A new assignment this year teaching middle and high school students in a new school in a new state has reinforced many of my long-held thoughts about two fundamental types of math students: 1) those who think they can’t do math without lots of help and 2) those undaunted to try, who see problem solving in math as an extension of problem solving in other arenas.
Painting in broad stokes, we might label these two types of students mathophobes and mathophiles, and the neurological, psychological, and performance differences between them are striking, fascinating, and perplexing.
Equally striking, fascinating, and perplexing is how the very same individual students often THINK completely differently about math and…EVERYTHING ELSE. My strong sense is that they use far less of their brain’s capacity while doing math than they normally use.
For decades, I’ve worked with students with a wide range of ability and learning—some hard-working and some not so much—many of whom exercise impressive critical thinking and cognitive firepower in other subjects—and in life in general—but as soon they encounter a math problem, their IQ falls off a cliff as their their body language screams, “I can’t!” If you look carefully, you almost see through their eyes the sudden appearance of blocks in synaptic pathways as they twist into whole new neural networks.
If you’re one of those students who doesn’t like math—or perhaps you’ve convinced yourself math doesn’t like you—I have good news:
No matter how math-phobic you are, you CAN do something about it.
I know, because I went through the process myself, and can bear witness to many struggling math students who made like a frightened caterpillar turning into a resplendent butterfly as they became competent math students, some of whom even now claim—gasp!—to enjoy the subject.
Why is it the same students who can immediately understand the most complicated sales in their favorite stores in the mall are completely befuddled when they sees questions like this on a math test?
A store is having a sale on shirts. If you buy one at the regular price, x dollars, you can purchase as many more as you like at z dollars off. If a customer purchases a total of n shirts, how much do they have to pay?
NEWSFLASH! SOLVING MATH PROBLEMS IN SCHOOL IS THE SAME AS SOLVING MATH PROBLEMS IN THE REAL WORLD. REALLY.
Look, there are basically THREE kinds of math students: those who can count and those who can’t. 😂 No, really, here’s a slightly more nuanced distinction between the two kinds named above, Mathophobes and Mathophiles.
Mathophobes:
Students who BELIEVE they’ve either been taught how to do certain kinds of math problems and have memorized formulas and problem-solving strategies they can apply, or, if they haven’t seen those kinds of problems before, immediately assume they don’t know where or how to begin. (One of the many fascinating features of this phenomenon is those very thoughts trigger lightning-fast reorganization of neural networks in their brains, somehow shutting down pathways and consequent cognitive abilities which these students normally and easily access everywhere else.)
Mathophiles:
Students who approach math problems the same way they approach every other problem in their lives—that is, with their whole brains—and exercise their full critical and analytical faculties. Neither their minds nor their brains apprehend a difference between a math problem in a book and a math problem in a store. I hypothesize brain scans would show neural networks firing to their full capacity as usual.
There are many reasons why students may identify as mathophobes, unhappily suffering through required middle and high school math classes. Most of them in my experience give up at some point, coming to believe math just isn’t their thing.
But the happy truth is in math, as in life in general, pain may be required, but suffering is optional! There is a solution. (See what I did there?)
Students can learn to apply the same common sense they use in almost every other activity, from tying shoes to figuring out an approach to a new relationship to sneaking a soccer ball past a defender. We humans all have the same basic cognitive ability, neurologically speaking, to solve the problems we encounter. Yes, all of us. Even YOU, even if a teacher or parent or you yourself have been known to say otherwise.
Let’s return to the problem above, an actual SAT problem rated medium to difficult on the Easy-Medim-Difficult scale. Put any high school kid in a store with such a sale, and very likely they’ll know exactly how many shirts they can buy for any given amount of money they have to spend. But put it on the SAT, and, well, out of nowhere, “I have no idea even where to begin.”
“Okay, so let’s say you want to buy 10 shirts. How many do you have to pay full price for?”
“Uh, 1.”
“Okay, and how many would you pay at the sale price.”
“Uh, 9.”
“So in terms of n, how many would you pay at the sale price?”
THINKING…suddenly thinking about and relating 9 and 10 to n from the problem, “O, I get it, (n-1), because you have to pay full price for just one shirt and the sale price for all the others.”
“Right, so how much does it say the full price is”
“x dollars.”
“And if you get a z dollar discount, what’s the sale price?”
THINKING again, “Well, x–z dollars.”
“Bingo! So how much do you have to pay in total if, again, you buy one at the full price, z, and all but one at the sale price, which you just said was x-z?”
“Oh, I get it now, x + (n-1)(x–z).”
I just love it when students say “Oh, I get it now.” It’s like the light comes on and what was once completely baffling is now almost intuitively obvious, or at least fully understandable, the way the amount of money they wouldd have to pay the cashier is fully understandable. No memorization of formulas or problem-solving strategies required! Taking students through problems like this enough times DEMONSTRATES to them they can actually THINK about math for themselves.
Math is not a set of rules and formulas and procedures to memorize as so many U.S. high school students believe, despite state-mandated curriculum frameworks intended to promote critical thinking. Solving math problems is applying common sense and logic that, in my experience, every student possesses in great enough measure to master primary and secondary math, at least in the U.S.
And if you don’t believe it, that’s half the problem right there!
YOU CAN LEARN TO BE A COMPETENT MATH STUDENT