What maths A-level doesn’t necessarily give you

I had a mathematical conversation yesterday with a 17-year-old boy who is in his second year of doing maths A-level. Although a sample of size 1 should be treated with caution, I’m pretty sure that the boy in question, who is very intelligent and is expected to get at least an A grade, has been taught as well as the vast majority of A-level mathematicians. If this is right, then what I discovered from talking to him was quite worrying.

The purpose of the conversation was to help him catch up with some work that he had missed through illness. The particular topics he wanted me to cover were integrating , or as he called it, and integration by parts. (Actually, after I had explained integration by parts to him, he told me that that hadn’t been what he had meant, but I don’t think any harm was done.) But as we were starting, he asked me why the derivative of was , and what was special about .

A trip to Watford Grammar School for Boys

As I said would happen in my post about a possible approach to teaching maths to non-mathematicians aged 16-18, I went last Wednesday to Watford Grammar School for Boys to try the approach out. The headmaster there, Martin Post, was remarkably helpful and assembled a usefully varied group of pupils, some from his school, some from the equivalent school for girls, and some from a nearby mixed comprehensive school (I wasn’t told which one) whose pupils receive some of their teaching in scientific subjects from Watford Grammar School. What’s more, some of the people there were doing maths and further maths, some were doing just maths, and some were not doing either. The one thing that was not representative about the group was that they were much brighter than average: for example, the non-mathematicians there had been chosen by their teachers as clever people who could have done maths but decided that they were more interested in other things. For most of the rest of this post, I’ll say what questions I discussed and how the discussions went. All but two of them were taken from the list in the earlier post.

How should mathematics be taught to non-mathematicians?

Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18. An obvious question follows. At the moment, there are large numbers of people who give up mathematics after GCSE (the exam that is usually taken at the age of 16) with great relief and go through the rest of their lives saying, without any obvious regret, how bad they were at it. What should such people study if mathematics becomes virtually compulsory for two more years?

A couple of years ago there was an attempt to create a new mathematics A-level called Use of Mathematics. I criticized it heavily in a blog post, and stand by those criticisms, though interestingly it isn’t so much the syllabus that bothers me as the awful exam questions. One might think that a course called Use of Mathematics would teach you how to come up with mathematical models for real-life situations, but these questions did the opposite, and still do. They describe a real-life situation, then tell you that it “may be modelled” by some formula, and proceed to ask you questions that are purely mathematical, and extremely easy compared with A-level maths.

Use of mathematics II

Today I had an experience that I have had many times before, and so, I imagine, has almost everybody (at least if they are old enough to be the kind of person who might conceivably read this blog post). I was in a queue in a chemist (=pharmacy=drugstore), and I knew that my particular item would be quick and easy to deal with. But I had to wait a while because in front of me was someone who had an item that was much more complicated and time-consuming. In this instance the complexity of the items was not due to their sizes, but a more common occurrence of the phenomenon is something that often happens to me in a local grocery: I want to buy just a pint of milk, say, and I find myself behind somebody who has a big basket of things, several of which have to be weighed, some of which don’t have their prices on, etc. etc.

Suppose you were a shopkeeper with just one till and you wanted to devise a good system that would allow people to jump the queue if they were only going to take a very short time. What might you do? And given that I don’t have a good answer to that question, what am I doing posting about it?

Help — I’m stuck in my ivory tower!

The UK Qualifications and Curriculum Authority is considering introducing a new A’level course (in Britain, A’level is the exam that is taken at the end of high school) called “Use of Mathematics”. As one might expect, this idea has not met with universal approval, and there is now a campaign to stop the idea in its tracks. (I should warn you that the preceding link is to a Word file rather than to a web page.)

The General Secretary of the National Association of Headteachers has this to say to the campaigners:

They should get down from their ivory towers. They should be out in the world where young people live and exist and they should be appreciative that young people have great skills in the use of technology and we have to latch on to that.

We cannot continue teaching an out dated 19th century curriculum. This is simply turning many children off education because it is completely not relevant to them at all.

Some sample papers for the new course have been made available, so let’s have a look at the up-to-date 21st-century curriculum that will enthuse a new generation of British schoolchildren. I’ll concentrate on one or two questions but if you want to see more, then the sample papers can be found at the bottom of this page. (Update: unfortunately, these sample papers have been taken down. I can’t help wondering why. Further update: at least some sample papers are now available at the bottom of this page.)

Why aren’t all functions well-defined?

I’m in the happy state of just having finished marking exams for this year. There is very little of interest to say about the week that was removed from my life: it would be fun to talk about particularly bizarre mistakes, but I can’t really do that, especially as the results are not yet known (or even fully decided). However, one general theme emerged that made no difference to anybody’s marks. There seems to be a common misconception amongst many Cambridge undergraduates that I’d like to discuss here in the hope that I can clear things up for a few people. (It is an issue that I have discussed already on my web page, but rather than turning that into a blog post I’m starting again.)

