I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.

We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof. We now do at least one proof a day in my Geometry classes; however, because of the increased frequency, we are doing them in ways that are scaffolded to promote fluency.

Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.

Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."

While this seems

*necessary*to me at times and for many students (especially during the early stages), it also seems dramatically

*insufficient*because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."

So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).

I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.

Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.

This has led to some amazing discussions of argumentation and logical dependencies.

An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:

-Sample proof to be sequenced & justified

Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been spectacular.

I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of developing intellectual need in his work!

Have you read RETHINKING PROOF WITH GEOMETER'S SKETCHPAD by Michael de Villiers?

ReplyDeleteMichael, I confess that I haven't read it, and so I am curious to hear what you value about it and why. I have read a bit about it and done some investigation of this kind of approach to geometric inquiry, but I've run into a few of problems so far. The first one is that my school is so far from 1-1 with the needed technologies it has been difficult to explore. The second is that I have two blind students (plus some visually impaired) who would be completely unable to access GSP given the state of available tactile display technology (though we are cheering on the U of Michigan and their effort to come up with a full-page refreshable tactile display!). Finally, there seems to be a difference between understanding the geometry interactively (as through GSP or Geogebra) and affirmatively constructing the written argument/proof that takes advantage of that interactive understanding.

DeleteCan you say a little more about your thoughts and/or experience with the de Villiers book/method?

- Elizabeth (@cheesemonkeysf)

Heya Cheese, this seems like really useful scaffolding for proof but I don't see the need angle. Based on this treatment, how do you hope students would respond to the question, "What's the point of proof?"

ReplyDeleteHi Dan, That's a reasonable question, but it's a bit more meta than we are working with right now. We're operating in a state of 'suspended disbelief' around the need for proof itself. My students here accept the a priori need for argumentation or proof. But they have trouble sometimes seeing why you would need a particular new theorem (for example). So the essential question might be, what would enable you to make a connection between congruent triangles, say, and parallel lines?

DeleteWe are working at a more modest, micro-level than justifying proof itself!

- Elizabeth (@cheesemonkeysf)

Gotcha gotcha gotcha. This makes sense to me now:

Delete"Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts!"