Geometry Is Useless. Teach It Anyway.

An accountant needs to understand division, but he doesn’t need the long division algorithm because he has a spreadsheet. An engineer needs to understand rates of change, but he rarely sits down to work a calculus problem by hand. And when will anyone need to know that triangles can be proven equal by SAS, SSS, ASA, AAS, or HL?

Never.

The idea that we study mathematics in order to function in the real world is absurd.

People learn the most useful skills on the job through training or experience. So if the content is useless, the value of mathematics must lie somewhere beyond utility.

I teach my students geometry, specifically Euclid's Elements, for three reasons.

1. Geometry exposes them to mathematical beauty.

2. It trains them to reason deductively.

3. It gives them the experience of knowing things for certain.

The image below is of proposition 47 from Book I of Euclid’s Elements, the same diagram as the one at the top of the page, but with color added for clarity. It is a proof of the Pythagorean theorem: the areas of the squares built on the small sides of a right triangle add up to be equal to the area of the square on the longest side. A student studying this proposition begins confused. That’s fine.

As the proposition unfolds in the text of the Elements, he begins to understand. The line AL is key, which divides the big square into two rectangles.

The green square is double the triangle FBC. And the triangle FBC is equal to the triangle ABD. But the blue rectangle is double the triangle ABD. So the green square is equal to the blue rectangle. Similarly, the red square is equal to the yellow rectangle.

The two small squares are squeezed into two rectangular shapes, which exactly fill the square on the longest side.

With effort, our confused student discovers an immaterial beauty in the proof.

Bertrand Russell, in his essay The Study of Mathematics, says,

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."

To teach mathematics is to train the palate of students so that they can appreciate that beauty. In elementary school, this training forms a child’s attitude towards math, getting him excited about what more it has in store for him. In middle school, this looks like getting students to say “wow” (they will never say, “that’s beautiful”).

Aquinas says in the Summa Theologiae, Prima Pars, Q. 5 A. 4:

"… beauty consists in due proportion; for the senses delight in things duly proportioned…"

What is geometry but the art of continuous quantity, finding proportionality among magnitudes?

The physical sciences train students in induction. Mathematics trains them in deduction. In the science lab, students look for patterns in the way a ball drops to try to guess at the underlying principles. In geometry, students learn the underlying principles and definitions, then they use their reason to determine particular properties of triangles and squares.

Look back at the proof you just read. The green square is equal to the blue rectangle because triangle FBC is equal to triangle ABD. But why are those two triangles equal? Because of proposition I.4, the ancient version of Side-Angle-Side, proved forty-three propositions earlier. And I.4 rests on the definitions, postulates, and common notions at the front of Book I.

Euclid doesn’t ask us to trust him that the Pythagorean theorem is true. There is a continuous chain of deductions throughout the book. Each proposition is a consequence of things a student has already agreed to, one at a time.

In class, this has to be worked on intentionally. A student looks at two triangles and says they’re equal.

“Why?”

“Because they look equal.” That’s not an answer, and my students eventually learn this.

“What’s your justification?” He has to cite a proposition, either by number or by describing what it says.

A student who has been made to justify every step of a proof has been trained to ask what follows logically from what, and to notice when a conclusion has been asserted rather than argued.

A friend of mine who teaches Euclid to sixth graders tells his students partway through the year, “I have something to admit. It was all a lie. Everything we’ve talked about so far in Euclid is actually not true.” The students revolt, and he loves it. “Just kidding,” he replies, “isn’t that incredible, though, that you know it so strongly that you are willing to disagree with your teacher about it?”

A student can be certain of the Pythagorean theorem in a way he cannot be certain of almost anything else he learns in school. This is why Brian T. Kelly, Dean of Thomas Aquinas College, argues that mathematics “frees the student from the grips of skepticism.”

Some students think knowing a proof just means memorizing it. Others think it means reading it and understanding it once through. Neither of these is true knowledge. To know a proposition means understanding it and being able to produce a demonstration for it. That is scientia, true knowledge. A student who has done this has experienced absolute truth. He knows it exists and cannot be convinced otherwise.

If you want to know whether a student really knows a proposition, give him a blank sheet of paper. Ask him to draw the diagram and talk through the proof without the book. Two days ago he was confused, but now he can confidently prove a timeless truth.

Our work as math teachers is to foster habits of thinking, and to train the minds of our students not just to know the truth, but also to love and desire it.

What Schools Forget is reader-supported. Paid subscribers get the Timeline of American Education in Primary Sources, an annotated reading list running from 1647 to 1987, with a note on each source explaining what it is and why it matters. I revise and expand it as I find more. They also get Euclid in Color.