In the first two parts of this series I hope to have shown that 1) real learning and understanding cannot be built on shaky foundations and that 2) math is a language that students can appear to speak perfectly even when they only have shaky foundations. Or: they can’t learn it unless they really understand it, and we can’t know if they really understand it.
What Convention Has Already Addressed
Even the most traditional math classrooms have word problems. The word problem can be as thin as a veneer of English over an equation: “If I have two apples and you have three more than I do, how many do you have?” instead of “find x if .” It can be as thick as a paragraph of English over an equation, or even, once you get in to advanced classes, two equations.
I mock word problems, but actually, they’re great. They go a long way towards eliminating that student strategy of memorizing and applying rules without understanding semantics. Even if you can solve by only memorizing “if-you-see-a-negative-number-on-the- side-with-the--then-add-that-number-to-both-sides,” you can’t solve “If I have two apples and you have three more than I do, how many do you have?” that way. At least, not until your teacher has given you ten or fifteen of the same sentence structure. Word problems require students to do more translations in their brains and so are more convincing evidence that a student understands. Word problems connect some English fluency (which we assume the students have) with mathematical fluency (which we assume the students haven’t).
What Are Way Better Than Word Problems And What Riley Feels Like He Invented Even Though Obviously He Didn’t
The problem with word problems is that the teacher must prepare all of the information for consumption by the student. While the student must connect some English fluency with mathematical fluency, he is still just using symbols at an abstract level. We can swap out “apples” for “grapefruit” (mmm, please do) and “three” with “seven” and the thought process is really identical. If the student had to prepare his own information, then we could be sure, proof positive, that he understands what he’s doing. In a word problem, doing random subtractions and additions on the numbers three and two, you can come up with like three different answers (1/3 chance of being right). If somehow you could present the problem without giving away these numbers, the likelihood of random success would drop by a lot.
A Slightly Better Word Problem (more information given)
Instead of asking, “If a rectangular field is 120 yards long and 50 yards wide, how long is the diagonal?” (1/3 chance of random success plugging numbers randomly into Pythagorean theorem) you could ask,
“Look at the diagram of the field below. It has a circle in the center of the field that has a radius of 5 yards, and there are stripes across the field every 10 yards. The lawn is cut with a mower 2 yards wide that takes 60 passes to cut the entire field. In total, the field is 120 yards long, and 50 yards wide.
How long is the diagonal line across the field?”
This is a good attempt at lowering the chance of random success (now something like 1/45) without raising the difficulty of the fundamental question. Throw in the surface temperature of the sun and the period of Haley’s comet if you like. These questions require more mathematical fluency – more basic understanding – to answer correctly.
My beef with this method is that your questions are really confusing and muddled and it feels like you’re deliberately trying to confuse the reader (you sort of are). And you’re still producing all of the information here – your students are still just consuming the numbers you come up with. The only difference is that now they have to have a little taste to distinguish between your good numbers and your fishy numbers.
A Way Better Word Problem (less information given)
“Here’s a satellite picture of our pond. What is the distance between the two red points?
To help you, I’ve put a traffic cone in each of the actual locations indicated by the three points on the picture. There are 150-foot tape measures in the closet.”
To answer this question, your students would have to make the ultimate connection: mathematical fluency with physical fluency. They already understand distance, and measuring it. They have to get that and in Pythagoras’ Theorem are distances, and can be measured.
The benefits of this type of problem go beyond lowering the chance of random success (now immeasurable, but obviously smaller). Since this problem connects to physical fluency, students are empowered to know about the reasonableness of their own answers. If they measure the two sides of the triangle to be 200 feet and 250 feet and come out with an answer of 1000 feet, their physical fluency will tell them there is a problem where their underdeveloped mathematical fluency could not.
An Aside: Riley’s Daring Stance on Relevance
When I talk about precision and deep understanding and other seemingly-pedantic characteristics, I often get comments like that in the first post of this series:
“If we want to encourage student precision, we need to have authentic situations where it matters. We can’t just demand it because some day it might make a difference. That’s just not relevant.”
I agree with the sentiment that we can’t just demand precision and correctness. Some people think that students need to be able to use math in their out-of-class lives for them to care about correctness in class, but with this idea I strongly disagree. Math doesn’t need to be “relevant” to be interesting – students simply need some degree of fluency to be interested. If you give someone something that they don’t understand at all they’ll feel totally helpless and confused (which presents as total apathy in students). If you give someone something that they feel total mastery of (like measuring distance) and then a question that they can almost answer with that mastery, their natural response will be curiosity and motivation. The connections they make between their prior fluency and their new learning will be stronger than any “relevance” you might have contrived in their real lives – just ask my students of five years ago who I forced through loan interest compounding formulas because it would be relevant to them soon.
And I mean, where are you going to find a relevant application for the Pythagorean Theorem anyway, you know?
The Big Conclusion
I recommend that all teachers try to explicitly connect new material to skills of which students already have complete, 100% mastery. It can be a basic skill like measuring distance, or a skill they learned last year like calculating distance given two points, or a skill they learned yesterday in class, but they have to be PERFECT at it, and they have to be able to recall it so fast that we’d be willing to call them fluent in it. When you connect new forms of expression with old fluencies, you give students the tools to 1) find flaws in their own reasoning, 2) extend their expression to include new meanings, and 3) remember their new skills with more conceptual connections. You extend that base of fundamental understanding I talked about in part 1 of this series, and you avoid the problem I warned of in part 2. And, to boot, your class gets more fun because everyone feels like they know at least part of what’s going on!
You can achieve these fluency connections by creating environments for your students to explore, or by properly crafting a word problem, or by following the WCYDWT example, and I’m sure there are many more. Whatever you do, as long as you intentionally include ways for your students to connect with your lesson on terms in which they are completely fluent, you’ll see interest, motivation, and test scores increase.