On applied math in high school

Walk into a university and you will hear that mathematics comes in two flavours: applied math and pure math. Applied math uses mathematical tools to understand the real world. For example, applied mathematicians created equations modeling how neurons activate in our brains, and compared their model’s predictions to experimental data. Pure math, on the other hand, is the domain of conjectures, proofs, and abstract ideas that live in our imaginations. For example, pure mathematicians have spent decades trying to prove that pairs of “twin” prime numbers are not just abundant, but infinitely abundant. 

But walk into a high school* and there is no distinction here: just, math. And yet, high school math is not applied mathematics — it implicitly takes place in a parallel universe rather than our own — nor is it pure mathematics — there are no conjectures or proofs in sight, nor mentions of aesthetics or beauty. In this post I argue that high school math is confused about its identity, falling into an abyss between applied and pure math and succeeding at neither. 

(Here, I focus on the applied side. In a separate post on “math appreciation” at VISST, I describe how pure math ideas are integrated into our curriculum.)

Consider this typical Grade 8 math problem: Yesterday Johnny ate 8 more apples than Lee, who ate 10% fewer apples than Johnny. How many apples did Johnny eat?

I abhor this question, not because of the calculation required but because the “correct answer” is 80 apples. Nobody eats 80 apples in a day! This is a common failure in attempting to teach applied math. A “word problem” is supposed to connect math to the real world, but word problems with nonsense answers achieve the opposite: they establish math as its own separate universe, rather than a tool used to describe our universe. Or, as Keith Devlin put it in his essay, The Problem with Word Problems, “The real lesson being imparted is that mathematics is a stupid, arbitrary subject having no relevance to the real world.” Sadly, such nonsense “correct answers” are all too common in math classrooms. 

One solution is to eschew made-up word problems entirely and apply math to real situations. A great example appears in Dan Meyer’s TEDx talk in which he has students examine a real water tank filling up with a hose, rather than a hypothetical one. I think this is a fantastic approach, though it can be resource-intensive to prepare just right. (The talk also highlights a key point: mathematics classrooms focus entirely on training students to find the right answers, completely missing an equally important mandate: training students to ask the right questions.) Another great way to apply math genuinely is interdisciplinary lessons via collaboration with other teachers. At VISST we are ramping up this approach next year with an integrated math/physics offering led jointly by two teachers.

In my own teaching, I connect math to the real world as often as I can. An exam question about buying McDonalds burgers required students to integrate their common sense knowledge and make some reasonable assumptions about how many burgers they can carry at once. A student (politely) grumbled that I didn’t provide them with all the information needed to answer the question. Aha! But this was exactly my point: math is part of real life, not separate from it. Building this mindset often requires un-learning some previously held beliefs about math.

In service of applied math, one of the main skills I aim to impart to my students is sanity-checking their answers. If a student finishes a calculation and arrives at an answer of “3000”, they often view the final result as just some squiggles on the page. Sanity-checking is the process of remembering that the “3000” represents something about our world; therefore, we can leverage our common sense knowledge to check if the answer makes sense. 

In one particular case, “3000” was a student’s answer to this exam question: 

Estimate how fast you walk in metres per second by estimating the size of your step, and the number of steps you take per second. Feel free to stand up and walk around if helpful. 

Does 3000 sound right? This particular student accidentally converted their measured step size of 60 cm to 6000 m (60x100) instead of 0.60 m (60÷100). That is completely fine! As I often tell my students, mistakes happen to all of us and cannot always be avoided. But what I want students to avoid is seeing 3000 m/s as their walking speed without objection and moving on to the next question. By contemplating 3000 m/s in the real world, we notice that this is much faster than an airplane! We can then start hunting for the calculation error and, usually, it is easy to resolve. The sanity check both helps us find calculation errors and also strengthens our mental connection between math and the real world. To my students I must sound like a broken record, repeatedly insisting on sanity checks, but they do build this sanity-checking muscle over time. 

Although I support teaching applied math by really applying it, in my classroom there is also a place for abstract problems, which serve as easy-to-craft extra practice for important concepts. For example, I do ask students traditional high school geometry questions like finding the third side of a right angle triangle:

Even here, though, one can find ways to keep the students alert beyond following memorized procedures. For example, on an exam I presented this triangle:

The only correct answer I accepted was, “That’s a ridiculous question!” 

Why? Because the triangle above has its longest side labelled 8 and a shorter side labelled 10. It does not make sense. I’ll admit I enjoy being irreverent for its own sake (and I think students enjoy it too!), but more importantly I am sending students the message that, contrary to typical school math, not all problems have an answer. The world is messy, and applied math is used to understand the world; therefore, applied math is messy.

Most high school math courses — even the highest level courses like calculus — are meant to be applied math. They do not enter the world of pure math by design, but unfortunately they do not truly enter the world of applied math either. To do so, we need stronger connections to the real world by going out and measuring things, using math to make predictions, and embracing ambiguity in our world. Students may need to un-learn their pre-existing relationship with math when they enter high school, but there is still time. High school is not too late to start objecting that nobody eats 80 apples in a day.

*It is hard to generalize about “high school math” as there are millions of such classes around the world and thousands just in BC. Here I am referring to the typical high school one might see in BC’s standard pre-university math curriculum, though likely one would find many similar classrooms across North America.

Thank you to Anupama Pattabiraman for several rounds of helpful feedback on drafts of this post.

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