I’ve always sorta suspected that explaining mathematics to students in very concrete ways wasn’t really a shortcut but it was like being offered a magic lamp with a genie who will grant your wishes. You know something about it is too good to be true even if you don’t really know what it is. Anyway apparently now there is some evidence that my intuitions are correct.

Of course I don’t think this means one should never use analogies to concrete situations or that math must be presented as uninterpreted formal manipulation. Nor is it really true that I didn’t know why I found the teaching of math in a super concrete fashion troubling. Ultimately math is the use of abstraction to solve problems. The power of things like group theory, analysis, recursion theory or even calculus is the result of abstracting the relevant features from the motivating examples. This is just as true for the subjects we try and teach our students as it is for the subjects we study ourselves. For example, in order for students to understand (not merely memorize) how to take the derivative of an integral using the chain rule and fundamental theorem of calculus requires the students have abstracted away from the idea of functions as rules they can right down in terms of powers,ratios, trigonometric functions and exponentials and can accept a define integral they can’t compute as just another function.

Ultimately, interpreted correctly I don’t think this study really says anything that shouldn’t have already been obvious. It doesn’t say that concrete examples can’t help students grasp the abstract concepts being presented (they can) it just tells us that there is no shortcut to mathematical understanding. If you want students to actually understand math that means teaching them to understand the concepts abstractly and if you take the easy way out and encourage your students to understand mathematics in a particular concrete fashion instead of offering the concrete examples/analogies and encouraging them to move beyond it their understanding will suffer.

As a side note the entire discussion around this study is a perfect illustration of the fundamental confusion at the heart of mathematical education today. Framing the study as showing that concrete teaching methods fail because they aren’t as effective in teaching students to solve other concrete problems seems totally backwards to me. Frankly in the modern world being able to compute when too trains will pass or find the symbolic integral of sin(x) isn’t an important skill for most people. Go ask anyone over 40 who doesn’t work in a technical field to solve a system of two linear equations, integrate a function, or even multiply fractions and they aren’t likely to be able to do it. If the particular skills we pass on in math class were so important to everyone how has our civilization manage to flourish up till now? It might be sad, you might not like it but the strategies of asking a technical friend, using a calculator, or just not dealing with mathematical problems work pretty damn well for most people. On the other hand the ability to abstract away particular features of a class of examples is a skill virtually all professionals require.

Ideally we would stop blindly teaching the traditionally expected skill set and ask what we mean to accomplish. If we want to give people practical skills to use in their life we need to take a look at some empirical data about what if any mathematical education translates to practical benefit. You might think that everyone should be able to calculate with percents and fractions but if most people simply use the calculator on their cell phone then we have to consider the possibility that we’d be better off helping them practice computing in that way. Certainly some people need to learn how to do things like multiply fractions, compute integrals and so forth but that isn’t everyone. On the other hand if the reason to teach mathematics is to impart quantitative reasoning and the ability to reason abstractly then why the hell is most of our mathematical education blind computation?