Scaffolding Proof to Cultivate Intellectual Need in Geometry

This year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.

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!

#made4math | Words into Math - Taming Troublesome Phrases with an interactive foldable translator

It's been busy here in the Intergalactic Cheesemonkeysf R&D Laboratories
(see trusty assistant hard at work, right). Ever since Twitter Math Camp 12, I've been working on implementing all the lessons and activities I learned about in person from my fabulous math teacher tweeps!

I'm using the Interactive Notebook structure that Megan Golding-Hayes showed us, and I'm also incorporating a lot of Julie Reulbach's foldables. The most helpful insight (out of many) I received from Julie was the idea of using a foldable as a way of getting kids to SLOW DOWN and trust the steps of the process as they're working on word problems. So I've made a nifty little foldable like hers that will go into an INB pocket the first week and will be usable on all quizzes and tests.

One of the reasons I like having students develop tools they can use on tests is that many of the discouraged math learners just don't trust their own learning. They have a habit of "collapsing" when they encounter a first speed bump. So from the perspective of encouraging students' courage in problem-solving, it is good to allow them to have tools they can use, even if the tools are sometimes nothing more than a security blanket — a talisman or a good-luck charm they can touch as a tangible reminder of their own courage and resourcefulness. So a four-step problem-solving foldable serves double duty: it acts both as a checklist (as in Atul Gawande's New Yorker piece and book) and as a reminder to have courage and perseverance in working through problems.

However many students have a habit of either not using the tools or finding the tools too complicated or frustrating. Nowhere has this been more evident than when I've given them approved lists of words and phrases they should stop, consider, and look up if need be. The charts and lists seem to turn into giant floating word clouds that signify nothing. So I wanted to come up with a slightly more interactive than usual foldable that students could use as a way of isolating and decoding some of the most troublesome words and phrases they get hung up on. Not only does it slow them down, it gives them a focal task that redirects an anxious mind.

After a lot of research on both blogs and on Pinterest ("PINTEREST!" #drinkinggame), I came up with the idea of a folded sleeve with a sliding chart insert, containing the phrases that often confuse kids or cause them to second-guess their translations from words into math. Here's what the finished product looks like:

Here is a close-up:

I used OmniGraffle to make the sleeve template and I used Pages, Preview, and Adobe Acrobat to make the insert. I'm linking to the Troublesome Phrase Translator sleeve, a generic sleeve you can customize for your own fiendish purposes, and a PDF of my exact insert (Troublesome Phrase Translator INSERT). 

If you want to make your own inserts, you'll need to set up your own table (Word, Pages, Excel, etc) making sure that your row height is exactly 1/4 inch. Your LHS cells should be 1 9/16" wide and your RHS cells should be 1/2 inch wide. You can have about 19 or 20 rows, depending on what you put in them.

Sometimes a little magical thinking is just the thing to displace a discouraged learner's anxiety (or freaked-out-ness) for that extra second it might take to recommit to the process of solving a problem. If that helps me hang onto just one extra student a day, it's a win. But usually I find that a tool like this will encourage multiple students to encourage each other's confidence as well, which is an even bigger win in my book!