“I’ve always been interested in using mathematics to make the world work better.”
– Alvin E. Roth, co-recipient of the 2012 Nobel Memorial Prize in Economic Sciences.
DESPITE changing curricula and classroom tech, the core debate over math instruction remains stubbornly unresolved.
More data science, less stuffy trigonometry?
Students are placed in separate classrooms by test scores, or are they doing differentiated work in the same classroom?
These questions are vexing, but I’ve got one suggestion for how we can improve: we can tell students that math is very, very hard.
It’s the truth.
The techniques of algebra, geometry and calculus were hard to create, and they’re hard to learn.
But saying so forthrightly doesn’t come naturally to many teachers or education commentators.
“Math is not hard: A simple method that is changing the world”, reads a headline in my inbox from a retired headmaster, extolling an approach that aims to ease kids into the subject.
It reminded me of early attempts at encouragement when mentoring a group of math students in Padawan.
(For the past two years, my friends and I have been guiding young minds overlooked by the system, yet capable of reframing problems in ways that most high scorers wouldn’t even consider.)
I always began problems by saying, “Now this is pretty simple”, thinking I was helping them build confidence.
But Andrew Gelman, my math professor who happened to be visiting on holiday and stepped in to help, set me straight.
When we say a lesson is “easy” or “simple”, and it manifestly isn’t, we are telling students that the difficulty isn’t with the mathematics – it’s with them.
And they will believe us.
They won’t think, “I’ve been lied to.”
They’ll think, “I’m dumb and I should quit.”
This applies to parents, too.
I’ve been learning math for years, and I still find myself telling my kids that a concept they’re struggling with is “not that hard”.
That’s not encouragement. That’s just my frustration with watching them struggle.
It’s not part of teaching.
One big problem is that math teachers mastered the concepts so long ago, they’ve forgotten how difficult they once were.
An oil and gas analyst once told me that secondary school calculus was as easy as following a recipe.
In one sense, that’s true: following a recipe is easy, once you know how to cook.
But recipes require tacit knowledge and substantial experience that novices don’t have.
How much salt is a dash?
What’s a rolling boil?
You learn to cook by cooking in the presence of someone who knows how.
At first, you flail; you make plenty of mistakes; you get results that are right in some ways but very wrong in others.
The outcome of all that work is that you become another person who thinks cooking is easy.
This isn’t just true of calculus, which most non-mathematicians accept is supposed to be hard.
It applies to supposedly easier things, too, like fractions, a fourth-grade standard.
When we first present fractions to children, we’re asking them to make a huge conceptual leap.
For their whole life until that moment, the definition of a number was something that answers the question: “How many?”
A fraction is a different concept, not so much a number as an amount.
Still, they’re supposed to add and subtract them, just as they can with “regular numbers”.
Even expressing a number as a string of digits is a deep, hard-won idea that takes time to grasp.
The idea that math is supposed to be easy gets in the way of the most effective learning tool students have: asking questions.
If math is easy, you should get it.
So students are afraid to ask questions in class because they’re afraid of looking stupid.
The situation is even worse for students who, because of gender or race or accent or household income, have a justifiable fear that their classmates or, worse, their teacher, will jump to exactly that conclusion.
I went through that myself, so instead of speaking up, I’d pack my bag and walk out of class each time.
If we were honest about how difficult and deep mathematics is at every level, this would be less of a problem.
We could move toward a classroom where asking a question means not “looking stupid” but “looking like someone who came here to learn something”.
That shift would help all children, not just those struggling.
Yes, some kids have no trouble picking up the basic rules of algebraic manipulation or geometric constructions.
They should still be asking questions of their teachers and themselves.
For example: “I’ve done what the teacher asked, but what if I’d tried this other thing the teacher didn’t ask of me? And for that matter, why did the teacher ask what they did and not something else?”
No matter how much math you’ve learned, there’s something adjacent to what you’ve mastered that you don’t yet know, and that’s where to point your eyes.
If math class is easy, that curiosity means you’re doing it wrong.
How do we build a classroom full of questions, where students engage with math not just to find answers, but to ask better ones?
Although math is often seen as a subject of answers, its real power lies in the questions we dare to ask.
This is how math has developed across history: by people asking more and more questions, and not being satisfied with superficial answers.
The questions might seem innocent, naive, vague or philosophical.
Any question stemming from curiosity, confusion, scepticism or even disbelief can lead to deep mathematics.
In that sense, mathematicians are a bunch of anarchists.
One day, a student asked me, “Is this really what they teach mathematicians at Columbia?”
Absolutely.
Believe it or not, mathematicians like making up their own rules that make sense for particular situations, and we hate having rules imposed on us.
Because we’re not just solving problems; we’re reshaping the rules to explore what’s possible.
That’s why the most powerful questions in math aren’t “What’s 84 times 107?” but rather:
Why does 1 + 1 = 2?
Why can’t we divide by zero?
What is zero?
What are numbers?
Where does math come from?
How do we know math is right?
How do we know math is real?
Is math real?
My kids are taught to ask why something is happening or in what context something is true, not merely arriving at correct answers.
One plus one is often two, though there are situations where something else happens.
If you mix one colour of paint with another colour of paint, you get one new colour of paint, not two colours.
If you turn a piece of paper over, and turn it over again, you get back to where you started – so in a sense, one plus one leads you to zero.
If someone impatiently presses the button for the elevator after you’ve already pressed it, nothing new happens: you don’t call two elevators.
Unfortunately, these aren’t the scenarios that usually come up in math class or math tests here, but they’re all interesting mathematical ideas.
They have fancy mathematical names like “the cyclic group of order two”, or “idempotents”, or “binary operations”, and you don’t need to know that to think about paint or elevator buttons.
It turns out that the questions mathematicians care about are the same questions asked by people who think of themselves as “not math people”.
Mathematicians and math-phobic people have many attitudes in common: we don’t think math is obvious, we don’t like just being told that things are true, and we don’t want to memorise answers to anything.
Lowering the barrier to question-asking means lowering the barrier to learning.
I get it: “Math is hard” can be discouraging.
But “Math is easy” is just false, and that’s even worse.
We can be truthful without being demoralising.
We can tell our students: Math is hard, and you can do it.
The views expressed here are those of the columnist and do not necessarily represent the views of Sarawak Tribune. The writer can be reached at med.akilis@gmail.com
