What do math and cookbooks have in common?
The problem with math education today and how we can fix it
“To solve this problem, use _____ formula. Then take the result and apply the ______ technique.”
Typical words from a maths teacher. For me (and probably most people), it sounds more like my lifeblood getting sucked out of me.
But don’t get me wrong: I love math!
During my spare time, I’ve ventured into finance, data science, psychology, etc and no matter which direction I turn, math always comes up. It has been pivotal for improving our lives in the last 200 years, and it is undeniably one of the most useful subjects for most careers.
This truth is not lost to most people — not to professionals who use math on a daily basis; not even to high schoolers who hate it with a passion.
But this hasn’t changed the fact that few enjoy learning math in high school. So what went wrong?
The problem with how we learn math.
We learn when we understand something.
What is a fetus? Oh, it's an unborn offspring of a mammal. We can say we’ve learned what’s a fetus when we know its definition (and of course, what the definition means).
But this isn’t how “learning” takes place in math lessons. Refer back to the very first sentence of the article. The problem is we never really understood what these formulas and techniques mean.
For example, when it comes to differentiation in high school math, several formulas are taught: the product rule, quotient rule, etc; along with techniques like implicit and parametric differentiation. The way we learn them is to solve equations that require us to apply these formulas.
And therein lies the problem. While we’ve learned to apply them, we barely learned how the formulas themselves work. Even if the latter is taught, they are quickly skimmed over by teachers or relegated to the appendix of worksheets
And that’s what math and cookbooks have in common. Cookbooks tell you the steps to take to achieve the desired result, but never how each step works. “Add eggs one at a time, beating well after each.” reads a website on how to bake a cake. But there’s no explanation for why eggs should be added.
This sounds awfully similar to formulas in math class doesn't it — where application is prioritized over understanding.
But math is nothing like a cookbook. Everything in math is provable, by definition. It isn’t just a laundry list of formulas or a catalog of techniques to choose from to solve problems with. Everything is connected under the rules of mathematics. In some sense, formulas are just shortcuts to help us deal with problems faster.
Cooking is not the same — it’s very difficult to understand exactly how things work. Why add eggs when making a cake? Well, it’s to give structure, color, and flavor. But how do eggs create structure? That becomes a very difficult question to answer — no wonder cookbooks don’t teach this.
But understanding the product rule in differentiation is much simpler; it only takes a few steps to derive the rule, yet this isn’t taught anyways.
Math education should help us understand math, rather than make us memorize a list of shortcuts.
The way out
The good news is: math education doesn’t have to suffer from this issue.
Rather than focus on applying formulas to solve equations, the formulas themselves should be taught. How did mathematicians derive the formula? In what cases is it useful? Why was it created in the first place? Applications, of course, have a place in the curriculum, but they shouldn’t be 100% of it.
And this idea isn’t new. Teachers far more experienced than me have realized the problem and tried to solve it. Unfortunately, they haven’t been successful because attempts have been constrained to enrichment classes and math clubs.
The fundamental problem lies with how we access students. Whether we like it or not, high school education is dictated by exams. Often, students only care to study what’s being tested in exams, while teachers are too busy and time-constrained to venture outside of the exam boundaries.
The current exam questions have thus far encouraged grinding math questions to pattern recognize what formulas to use for what questions. Memorizing equations is incentivized, making genuine understanding a “bonus”.
So how might we design math exam questions that test for genuine understanding?
But education systems are rigid. Pedagogy takes years to change, if not decades. Even if changes are made, the wait for them to get implemented in schools would’ve deprived millions of quality math education.
As such, taking more immediate steps is necessary. What teachers have currently tried have limited impact, so we need something completely different. But I must confess — I don’t have the answer to how this can be done.
Here are a few thoughts:
- How can we help students understand formulas while allowing this knowledge to aid them in today’s exams? Students are motivated by exams — this aligns their motives with our motive of teaching how formulas work.
- How do we make teaching formulas interesting in itself? A struggle constantly faced by teachers — simply throwing a bunch of proofs at students is the worst way to teach. Perhaps implementing some kind of educational game, involving the history of math formulas when teaching them, or even using VR to visualize math concepts like vectors — these are ideas educators and entrepreneurs can explore.
Would love to hear your thoughts in the comments section below!
