Introducing Proof Using Formal Systems

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.

Shadow puppets were clearly a key element to my presentation.

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.

WORDs sheets (PDF)

WORDs sheets (DOCX)

Finally, should anyone be interested, here are my slides from the presentation.

ATMNYC slides

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!


17 responses to “Introducing Proof Using Formal Systems

  1. I do this sort of thing before doing proofs of trig identities. It even matches the process pretty closely of doing trig substitutions.

  2. Great stuff. Formalism is so important to mathematics. It also brings to mind the idea of “generators” which we see everywhere in algebra. From basis elements to cyclic groups and so on. Finally, playing with formal systems like these is the start to an incredible story about Godel incompleteness.

    Thanks for sharing! Proud of you for your first presentation.

  3. Hi Justin,

    I just found your blog because of your tweets. Glad to see you doing both!

    Very nice material. I’ve also considered using the Godel, Escher, Bach formal systems to introduce formal proofs. Seems like a good transition step.


    • Hi Japheth!

      A stray thought about what makes formal systems a good transition step to proof: they sidestep the belief-and-truth aspect of proofs. Instead of getting embroiled in conversations and mindsets that say “Why do we have to prove this? It’s obvious!”, we can focus on how a system that superficially seems to have a small amount of power can actually be rather potent. Bringing this, and the notion that individual facts can be brought together through reasoning, to a geometry class–or whatever mathematics one happens to be teaching–has some really nice consequences.

      See you around!

  4. This article is superb! I shall look at the slides and the worksheets when I have some time.

  5. Great Idea for an activity. I’ll have to try it. The question of what end sequences are possible is a pretty compelling one.

  6. I like these, will have to try them. I bet the kids can come up with good ones too. Thank you so much, Justin.

  7. I am exploring the idea of teaching math without numbers, and I believe the introductory method of teaching proofs that you use would fall into this category. I also like it very much and would love to use it!
    It also feels like logic problems, which I use in my geometry classes to introduce the idea of formal statements. I had not really made the connection on an obvious level, but your examples give me an idea of how to segue. Thank you! I will think on this and let you know- would love some feedback at that point.

  8. I know I’ve told you this but I really love this, and I can’t wait to use it this year. I’ve already incorporated it in my materials!

  9. I don’t teach maths now but I teach computational thinking. This would make an awesome activity. cheers!

  10. Just tried some of these in class today. I explained the directions, barely nudged them toward solving the first one, when I realized some of them were already on number 2! Although some of them wouldn’t admit it, they loved this activity. THANK YOU SO MUCH. I am about to start introducing proofs and I believe this will provide a huge stepping stone. Thanks so much for sharing.

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  12. Somehow I was not aware of this early in the year. Definitely will include this idea early in the year next year. Editing text and activities/HW/handouts soon.

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  14. I love this idea and am going to use it with my Geometry students. They’ve been working on intuitive proofs to problems, and this will be a nice way of introducing them to demonstrating one single step/idea at a time.

    Thanks for sharing this!

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