Telling and Showing
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education

It is best to both tell the students the rules in play and show them how the rules are used in practice.

A mathematical word problem is an encoding of a set of equations. Consider, for example, the following:

If a dongle costs \$35 to create, \$4 to ship, and is to be marketed at a 25% discount while still making the manufacturer a 10% profit and the retailer \$2 per item sold, at what retail price should it be marked to before the discount?

This sentence is an encoding of the equations

• cost = 35 + 4   (“‍costs \$35 to create, \$4 to ship‍”)
• revenue = (1 − 0.25) × retail price   (“‍markted at a 25% discount‍”)
• profit = 35 × 0.10 + 2   (“‍making the manufacturer a 10% profit and the retailer \$2 per item sold‍”)
• profit = revenue − cost   (definition of profit)

which equations are to be solved for retail price. There is nothing else magical about a word problem; it is just a way of encoding a number of equations.

I recently had occassion to make a statistically-unsound sampling of math texts and none of them presented word problems as an encoding of equations. In every case all they did was present a few examples, most of them noticably light on detail. That was it. There was no discussion of why or how to work these problems in general.

I understand the value of example and of letting students discover. I continually push for these things in working with other instructors. Teach by what people know. But I don’t understand why you would hide the basic structure of a problem from the students. Word problems are, in whole and in every part, encodings of non-word-problems. There are simple rules for translation, like “‍‘‍of‍’ means ‘‍×‍’‍” and “‍‘‍y percent off of‍’ or ‘‍y percent discount‍’ means ‘‍(1 ‒ y ÷ 100) ×‍’‍”. Why not give the students these rules directly?

One of the long-term problems with discovery learning and with building from the simplest to the most complicated ideas is that along the way students may develop incorrect mental models that they need to replace later on. You can easily waste a lot of memory learning how to solve each kind of word problem in a textbook if you don’t realize that they are all basically the same; and why would you realize that if no one tells you how they are the same? One class I assisted taught programming by first teaching straight-line programs, then control constructs, then methods, then objcets. It made sense on one level; each topic builds on the ones before; but with each new topic we had to knock down the students’ mental models of how the computer was working to make room for the new topic.

So what’s the “‍right‍” balance of telling and showing? How do you maximize student’s creativity and intuition without letting them get too far from productive thought patterns? I don’t pretend to know. Too much telling is dry, painful, and confusion while too little makes things difficult and frustrating.

As with most topics worth discussing, the right solution, the right point in this balancing act, is not obvious.