High School Level Math: What is it?

There's a pretty decent discussion going on over at Joanne's about an EdWeek article that recently came out, itself about a study that recently came out about "dyscalculia". Now, this isn't anything new. I remember blogging when this same issue came up back in January of 2003, although at the time it was something that schools were facing in Italy. Now it's a 10-year study in America (which means the study had started back in 2001 -- so maybe it wasn't just an Italian thing back then).

But that's not really what I wanted to talk about. That's just by way of introduction. Some of the comments over at Joanne's site have suggested that maybe this dyscalculia thing opens the door to a way to get rid of the Algebra I requirement for a High School Diploma. Which left me thinking:

What exactly does a high school diploma stand for, in terms of mathematics?

We know what various exit exams call for -- and it's something on the scale of complex arithmetic with fractions (see the study guides for California's test here). Ostensibly, tests such as the CAHSEE cover Algebra I and Geometry; but it's usually non-logic-based Geometry (i.e., it's just calculations of the kind you do in 6th and 7th grade, not proofs and theorems) and the threshold for passing is pretty low; it's not clear that you actually need to know Algebra at all to pass them (I'm sure some states have stricter standards, but I'm also sure some states have lower standards).

So what is high school level math? What degree of proficiency should a high school degree convey? Or should it merely be a marker that someone sat in a chair for X number of hours studying some kind of mathematics?

Because that's kinda what it is right now, I am afraid. For what follows, I'm going to use the California exit exam as my whipping boy; I want everyone to know that I recognize I'm picking on one particular test, and that other states might have better measures. (Some might have worse.)

So the exit exam helps, some, with pinning down what counts as high school math. If it's on the test, then it's high school math, right?

Well, not quite.

If they were serious about it, they'd have each section graded (and passed or failed) separately: Arithmetic, statistics, geometry, and Algebra. Strength in one area wouldn't be able to overcome weakness in another, and a passing grade wouldn't be 350 out of 450, which sounds impressive until you know that the lowest possible score is 275, making the passing score essentially 75/175, or around 42%. I should note that this is a "scaled" score, so it's not as if getting 42% of the questions right means that you passed; it's closer to 53%, or at least that's what I got after running a weighted average on Table 4 of this 2010 scoring document.

But it's also a multiple choice test where you can guess. FOUR-ANSWER multiple choice, for that matter. Here's a math problem for you:

Let's say there were 80 questions, and you needed to get 43 of them right to pass the exam. Let's assume that you can get 25% of the questions on which you guess right without knowing the material. Holding that guess-yield rate constant, how many questions would you actually need to really know to pass the test?

Answer:
Let K=Number of questions really known, G=Number of Questions guessed

We have two equations:

K + G = 80
K + .25G = 43

So: G=80-K
So: K + .25(80-K)=43
So: K + 20 - .25K=43
So: .75K + 20 = 43
So: .75K = 23

K = 31 (when rounded up).

31 questions out of 80. If you know 39% of the math on that test, you know enough "high school level math" to get your diploma.

Why am I going over all this? For exactly the same question I was asking the question above: just so we can all be clear about what it is a high school diploma actually means with respect to mathematics knowledge, if it means anything at all.

Which maybe it doesn't.

And maybe it shouldn't.

But we need to be clear about it.