books, papers, web pages, videos, whatever) that teach students how to use pictures in proofs?

• I can point to a not-so-great resource [in my estimation/memory] related to pictures and proofs: The book Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Picture underwhelmed. Commented Oct 8, 2015 at 23:30
• Hmm. I'll think about it. I remember my finite geometry lecturer / PhD supervisor talking about the need for "generic" pictures. I'll see if his textbook mentions it at all -- might take me a while to look it up though. Commented Oct 9, 2015 at 8:54
• Hardly surprising, though, considering how strongly the idea that "a picture is not a proof" is pushed at school. Commented Oct 9, 2015 at 8:54
• Teach by example. I remember my foundations professor (who I had for subsequent analysis courses) writing proofs with illustrations on the chalk board. I certainly picked up my methods for drawing general figures from those lectures. Commented Oct 13, 2015 at 12:40

The first thing comes to (my) mind is Proofs without Words I (Exercises in Visual Thinking) and II (More Exercises in Visual Thinking).

Also you (not your students) might find interesting the section "Visualization and Diagrammatic Reasoning in Mathematics" of the chapter "Proof: Its Nature and Significant" of the book "Proofs and other Dilemmas: Mathematics and Philosophy".

I'll try to add to this list later on.

Again you might find this one worthy of reading: "Visual Thinking in Mathematics An Epistemological Study"

The videos used in this personal website about projective geometry might be handy to show your students some pictures hoping that they use them in proof.

This is not an answer to your question (sorry!), but rather two example books that to me illustrate(!) the power of pictures in proofs:

(1) Tristan Needham, Visual Complex Analysis, Oxford Univ. Press.

(2) Nathan Carter, Visual Group Theory, MAA.

• I have thought about buying the Visual Group Theory before. Is it worth the money? I have an old Frayleigh book.
– Karl
Commented Oct 9, 2015 at 19:37
• I love it, but I'm a fool for illustrations. :-) Commented Oct 9, 2015 at 19:49
• I love Visual Complex Analysis, but I wouldn't think it is a good place to learn about the use of pictures in proofs, because it is deliberately written without any attention to rigor in proofs at all. Commented Oct 9, 2015 at 23:09