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On the test-taker's failure:

In order to perform well on a math test you don't always have to really understand the fundamental principles behind what you're doing. Sometimes you can get away with wrote memorization. e.g. You can come up with equations for a lot of things yourself if you understand Gaussian distributions but, if you're being prepped for an exam by teachers who know roughly what's on it, they might just give you equations to memorize for the things that are likely to be asked. You may perform nearly as well as someone with deep understanding of the material, but you are unlikely to remember those equations long after you've taken the test!

It is quite likely that the person who took and failed this test was the sort of math student who was able to get by memorizing what he was told to without really understanding things. If he had written the same test in high-school he likely would have done much better because he would have been prepared for it with memorized methods and equations that he never understood and has long since forgotten.

On teaching methods:

When teachers teach students to pass tests, short-cuts like wrote memorization tend to happen. The useful knowledge learned from this kind of teaching is minimal. Unfortunately, many teachers are unable to teach the deep meaning behind mathematics because they were taught by wrote memorization themselves. When they were assigned to teach math class they probably had to look everything up themselves since, like our test-taker, they never understood the basis behind it and have forgotten most of what they memorized.

It seems that we must overcome the problem of educating teachers before they can overcome the problem of educating students.

Or he's in a profession that doesn't use much math at all, and has just forgotten it because his brain deemed it not important--even if he understood the principles behind geometry. There are formulas and theorems that those tests ask about specifically.

I think the more damaging thing about this article was his (and the author's) insistence that the math is not required for the majority of professions, and therefore, should not be required in high school. Basic education is liberal arts education, and, setting aside barista jokes, liberal arts education is about understanding a wide breadth of subjects. It is not, nor has it ever been, vocational training. Moreover, this type of math is absolutely imperative to some of our most needed (and oldest) professions, like engineering and construction, and some of the newer ones, like software engineering.

That aside, I agree with your point, and what you highlight is a problem with standardized test per se. People will always optimize for the variables by which they are measured. The trick is to align those incentives with the desired outcome of education.

As a postscript, the word you're looking for is "rote," not "wrote"--the former means learning by repetition, while the latter is the past tense of write. Since we're talking about education, this paragraph seemed appropriate.

You hit upon two important points, but I'd take the conclusion further:

A) Our current evaluation methods don't tell us whether a student is simply memorizing steps or learning the subject matter.

B) When your primary evaluation critera is to pass a standardized test, the system will optimize to achieve that goal. Since we know that standardized tests don't really evaluate understanding, we have a vicious cycle on our hands.

I would add that it is possible to make standardized tests that teach more than memorization, but those tests will cost more to construct, norm, and grade. Schools are caught between the imperatives of testing and budget-cutting.

Good point. I for one can't wait for the day schools can afford to hold the Kobayashi Maru test.

To add an anecdote to your point (A), a former workmate of mine used to tutor maths at a university level. He had a student that was very good at rote learning, they could do any of the set problems quite easily by just going through the steps to solve them. But when faced with an unusual presentation of the problem they were stumped.

I'm pretty sure I was guilty of that sort of rote learning more than once, and it seems to be the first thing that disappears once exams are over.

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