Mathematical Structures: the aftermath

The academic year finished off with two rather different events: my LMS–Gresham lecture about the Mathematical Structures course, and marking the approximately 270 scripts.

The LMS–Gresham Lecture

Last week I gave the annual LMS–Gresham lecture. When they asked me to do it, a year ago, I was writing the course material for Mathematical Structures, and couldn’t think of anything else, so I said I would talk about that. In fact, I believe that a cohort of talented and passionate mathematicians is important – why else would I be doing this job?

Anyway, the lecture went well. You can see the slides here; the slides don’t have all the ad libs, or the questions afterwards. After the lecture, we went for a pleasant meal in the nearby Indian restaurant.

Incidentally, I went to the LMS–Gresham lecture last year to see how someone else (in this case Bernard Silverman) did it. But this year I have no idea who my successor is.

Marking the exam

The good news here is that the students seem to have done rather well in the exam. Of course there is lots of monitoring and standardization to be done yet, but it looks as if both the median and the average mark are just above the B/C borderline. The students have done me proud. (I did wish, though, that I had a little stamp saying “An example is not a proof!”)

However, I noticed a couple of curious tendencies:

• Having given a counterexample to one statement, people tend to hunt for a different counterexample to the next, even if the same one works (for example, if the second statement is the contrapositive of the first). It’s as if the counterexample has exhausted its potency on one statement.
• When asked whether several properties hold (e.g. is a relation reflexive, symmetric, transitive?), there is a tendency to mention only the ones that do hold. We all have a mindset that inclines towards the positive, but to test a hypothesis you have to look at possible negatives.

Here is a selection of things written by students which caught my eye. These are not in any sense a representative selection. Some of them show a lack of understanding which you might find worrying; I would excuse a lot of this on the grounds that in the stress of an exam you will almost certainly write things that with calm consideration you wouldn’t. Some contain good sense hidden under poor expression. Some show original and creative thought. Some of them show that the students have picked up my passion for mathematics! I have sometimes lightly edited these.

• √2 must be irrational since it falls between √1 and √3.
• “Clearly” cannot be used in a proof since nothing is clear without a proof of it. [This was a common reaction. One candidate said “You cannot be lazy and write “Clearly P(0) is true”, you must prove it. It may be clear to you, not always clear to the reader”. Bravo!]
• A set with no elements contains a single element which is the empty set.
• [After working out the cases n = 1,2,3] So it seems the formula is correct. However the above is not a proof.
• m² is also even, since anything squared becomes even.
• If “less than” is related to ℕ, then ℕ is related to “less than”.
• Hence it is shown that P(n) is true for P(n+1).
• The proof does not include a conclusion box ☐ to indicate the end of proof [and is therefore invalid].
• Therefore A is infinitely countable. [This quite common.]
• The relation “less than” on ℕ is not reflexive since a is not related to a for most natural numbers a.
• We need to prove by rejection.
• Suppose √2 is a rational number. Then it can be written in the form m/n. But we cannot write √2 in this form, so contradiction.
• The statement is not true as this cannot be proven. The contrapositive is not true as the statement is often true.

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