re: the a in f(x) = a - x2 (question below), which appears in the College Board's Online SAT Test 5 and is designated "medium difficulty"
I should have made clear in the post I wrote about the post Debbie wrote* that this question is not "tricky" for me. This question is easy for me, and the fact that this question is easy for me tells you nothing about whether I have mastery of quadratic equations and their graphical representations. At this point, I do not.
This question is easy for me because I have some basic understanding of shifts, because I have memorized the rules about shifts listed in all of the SAT test prep books (and most notably in Phillip Keller's book), and because I see the intended trick of the question the minute I look at it.
The fact that this question is easy for me probably does tell you I am pushing 700 on SAT math, which I am. I scored 680 on the real test, putting me at the 90th percentile for all test-takers; on sample sections my range is well into the 700s. I don't know the exact percentage of test-takers in the 90th percentile who get this question right and find it easy, but it's going to be very high. The top 10% of test takers is the group for whom this question is not tricky.
This question is tricky for test-takers scoring in the 500s, and the College Board knows it. They tell us they know it; they're not keeping it a secret. When the College Board assigns a "Medium" level of difficulty to an item, they are telling us that x number of kids scoring in the 500s will reliably get the question wrong, and x number of kids scoring in the 500s will reliably get the question right. That is the meaning of the words "medium level difficulty."
Kids scoring in the 500s are the ones you can depend upon to see a in f(x) = a - x2 and think, in a certain percentage of cases, a-the-coefficient-of-x2-in-ax2+bx+c=0. Those are the kids getting tripped up by this question.
Test-takers scoring in the mid-500s do not have mastery of quadratic equations and their graphical representations, and neither they nor anyone else is claiming they do. So when a designated percentage of kids scoring in the 500s get this question wrong, we learn nothing about them we did not already know. By the same token, when a designated percentage of kids scoring in the 500s get this question right, we also learn nothing about them. As a general rule, 500-scoring kids do not have mastery of quadratic equations and their graphical representations. I think that is a safe assumption to make.
Ditto for me. When I easily get this question right -- and, again, I stress the word easily -- no one knows anything about my level of mastery of quadratic equations and their graphical representations. I do not have mastery at present, yet for me this question is so easy it's a gimme. In Steve H's words, I have (near)-mastery of the test, not the math.
So why is this question -- or, rather, questions just like it**-- on the test?
Questions like this are on the test because they have appeared in the experimental sections of previously-administered SAT tests and have been found to reliably produce a certain level of error. The College Board needs test items that produce reliable levels of test-taker error in order to keep raw scores stable from test to test. They can't have everyone getting all the questions right; they can't have everyone getting all the questions wrong; and, in the big, mooshy middle, they need items that reliably produce a certain percentage of right and wrong answers.
Hence the a-question.
* this is the mouse that lived in the house...
** I don't know whether items in Online Course have been tested in Experimental sections. As I understand it, all items in the tests actually administered to real test-takers have indeed been tested in experimental sections taken by other test-takers.