Saturday, April 26, 2008

Checking What "Everyone Knows"

An interesting item in the New York Times tells that the method our teachers used to teach us arithmetic may not have been as useful as everyone thought:

One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other?

Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn.

That idea may be wrong.

I liked this one for various reasons.
1. The idea that when "everyone knows" something it's also true, was never very convincing to me. My reading of the story of humanity is that when "everyone knows", it may be true, and it may not - no more than that.
2. Kudos for the researchers who decided that accepted wisdom ought be questioned.
3. The honorable researchers seem to have discovered something I could have told them more than 40 years ago: That when faced by questions about trains in the night, at least some students are distracted from the maths into more important questions: Who was on the trains? Where were they going? What was the hurry? Why didn't they fly? Were they tense? Relaxed? Could they even have noticed the other train, or were they too involved in their own thoughts?


Lydia McGrew said...

The "teach them by real-world example" stuff has become another educational gimmick. In a more traditional approach, you _teach_ the child the equations, but at the more advanced levels, you ask him to _apply_ the equations to a few examples using real-world objects to get him to recognize the underlying math in otherwise distracting situations. Using the trains in the night to teach the equations in the first place is foolishness, educationally speaking, because he doesn't get it to begin with, which leaves him helpless to apply it anyway.

Similarly, a new and ridiculous idea is to refuse to teach children their multiplication tables but rather to have them use various forms of manipulation to understand multiplication. There's nothing wrong with understanding multiplication, but of course they should learn their times tables. Looking at grids and such is no substitute for knowing the math facts.

Anonymous said...

Well, if you wanted to know how many people were coupling on the train, you'd have to ask the conductor.

On the other hand, if you wanted to know WHERE the two trains passed in the night?

Ah, then, go to Al-Ja-bar's bar. And, set 'em up. As 400t + x = 500t

This formula would tell you (by using the differences in speed, each train was traveling ... to the UNKNOWN "X" ...

You can never be too early in explaining to kids that "algebra" is really a way of finding unknown numbers. By using those early skills of addition, or subtraction; multiplication and division.

While the long sentences bore most people to death. And, they're fast asleep by the time they reach the equal sign.