Using examples to teach math

[C]ollege students who learned a mathematical concept with concrete examples couldn’t apply that knowledge to new situations. But when students first learned the concept with abstract symbols, they were much more likely to transfer that knowledge

In a third experiment, the researchers presented 20 students with two concrete examples and then asked them to compare the two examples and write down any similarities they saw. After this experiment, about 44 percent of the students performed well on the test concerning the children’s game, while the remainder still did not perform better than chance.

If I’m reading this correctly, my response is, “Duh!” Maybe it’s just a bad press release, but it sounds like teaching by giving an example isn’t as good as teaching by giving the general concept, and then perhaps reinforcing it with an example. So we look at the paper

The belief in the effectiveness of multiple concrete instantiations is reasonable: A student who sees a variety of instantiations of a concept may be more likely to recognize a novel analogous situation and apply what was learned. Learning multiple instantiations of a concept may result in an abstract, schematic knowledge representation (1, 4), which, in turn, promotes knowledge transfer, or application of the learned concept to novel situations (1, 5). However, concrete information may compete for attention with deep to-be-learned structure (6–8). Specifically, transfer of conceptual knowledge is more likely to occur after learning a generic instantiation than after learning a concrete one (7)

Perhaps I’m missing something, but that belief isn’t obvious or reasonable to me at all. I’m not aware of a “long-standing belief” that you only teach specific examples, as is implied in the press-release article. When you teach physics, the standard method is to give the general material — the abstract equation — and then give an example of how that’s applied. But that’s not what is being done here (though they are discussing math rather than physics). A “bottom up” approach of giving a concrete example does nothing to identify assumptions that might go into the generic formulation. That may be something more important in physics as we tend to approximate, e.g. cows as being spherical, so you may not come up with a relationship that applies in the point-cow case. But it’s still an extrapolation of specific knowledge to another situation, and requires pattern-recognition, while the top-down approach states the pattern at the outset.

I recall the frustration of learning some equation, very general in its format, and not understanding how to put it into a form that I recognized and could solve, because I hadn’t yet had enough practice in doing so. There is some higher-level connection that has to go on (i.e. one has to actually learn something) to be able to apply the equation to other problems, but going the other way? A specific example, and trying to apply that to another specific example without formulating the general concept? I just don’t see why anyone expected that to work.

I see that over at Uncertain Principles, Chad has a post on this, as well, highlighting the issue of students “compartmentalizing” what they learn:

When we present new mathematical apparatus for solving physics problems, students will often get tripped up on basic elements of calculus. When asked “Haven’t you done this in your math classes?” they often respond with an answer that amounts to “Yeah, but that’s math class. It doesn’t have anything to do with physics.”

That raises the question of whether the math class is doing any examples of the things learned in the class, or relying on the physics prof to do so. But this is where the connection gets made — Oh, this is why I learned derivatives! — and you probably have to do that before the knowledge gets dumped out of the buffer from non-use.