Is Math Geometry?

I think my earliest memory of mathematics was learning geometry. Yes, as a young child I learned to count and numbers and 1+1=2. But from the perspective of learning math in a math class or math lesson, the earliest memory I have is geometry.

The Joy of Mathematics

I grew up in Kitchener, Ontario, and went to Catholic school. For reasons that are out of scope here, Ontario has had a "separate" Catholic school system alongside its public school system, both of which are publicly funded and have standard curricula that are fairly similar across academic subjects like reading, math, science, and so on. My first encounter with the mathematics of geometry was likely shared by many, many students all over Ontario, likely the rest of Canada and many parts of the world. As a point of reference, I think we learned geometry just before (or concurrently with) learning times tables.

When I mean we learned geometry what I mean is we learned Euclidean geometry. This is the mathematics of what can be created and proven with a compass and a straight edge on a flat piece of paper. This is literally an ancient subject of mathematics; Euclid's Elements is thought to be one of the first textbooks ever created. It appeared in roughly 300 BCE. Many of the concepts found in the Elements are, almost line for line, what was taught to me in grade school.

I can remember having a small compass and straight-edge set. It came with a tiny wood pencil, a protractor for measuring angles, and the "straight-edge" was actually a small ruler with measurement in inches and centimetres. Along with pencil crayons, stacks of lined paper, pencils and pens, it was a standard item I needed at the beginning of the school year during elementary school until around grade 6 or so.

As a refresher on Euclidean geometry and what it entails, consider two straight lines on a page. Euclidean geometry makes some fundamental assumptions about the structures of lines and shapes (called axioms) and then uses these assumptions to logically try to prove some results.

In the above image, you can see two horizontal lines and some dots with letters. The two lines with arrows are assumed to be parallel, and so will never meet in either direction. There’s another line that is not parallel and crosses both of these parallel lines. The dots represent points on the plane, and these points are labelled A, B, C, and D, respectively. By using Euclid’s axioms, we can show that the angle defined by ABC (the top semi-circle) is equal to the angle defined by ADC (the bottom semi-circle). Such a proof can be found in various places.

Since Euclidean geometry was a product of the classical period of ancient Greece, I'll refer to this kind of geometry as either Euclidean or classical interchangeably throughout this post.

Euclidean geometry taught in elementary schools is a good introduction to mathematical thinking. Basic logic skills, formulating a problem into a solvable form, using standard methods to solve similar problems, and more are developed from geometry. It also introduces concepts like triangles, circles and parallel lines, which are incredibly useful not just for further mathematics studies but in all sorts of parts of life. For teachers, Euclidean geometry is also a good topic to introduce because as a field of mathematics it is "done", and in turn is highly consistent and well understood. No rogue student is going to draw a triangle with angles that add up to 169 degrees on a flat piece of paper.

What I remember most about learning geometry is that it was, well, enjoyable. Most students in the class enjoyed learning geometry, whether from the logical deduction aspect or because students could draw shapes to learn things. Concepts were logical and made sense, particularly since you could see them visually in front of you. I recall almost everyone in my class at least somewhat enjoying geometry. Even if students didn't like working with circles and lines, they didn't appear to outright hate working with them either. In geometry class, life was not too bad, from a pedagogical perspective.

Geometry, Interrupted

Then a funny thing happened: sometime around maybe grade 6, we stopped doing geometry. Compasses and straight-edges were no longer needed, and we moved on to other topics in mathematics. Eventually, just before high school, algebra was introduced. Occasionally we would make use of facts from geometry such as angles in a triangle adding up to 180 degrees but overall, there wasn't much geometry. This changed slightly in high school where in grade 10 we learned trigonometric concepts, and later in my final year of high school where I took advanced math courses and learned techniques of proof (not from a required math course). Compared to other topics we learned in grade school like multiplication and division of integers, basic algebra and factoring, geometry more or less disappeared in the classroom.

Geometry simply stopped being taught. The only way you could learn geometry formally was to pursue a mathematical education. In my undergrad there were courses offered in Euclidean geometry, which was followed by Non-Euclidean Geometry, as upper-year math courses, but that was it. Geometric concepts arose in other courses but were either assumed or used to illustrate other mathematical concepts such as proof techniques. Even more generally, Euclidean geometry is seen by mathematicians as an inactive area of math research. It's not quite "dead" in the sense that Latin is a dead language, but it's not far off from that state either. Keeping in mind that mathematics research overall is vast, it's kind of interesting that there's not much "active" work being done in classical geometry.

There's also the aspect of logic and fun. Euclidean geometry is a relatively accessible subject. It is a topic that is visual, lending itself to drawing and painting. It's intuitive, as classical geometry is based on logical thinking. And it seems to be enjoyable. I don't recall much mathematical anxiety in any geometry classes I've taken. Math can be fun, it appears.

Is Mathematics Geometry?

If geometry encompasses logical thinking, visualization of abstract concepts and making shapes, mathematics could well be geometry overall. Geometry can be a neat introduction to mathematical thinking and even to mathematical rigour, without weeping and gnashing of teeth. Perhaps this is really what math should be: a way to learn clear logical thinking in an accessible way.