“When will I ever need Pythagoras?” — an honest response

It is so much more than a formula

Why do we have to learn Pythagoras?

It’s a question that stretches back almost as much as the Greek maestro himself, and one that hip-hop artist and activist Akala recently challenged educators with. For maths education to find relevance and resonance with students, we have to interrogate the stock response of ‘because it’s useful’. Is utility really the goal of mathematical study and, if so, is the current brand of school maths even serving this purpose?

As currently taught, the usefulness of Pythagoras’ Theorem is far from obvious. The procedural diet of the curriculum elevates the Theorem as a divine formula that must first be accepted without question, and then applied to countless examples to resolve missing triangle lengths. The joyous ritual of symbol pushing leaves the question: when will I ever need this?

The mathematician knows that applications of Pythagoras are to be found in all corners of their subject, but learn this inane formula so that you may learn a few moreis the most circular of rationales. Perhaps then, we look to the real world. Surveyors, architects, engineers all rely on Pythagoras, right? Kite flyers too, for that matter. The problem with appealing to career prospects is that it asks students to delay their reward: learn this today because it will help you several years from now.

Seeking to make Pythagoras relevant today, textbooks are riddled with botched attempts to retrofit concepts into students’ everyday lives. It’s not like students have struggled through life on account of not having Pythagoras at hand. And even in the most plausible real-world applications of Pythagoras, the rigid focus on cold calculation remains.

As a blunt formula, Pythagoras’ Theorem is too high a price for students to pay for the paltry benefits it returns — no wonder so few are buying. Can we do better?

How about Pythagoras as a gateway to mathematical reasoning? The greater significance of Pythagoras’ Theorem is that it holds for all right-angled triangles, large or small — all infinitely many of them. To proclaim any statement for an infinite collection of objects is audacious; such assertions demand the highest form of argumentation, or proof. Mathematical reasoning is powerful enough to cut through finite bounds; a single argument can qualify Pythagoras’ formula as true for every conceivable right-angled triangle. Here is a sample of just one available proof:

By comparing the white space in both pictures, we arrive at the treasured formula, a²+ b² = c².

You can find heaps of alternative proofs — Elisha Scott Loomis gathers no fewer than 344 in a single volume. Every proof invites you to probe; to interrogate each assumption and every leap in logic. Your reward is permanent truth: faultless logic cannot be undermined by any experiment, now or in the future. With experience, you will learn to tweak the parameters, giving rise to all sorts of interesting problems and solutions — things get especially fruity in higher dimensions.

Mathematics is conditioning for the mind. Pythagoras’ Theorem is a proxy, one of many, for helping us to develop our reasoning skills. Its substance lies not in number crunching, but in asking why? until the most uncompromising standards of rigour are met. In an age where reason and truth are under attack, what better way to market mathematics than as the most creative system ever invented to help humans think critically?

Constructing proofs, or even combing through ones handed to us, can be immensely satisfying. To see the rhyme and reason of a mathematical argument fall into place can trigger the most delightful of human sensations. The most profound reason for studying mathematics may be that it does not demand a reason other than its intrinsic joy and beauty. In this regard, mathematics must also be considered an art — we study maths because, quite simply, we can. And while the beauty of mathematics may not be immediately apparent to all, as educators we can strive to make it so. If Shakespeare can be renovated for modern times then so too can our representations of maths.

No study of mathematics can be considered complete without attention to its historical underpinnings. As Akala is aware, the non-European roots of mathematics run deep. Pythagoras’ Theorem is a case in point. Right-angled triangles, and the triples they give rise to, were the subject of intellectual inquiry across multiple civilisations. The formula now accredited to Pythagoras was uncovered several hundreds prior to his own work. So there’s another reason Pythagoras matters — because the prominence of one mathematician can draw our attention to the countless others neglected in our curriculum.

Pythagoras’ Theorem, then, is more than a formula: it exemplifies how mathematics is bound to logic, art and history. Three decent reasons to engage with those pesky triangles.

I am a research mathematician turned educator. Say hello on Twitter or LinkedIn and sign up below to receive more content like this.