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?
The answer is that a few months ago I wrote a post in which I criticized a proposal for an A-level in Use of Mathematics. Amongst the responses I got to that post was an email by Joseph Malkevitch, from the mathematics department at CUNY, drawing my attention to a thought-provoking article he had written. In this article he defends the idea of interesting children in mathematics by showing how it can be used. My instinctive reaction before I read the article was to expect not to like it (except that there was something promising about the tone of his email), but I actually found myself liking it rather a lot, and have meant for some time to mention it.
His basic idea is that there are certain problems that are not phrased as maths problems, that are engaging to just about anyone, and that gently force you (if that’s not too oxymoronic a way of putting it) to think mathematically. I started this post with an example of my own, but it is very much inspired by the kinds of examples he gives.
Returning briefly to my example, let me say slightly more about why it might cause one to think mathematically, since it may not be obvious (though to most mathematicians I think it would be). I think one has to start with some non-mathematical thoughts, just to understand better what the properties would be that one would look for in a good queueing system. For instance, it won’t do to say that whoever is going to take the least time should go next, since then somebody with a big basket could end up never getting served. Perhaps some principle should apply that said that the time you wait should be comparable to the time you will take. But that cannot always apply, since someone with a huge basket will eventually have to be served, and if there is a constant stream of people with single items, then some of them are going to have to wait a long time. So perhaps the system should be that the person who has the greatest time-waited-to-time-needed ratio is the next person to be served. But what would the effect of that be in various situations? Would it do what one wanted? We have now got to the stage where we have a mixture of straight mathematical questions and more nebulous questions about fairness, what would irritate people, etc.
I’m not saying that the usual more formal style of teaching things like algebraic manipulations should be swept away — far from it — but I do think that Joseph Malkevitch’s idea could, in the hands of a good teacher, be a wonderful supplement to a traditional mathematics course, and it shows up the poverty of imagination of those who devised the use-of-mathematics syllabus that was in the news a few months ago.
It would be interesting to try to come up with a list of real-world problems of this kind that forced one to come to grips with specific mathematical techniques, such as matrix multiplication or solving quadratic equations. It’s important that the problems shouldn’t be stupidly artificial: e.g. you’re not allowed to say, “I have a square piece of paper, and I notice to my surprise that the sum of its area and eleven times its side length, when measured in centimetres squared and centimetres, comes to precisely 24. How long is each side?” The problems have to be engaging before they are recognised as mathematical.