On a Common Criticism of Common Core, By Someone Who Knows Nothing About It

Caveat: I am probably the last person who should think about math education. Not only do I have little experience teaching the subject — help-session-tutor TA-ships in algebra and calculus in undergrad — but through most of my own schooling, math came easily to me. There’s a real sense in which I can’t understand what it means to struggle with the elementary school curriculum.

That said, it bothers me to see images like the one featured in this article on humour sites. (Go ahead, check the link. I’ll wait.)

It bothers me because my first reaction is: whoa, you’re an engineer who uses math every day, and you don’t know how to answer this question? How do you make sense of the math you use? What understanding of it do you have?

That’s not the intended conclusion of the image. It’s most often used as evidence that this type of math curriculum, teaching the fundamental concepts of number and operations that underlie the more familiar tasks of addition, multiplication, and division, is inane and useless. If an adult who deals with numbers and formulas in his or her working life can’t answer this question, the question must be a foolish one.

I disagree.

The reason I disagree is because — again, in my limited experience — what held back so many of the students I met during my help sessions was an approach to mathematics that emphasized memorization rather than comprehension and taught students to identify an algorithm instead of a process of reasoning.

For example, I remember helping a student work through an exam question she’d got wrong. After some discussion, we located the error: she’d simplified x squared over x as one over x rather than x. To use numbers as an example, her error looked like this:

4/2 = 1/2 instead of 4/2 = 2

“Oh,” she said with a sigh of exasperation, “I always forget which way that goes!”

I was taken aback: in the context of our discussion, the student clearly meant that she couldn’t remember whether one subtracts or adds the exponents when dividing by powers of the same base.

The way I’ve phrased it there may make it sound complicated, but the concept shouldn’t be, not if it’s taught with emphasis placed on understanding the concept rather than memorizing the rule. It’s no different than being able to remember whether 9 [3×3] divided by 3 is 3 or a third. How many times can you subtract 3 from 9? 3 times or one third of a time?

Another student, who often attended the help sessions, admitted he was frustrated with his grades in university.

“I was the best math student in my high school,” he told me.

I don’t doubt that he was; he was studious and determined to do well. But he struggled with calculus not just because the concepts were new and different (something everyone has trouble with) but because the strategies that worked for him in high school math didn’t translate to the new subject.

He’d show me a homework problem he couldn’t solve. I’d start to go through it with him to explain where he went wrong and help him come to the right solution.

“No, that’s okay,” he’d interrupt. “I just want to know how to do all the problems like this one. What’s the formula I can follow?”

Through conversation, it emerged that he didn’t have a strong grasp of mathematical concepts. In high school, he explained, his teacher had taught him to sort problems into categories by examining them for key phrases and features. She then assigned each type of problem a set of steps that students could follow to get the answers.

While this seems like a good way to help students master lower-level mathematics skills, it resulted in this student’s dilemma. Because he didn’t understand why he was applying each step to solve his problem, novel problems (or familiar problems that omitted the key phrases he’d learned) stumped him. In subjects like calculus, where problems are harder to classify into types that can be solved through a set procedure, his approach left him high-and-dry.

Now, this student intended to go into a job that required math. If he did (he was only in first year, and minds change in undergrad), I’m sure he’s doing well at it. It wouldn’t have required him to assess novel problems and come up with unique solutions.

But it worries me that, despite his ability — and despite the abilities of people taught like him — he doesn’t understand why he’s doing what he does.

So what? you might ask. If he prescribes the right dosage/builds a solid bridge/programs the correct algorithm, who cares if he has mathematician-like concepts of what he’s doing?

That’s just the thing. Learning mathematics by rote — procedurally — is good only so long as the problems stay the same and nobody makes an error. It terrifies me that some of the best math students graduating from the high-school system can’t spot or understand their own mistakes and have trouble adjusting to new problems.

Heck, I’m kind of the same way. I’m very good at applying what I already know to solve diverse problems, especially when I know roughly what argument is expected: word problems and exam questions hold no terror for me.

But show me an unsolved question or an open-ended puzzle, and unless I can attack it one of my usual ways — unless I can make it fit a familiar pattern — I get bored and frustrated.

My selection of patterns isn’t as narrow as that of students who don’t understand the underlying concepts. All the same, it keeps me from exploring further, tackling key issues, creating new solutions.

Not everyone wants or needs to create new mathematics. But even if we lived in a society where nobody became engineers, doctors, programmers, or scientists, we’d still need people who can tell the difference between an error in approach and arithmetic — who can do their own taxes, split their own bills, and estimate their own retirement funds even when the problems aren’t phrased a certain way or the parameters change.

So even though the above homework example may not be the best way to teach the concept of number (what do I know?) — even though the system may not be able to overcome the problem of teachers and parents who learned math the “old” way and may not grasp the new system — I can’t fault what it’s trying to do. Nor do I think the parent’s complaint is a strong argument against its methods.