Moving from concrete to abstract

In reading through Cédric Villani’s Birth of a Theorem, I definitely enjoyed his descriptions of the math but had a much more difficult time following his presentation of the actual calculations. Not really that surprising of a situation since I’ve been out of academic math for so long, but it still gave me something to think about.

Being able to provide accessible descriptions of complex (and in this case Fields Medal winning) math is a skill that not everyone has. Being able to translate back from the simplified descriptions of the math to the more complicated (and precise) equations is another incredibly impressive skill.

The memory of this these skills on display that will live in my mind forever comes from my undergraduate physics advisor, Ed Bertschinger. I was working on a project that was trying to explain some of the new data about the cosmic microwave background radiation that had just come in from the COBE telescope. The angle that Bertschinger wanted me to take a look at was what would that data look like if the entire universe was rotating. I don’t remember the details of the research so well now, but somehow (and I’m sure “somehow” involved many hours of help from Ed) I was able to write down an equation that would describe the situation.

The problem for me was that the equation was impossibly complicated and I had no insight at all about what the solutions would look like. Ed, however, looked at the equation and made some good old physics-y, hand waving arguments and drew a graph of the solution on his chalkboard. About a month later a computer simulation I put together drew pretty much the same graph.

It was an eye-opening moment for me – how in the world did he draw that graph?

Fast forward to today. I really enjoy watching my sons learn and talk about math. Generalizing a bit, obviously, my older son’s natural instincts are to calculate and my younger son has some really nice hand waving / intuitive approaches to working through problems. Working with them often involves very different teaching strategies.

This morning a problem about rectangles and triangles highlighted my younger son’s intuition:

My attempt to help him see an approach to this problem that was a little more abstract and involved a bit more calculating did not go nearly as well – but the struggle here definitely motivates me to do a better job helping him (and his brother) see problems from other perspectives:

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