After seeing a recommendation online, this book arrived at the house in the mail yesterday. I started reading it after dinner and was immediately inspired. Enough so, that I'm going to write up some first impressions even though I'm only into the first chapter.

The preface described the Berkeley Math Circle's history and a little bit of how it runs. So often discussions of math run between either impassioned defenses of direct instruction vs. constructivism/progressivism. There's no sense of problem solving in the first camp (students don't have enough knowledge, their working memories get overrun and they only engage in means-ends analysis that doesn't stick.:David Didau's argument along these lines. On the other hand, the other side is passionately concerned with equity and breaking down narratives of who is capable of doing mathematics. In the process, the curriculum mostly takes second place to the pedagogy around it.

The paragraph above speaks to the third way I'm always looking for: focused on problem solving yet rigorous and above all fascinating. The kids in the circle were engaging in really interesting problems and getting exposure to a much richer view of the subject. Computation was not important but proofs took central place.

At the same time, this model raises a lot of questions for me. There's a certain elitism present from the start. The Berkeley Math Circle is described as a "top tier" math circle. This is not idle boasting, it took a prestigious university and multiple mathematicians to create this group. Each session drew on a small slice of the Bay Area (20-30 students) able to discover it through the web site or word of mouth and lasted an intense 2 hours. Here in Seattle which is another tech node we have a similar Math Circle run out of UW https://sites.math.washington.edu/~mathcircle/circle/ aimed at the Middle School level. At the same time, I wonder how can we replicate this model on a larger scale? There are challenges both in finding kids and instructors and covering the full range of MS through HS.

I think part of the solution probably lies in virtual math circles and the online world since even if most research universities setup a circle (and that would be a tall order) there still would be huge swaths of kids out of reach. Can you infuse any of this into a classroom? But this train of thought probably deserves some more work.

#### The Meat (Inversions in the plane)

After thinking about the preface I finally started the first chapter. At this point my mind was blown away.

I'm only about 12 pages into a discussion of circle inversions. For me, this is a revelation at the same level as learning about cyclic quadrilaterals (which I hope would be shared by kids).

The basic idea is you apply a transform based on projecting in or out of a given circle and the geometry problem can be solved in the new space more easily and then translated back.

Example Ptolemy's theorem:

More info: http://mathworld.wolfram.com/Inversion.html As I said, its fascinating so far, and I'm going to work through the exercises to see exactly how powerful it can be. As a bonus: geogebra already has a built in inversion tool (reflect about a circle) which I never even noticed.

The book is focused on material as opposed to methods of instruction. So unfortunately, you won't find how this material was presented and the student's reaction to it. For me that's the next frontier: can I build something around geometric circle inversions that will work for the kids I'm going to have?

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