Mathematics Education: Taking a Clue
From the Recent Technological Revolution
Problem Posing vs Problem Solving
Looking back - the fourth and last stage in Polya's approach to problem solving - is crucial. Tinkering with attributes of a problem may, eventually, result in a problem of a new class. (From Pythagoras to Fermat.) More importantly such tinkering reinforces skills and knowledge acquired previously.
Ralph P. Boas, Jr. wrote (Monthly 64(1957), 247-249):
... the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in the calculus course, either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on through the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it.
This must not be so. Inventing problems even the dull way is a great tool for learning the current course and mathematics in general. How much can be learned by tinkering with a problem? Let's look into an example. 1 + 1 = ?
Assume the student has already figured out that the result is 2. By increasing a term on the left by 1 the total on the left is increased. In order to get the equality back, one should increase by 1 the right side as well. Applying the enlightening idea repeatedly, the student may even conceive of the notion that adding any number on the left can be balanced by adding the same number on the right. The left hand side is a sum of two terms. A second observation can be made to the effect that it does not matter to which of the terms the number has been added. And then a third one that the number does not have to be wholly added to one of the terms - it can be split in any way imaginable. It is also possible to add 1 to the first term and reduce 1 from the second without changing the result. Is this only true for 1 or, as before, for other numbers as well?
With an attentive and knowledgeable teacher near by, the student has a good chance not only to learn his arithmetic but also get a glimpse of what is lying ahead in an algebra course.
Can anybody learn that much after filling in a 10x10 worksheet? I doubt this.
I would go a long way to make students invent their problems even the dull way. Here is a problem to which I immediately volunteer a solution:
One of two brothers is 7 years old while the other is only 5. As you can see, the sum of their ages is 12. Please invent another problem and solve it.
Even if the following was obtained as (7 + 1) and (5 - 1)
One of two brothers is 8 years old while the other is only 4. As you can see, the sum of their ages is 12. Please invent another problem and solve it.
a valuable knowledge has been acquired. I would accept the following result as very valid:
One of two sisters is 7 years old while the other is only 5. As you can see, the sum of their ages is 12.
A less creative is even better:
One of two sisters is 7 years old while the other is only 5. As you can see, the sum of their ages is 12. Please invent another problem and solve it.
The notions of problem attributes, problem class come up quite naturally. May the fellow have fewer troubles with word problems years hence? What about recursive definitions?
(There are several dynamic tools that implement the strategy discussed above.)