Good Mistakes, Constant Mistakes

I’ve spent the last two days going over my problem, going over my approach, finding new gaps in my proof, fixing them… wash, rinse, repeat. It’s amazing that this vision of math as “getting to the right answer on your first try” even exists. I have to make, unmake, remake so many mistakes to get where I’m going. I think all mathematicians work that way.

Einstein famously said, “Do not worry about your problems with mathematics, I assure you mine are far greater.” I don’t think he said this to brag about how advanced he was: I think he genuinely had a lot of trouble with math. Somehow, a big part of the experience of math is trouble. Frustration is the status quo. But when you get something—the thrill!

Speaking of which, I’m feeling pretty good that I seem to have my argument at least mostly patched up at this point. I think I benefited quite a bit from certain mistakes I made along the way. To quote Simon Singh’s Fermat’s Enigma:

While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surpisingly this was a trait that Shimura admired: ‘He was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried in vain to imitate him, but found it quite difficult to make good mistakes.’

Math doesn’t happen in a straight line. If I hadn’t made as many mistakes in my thinking about this problem, I don’t think I would have solved it.

Of course, I should sleep on it to see if I have a real solution. Can’t sound the fanfare too quickly.

So Close–Update from MSRI

I’m attending a conference in Berkeley, CA right now, at the Mathematical Sciences Research Institute, lovingly dubbed MSRI (pronounced “misery”). When I first started attending conferences I felt good if I understood the first half hour of a one hour talk; that was a very rare occurence in the beginning. It was common for the speaker to lose you in the first 5-15 minutes. Maybe that’s why conferences were so tiring—seems like I spent most of the time groping to understand.

Now that I’m (somewhat) more mature as a mathematician, I’m noticing two things. First, the talks that are in my area are comprehensible. It’s deeply gratifying to know what’s going on. Second, when a professor lectures on an area that I don’t understand and am not interested in, I have no qualms about ignoring him and doing my own work. I would estimate that two thirds of a typical audience at a math conference isn’t paying attention to the front of the room.

More exciting than the talks were the fact that I got engaged in a problem I’ve been working out: a new thought on how to proceed with it and a pair of good papers to adapt to my own needs, and suddenly I’m putting all my time into making it work. I thought I solved a subproblem yesterday, and went so far as to write it up and email it to my advisor. When I revisited it the next morning, though, I caught a mistake.

So close…

Now I’m trying to see if I can make the thing work. It’s productive work, even if it doesn’t work out. I’m starting to understand how (technical gibberish alert) the K3 surfaces relate to their lattices and to their Fourier-Mukai partners (end alert). I think I may have figured out how to close the gap in the proof, but I’m a little hesitant to declare victory. In fact, I’m thinking there may be a few more wrinkles ahead. Hopefully it will come through.

So far…

In the meantime, though, I’ve got plenty to do when the speakers veer into topics that I don’t care for.

What we learn in school

Link: What we learn in school

I recently took on a remarkably talented tutee. Seeing the natural ease with which he, at 12 years old, sees deeply and sharply into all kinds of mathematical problems is a pleasure. For me, making sure this kind of student gets proper guidance is essential. Part of the reason, I’m sure, is that I was accelerated in math when I was younger… and I’m not sure anything in my education gave me any kind of idea what math was about. My first real exposure—and one of the reason I ended up as a mathematician today—came when I went to math camp summer after my ninth grade year. It’s hard to communicate how profoundly my relationship to the subject changed as a result of that program.

But what does school offer? Here’s an article from London that gives a sense of what school math is like for gifted students (and one in particular, named Joe). I’ll post the last half of the article here:

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“In the National Numeracy Project’s Framework for Teaching Mathematics (DfEE, 1999) the suggestion is that there are three ways of making the content of mathematics lessons more challenging:

  • by offering the same content within a shorter time (acceleration)
  • by offering additional content outside the regular curriculum (broadening)
  • or by offering increased cognitive challenge within the same curriculum area (deepening).

