Tuesday, August 6, 2013

Proof in Geometry

When I was a high school student I hated the sections in Geometry on proof. My teacher frequently said that "implied" too much. It was hard for me to wrap my brain around the idea of proof and how to prove something. After high school I went to college where I completed a bachelor program in Mathematics and Mathematics Education. My coursework in Mathematics has taught me that higher math is more about logic and proof than solving equations. So even though I hated proof as a student, I feel that it is really important.

Previous Status Quo
I am about to start my fourth year of teaching Geometry. In the past my students have always complained about proof (like I did when I was in high school). Because I feel that proof is important, I want to find a better way to introduce it and teach it. I have an idea, and I am interested to see what will happen this year.

The book we use (Prentice Hall (C) 2011) introduces proof in Chapter 2 and immediately has the students do Algebraic Proof and then proofs about angles. As if solving an equation on one side of a two-column proof and the reasons on the other is enough of an introduction to proof that students can jump into some complicated Geometric proofs in the next section. For example the very first proof in the book is The Vertical Angles Theorem, which is not simple. It requires students to think outside the box and use an angle other than the two vertical angles, notice that they are supplementary, and then do some pretty complicated Algebra.

It is during and after these first two sections of proof that my students started complaining about proof. I would then tell them that we have at least two more chapters of proof and they would grumble.

In Chapter 3 we would do proofs about proving lines to be parallel. Students got pretty good about recognizing the angle pairs and knowing which Theorem or Postulate needed to be used (ex: Corresponding Angles Postulate, or the Converse of the Alternate Interior Angles Theorem). But when it came time to put those ideas into a two-column proof, they would get so confused about where to put statements and reasons that they couldn't write the proof. The same thing happened in Chapter 4 when we would prove that triangles are congruent.

My Idea
My current thought is to not introduce two-column proofs until Chapter 4 when we prove triangles to be congruent. Before then, require that the students give a well thought out justification that includes Theorems when applicable. They could do this for triangles in Chapter 2 and parallel lines in Chapter 3. Then after students have been working on written justifications for two months, they will then be introduced to "formal proof" meaning, two-column proofs. We would only do two-column proofs for triangle congruence, CPCTC, and later in the second semester, proving that triangles are similar.

In essence, I am trying to encourage students to understand the concepts behind proof without getting confused by the formality of a two-column proof. I don't want the students to get lazy, but I do want students to think through why something is happening instead of guessing at statements and reasons.

No comments:

Post a Comment