I work in a fast-breaking, exciting field, with new results constantly announced. The preprint server arXiv lists hundreds of articles in mathematics each day, and about 34,000 articles were posted there last year. Mathematics has found application to every discipline, from science to engineering, to medicine and the social sciences, and even in fields such as the visual arts and literature. I find my mathematical research exhilarating, and it’s always glorious when work with my research teams or students leads to the discovery new theorems, models, algorithms, or techniques.

Put simply:

*Math moves the world*.

### A conundrum

Despite the importance of mathematics to every aspect of our lives, I’m stumped.

I’m stumped not just on my latest research problem but by the widespread practice of teaching the subject as dull and lifeless. Worst still, we teach it as if all the important parts of mathematics are settled. Mathematics is taught largely the same way as it was when I was undergraduate, and no doubt long before that. Yes, we use flipped classrooms and teach reformed Calculus, but the fundamentals of math curriculum and instruction never seem to change.

My critique is applicable at all levels, from grade school to grad school. I’m a university professor with twenty years of lecturing experience, so I’ll focus on that part.

The good news is that we can do better. Easily and tomorrow. Read onwards!

### Case 1: Sterile linear algebra

Linear algebra, if you randomly peruse the dozens of texts on the topic, is something to be suffered through, much like having your wisdom teeth removed or having a colonoscopy. As you read this, tens of thousands of students across the disciplinary spectrum blindly row reduce an endless variety of 3 x 3 or 4 x 4 matrices, drearily solve systems of linear equations, and struggle universally with concepts like spanning sets and linear independence.

What usually isn’t taught in our first-year university courses is the extraordinary power of vectors and matrices, and how, for example, they are fundamental to our understanding of search engines like Google or Bing. PageRank is one of the many tricks in Google’s arsenal, and one that was disproportionally effective in the early days of the company.

PageRank is the probability a random web surfer will visit a page, modified to allow for the occasional random jump to a new page. It can be described with a matrix that you can associate with any directed graph, and you can solve for it as a system of homogeneous linear equations. Even better, you can solve it efficiently with numerical techniques such as the power method, which requires only a knowledge of matrix multiplication.

Rather than teaching dry exercises on row reducing matrices, we would be better served to teach our students about PageRank. We could use it to explain to them what a matrix is, what matrix multiplication is, and how to solve a matrix equation. The topic is a gateway to eigenvalues and eigenvectors and makes a direct connection between linear algebra, probability, and discrete mathematics.

### Case 2: Calculus overload

One of the most profound experiences I had as an undergraduate was learning about uncountable sets. It was as if a doorway was unlocked, and I had my first glimpse at the real power of mathematics. This experience was ushered in by Bernard Banachewski at McMaster University, who fit the stereotype of the wizard mathematician, pointy eyebrows and all. As a twenty-year-old math undergraduate, I thought my brain would explode from excitement.

Up to that point, I’d taken a year and a half of university mathematics and had learned Calculus, linear algebra, Calculus, differential equations, and then more Calculus. Indeed, there was too much Calculus! Don’t get me wrong: Calculus, the mathematical engine behind the study of change, volume, and variation, is incredibly beautiful. I’ve taught it many times and was nominated for an award teaching it to engineering students. Real numbers are fabulous, mysterious, and present all kinds of open questions in number theory and computation.

But to make our students endure course after course in Calculus without a peek into how the universe of sets works is tragic. The reason why Calculus itself works rests on properties of real numbers and that number system was understood historically very late in the game.

Perhaps we can temper our obsession with Calculus by including…

### Case 3: More networks, please?

There’s a pervasive view that discrete mathematics and network science is a subject relegated to computer science programs, and of little import within pure and applied mathematics.

*Utterly. Hopelessly. Wrong. * Hearing that kind of stuff, I get that sinking feeling like when Officer K in Blade Runner 2049 flunks his Baseline Test: “Within cells interlinked. Within cells interlinked. Within cells interlinked.”

Discrete mathematics and graph theory, in particular, is at the heart of understanding social networks like Facebook, how proteins interact in our cells, and how information flows through our e-mails, phones, and websites. We use it to model stock correlations in financial mathematics and in our understanding of how molecules fit together. Discrete mathematics interfaces with all of pure and applied mathematics, from modelling to coloring, to searching networks in fun and deep games such as Cops and Robbers.

We need to bring discrete mathematics much earlier into the undergraduate curriculum, and it should play an equally important role as a topic like Calculus and linear algebra. Basic proofs in graph theory are great for beginner theorem-provers, and applications of networks work wonderfully within any applied mathematics program.

One of my quibbles: if you teach introductory graph theory, then don’t start the course with the Bridges of Konigsberg problem and Eulerian graphs. Hundreds of years have passed since the topic was introduced by Euler, and I can think of about a million things I would teach before that topic.

### Where to then?

Students don’t avoid the mathematical sciences because the topic is too hard. Or because it won’t get them a job. Students avoid the topic or quit it because we teach it as boring and lifeless.

I don’t blame them.

Mathematics, in its pure and applied forms, is the exact opposite of lifeless. If you are teaching mathematics and your students are bored, then *you are doing it wrong*. Take a good look at your curriculum, how you teach it, and how you engage your students. Can you weave in applications, prove beautiful theorems, or recount the history of the subject?

I don’t have the definitive answers on how to fix the problem, and as a caveat, I’m only shining a light. The consequences of doing nothing are disastrous, however. Can we afford to lose a whole generation of bright young minds because of our obsession with the way we’ve always taught the subject?

It’s time for bold new ideas. The time is now to become excited about mathematics and let that excitement shine through your teaching.

Anthony Bonato

Really interesting post! Could you please suggest us a few topics and book references to teach introductory graph theory at the undegraduate level and at high school level?

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There are no texts at the high school level that I know, but West’s book Graph Theory is perhaps the most straightforward reference. Much of my book with Nowakowski on Cops and Robbers should be accessible to undergrads or strong senior HS students.

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Thank you!

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Looking forward to you teaching MTH 108!

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