# Teaching Probability in China

As I’ve mentioned in passing in a few recent posts, which have tended to focus on Markov chains more than might perhaps be expected, I’ve just finished a two-week trip teaching at Linyi University in Shandong province, about 600km south of Beijing. I’ve been describing some of my touristic and social observations of China on my personal blog, but it has been mainly a mathematical adventure, so I thought I would describe some of the main cultural differences in the approach to science here. I was surprised to find these a far larger obstacle than the more obvious language barrier in the classroom.

1) I think many of us were surprised by how different the gender divide was at Linyi to what we are accustomed to in Cambridge. Although of course it varies from year to year, from course to course, and in select cases from lecturer to lecturer, in general there is a significant male majority in lecture halls. In China, amongst my students (who had just finished their third year), the ratio seemed to have been reversed and then further extremised. I had about twelve students in attendance during the middle of the course, and only one was a boy.

2) Perhaps the most clear reason for this reversal of the Western trend became apparent when I asked some of my group what they were planning to do after leaving university. As most of them have just a year left, they all seemed to have a fairly concrete idea of what their plans were, and the answers were perhaps unexpected. One of the most able girls said she wanted to continue with postgraduate mathematics, but all of the rest wanted to become teachers. And not just high school either. In fact, all bar two or three planned to teach middle or even elementary school. I don’t want to get burned by talking about things I don’t much understand, but I understand this is very different to the UK. Here, I do not believe many maths graduates from top universities (in a national context) have ambitions in primary education. I do not know which way round the causality works. Are they encouraged towards a pure science degree as a route into schools, or are teaching jobs viewed the natural job to follow a hard quantitative course? In either case, it sounds like an excellent situation to be in from the point of mathematical education.

3) The role of the lecturer in China evidently carries a great deal more gravitas than comparable positions in England. Clearly his task is to stand at the board and write down the notes, which the audiences copies politely. So far, so like Cambridge Part III. But then it seems that the notion of interacting with the class doesn’t really apply here. Even an utterly uncontentious question such as ‘what is your name, and which year are you in?’ produced a flurry of nervousness, and then each student stood up stiffly in turn to answer. This continued even as the class size reduced, and my manner became more informal, for example starting to explain ideas while perching on a desk in the front row. I guess old habits die hard. It’s a bit of a shame because I feel the purpose of a lecture is not just to transcribe course material, but also to give motivation and a flavour of the mathematical landscape, as well as to inject some enthusiasm into an appreciation of the topic. And it’s very hard to start such ideas in a culture where it is totally unexpected.

4) This was even more evident in some of the research talks. Each member of the Cambridge group and various Linyi staff, postgraduates and masters students gave a talk outlining some aspect of their research interest. The motivation had been to try and recreate a seminar series, at least in environment if not in totality given the expansive range of areas of study represented. I was struck by the fact that what seemed like the entire maths undergraduate community had been forced to attend, but many of the talks were completely inaccessible to a non-specialist audience. A paper on a technical property of spectral theory read out at speaking pace from a powerpoint is not going to get younger students excited about research maths.

5) As suggested by their enthusiasm for PDEs and complicated graph theory construction problems in the research talks, the students seemed to relish calculation over theory. I found it very difficult to get across the idea that it was sometimes better to use the reuslt we had just shown rather than immediately attempting to diagonalise all visible matrices. I think some of the undergraduate teaching mirrors sections of Maths A level in this country, where the emphasis is on deploying solution techniques rather than ideas. For example, the class looked absolutely lost when I tried to use recurrence relations, to the extent that I decided just to start my explanation from the basics. It turned out that I had notated the general solution as $h_i=\alpha \lambda^i+\beta \mu^i$ whereas their teacher had used $h_i=\alpha \lambda^{i-1}+\beta \mu^{i-1}$, and this had caused the major confusion. The problem is compounded by the fact that despite constant encouragement, they feel mortified at the thought of asking a question or seeking a clarification. Again, it seems to be a matter of respect, but of course this spirit can remain in a more enquiring environment.

Overall, it was a very interesting insight into a completely alien mathematical culture. I know which one I prefer, but I think my perspective is richer for having experienced something of mathematics in China. And hopefully any teaching I do in the near future will seem straightforward by comparison, if only for the property of sharing a language with the students…!

# Approaching Infinity

Last summer, I worked at and gave some lectures at the National Maths Summer School. The students submitted feedback forms, and a surprisingly large number mentioned that they would have liked to have a session about ‘infinity’. I was reminded of this by a post on an interesting blog that I’d seen linked to by, of all people, Stephen Fry. It is easy to forget, a full three years after a first university course on analysis, that the infinity had once seemed so confusing.

The problem is as much one of presentation as of mathematical content. The impression often given is that mathematical statements concerning infinity are not properly defined, or can’t be understood in a ‘real world’ setting. Unqualified and often rather misleading explanations are absolutely rife. And even some well-qualified scientists have put forward theories that are questionable at best. First we talk about some of the usual problems, and why they might not be so significant after all.

• No-one can imagine what infinity is: I’m not sure whether this is true – I personally feel I have a reasonable idea. But even this doesn’t matter. Arguments like this often reference the fact that there are $10^{80}$ atoms in the universe (or something similar) and how this doesn’t even compare to infinity. This is true, but it doesn’t affect our ability to understand and make deductions about a concept. I can’t imagine what 5-dimensional space looks like, but with five co-ordinates (x,y,z,w,v) I can describe it in mathematical terms that are entirely reasonable. This allows me to start working out properties of the object even if I can’t visualise it.
• Infinity is about philosophy: This might well stem from its appearance in popular culture (‘to infinity and beyond’) and the metaphysical (‘the Father of an infinite majesty’ etc). I would suggest that if you are worried about coming to a philosophical understanding of infinity, first you should question whether you have a true philosophical understanding of seven. I can picture seven oranges in my mind, but does that alone really explain all the seven-ness of seven? In any case, we can learn some simple rules to deal with seven (like 3+4=7) in a concrete way, and though the rules aren’t as ‘obvious’, we can do the same for infinity.
• Infinity is not a number: Again, this is in some sense true (see below). But it doesn’t make any difference if you use it correctly. At various points in time 0 has been considered ‘not a number’, as have negative numbers. If you build up the world of complex numbers by defining the square root of -1, is this a number? As with many words, infinity means different things in different contexts. This is actually often really about the following: Continue reading