Reevaluating examples

The New York Times reports on a recent study just published in Science:

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”

Instead of promoting better understanding, the concrete examples might only distract and confuse the students:

In the experiment [conducted by Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler], the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. […]

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

Let’s first define our terms:

• Technically, if I explain a math procedure in class and then go on to applied examples a few minutes later, I could claim that the students “learned the math abstractly” first. Assuming, of course, that they followed every word of my lecture and that they were learning in real time as I was speaking. That’s a little bit unrealistic. Students don’t learn a math procedure just by watching someone else do it – they need to try it out on their own, which is usually done after class. In practice, separating the “theory” from the “concrete examples” would mean teaching them in separate lectures and perhaps including them in separate sections of the textbook.

• The word “examples” can refer to several different things, from “maximize the function $f(x)=\sin x+\cos x$ on the interval from 0 to $\pi$ to “a bacterial culture grows at a rate proportional to its size. Experiments have shown…” The first kind is absolutely necessary; some of my least favourite textbooks earned that distinction by not having enough of them. It is the latter kind, with concrete objects and backstories, that I think can mess with students’ heads when used excessively or at the wrong time.

A typical calculus book is full of examples with backstories. In some cases, they are so “integrated” with the text that it takes me a while just to filter out the frills (pictures, backstories, other such) and figure out what the mathematical content is. Especially in those cases where the mathematical content is never even stated explicitly (yes, this does happen). And I’ve known the relevant mathematics for decades. I’m not surprised that students can be confused.

I’m guessing that most textbook authors start with the right premise: there’s no “best” explanation that works for everybody, different students need to be approached differently. Thus they try to come up with as many different explanations and interpretations as possible and include them all in the textbook. The upside of this is that, if the author got it right, most students should find at least one interpretation that they connect with. The downside (at least if the textbook material is required on the exam) is that every Bob is also forced to learn the explanations that appeal to Dick or Harry, but not to him. Based on what I have seen, it does not promote understanding. It promotes memorizing Dick’s explanation and Harry’s example. Also, the textbooks get heavier.  I wouldn’t want to have to carry several of them around campus.

In my experience, it’s a good idea to learn one thing at a time. If you want to learn a mathematical procedure, such as maximizing a function, why not focus on doing just that. Learn the procedure, then work it out for various specific functions. Those functions don’t need to have a backstory or be motivated by real-life examples. They just need to illustrate the procedure in question.

Once you’ve learned the procedure, you might be interested to see its real-life applications. This comes with a caveat, though. Real life is, well, real life. It’s multifaceted and messy. You shouldn’t expect to be able to work out realistic examples from another field of science, such as physics or economics, unless you’re also studying that field of science. You might see simplified versions of such examples in calculus courses – some textbooks actually do a good job explaining them. However, since calculus is often a prerequisite or corequisite to physics courses, and not the other way around, we can’t expect all math students to have a physics background, and that’s why you should be getting your real real-life examples from physics in your physics class. (Incidentally, technology can help with the computational side of the real-life examples, but not with the conceptual side.)

But where was I? I’ve already mentioned that when I’m faced with a textbook section full of stories and pictures, I start out by filtering out the frills. That’s how mathematical abstraction works. Complex objects are represented by symbols, with all the “frills” discarded (what can and what cannot be discarded is another story). They are then manipulated according to abstract rules. Those rules have nothing to do with the original problem. That’s why they apply to many completely different situations. That’s also why they can be taught, at least to a student so inclined, independently of real-life examples.

Remembering a math procedure and using it correctly is one task. Setting up a mathematical model for a real-life problem is quite another. It makes sense to me to teach them separately and to start combining them only after each component has been more or less learned.

I’m guessing, too, that Dick and Harry may like their examples and applications, but there might also be a Bob (or a Roberta, if you prefer) who likes his math straight and isn’t that interested in models from economics or math biology. I don’t really think that we are catering to this group of students.