Why “For Example” Is A Bad Way Of Explaining Things
Niklas Goeke
29715

I disagree 100%. Based on my personal experience:

IMO, there are two kinds of example-collecting in the classroom. There’s “example-collecting for future reference” and there’s “example-collecting for pattern-finding”.

“Example-collecting for future reference” means hoarding a large stockpile of problems that, in order to be solved, require a certain principle or algorithm (such as the Pythagorean theorem, or x=vt+½at², or integration by parts, or whatever). You hold on to those specimens hoping that any future problems, e.g. on the homework, will parallel one of the examples in your collection, and so the problem in the assignment can hopefully be solved by copying the example but changing the values of the numbers. This is bad, you don’t really learn… and I suspect that this is what the article here is about.

But “example-collecting for pattern-finding” is great and leads to learning. Maybe you see one example and you get a rough idea of how the principle is applied but you’re not sure whether this one number came from this step or from that step, or maybe you’re not sure whether this step influences that step or if this step was a guess based on the expected outcome of that step in the future. So you ask for another example. A-ha! Now it’s clear where that number came from. Now it’s clear that the input for this step was a guess about the outcome of the next step. So now I wonder what would happen if that guess was way off. Then you want to start experimenting (i.e. making your own examples).

Let me give you an example ;] Say that you’re learning to multiply numbers with two digits: AB times CD (where “AB” is B+10xA). I show you

12x34=8+40+60+300=408

And you go “I think I see what’s happening here. 4x2 + 4x10 + 30x2 + 30x10, right? Show me another example!”. I show you

56x78=48+400+420+3500=4368

Now you go “Yep, I got it now. ABxCD=DxB+DxAx10+Cx10xB+Cx10xAx10”.

And if I show you some examples multiplying numbers that have more digits, you can probably generalize the idea that the product is the sum of each digit (times its corresponding power of 10) in one number times each digit (times its corresponding power of 10) in the other number. Maybe it occurs to you to build a 2D table. And maybe you could group the digit multiplications by the power of 10 (“number of zeroes”) that each one ends up with, and add the digit multiples together before slapping all the zeroes onto them…

This conversation leads into another argument: It seems like most teaching (in the maths / Newtonian physics / engineering fields, until you get to college) follows the outline “Here is the formula. Let’s do some examples, then you do some exercises”. But wouldn’t it be more fun and interesting and educational (i.e. Wouldn’t we get better learning) if instead we did “Here is a problem. How do you solve it? I don’t know. Try something. Can you get even an approximate answer? Can you break it down into smaller pieces that you can solve? Can you think of an experiment in the real world where you could gather data to learn how to solve this general kind of problem? Try drawing a picture of the problem, does that help?”. That way, the students could find the analysis technique themselves, or at least most of its pieces. They would then “own” it more thoroughly, and have a real appreciation for the importance of each step, for why it’s there. I won’t repeat all of “Lockhart’s Lament”, but I agree with what Paul Lockhart says. What’s the formula for the area of a triangle, or the area of a circle? A student that finds the formula (or a crude and slightly incorrect approximation of the formula, which then gets corrected) will probably remember and apply it better than a student who is simply told the formula at the very start and then led through examples/exercises. Right?

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