This speaks more to my naiveté as a first-year teacher than anything else, but I was shocked to find how fervently my students despised the things they called “word problems.”
“I hate these! What is this, an English lesson?”
“Can’t we do regular math?”
“Why are there words in math class?”
Their chorus: I’m okay with math, except word problems.
They treated “word problems” as some exotic and poisonous breed. These had nothing to do with the main thrust of mathematics, which was apparently to chug through computations and arrive at clean numerical solutions.
I was mystified—which is to say, clueless. Why all this word-problem hatred?
To me, the very phrase “word problems” sounds bizarre. It’s like “food meals” or “page novels.” Of course meals have food. Of course novels have pages.
And of course math problems have words!
One half of mathematics is “pure math.” It deals with abstract ideas, pursued for their own sake. It demands clear definitions and logical arguments. In short, it’s chock full of words. You don’t have trust me on this: take a look at a random research paper in pure math. There are more sentences than equations.
The other half of mathematics is “applied math.” It aims to solve real-life problems, working its fingers into fields as diverse as science, finance, design, and government. If you want to communicate about the quantitative problems in these fields (let alone solve them), well, you’d better get ready to use some words.
Word problems aren’t an invasive species. They’re the whole biosphere.
My first year, I taught students who had come through a tough, testing-driven middle school. They had bought into a simple vision of education: school is work. To them, cranking through math problems was like hammering nails or folding laundry—not necessarily a barrel of laughs, but satisfying in a familiar, workaday way. You’ve got a task. You do it. Then it’s done.
Willing to work hard? Yes; admirably.
Willing to think hard? Not necessarily.
I suspect many students share this rather joyless, overcast view of education. They may not love math, but at least they know what to expect. Word problems violate that contract; they interrupt the flow.
To see why, take an example:
Now, I look at this and think, “Ah, a problem about traveling objects.” This conjures up a whole mental map of ideas:
Brushing aside the clutter, you’ve got this basic structure to my knowledge:
The concept forms the foundation. Everyday words, mathematical symbols, whatever—those are just different languages for talking about the underlying idea. With this kind of understanding, solving the problem is pretty straightforward:
But if I’m a typical word-problem-hating student, I look at things a little differently. My knowledge is structured like this:
For a student like me, “doing math” means learning which symbols trigger which procedures. For example, the symbols “4 + 2x = 18” triggers the procedure “18 take away 4, then divide by 2,” giving an answer of 7.
These connections between symbol and procedure are mostly arbitrary. I take away 4 because… well, because that’s what I’m supposed to do. There’s no deeper reason.
And uh-oh… here come word problems.
My old triggers (the symbols “x” and “+”) are nowhere to be seen, so I don’t know which procedures to execute anymore. I need to start from scratch, learning a whole new set of cues.
And even worse, these new triggers aren’t clear and unambiguous like the old ones. They’re subtle and tenuous. They’re embedded in that messy human creation called language—full as it is of paraphrase, omission, and implication.
My teacher thinks that these “word problems” are a lot like the numerical ones, but to me, they’re a whole new breed. All my old knowledge feels useless in this unfamiliar verbal swamp.
Is it any surprise I’d rather stick to what I know?
As I wrote last week, procedural thinking is often useful (even crucial!). But it cannot replace real understanding, the kind of conceptual thinking will keep bearing fruit. Moreover, success demands connecting the two, linking concepts to procedures, ensuring always that the how is rooted in the why.