I would predict that if you talk to any group of maths teachers in any school you would find that acceleration is the predominant strategy to increase challenge. Broadening, after all, requires additional preparation by hard-pressed teachers so is unlikely to find universal favour. Non-specialist colleagues may lack the confidence (or competence) to ask deeper questions requiring higher-order levels of thinking, so acceleration by default becomes the most common strategy. Certainly for Joe it was the most frequent offer.

Of course, there are advantages for a teacher in accelerating pupils through the curriculum. There is no need to invent activities or acquire different resources as everything is already mapped out for the older pupils. It’s an easy option but can set up possible problems for the subsequent teacher and, as we saw from Joe’s experience, unless carefully planned can lead to repetition or, perhaps even worse, to gaps. For the student it may well offer increased self-esteem but it may also be lonely and seem like a random and disjointed set of experiences.

Thinking like a mathematician
For me, the saddest part of Joe’s story is that he is not inspired to continue his mathematical learning. He has no idea of what it is to behave mathematically because his perception is that maths is the acquisition of a set of skills, a set of exercises to be completed, a speed test. Essential though these skills are, they are just the starting point. I want able students to ‘think like mathematicians’ – to enquire, to generalise, to question and seek proof. I don’t want them to accept everything I offer or be able to regurgitate pre-digested algorithms. I want them to engage with the thinking and find some things difficult. I want them to know that mathematicians see things in lots of different ways: that maths is creative and although there may be accepted elegant methods of solving particular problems or calculations, there are other equally acceptable ways of doing the same. I want them to take time to deepen their understanding rather than skating through the curriculum at high speed.

Writers on the subject (Krutetskii, Kennard) say that these forms of behaviour (generalising, seeking proof and so on) are characteristic of able mathematicians. This is what they do naturally. The teacher’s role is to offer opportunities, activities, investigations for able students to display these characteristics, and to model such behaviour themselves and hence help students to refine their thinking. Spending time deepening understanding in this way is time well spent for able pupils, and can have knock-on effects on the rest of the class who are exposed to behaviour that they might otherwise never experience.

I want able students to think like mathematicians– to enquire, to generalise, to question and seek proof.”

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My new tutee is already accelerated. I’m going to try to offer the depth and breadth that, sadly, he may never get in school.

"I was told there would be no math"

Link: “I was told there would be no math”

When Sudoku came they said it didn’t involve any math, just logical thinking and pattern recognition. (No matter how we mathematicians claimed that logical thinking and pattern recognition are much closer to the heart of mathematics than arithmetic.)

And here comes the cousin of Sudoku: KenKen! Is it possible that everyone will start doing this one too, even though they can’t pretend they’re not doing math in this case?

I guess we’ll find out.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs.

Paul Halmos

Teaching with the Socratic Method

Link: Teaching with the Socratic Method

A professor of mine once said, speaking about his education, “If they had actually taught us anything, we’d all be geniuses!” And indeed, when I talk to people who hate/fear math, a bad teacher is sure to come up in the conversation somewhere.

This conversation with a group of third graders about base 2 is a lovely counterpoint to what many of us experienced in our own educations. Read the question and answer section (at least—the whole thing is good), and see if he doesn’t make you think differently about numbers, and education.

In teaching as in math, there are many options.

Hilbert’s Hotel

Link: Hilbert’s Hotel

I love this Waylay comic about Hilbert’s Hotel. She does a beautiful job of explaining the paradox succinctly, and of tapping into the sense of human (nonmathematician) frustration with the whole deal at the same time. It’s an old favorite of mine.

There’s more on Hilbert’s Hotel at wikipedia. The classic version is: a hotel has an infinite number of rooms, and a person is staying in each room. A new guest arrives and is given a room. How is this possible?

Here’s a harder version: the infinite hotel across the street has a power outage, and sends all their (infinite) guests over to Hilbert’s hotel for the night. How do you give all of them rooms as well?