I meant to make this year’s edition of my materials on how and why we do proofs available from the page
some time ago, but I have just realized that a lot of the links were broken. Hopefully this is now fixed (please let me know of any problems).
So what are these sessions about? These are some optional extra sessions I run for the second-years to partially address what I perceive to be weaknesses in their motivation, confidence and fluency when it comes to writing proofs. Of course, students really need to follow this up by putting in a lot of work understanding proofs from lectures and constructing their own proofs, using question sheets to help them to practise.
In Why do we do proofs?, the idea is to address some of the motivational issues. In particular, the idea is to convince students that the answers to mathematical questions may be neither obviously true nor obviously false, and that clear definitions are important: indeed, the answers to your questions will depend heavily on which definition you are using. On this point, I quote the last example from Why do we do proofs?
- Are squares rectangles?
- Are equilateral triangles isosceles?
- Which of the following numbers are prime?
If you listen to the audio, you will notice that the students were not happy with my answer to 3(c). In my view, is not, officially, a prime number, but it is a prime element of the ring of integers. Thus the answer depends on how you interpret the question.
The next two sessions (How do we do proofs? Parts I and II) concern how to do proofs. My primary focus in these sessions is on the “routine” portions of doing proofs, especially use of the word “let” and making deductions from definitions. This may not be the most exciting part of doing proofs, but unless students are fluent with these routine bits, they will struggle to get far enough in other problems to have a chance to be imaginative.
I asked for feedback after these sessions. The last proof in the first of these two sessions on doing proofs was too easy this year (2008-9), so I’ll need to put something else in there. Otherwise, the students claimed to have more confidence than before after the sessions. In particular, they claimed that they had a better idea of why lecturers did what they did during proofs. However, some of them would have liked more sessions, and others would have liked some booklets full of similar examples to take away and look at.
These sessions have not “solved” the problem. Students are still not fluent in the appropriate use of definitions and “let”. Nevertheless, I feel that I have made some progress on the motivation side. I used to receive a lot of feedback on my evaluation forms about how pointless and boring all the proofs were in second-year mathematical analysis. I get very little of this now. Instead, I mostly get positive feedback on the additional comments I give during proofs in lectures explaining WHY we are doing what we are doing at a particular point of the proof.
Ideally students would already be fluent in these routine parts of mathematical reasoning by the end of their first year. However, there is always competition for the available time between covering a large amount of essential background material, and practising various specific aspects.
Three sessions on how and why we do proofs is, of course, not very much. However, the requested take-away booklets of additional problems might well help some students.