# Improvement under the common core

An introduction to mathematical proof

There are a lot of reasons that I am happy with my school’s transition to common core (I say that as if we had a say in the matter. We didn’t. But we have chose to make the best of it, at least in Geometry.)

In previous years, we have stuck to a textbook that was aligned to the Michigan Merit Curriculum. We used the book as the backbone and central nervous system of our class. Vocabulary, notation, structures, and other things were decided by what the book used. The book wasn’t bad.

The Common Core doesn’t align at all. It forced us to scrap the book.

It may have been the best decision we could have made.

Our recent liberation from the textbook has been a huge benefit to our students. “The proof” is a hallmark of most geometry curricula. Most text books push the structure of “two column proof, ” (you can see an example here if you aren’t familiar with two-column proofs).

The biggest problem with two-column proofs is the lack of engagement. They seem unapproachable, unnatural to students. The students aren’t sure how to work with them and because of their rather unique look (can you think of any other form of writing that looks like that?) it is tough to connect the two-column structure to any of their previous experiences.

But what is the structure doing? It is attempting to mimic a deductive reasoning structure. But, see, students are familiar with deductive reasoning.   Most students who try to convince me that I shouldn’t make them take a test because of an absence the previous day will use a deductive argument to do it. (“Mr. Shauver the school rules say that I get a day for every day I missed. Today should be that day!”)

So, the problem with students not being able to “do a proof” could be more of a problem with the two-column structure than with their ability to support a mathematical statement.

Today, I gave them the problem at the top of this post. I gave them an image with four facts. Their task was simply to explain why the two triangles were congruent, but to write two different explanations. The first used rigid motions to support their argument and the second used the definition of congruent triangles. I told them that they could write “paragraph-style.” I have some who choose bullet points, but not many.

Engagement was over 90% across the board. Students weren’t all confident in their answers, but they all had answers. They had all written a few sentences for each. And it didn’t take a lot of wrangling to get those sentences. In previous years, I might have had half the class try and the rest practice their avoidance behavior, claiming they didn’t know how, “collaborating” with neighbors, asking to use the restroom.

Here’s to hoping those days are gone.