I recently received a request for help with an exercise in a text book (I guess this happens when your profile suddenly becomes higher). The exercise was a dressed up version of a children’s game. The idea is to take one word and transform it into another by changing one letter at a time. For example, to change DOG to CAT we can do the following: DOG COG COT CAT Obviously longer words make the game more complicated but my interest in this is that it is a great way to get students talking about proof and doesn’t involve any new mathematics getting in the way of their understanding. I got the idea from Ian Stewart’s Nature’s Numbers. In Chapter 3 he talks of the necessity of proof in mathematics and to show this uses the above game played with changing SHIP to DOCK. (How this game is connected to proof is given at the bottom of this post.) You can try this yourself. One possible answer, given by Stewart, is SHIP, SLIP, SLOP, SLOT, SOOT, LOOT, LOOK, LOCK, DOCK. One year I set the problem of getting the SHIP to DOCK to my group of six tutees during their first tutorial in their first week of university and left them to play with it while I located some chalk. When I returned some of them had done nothing – not one letter had been changed and progress had not even been attempted. The reason given by one of the students has stuck in my mind ever since. He shrugged in bewilderment as if I had asked him to perform brain surgery and said “But I’ve never seen this before”. He honestly could not see how he could solve such a problem without having...

## How to get a good maths degree...

posted by Kevin Houston

I recently taught complex analysis to our second year students. One particular problem jumped out whilst marking the exam. One question was “Define the length of a contour.” This was only worth about 2 marks and the bulk of the students got it mostly right. My point is that students’ responses can tell us something about how they see mathematics and perhaps how they do mathematics. The main mistakes were 1. Not giving enough information. 2. Not being mathematical. 3. Giving the procedure. 4. Trying to memorize without understanding. Let’s deal with these in turn (the most important is number 4!). Number 1. Not giving enough information. A good answer to “Define the length of a contour” is “Suppose that is a contour. The length of the contour is .” Instead many students slapped down . They lost a mark because they didn’t tell me what , and were. This happens a lot, students focus on the equation and forget about the surrounding information. If I did not know the definition of a contour, then the equation doesn’t tell me enough. I wouldn’t know where the , and were coming from and their relevance. 2. Not being mathematical. Another problem with definitions in general, not just this one, is that students give a hand waving definition, e.g. “It’s the actual distance that the curve moves.” This is not very mathematical and would not help anyone understand length except in an intuitive way. (In this case you could probably guess from the name that length is to do with distance!) 3. Giving the procedure. Another very common mistake with definitions is confusing the definition with a procedure used to calculate the object defined. For example, “Define the order of a pole” is often incorrectly answered...