Mathematics is one of humanity’s most creative and poetic endeavours.
And it is a disaster that it isn’t taught this way to students.
“A Mathematician’s Lament” is a classic polemic (later expanded and published as a book) written by maths teacher Paul Lockhart. The essay is a devastating and passionate assault on the mechanistic way mathematics is taught in most of our schools.
A Student’s Nightmare
Lockhart begins with a vivid parable in which a musician has a nightmare in which music is taught to children by rote memorization of sheet music and formal rules for manipulating notes. In the nightmare, students never actually listen to music, at least not until advanced college classes or graduate school.
The problem is that this abstract memorization and formal method based “music” education closely resembles the “maths” education that most students receive. Formulas and algorithms are delivered with no context or motivation, with students made to simply memorize and apply them.
Part of why many students end up disliking maths, or convincing themselves that they are bad at maths, comes from this emphasis on formulas and notation and methods at the expense of actually deep understanding of the naturally fascinating things mathematicians explore. It’s understandable that many students (and adults) get frustrated at memorizing context-free strings of symbols and methods to manipulate them.
This goes against what maths is really about. The essence of mathematics is recognising interesting patterns in interesting abstractions of reality and finding properties of those patterns and abstractions. This is inherently a much more creative field than the dry symbol manipulation taught conventionally.
Playing With Triangles
Lockhart uses a geometry problem to illustrate. He draws a triangle inside a rectangle:
How much of the rectangle does the triangle take up? Lockhart notes that mathematicians are interested in shapes in the abstract:
“I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what maths is — wondering, playing, amusing yourself with your imagination.”
We’ve come up with this imaginary triangle, and now we want to better understand it. The way to do this is to try different things and see what they tell us about the triangle.
Lockhart presents one possibility that turns out to be useful in answering the question: drawing a vertical line from the top of the triangle to the base:
This answers our question:
“If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box!”
Drawing a triangle and playing around with it and eventually realising something about the relationship between the triangle and the rectangle is much closer to the spirit of mathematics than simply being told a formula.
In this example, we’ve actually figured out and proven the triangle area formula written in the front cover of any middle or high school geometry textbook: Area = (1/2) × (length of base) × (height). The length of the base times the height gives us the area of the rectangle, and we just observed that the area of the triangle is half of that.
Challenging students to think about shapes, numbers, symmetry, or motion are more fun than the standard practice of memorizing techniques and applying them over and over again. Allowing students to explore these concepts and figure things out for themselves also builds up the critical thinking and reasoning skills that we supposedly want our children to learn rather more effectively than applying a handful of memorized, unmotivated, and unexplained formulas dozens of times.
The Soul of Mathematics
At the heart of mathematics is a need to understand structures, real or imagined. This is a profoundly speculative and creative exercise: a strange type of higher dimensional shape might hint that it has some interesting properties; a data set describing Ebola infection rates could roughly fit the same pattern as uranium atoms undergoing atomic decay. The job of the mathematician is to find and, far more importantly, explain these kinds of properties and relationships.
While it is important for students to work through a few basic problems at every level of mathematics they encounter, we live in an era when, once an understanding of the underlying concepts is mastered, one can turn to calculators or computer programs to do the mindless symbolic manipulations needed to get an answer. Pedagogy needs to move away from finding the answer, and towards understanding why this is the answer and why we care about the answer.
Mathematics is unique among human endeavours because it combines our most “right brained” creative, abstract, imaginative instincts with our most “left brained” logical, evidence based, focused instincts. Maths is about making a poetry out of pure reason and about abstractions based on seeing patterns in our world, and it is very sad that so few people ever get to experience this.
Lockhart’s entire essay is a beautifully and passionately written plea for a better way of educating students to truly understand the wonderful world of mathematics. Anyone who has any interest in maths and maths education should read the whole thing.