I was raised in the math field to believe that measuring with ruler and protractor have no business in deductive reasoning. “Don’t trust the picture,” they’d say. I once sat at a PD session instructed by a man who told me that he would purposely give his students ridiculous pictures that had no bearing in reality so they, “learned to only trust the given information.”
This bothers me.
It bothers me because the official stance involves deliberately providing a misleading visual to students.
It bothers me because it excludes students whose logical sense is still developing.
It bothers me because it sends the message that there is no place for the skill of observation and intuition in the world of deductive reasoning. It is as if observation is peasants work while the real men and women learn to reason deductively.
I teach geometry (as the web address of my blog might suggest) and so, I teach the students as we develop the art of “The Proof,” which is treated like some secret membership card. Those who can “do proofs” are those who are deemed worthy to enter the realms of upper level mathematics.
Up until this year, I’ve always taught proofs the same way I was taught. Start with the “two-column” method and flood them with theorems, properties and postulates. Then teach them to jigsaw them together in ways that make the learning look more like classical conditioning than deep understanding.
This year, I scrapped that. I started with what it means to “prove” something first. Let them write in their own language. Lots of words; paragraphs, arguments like that.
Then, when they get tired of so many words, I teach them the shorthand and the notation.
Then, when they get tired of making the same arguments over and over and over again, then I teach them about theorem and postulate.
You know what? Engagement is soaring! Students aren’t afraid. They are willing to try. They are willing to listen to the feedback. They are willing to learn. Pieces of this story are documented in several previous posts here and here.
So, if the model that I am using is indeed the improvement that it seems, then what do the students do before we get to theorems?
The students have to be able to use some sort of evidence to support their developing sense of deductive reasoning. We are going to start them where they are. And use need and convenience to justify making changes. We start with rulers and paragraphs, because deductive reasoning is hard enough to pick up new. Why throw in new structures, notation, and vocabulary in with it? Once they begin to understand deductive reasoning enough to begin worrying about style and quality, then we can move to notation… and on and on.
The ruler and protractor are already starting to become obsolete to some of my students. Why? Because protractors are a pain-in-the-you-know-what when all you’d have to write is “Vertical Angles Theorem.”
But I didn’t tell them that. They are making those decisions on their own. That sounds like understanding to me. The understanding I could never give them in previous years as to why rulers and protractors weren’t used much in deductive reasoning.
But, it required me letting them use them at first and still being okay calling it proof.