Two-Column Proofs that Two-Column Proofs are Terrible

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

Statement

Reason

1. In an argument, all steps must be justified. 1. Definition of Argument
2. In real, adult arguments, such justifications often take the form of cogent explanations, appeals to agreed-upon facts, and clear, explicit reasoning. 2. Definition of Justification
3. High schoolers are too simpleminded for such techniques. 3. Fundamental Axiom of Condescension Towards Young People
4. Besides, it would take too long for instructors to grade such arguments. 4. Overworked Teacher Postulate
5. Instead, high schoolers ought to justify their arguments by reciting the names of theorems and axioms, invoked as if they were not logical statements but magical spells. 5. Property of Nonsensical Schooling
6. Logic ought to be learned through two-column proofs. 6. Theorem of Maximal Damage 

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.

Statement

Reason

1. In a good proof, each individual step is obvious, but the conclusion is surprising. 1. Definition of a Good Proof
2. In many two-column proofs—especially those taught earliest in a geometry course—each individual step is mystifying, while the conclusion is obvious. 2. Definition of a Stupid Proof
3. Experience with such proofs will help students see logic as an alien enterprise, foreign to common sense and deaf to the simplest realities. 3. Definition of Students Not Being Idiots
4. Logic ought to be learned through two-column proofs. 4. Definition of a Bad Idea

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:

Statement

Reason

  1. Angle 1 and Angle 2 are right angles.

1. Given

  1. m(Angle 1) = 90o, and m(Angle 2) = 90o

2. Definition of Right Angle

  1. m(Angle 1) = m(Angle 2)

3. Transitive Property of Equality

  1. Angle 1 and Angle 2 are congruent.

4. Definition of Angle Congruence

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.

Statement

Reason

1. High school ought to prepare students for their remaining years as scholars, and their future decades as citizens. 1. Definition of Education
2. Real mathematicians employ two-column proofs all the time! 2. Theorem of Lies
3. Two-column proofs are also used in the workplace and the political sphere. They’re everywhere! 3. Theorem of Even More Lies
4. I mean, we wouldn’t be using such an artificial, opaque system for teaching logic if it didn’t have some real-world utility, right? 4. Definition of Wishful Thinking
5. Logic ought to be learned through two-column proofs. 5. Axiom of Systemic Stubbornness in Education

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”.

100 thoughts on “Two-Column Proofs that Two-Column Proofs are Terrible

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