Two-Column Proofs that Two-Column Proofs are Terrible

Theorem #1: “Justifying steps” ought to be an opaque, frustrating process.

Before sarcasm carries me too far down the rhetorical river, let me plant an oar and explain my stance.

I see the appeal of two-column proofs. They’re clean. They’re easy to grade. They offer a scaffold, a structure, a formal framework for students to lean on. Properly understood, they function almost like diagrams of arguments, and can serve as useful tools.

But in practice, they often obfuscate more than illuminate. A good proof contains not only bare statements of fact, but connective tissue of explanation. In a two-column proof, the organic matter that holds the argument together is flushed away, and replaced with a right-hand column full of terse bullet points that students may use without understanding at all.

Theorem #2: A proof is just an incomprehensible demonstration of a fact you already knew.

My beef here is that a geometry course often begins with totally mystifying two-column “proofs” of elementary facts. For example, take the “Right Angle Congruence Theorem,” which states that all right angles are congruent to one another:

When I run such arguments by math PhD friends, they look at me dumbfounded. “Why would you prove that?” they ask. There are important lessons here, surely—for example, that even seemingly obvious truths demand justification. But the argument hinges on the fussy technical distinction between the angle (a geometric object) and the measure of the angle (a number describing the size of that object). If you’ve ever been a 9th-grader—or even met one—you know that such a distinction isn’t the most inviting welcome mat to geometry.

Theorem #3: Two-column proofs are great preparation for the future.

Finally, there’s the fact that no mathematician—indeed, no adult human other than a geometry teacher—uses two-column proofs. Flip through a math journal, and you’ll find only what geometry textbooks call “paragraph proof”—that is, English prose laying out an argument. Sure, the proofs are dense, and often interrupted by equations spanning the width of the page, but they’re written to be understandable, not to adhere to an artificial boxed-in format.

How do we fix the system? Here are some imperfect suggestions:

Begin with better proofs. Start a geometry class with a unit on proof structure, and don’t worry about what you’re proving. Prove that there are infinite primes. Prove the Pythagorean Theorem. Prove that there’s no school on Saturday. Prove that a bear would defeat a lion in combat. The axiomatic development of Euclidean geometry can come later. First, the students need practice playing the game.

Don’t let students give “Definition of A” or “Theorem B” as reasons. At least, not at first. They need to understand that a reason is a truth, not a phrase. The current format of two-column proofs obscures the logical content that underpins arguments. Instead, of letting students invoke “Congruent Supplements Theorem,” make them write out, “If two angles have congruent supplements, then they themselves are congruent.” (You can wean them off of such wordy explanations later.)

Use two-column proofs like spice or seasoning: Sparingly. They can be helpful clarifying agents. But two-column proofs ought to occupy a place in geometry similar to the “flow proofs” that some textbooks like to invoke. They should be a “sometimes” food, not a staple of the diet.

Consider adding a third column. My own geometry teacher started the year with three-column proofs: (1) Statement; (2) Reason; (3) Previous Steps on which This Step Relies. This way, a “reason” does not feel like a secret password, but what it is: a link between statements that have come before and the current statement.

I’d love to hear other geometry teachers’ take on the two-column proof, especially anyone who’s got a passionate (or dispassionate) defense for their pedagogic value.

For related ramblings, check out Black Boxes (or: Just Say No to Voodoo Formulas) and For #9, I got “snake”.