Teaching

Completing the square is one of those mathematical techniques that should be taught but probably shouldn’t be assessed on state exams.

First, when you insist that a student use a specific technique to solve a problem, you penalize flexible and creative thinking. Second, completing the square probably isn’t important enough a problem solving technique to warrant its yearly appearance on the New York State Algebra 1 exam.

Of course, the one situation in which completing the square is absolutely indispensable is in deriving the quadratic formula. Which makes this sample student response from the exam’s official scoring materials a bit puzzling.

The student lost a point because, instead of completing the square, they used the quadratic formula to solve this equation. But the whole point of the quadratic formula is that it completes the square for every trinomial. The quadratic formula is completing the square.

As I clarified in comments on Twitter, I find this more amusing than objectionable. But these little windows into the testing process often tell us more about what is valued and understood than test scores themselves.

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This type of problem frequently appears on New York State Algebra 1 Regents exam.

There’s really no way for an Algebra 1 student to properly “explain” their answer to this question. Proving that a number is irrational is a concept from elementary number theory and is not part of the Algebra 1 course. What the test makers expect is for the student to simply state that a rational number times an irrational number is irrational. Not only is this not an explanation, but such a question reinforces the idea, for both students and teachers, that mathematics is a collection of facts to be memorized and regurgitated.

I’ve written about this issue before, but I didn’t think this kind of problem could get much worse. I was wrong. Here’s an example of a full credit response from the official scoring materials for the 2020 Algebra 1 Regents exam.

In this exemplar full credit response, the student erroneously represents as a number with a terminating decimal expansion, i.e. a rational number. Then the student incorrectly claims that the number 5.19615243 can not be expressed as a fraction and thus must be irrational.

The student has demonstrated some understanding of the situation, but doesn’t grasp the fundamental issue of what an irrational number is. This response shouldn’t get full credit. More importantly, the official scoring guidelines should not communicate erroneous mathematics to those who use them. How many teachers will walk away thinking this is valid? And then teach their students that to show a number is irrational, all you have to do is plug it into your calculator and see if it has eight digits past the decimal point?

This is kind of work that makes this other exemplar full credit response seem not so bad by comparison.

A lively Twitter thread on this problem can be found here.

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