The question where the misconception made itself felt was one about functions, injections, surjections, etc. I noticed that a lot of people wrote things like, “If then so is well defined.” Now if you fully understand what a function is, then you will find this quite amusing: if then trivially by the very basic principle that you can substitute something for something else if the two things are equal to each other. (A famous type of counterexample to this from philosophy: two years ago, Michelle Obama was the wife of Barack Obama; Barack Obama is the president of the United States; two years ago, Michelle Obama was not the wife of the president of the United States. Yes yes, there are ways of explaining why this isn’t a real counterexample.)

But it seems only fair, if one is going to laugh at such sentences, to provide examples of functions that are well defined and functions that aren’t, so that the difference can be made clear. But now we have a problem: any putative example of a function that is not well defined is not a function at all. So it begins to seem as though all functions are well defined. But in that case, what are people doing when they check that a function is well defined?

A solution to an exposition problem

Let me explain the title of this post by quoting from Timothy Chow’s highly recommended expository article A beginner’s guide to forcing: “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.” He goes on to claim that forcing was an open exposition problem, since there was no explanation in the literature that had these qualities.

I have just finished giving a graduate course in Cambridge on two highlights of theoretical computer science: Razborov’s lower bound for the monotone circuit complexity of the clique function, and Shor’s quantum algorithm for factorizing integers. Because nobody was taking an exam on the course, I was free to be somewhat informal in the lectures, but at one or two points it got slightly too informal so I rashly promised that I would produce notes on Razborov’s theorem. However, the document that resulted ended up being less a set of lecture notes in the usual sense and more an attempt to solve an exposition problem in Chow’s sense. Very briefly, Razborov produces a lattice of a certain kind, with a rather strange definition, and it goes on to do the job it is supposed to do in a seemingly miraculous way. What I have written is an attempt to solve the problem, “Where did Razborov’s definition come from?”

Tricki now fully live

Update (25/4/09). Since the launch, the number of pages on the Tricki has doubled (from 104 to 208), and is increasing fast.

Main post. If you have visited the Tricki recently, then you will already know that it has gone live. I’ve delayed posting about it until we were sure that everything was fully transferred: if you visit the prelive site you are now automatically redirected to the proper site, which you can also get to by clicking here. The URL is http://www.tricki.org.

A few small points to note here. In response to comments, we have introduced some new features. One is a feature for marking an article as a stub. Our working definition of a stub is that it should have no substantial mathematical content, and should not link forwards to any articles with substantial mathematical content. (That is, a parent of a non-stub is always a non-stub.) The thought behind this is that there are two directed graphs of interest: one with all articles, whether written or unwritten, and the other the set of all ancestors of articles with interesting content. The stub feature allows one to explore either of these trees with ease, because if an article is marked as a stub, then all links to that article are clearly marked as well, with a little leaf symbol.

Tricki available for viewing

It’s been a long time coming, but the Tricki is now on the point of going fully live. If you need convincing that this is a stronger statement than earlier and almost identical statements I have made on this blog, then click here to be taken to the site.

At the moment the site is read-only. This is for two reasons. First, we would like to give people a chance to spot flaws with the site as it now is, while it is still relatively easy to correct them. These can be anything from technical bugs to the content and organization of the articles. Any suggestions for improvement will be greatly welcomed: the best way of making them is to click on “Forums” at the top of any page on the site and to start or continue a forum topic. Of course, you are also welcome to make comments on this blog post.

The second reason is that I will be on holiday for the next week or so, and I want to be on hand when articles start coming in, in case work needs to be done in fitting them into the organizational hierarchy of the Tricki, or making sure that they are consistent with the Tricki house style.

An advantage of this final delay is that if you will have a chance to browse the site and get an idea of what it is like before contributing an article, if you have a topic that might be appropriate. If you click on “Help” and then on “Formatting on the Tricki”, you will discover that writing an article is extremely easy (at least if you know what you want to say). In particular, if you want to type in mathematical symbols, you just have to write them in TeX or LaTeX and enclose them in dollars. I hope you will agree with me that Alex Frolkin and Olof Sisask have done an amazing job and will enjoy using and contributing to the site as much as I have.

Just-do-it proofs

This post is another sample Tricks Wiki article, which revisits a theme that I treated on my web page. Imre Leader pointed out to me that I hadn’t completely done justice to the idea, and that the notion of a “just-do-it proof” had some more specific features that I had not sufficiently emphasized. His opinion matters to me since he was the one who told me about the concept. He himself got it from Béla Bollobás: I don’t know whether it goes further back than that. This is a second attempt at explaining it. Imre, if there’s anything you don’t like about this one, then please edit it when it appears on the Tricks Wiki.

Title: Just-do-it proofs.

Quick description: If you are asked to prove that a sequence or a set exists with certain properties, then the best way of doing so may well be not to use any tricks but just to go ahead and do it: that is, you build the set/sequence up one element at a time, and however you do so you find that it is never difficult to continue building.