July 2007

I've read one or two things lately about preferences for sequential learning vs. global learning i.e. learning things in a step-by-step manner vs. understanding how different pieces fit together. This reminds me of understanding new mathematical proofs when I was at university. In books, proofs are always written out as a sequence of logical steps. A proof not written out in logical steps isn't considered a proof. However, understanding a proof is much more than being able to reproduce the logical steps of the proof. I'm not sure that I could say precisely what it means to understand a proof, but just knowing how one line follows from the previous one definitely isn't understanding a proof.

One of the first 'real' theorems you come across when you study mathematics at university is called Lagrange's Theorem. It says that 'the size of any subgroup of a finite group divides the size of the group'. That obviously won't make any sense at all if you haven't had the delight of university-level mathematics and don't know what a group is, but try and bear with me and get the gist of what I'm saying if not the details.

If I'd never seen the proof of Lagrange's Theorem before and was trying to understand its proof, what I'd do is first try and figure out the overall structure of the proof e.g.

First we prove that the cosets of the subgroup are equivalence classes, then we show that each coset is the same size. It follows from that the size of the group is the number of cosets times the size of a coset, so the size of the subgroup divides the order of the group.

Then I'd go down a level and look at how each of those bits is proved. So for instance I'd make sure I understood why each coset has to be the same size. Maybe at one of these points, I'd get stuck - something that was obvious to the person who wrote the proof wasn't obvious to me in which case I might have to go away and figure out why it was true.

Sometimes, with more complicated proofs, I might have to go down more levels. This is a bit like if you're trying to figure out the route from A to B with something like Google maps. You'd start with a large scale view where you could see the beginning and end of your journey. For some bits like a long stretch on a motorway you wouldn't have to zoom into very much whereas other bits where you were going through a town, you might want to zoom into with a lot of detail.

When I was reading the proof, I would be skipping backwards and forwards a lot while I went through this process, working the overall picture and then zooming in on particular bits.Just to make things more complicated, while I was doing all this, I'd probably have a specific example of a group in my head and be simultaneously running through whatever part of the proof I was on with that example. I wouldn't believe part of the proof because it obviously true for the example, but it would probably help me visualise and remember what was going on. Sometimes, with a new proof, I wouldn't even do any of this at all - I would pretty much ignore the proof in the book and just take the statement of the theorem and try and prove it from scratch for myself. When I got stuck sneak a look at the proof for a hint. But in practice, there just wasn't time to do this for every proof that you came across.

What I definitely didn't do was read through a proof in a step-by-step way. But despite this I would absolutely have hated it if a proof **was not** written out sequentially. I didn't want the proof to be written out in the same way that I understood it. I think it would have made it much harder to understand. Why is this? I'm not sure. Maybe, it's because it was the actual process of understanding the proof that was important and to have it written down would have made it too easy to skip this process. Maybe because it would have been difficult to get it right - something I might have struggled on, somebody else might have seen immediately and vice versa. Maybe it's because when you're stuck on a sequential proof, it's easier to pinpoint where you're stuck. Maybe, it's because, once I understood a proof, I actually did think of it sequentially so having it written down sequentially made it a more useful reminder afterwards.

Very occasionally, you'd find a book that did try to explain a proof in a backwards and forwards type way, but I never really liked that unless it was clearly demarcated with a separate 'proper' proof. With one semi-exception, the lecturers I preferred at university were the step-by-step ones who also gave overviews rather than the handwavy ones. I guess what I liked most was a standard proof with either an overview of how the proof worked at the start on with extra lines thrown in like 'Now we prove that all the cosets are the same size'.

Is this how other mathematicians understand proofs? I'm not certain, though it worked for me. It would be interesting to know what correlation there is between ability at mathematics and how folk go about understanding proofs. What appears a learning preference could just be a case of not knowing better. Alternatively, it could turn out that maybe it genuinely is a personality thing. Or it could be more complicated still - the Fields medallists of this world might approach things as differently from people like me as I maybe did from the many people who struggled with mathematics at university. The one thing I am sure of is that it's a bit more subtle than being either a sequential learner or a global learner.