In many school districts, including Weston, we try to resolve the Math Wars by promoting a balance between skills and concepts. We tend to adopt the party line as promoted by former Education Secretary Richard Riley:
We are suffering here from an “either-or” mentality. As any good K-12 teacher will tell you, to get a student enthused [sic] about learning, you need a mix of information and styles of providing that information. You need to provide traditional basics, along with more challenging concepts, as well as the ability to problem-solve, and to apply concepts in real world settings.
I used to agree. But my sister, Ellen, has recently convinced me that balance is the wrong goal: what we need is integration of skills and concepts. Here are two recent true stories to illustrate the point:
- Mary buys a large rug and decides to have it shipped home, since it’s too big and too heavy for her to carry it in her car and then get it into the house. So she arranges with the store clerk to have it shipped. They roll up the rug and place it on the scale. Oh, no! It’s three pounds about the maximum weight for this shipper.
So the clerk instructs Mary to fold the rug in half and then roll it up again. “What good will that do?” asks Mary.
“It will weigh less,” replies the clerk.
Mary points out that it will indeed be smaller in one of its dimensions, but its weight won’t change. “Yes it will,” insists the clerk; “if it’s smaller, it will weigh less.”
An ill-advised attempt to explain conservation of mass turns out, of course, to be of no avail.
While the clerk’s misconception is more about physics than math, it’s surely tied in with the general belief that perimeter is always directly related to area. If you “increase” a rectangle from 6-by-8 (perimeter 28) to 3-by-12 (perimeter 30), you have surely increased its area. Right?
- On a recent test, one of my better students, whom we’ll call Matilda, consistently gets slightly incorrect answers in calculating outputs of an exponential function. The instructions said to “round off correctly to the nearest tenth of a pound,” and the calculator gave answers like 3.24312237 and 4.724646398 pounds, but this student got 3.1 and 4.6 pounds respectively. What was Matilda doing wrong? I scratched my head.
You’ve probably figured it out already, but I asked Matilda to show me her calculator work and explain her answers. She indeed got 3.24312237 and 4.724646398, and then explained that the hundredths digit was less than 5 in each case. The rule, she said, told her that when a digit is less than 5 you need to round down, so she rounded the tenths digit down…
What do these anecdotes have in common? In both cases there was a disconnect between skills and concepts. No amount of “balance” would help, but a successful integration of the two would reinforce the necessary connections and prevent belief in a rule that doesn’t make any sense.