An article on Ashoka University’s Foundation Course in Mathematical Thinking. I wrote it in 2016 for the University newsletter for donors.
Ashoka sets a high bar for admissions – we did so from the very beginning. So we can assume that even those of our students who officially hate mathematics are at least numerate. At the other end are those who officially love it, are going to major in it, and can’t wait for the good stuff. Why do we lump all these students together and stick them in a required math foundation class – and what on earth do we do with them there?
The truth is that what these students are loving or hating is not mathematics. What they experienced in school isn’t mathematics – at least not in any form recognizable to a mathematician. Which is good if I’m teaching the foundation course, because it means that all my students are beginners – the apparent disparity is irrelevant. Mathematicians do not do sums! Or even trigonometry or calculus. They think about abstract objects, and they don’t apply algorithms – they solve interesting puzzles.
Now here’s a puzzle. Suppose that you have to divide a slice of cake for two children. They are watching you carefully. You want to do more than just divide the cake overall into two parts: you want to make sure that the cakey part, the layer of icing and the blob of cream on top all get divided equally. (Of course you know that no matter how you do it, they’re going to squabble, but never mind, you want to try.) With a single straight cut of a flat knife, can you do it?
If the slice of cake is perfectly symmetrical, and the icing perfectly level, and the cream blob perfectly round and centred (etc. etc. – you get the picture), then of course you can. But what if the icing is thicker on one side, the cream is starting to drip, and the whole slice is madly uneven? Can you still do it? Can you always do it?
Now that last is a yes/no question. Yes means: no matter what the configuration of the components is, you can always do it. No means you *can’t* always do it: there is a cake, with a layer of icing, with a blob of cream, that is put together in such a way that you can’t possibly divide all components equally with a single slicing plane. Highly irregular, I say.
So now. What do you need to start thinking about this? Nothing really – just start imagining and playing around with it. Poke at it. Sleep over it. Draw some pictures. Draw lots. They’ll draw themselves, after a while. And does it matter if you forgot your binomial theorem? No. Does it help if you are ace at trigonometric identities? No. So, plus-two math or no, lovers of algebra or haters, we’re all on the same page. We just have to think about it.
As it happens this is not a trivial problem. It’s juicy. It’s big. It’s interesting. It allows us to make use of our experience in 3-dimensional space, but will then take us further. And it’s not going to take five minutes. This is perhaps the greatest difference between school math and university math: problems don’t take five minutes. Or perhaps even five days. And what real problem does? What real interesting problem do we prepare students for when we make them proficient drillers and formula-appliers, and nothing more? Leave behind your fat problem books, students, your bread and water, and sink your teeth into some cake.