When it comes to teaching geometry, the most difficult part for me is teaching proof writing. For the sake of grades it can be easy to fall into the trap (as I have done) of making the proofs that students are asked to complete very similar structurally. This usually relieves some of the tension that proof-writing can cause, making some students feel more successful at it. This does not, however, guarantee that those successful students could write as good a proof for more non-routine problems.
Big Idea for this chapter: Proof is part and parcel of doing math and should be a regular and ongoing part of the learning of mathematics. Two other essential ideas:
- The processes of proving include a variety of activities, such as developing conjectures, considering the general case, exploring with examples, looking for structural similarities across cases, and searching for counterexamples.
- Making sense of others' arguments and determining their validity are proof-related activities.
(These are quoted from the text.)
As I've stated in prior entries in this book review, the process of inquiry and making conjectures is one of my goals for revamping my courses. The first essential idea stated above also ties this into my desire to incorporate more proof-writing experiences for my students. The whole process has a natural flow to it, but will take work because its entirely different than what I've previously done. "Tasks that create a need for proof and invite students to participate in the process of proving are not difficult for teachers to devise, though facilitating them can be challenging."
5 Purposes of Proof (I want this on a poster!)
- To verify results
- To explain results
- To promote discovery
- To communicate
- To formalize
Challenges for Students when Writing Proofs:
- Grasping the Process - I have many students each year that get stuck getting through step 2 in a 2-column proof.
- Dependence on Visual Diagrams
- Readiness for Proof - This goes back to the van Hiele level of my students.
Ways to Overcome these Challenges:
- Increase focus on conditional statements. (The book suggests a greater emphasis on converse, inverse, etc. We only cover conditionals, converses, and biconditionals.)
- Allow more time to practice finding counterexamples. Build the understanding that one counterexample is sufficient.
- Use a variety of proof-writing techniques: visual, algebraic, transformational, written (2-column, flow, paragraph)
- Stress to students to trust markings in diagrams, not just what appears to be true. (This is already something I stress in class.)
Here's the general process I will try to incorporate more frequently:
- Investigate/Inquire (notice and wonder, teacher led discussion or activity)
- Make Conjectures (Teacher and students will refine statements to be concise, precise.)
- Validate results. (Prove conjectures or find counterexamples.)
- Reflect (Discuss the proof quality. Discuss alternate proof methods. Discuss if the proof could be made more precise. Have students reflect on what they learned via the process.)