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This Tool Fights Fascists: The Power of the Two-Column Proof

Many of the students at our school have a rebellious, social-justice streak which has only grown louder this year. "How can I learn Geometry," a student once cried out in class, "when all over the world there are people starving and in pain?"

I figured out how I could leverage this characteristic in introducing a new unit of study last week. I started class by telling them that we were about to learn Woody_Guthrie_NYWTS.jpgsomething that has the power to change the world: "Whatever progress we make as a society will rely upon us mastering this skill. Tyrants, despots, and fascists the world over depend on us failing to learn it in order to remain in power."  

This was certainly not the introduction my geometry teacher gave on the day we started studying two-column proofs. Based on the public perception of the concept, I'm willing to bet that your teacher didn't say anything similar either. At almost every social gathering with non-math teachers I attend, I hear some version of: "Oh proofs, I hated that when I was in high school" when I share that I teach high school geometry.

And while it may not be the most obvious approach, I stand by my characterization. Evidence-based reasoning provides a bulwark against superstition and demagoguery. It forms the basis of most of the life-changing institutions humans have developed: from science to law, journalism to business. This foundational principle is beautifully encapsulated in a simple form that starts with a "given" and moves, evidenced step by evidenced step, towards a desired assertion.

So why have "two-column proofs" gotten such a bad wrap?

My hypothesis is that teachers tend to focus on form over substance. "Doing a proof" becomes a series of rote steps, blanks to fill in with a red check next to the correct use of "reflexive property" or "CPCTC" and otherwise, a red X.

When I started teaching formal triangle proofs, I threw away the two-column structure in an effort to free students of the shackles that were turning the inquiry process into a sisyphean boulder.

Unfortunately, my attempt to open different paths for students created a situation where many were no longer reasoning with the precision required by this task. I value the processes of exploration and conjecture, and I create space in lessons for students to do this, leaning on their intuition about math structures to guess and suggest. But if the reasoning process doesn't progress beyond these informal stages, students do not really have the opportunity to "know" anything in math. To paraphrase the World War II era adage: if they don't know something, they'll fall for anything.

My speech about how despots don't want you to learn about two-column proofs represents an attempt to bridge the gap between the two extreme ways of teaching two-column proof.

220px-Twocolumnproof.pngProof is an expansive process, one about big ideas with life-changing, world-shaping implications. The two-column structures is a useful way to get at the level of rigor and precision required by mathematicians the world over and throughout history.

We will spend a fair amount of time during this unit focusing and minute details, but we do so only in service of the larger goals which we discussed on the first day of the unit. (It is worth noting that we have and will also discuss flow chart, paragraph, and diagram proofs in this unit as well, there is more than one way to show precise reasoning).

What did you do today to help make the world a better place? I taught proof. Take that, fascists.


Photo 1: Folk singer Woody Guthrie with his guitar which read "This Machine Kills Fascists." https://en.wikipedia.org/wiki/Woody_Guthrie

Photo 2: A traditional two column proof with statements and reasons proving that corresponding angles are congruent when lines are parallel (assumes Euclidean parallel postulate) https://en.wikipedia.org/wiki/Mathematical_proof

 

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