In literary writing, we are taught to provide an introduction, a body and a conclusion. We are encouraged to—inefficiently—relate parts of an essay to a central idea. In mathematics, our writing is at times too efficient. Perhaps the teacher can allow the student to be a little less efficient. Here is an example.
A little calculation
“10 is the greatest common divisor, or greatest common factor, of the two numbers 130 and 50.” We may abbreviate this as:
We see this by inspection, by factoring our numbers. There is a harder way to find the answer, called the “Euclidean Algorithm.” It turns out that this method will usually be much easier for large numbers where factoring is difficult. Let us use this example to introduce the Euclidean Algorithm. Consider this little calculation:
130=2(50)+30 … gcd(130,50)=gcd(50,30) (why?)
50=(30)+20 … gcd(50,30)=gcd(30,20) (why?)
30=(20)+10 … gcd(30,20)=gcd(20,10) (why?)
and finally, gcd(20,10)=10, since 10 divides 20.
What is the main pedagocial point?
The question “why?” appears three times in the little calculation. If we’re clever, we can answer all three questions all at the same time, with a little lemma—and we should do this (if you don’t know the answer, try it!). The answer constitutes the central idea behind the Euclidean Algorithm. The pedagogical problem is that the required lemma is usually discussed first; so that the chain of equalities: gcd(130,50)=gcd(50,30)=gcd(30,20)=gcd(20,10) can be suppressed!
My suggestion to the teacher is to introduce the lemma only after a few examples have been worked by the student in this inefficient way. Thus, the lemma is an eventual recognition of a pattern—a celebration of just how useful it appears to be, rather than an efficient shortcut in order to avoid it.