Reasoning and Sense Making in Geometry

I've finally caught my breath after returning from a fantastic 3-day Institute in Orlando about motivating students to learn mathematics through Reasoning and Sense Making (the link goes to a page with handouts from the many presenters.) What a wonderful experience being together with about 700 other math teachers from all over the country (including Puerto Rico and the Virgin Islands - but only one other person from my neck-of-the-woods) who are also concerned about the state of math instruction these days.

Funny thing is, there has been concern about the way mathematics has been taught in this country since way back in 1833, when a guy name Colburn wrote about using the Pelozzi method for teaching arithmetic. He complained that arithmetic was all drill and memorization, not reasoning. Sound familiar? The NCTM president Mike Shaughnessy went through a long list of early quotes, including 1923, 1935 and more recently since the '80s. I hope he'll upload his PowerPoint, because I'd love to have those quotes!

I mostly followed the sessions about Geometry, since I couldn't figure out last winter how to get students interested in doing proofs, which is what I think is the fun part of geometry. It turns out that kids are turned off by having to prove the obvious, when we ask them to prove things like vertical angles as congruent.

Michael Battista, in his presentation, The Role of Proof in Geometry, said that proof in Geometry is a caricature, since we are teaching the form of proof, rather than the content. We start too low to teach the method of proof at a point where it just doesn't make any sense to prove things. Kids experience that as busy work, which just doesn't make sense! We should start where it takes some thinking, reasoning, struggling, to figure out how to get from the given to what is to be proved. When the students have figured it out and can explain it - that is when they will be open to learning how to do formal proofs. The proof is just a final written justification of the work they have already done. No mathematician would start writing the proof before having reasoned his way to a solution. We shouldn't expect our students to do so either. Proof is a personal sense making, he said, where we go from saying what is "true" to "why" it is true. We are explaining our reasoning to others. Students move through 5 levels of geometric understanding, he said (known as the van Hiele levels:)

1. A visual, holistic examination of the shape
2. A description of the parts and their relationships
3. The interrelating properties (like vertical angles and the properties of parallel lines)
4. Conceptual proofs (explaining verbally)
5. Formal proof.

We have been trying to get students at level 2 to write formal proofs. They need to work with geometric shapes a while to get to level 4. At that point we can introduce formal proofs.

Battista has written a lot about using the software Geometer's Sketchpad to help students reason about geometry so that it makes sense, and has contributed to several books published by the NCTM. The man behind Sketchpad, Michael Serra, honored us with a wonderful collection of Investigations in Geometry from his textbook Discovering Geometry. We had a nice break in all the talk working in groups to figure out a variety of geometric problems. I worked with Origamics problems, developed by Kazuo Haga. (I just ordered the book. What a fun way to work with geometric reasoning!)

Jeffery Wanko provided a fun session, Developing Proof Readiness with New Logic Puzzles. He uploaded his materials to the Institute website, so you can enjoy them, too: Presentation (PDF)and Handout (PDF).We started with puzzles and their solutions, which we studied to figure out the rules. Then we did a big one together (this was a whole roomful of Sudoku addicts, of course,) and worked individually and in pairs to solve some smaller ones. He provide several pages of puzzles, which gave me something to do besides reading on the long plane trip back home! He recommended the Japanese puzzle magazine, Nikoli.com. Students can become ready for writing formal proofs through talking about puzzles like these with each other, getting to at least levels 3 and 4 listed above.

All that confirmed my previous experience that geometry is fun. I hope I can inspire my students the same way!

I will try to find time to write more about my experiences at the Institute in another blog. Besides swimming every day,of course, the most valuable part of the Institute was the many discussions with teachers from all over the country with many different school experiences.