Your typical high school student, I presume, has spent several years setting up equations and

*solving for x*. At least, let's hope so. I certainly did.

The SAT uses this fact to elicit many wrong answers from test-takers who have worked a problem correctly. The student gets the solution right but the answer wrong because the answer isn't

* x*. The answer is 3

*x*, say, or

*xy.* I seem to recall a problem or two where the answer was

**-***x*, for god's sake, but I might be making that up.

Other times the test will give you a value for

*x* +

*y, *say, and you're supposed to see that you should simply insert

*that* value some place else in the problem,

*et voilà*: the answer they're looking for pops up.

Here's a typical problem,

*medium difficulty* (according to the College Board):

If 4(*x *+ *y*)(*x *-* y*) = 40 and (*x *-* y*) = 20, what is the value of *x* + *y*?

A kid who's had no test prep at all will likely miss this question -- either miss it outright or take too much time spotting the solution, thus leaving him too little time to finish the test and increasing the likelihood he'll make "careless errors" on the questions he does get to because now he's working too fast trying to make up for the time he lost on the

*x* +

*y* problem.

For what it's worth, I think using

*x* +

*y* as the value, instead of

*x* or

*y* alone, is an interesting and instructive way to write a problem. (I'm curious what math people think). It seems to me that writing problems in which

*x *+

* y* is the salient unit may be a way of teaching what

Ron Aharoni calls the fifth fundamental operation of arithmetic:

In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication. What I Learned in Elementary School by Ron Aharoni

Maybe I'm wrong, but it seems to me that the

*x*+

*y* questions test math as opposed to obedience under pressure, which is what the

*Find xy* questions test.

Still, there is no doubt in my mind that these questions elicit wrong answers from test takers who know the math involved, can do the math involved, and have a reasonable understanding of the math involved. Students who have spent years of their lives solving for

*x* aren't going to break the

*Solve for x* habit for the first time ever when they're working at breakneck speed and their eyes are bleeding from

the Ella Baker passage.

Which brings me back to

extinction learning. Test prep for SAT math involves spending a fair amount of time building new habits that conflict with ingrained old habits. You've been conditioned to solve for

*x*; now you have to condition yourself

*not* to solve for

*x*. Also, you have to build as much speed as possible at not solving for

*x* because you are never going to

*forget* solve-for

*-x*. The two impulses are inside your head, competing with each other, and the competition takes time (and probably eats up some precious working memory resources to boot).

Funny thing: during the time we spent doing SAT math prep around here, I overlearned

*don't solve for x* to the degree that a couple of weeks before taking the real test I came across a practice problem that

*did *ask the test-taker to solve for

*x*. I was so surprised that I wasted several seconds reading and re-reading and re-reading again to make sure I hadn't misunderstood. You can't win.

For parents: your child needs to spend enough time

*not solving for x* that he or she gets to be really, really fast at

*not solving for x*.

Then he should be on the lookout for problems that say

*Solve for -x*.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT