I’m trying a couple of new things in the triangle congruence / quadrilateral properties quarter of my geometry class this year.
First, I’m implementing CME Geometry’s methods of analyzing proofs, which uses flowcharts as one way to figure out a proof. I’ve always used flowcharts more as a way to actually present a proof, as done in Discovering Geometry, but I think I like this better.
To help with the flowcharts, I’m planning to use proof blocks, as described at http://www.proofblocks.com and in so many wonderful blogs. These can be cut out and used on white boards or paper to provide visual feedback as to how various definitions and theorems can be connected. I decided to make my own slight variation from arrows. Here are a few of them:
The block in the upper right is the “statement” block, in which you actually write geometric statements. Those always fall between two of the “reason” blocks. “Given” and “Picture” are the ONLY blocks that can start a line of reasoning.
Here’s a picture of a sample proof using my proof blocks (click to make it larger)
The definitions are double-arrowed so that you can tell they go either way, and are oriented vertically so either the top or bottom can be connected to either side of the mostly-horizontal flowchart.
You can access the entire PowerPoint here, with all of the blocks I made, a few instructions, and a sample proof. Most of the blocks are editable – a few of them got copied as images from my initial Google Drawing attempt and would need to be redone if you want to change them.
I intend to print out a copy for students to use, though I don’t think I’m going to laminate-and-tape them this year. Not enough time. For this year, they will use the printout as reference and draw the blocks. I am also providing them to the students virtually, as a series of images in Microsoft OneNote. Here is a silent screen cast documenting the process I used to work with them virtually in OneNote (on Windows, with tablet PCs). You could probably modify this method to work with may different pieces of software, PowerPoint itself for one, or even something like Google Sheets.
The second new thing I’m trying is having students keep a Proof Portfolio. This arose out of a realization that in the past I (and the book) have provided them with nearly every proof of an actual interesting theorem. The proofs they have had to do own their own have always been more limited, specific-case theorems, using the theorems “we” proved together in class. This year, I’m intent on changing that. I want them to prove the vast majority of the triangle and quadrilateral theorems themselves, in groups of at most two, usually in class but some on their own time. We will keep a full portfolio of proofs of the theorems on OneNote. I’ve provided them with a four-step template they will copy and work on.
- Write the theorem you intend to prove in English (Example: “One diagonal of a parallelogram divides it into two congruent triangles”). This will be done following investigations in class.
- Convert the theorem into “Image-Given-Prove” form. This is a step I have nearly always skipped in the past, doing for them. I think it’s a hard and important step, so I’m spending more time and effort drawing it out of them this year.
- Analyze the proof and figure it out. Room for scratch work provided. They can use Visual Scan, Flowchart, Reverse List, or whatever-else-they-want to do this. If they see the “answer” quickly, they can jump straight to…
- Write the proof in formal paragraph or two-column form.
They will hand the entire portfolio in just before Thanksgiving, with 18 proofs included. I hope this will give them more ownership over the theorems, more understanding of the power of proof, and a better sense of the real mathematician’s process.
I’m sure others have done this, but it is new for me. I have usually under-emphasized proofs, but this feels more authentic to me than the versions I have tried in the past, so I’m swinging the other way a bit. I’m hoping that by making them prove the important things, and not as many meaningless things, I will feel like it is more worth the trouble.
I’l let you know how it goes.