How I teach students to do proofs (and it’s not how I do proofs).
As in many math departments we have a ‘transitions’ course in which math majors are introduced to logic and mathematical proofs. The key feature of such courses is the students get to do lots of proofs and, hopefully, are prepared to do abstract math/proofs in the rest of their advanced math courses. For my own sake mostly, here’s a description of how I approach teaching students to do proofs.
We start with proofs after we’ve done some work with logic: truth tables, or, and, negative, statements, variable statements, contrapositives, etc., and discussed definitions. We’ve actually done some proofs already via truth tables, but those are so exceptional that I don’t really count them as proofs. Here are the main ideas/steps I use, in approximately the order we work through them.
Step 1: A proof is an ordered sequence of true statements (definition). You might be expecting more, but that’s where I start, as the first point I want to make is what it takes to not be a proof. One false statement, and you are out. I also make the point that every proof, proves something, just maybe not what you want it to prove. So now I’ve set the stage for reading and evaluating proofs: (1) is every statement true? and (2) what does the proof prove?
Step 2: Every statement in a proof is true for a reason. Initially I have them use two column proofs with every statement (column 1) having a reason (column 2). These are simply proofs with sets or basic number theory. You’d be surprised how hard it can be to come up with reasons (or at least names for reasons) some times. As the frustration with the tedium of 2-column proofs builds, we begin discussing the purpose of a written proof (to convince the reader of the truthfulness of some statement) and how different communities establish the standards for a complete proof. We soon move to paragraph proofs, but the students know that they have to be able to support every statement they make.
Step 3: A proof communicates its results clearly. As we move to paragraph proofs, the emphasis shifts to clear communication. At times I’ve used a 3 part rubric with one part being purely on the quality of the communication, grammar, etc. I’ve also had them read different proofs and evaluate and discuss their clarity. For practice, they can also rewrite some of their two-column proofs in excellent paragraph style.
Step 4: The structure of the proof reflects the statement being proved and vice-versa. By this time we’ve moved past sets to field axioms, functions, and other topics and so they are seeing many different types of statements to prove and many different types of proofs. So I start asking them to look for the connection between the statement, the definition involved and the form of the proof, and I expect them to make the match. For example, when we do proofs for equality of sets, we know, by definition, that two sets are equal if each is a subset of the other. Thus every ‘set equality proof’ is structured as two subset proofs (‘subset-subset’), and a subset proof is a class if-then proof, which has it’s own specific structure. In the big picture, this is a fairly limited idea, but for the proofs we do, it is adequate, and, in any case, it is always a reasonable place to start. It also gives us a new way to look a definitions, to try to determine what proof structure the definition evokes.
Step 5: A proof is read and judged within a community with its own standards. This first came up in Step 2, but I bring it back towards the end and spend more class time discussing all the aspects that we’ve covered and what we as a class (our community) has set as the standards for proofs. This step helps them begin the movement from highly technical and verbose proofs, to more concise proofs that tell the true (no pun intended) story. I try to move out of my position as ‘arbitrator of truthfulness’, but am not always that successful at it.
Step 6: A perfect proof is a thing of beauty and you should be proud of having created it. The last part of the course includes each student creating a portfolio of 20 or so, ‘perfect proofs’. The perfection reflects what we’ve done in class in clarity of composition, correctness, and the telling of a good story. They can submit versions of a proof until they get it right, and, at the first, I am very picky about even the tiniest of punctuation or grammar errors. The reward for a complete proof is it gets stamped ‘Completed’ and initialed and it goes in their notebook. There is often much celebrating when the 2nd (or 3rd or ..) draft gets approved.
I do other things, and I have ideas for the next time I teach the course of other things I want to try (like including more experimentation and conjecturing for forming statements), but this is the essence of my course.