In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs:
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Students see proofs in lecture and in the textbooks, and proofs are explained when necessary, for example, the first time the instructor shows a proof by induction to a group of freshman, some additional explanation of this proof method might be given. Also, students are given regular problem sets consisting of genuine mathematical questions – of course not research-level questions, but good honest questions nonetheless – and they get feedback on their proofs. This starts from day one. The general theme here is that all the math these students do is proof-based, and all the proofs they do are for the sake of math, in contrast to:
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Students spend the majority of their first two years doing computations. Towards the end of this period they take a course whose primary goal is to teach proofs, and so they study proofs for the sake of learning how to do proofs, understanding the math that the proofs are about is a secondary goal. They are taught truth tables, logical connectives, quantifiers, basic set theory (as in unions and complements), proofs by contraposition, contradiction, induction. The remaining two years consist of real math, as in approach 1.
I won't hide the fact that I'm biased to approach 1. For instance, I believe that rather than specifically teaching students about complements and unions, and giving them quizzes on this stuff, it's more effective to expose it to them early and often, and either expect them to pick it up on their own or at least expect them to seek explanation from peers or teachers without anyone telling them it's time to learn about unions and complements. That said, I am genuinely open to hearing techniques along the lines of approach 2 that are effective. So my question is:
What techniques aimed specifically at teaching proof writing have you found in your experience to be effective?
EDIT: In addition to describing a particular technique, please explain in what sense you believe it to be effective, and what experiences of yours actually demonstrate this effectiveness.
Thierry Zell makes a great point, that approach 1 tends to happen when your curriculum separates math students from non-math students, and approach 2 tends to happen math, engineering, and science students are mixed together for the first two years to learn basic computational math. This brings up a strongly related question to my original question:
Can it be effective to have math majors spend some amount of time taking computational, proof-free math courses along with non-math majors? If so, in what sense can it be effective and what experiences of yours demonstrate this effectiveness?
(Question originally asked by Amit Kumar Gupta)
Best Answer
This is a great question. In fact, I hope people won't think it over-dramatic if I call it one of the great math education questions of our time.
At the University of Georgia, we have decided as a department to follow the second approach: we offer a course Math 3200: Introduction to Advanced Mathematics. This is one of our three "transitions" courses, the others being (Math 3000) linear algebra and (Math 3100) sequences and series. But this is not to say that the faculty here are unanimously enthusiastic about approach two: in fact I have heard more dissent than agreement among the (mostly young, as it happens) faculty with whom I have discussed the matter.
I myself taught this Math 3200 course twice in recent years: here is my course webpage (don't get too excited: it only gives a very limited picture of what the course was about). I was somewhat bemused when I taught this course for the first time, since this is not a course I have ever taken. For instance, we spend about three weeks of the course on mathematical induction, a topic which I learned in high school. (More precisely, I learned about it during a self-paced summer Algebra II course I took through the CTY program after my freshman year of high school. It wasn't until years after that I began to understand that -- in that I actually read, did problems on and was tested on the entire Algebra II book -- I actually learned rather more than what takes place in an actual Algebra II course even at my (very good) high school.)
And yes, the course began with a chapter on logic: truth tables, contrapositives, negating statements, and so forth. I was surprised to discover that many of my colleagues found this material to be dry, pointless and difficult to teach. (Some of them even affected not to be able to easily solve some of the logic problems that appeared on later exams. I do think this was an affectation, and a curious one.) But for my part I very much enjoyed teaching the course and most certainly did not find it a waste of time: spending say, two weeks setting up logic is a small price to pay for being able to expect that students will not confuse the converse with the contrapositive for the rest of their careers. And I confess that I did not in fact find it boring: I remember deciding at one point to draw one big table with all $2^{2^2}$ different binary connectives and ask the students to give the simplest description they could for each one. This took most of a class period, but compared to, say, finding the rate of change of the length of some guy's shadow at the instant he is 10 meters away from a lamp post, it was great fun.
I believe this course was very useful for the students: it is nice to have one course where one can spend as much time as one needs concentrating on the processes and methods of proofs themselves, rather than on proving particular theorems. (Which is not to say that we didn't prove anything at all: there was a unit on divisibility and another on modular arithmetic, for instance. When I have taught undergrad number theory, I assume that students have seen this material twice over: in this course, and then again in the required semester of abstract algebra, and I really don't cover it again.) Moreover there was time to concentrate on the students' writing in particular, and may Gauss strike me down if the writing didn't improve from horrible to halfway decent throughout the course of the semester.
This course is certainly not appropriate or helpful for all undergraduate math majors. For instance, we offer one section of Honors Calculus a la Spivak per year (I have the good fortune to be teaching this course next year: a year free of lamp posts!) and I think that students who do well in this course learn everything that we would like them to learn in the Math 3200 transitions course and more. But for a certain level of student -- a level that can be trained to do well as an undergraduate math major -- this course works very well.
Added: after rereading the question, I want to make clear that the above long answer is not an argument for option 2. versus option 1. Option 1. -- i.e., include proofs in all university level math classes, presumably in an increasingly sophisticated way as the classes progress -- which in my understanding is standard in most European university curricula, is not even an option on the table at my (and, I think most) American universities. (I was an undergraduate at the University of Chicago, and that was definitely an exception to the rule. Not only did I have classes which concentrated somewhere between primarily and exclusively on proofs from the very beginning, but in fact all calculus classes there insist on treating some theoretical aspects, including about a month of class time on epsilon-delta proofs.) So my answer takes as a given that there is a transition being made from almost exclusively computational courses to somewhat theoretical courses. Given this, the question is whether this transition should be done in exclusively in the context of content-based courses (e.g. linear algebra with careful definitions and proofs), in the context of an "introduction to proofs" course, or both. At UGA, our answer is "both". What I am saying that in my opinion the "introduction to proofs" course is not a waste of the students' time. Some others feel differently.