Problem-Solving and the Unparalleled Euclid
Soon after preparing a discussion for a geometry class, I read a recent article about the future job market. It noted that, according to a new report, 40% of jobs in Australia are likely to be automated in the next 10 to 15 years. To thrive in the future workforce, employees “will need to acquire a different set of capabilities than what is currently prioritized,” claims the authors of the report. Which capabilities do they suggest? Innovation. Creativity. Problem-solving. I smiled as I returned my focus to parallel lines and right angles, for the report had just identified one of the benefits of learning geometry. More accurately, it identified a benefit of learning geometry the way students do at the Millis Institute.
You’ve likely heard that a2 + b2 = c2. But do you know why this is the case? Most geometry courses simply hand this formula to students to memorise and utilise in calculations. But at the Millis Institute, we have our students explore why the Pythagorean Theorem is true. That is, they must prove why, in a right triangle, the square of the hypotenuse always equals the sum of the squares of the other two sides.
What’s the point of this exercise? Surely, a student will never actually have to justify the Pythagorean Theorem in “real life.” After all, it’s just an accepted equation in math, and computers can do the calculating for them. The reason we have our students prove this famous theorem is that it trains them to think in a certain way. It trains them in the habits of justifying their claims, tracing the logical links between the steps of an argument, and solving problems (which is one of the skills required in the future workforce). In other words, it trains them to think like Euclid.
Euclid lived around the time of Aristotle (300 BCE). In his book, The Elements, Euclid provides a very complex proof for a2 + b2 = c2. His genius was to base all his geometry on the fewest possible number of assumptions, called “axioms.” He narrowed these self-evident starting points down to only five, having to do with basic claims about points, lines, circles, right angles, and parallel lines. From these building blocks, he proved over 460 different truths in geometry--including the Pythagorean Theorem--each one relying on the proofs worked out before it.
This sort of axiomatic system is a gymnasium for rigorous thinking. You can think of geometry as a logic class using shapes—a mental training ground of triangles.
Abraham Lincoln understood the value of thinking like Euclid. While studying to be a lawyer, Lincoln realised that he needed to become an expert in what it means to “demonstrate” a point or argument. What did he do? He went to his father's house and “staid there till I could give any propositions in the six books of Euclid at sight. I then found out what 'demonstrate' means, and went back to my law studies.”
Today, many people consume “fake news” without being able to discern the difference with “real news.” Many employees hit a brick wall when asked to tackle a problem that wasn’t part of their formal training, and many home-owners get stuck when something goes wrong and requires a work-around. And, sadly, many people of faith do not comprehend the underlying reasons for their basic beliefs. Which raises the question as to whether modern education has given up teaching units like Socratic Logic and Euclidean Geometry too quickly. Even though most of the students at the Millis Institute will not go on to become professional mathematicians, they will enter a world in which problem-solving is a necessary skill. To paraphrase C.S. Lewis, whatever they do, they will think … and if they do not think well, they will think poorly. When it comes to guides to thinking well, Euclid has no parallel!
