Over the weekend, I gave a talk at ATMNYC‘s fall conference. I talked about formal systems problems, which I use both with my middle schoolers and my geometry students. If you attended my talk, thanks for coming and for dropping my blog! For everyone else, I hope you find something here worth using or thinking about.
In brief, by a formal system I mean a set of objects and a set of transformation rules. A formal system that I use in my classes goes like this:
Briefly, in middle school classes I generally have students work through some sheets like this, create some puzzles of their own, and share these with each other. We talk about “skipping steps” and impossible problems. I have them add rules to this first system that they think might make it more flexible, interesting, or fun. Sometimes we come back to the idea of formal systems later on during discussions of vectors or symmetry groups or modular arithmetic or permutations. Once you have this formal-system model in your head, it’s really easy to put many other topics under its umbrella.
In my geometry class, formal systems serve a couple of functions. They’re a nice introduction to proofs, as well as to the notion of a mathematical system of that hangs together as a logical whole. Discussing these without having to juggle geometric content at the same time allow students to build up a model of proofs and systems that they can later connect back to. Proofs with justifications for each step, independence of axioms–these and more can arise from discussions of really basic formal systems.
A second reason I use formal systems with my geometry students is that they provide a nice parallel to geometric transformations. When taking Felix Klein’s approach to geometry–that geometry is about transformations that leave certain properties of geometric objects invariant–it’s nice to have formal systems in the background. Invariants, shortest-paths, and just the basic notion of a step-by-step transformation all can arise from discussions of simple formal systems.
So for my geometry class, it’s like killing two birds with one stone.
One thing to have in mind is that formal systems activities are very modular. It’s an idea more than a set of lessons, and a flexible one at that. You can drop it into a class for a day or for a week or scattered throughout the year. Bringing them into a classroom can have an impact regardless of the duration or what your particular goals are.
Here are the little charts that I had in my slides about my reasons for doing these kinds of activities.
Here are links where you can download some worksheets that illustrate this sequence.
Finally, should anyone be interested, here are my slides from the presentation.
As I said at the end of my presentation, I’d love to have your questions and thoughts, and I’d be crazy interested should you try this or something like it with your students!