Yesterday in class we were talking about similar triangles and indirect measurement, so I brought several tapemeasures to class, and got groups to work through the process of measuring a distance by measuring the apparent length of a meter stick placed at the end of the distance to be measured. I took the activity out of the activity manual that's one of the prescribed texts for the course, and modified it so that they did the calculation in 3 different ways. I was mainly curious to see what they thought of it - I didn't think it was a particularly well thoughtout activity. But the response was quite positive, with lots of discussion around getting the correct sketch of the problem, and how to solve with each method. Also a good discussion afterwards about how to make it a better posed activity, and ways to make it appropriate for different age groups.

Today we talked about the link between geometry and algebra. What I found interesting was how absorbed they became when I was telling the story of the ancient Greeks, and how they felt about mathematics - that for them mathematics was a part of their religion and philosophy, and ideas about 'ideal shapes' influenced how they thought about and did mathematics. I mentioned the platonic realm, where 'ideal shapes' live, and one student suggested that that's probably where good mathematicians go when they die :-)

There's a very different look on students' faces when they're absorbed as they were today, and when they're concentrating but not really engaged in a class. I always try to make every class a story, with exciting bits, and unexpected twists, and an overall story arc. I've been more concious of doing this since reading 'Mathematics and humour' several years ago. The guy who wrote it does an interesting analysis of why you'll often see mathematicians smiling or laughing at the end of hearing a proof; essentially, a good proof has the same structure as a good joke, with an unexpected or pleasing punchline. I've tried to build on this idea when planning my classes, so that there's a logical and compelling structure underlying the material that will pull the students in without them even realising it. Sometimes it even works! Today we'd got as far as building a number line with integers, only using geometric construction. I asked the class how we could put fractions on the number line. There were no ideas, but one of the students said excitedly 'Tell us, tell us! I can see you have a cunning plan! What is it?'. He really was excited to know! And when I showed them, there were a lot of ... giggles ... really, because it was a cute construction.

One of the students was a maths major, so he's pretty comfortable with the material. We've been doing some rather cute proofs with unexpected twists, and he has the endearing habit of giving a fencer's 'Sa-sa!' when we get to the punch line. I really love this; I'm not sure that the rest of the class gets appreciates it, but I love that there is one person who is seeing the elegance of the game.

After class one of the students stayed behind to tell me that she gets very frustrated when the maths major answers questions in class, because she can't understand what he's saying, so she has to ask me to answer the question again, which makes her feel stupid. It's only relatively recently that he's started answering questions in class, and he's obviously trying hard to be comprehensible, but still hasn't quite geared down to elementary math from university math. When he gives an aswer I try to guide him to be more comprehensible by asking appropriate questions, but obviously it's not helping enough. I don't want to discourage him from speaking - after all, he needs to learn how to answer question appropriately, so I'm not quite sure what to do to avoid frustrating the other students. This is a tricky course to handle: teaching the mathematical content, the methods of teaching the content and on top of that how to actually teach the content! I wish there was a course I could take on the maths education education!