NAEP revisited: the math exam

[tahnan]

Apologies for two closely-related posts in quick succession, but: I've decided to skim over some of the math questions, since that's a field where I have more expertise than in history. One of the first questions I looked at was the following, a "hard" question from the 2009 12th-grade test (block M2, Question #7):

x	y
-2	3
-1	0
0	-1
1	0
2	3
3	8

The table above shows all the ordered pairs (x,y) that define a relation between the variables x and y. Is y a function of x? Give a reason for your answer.

The NAEP's answer: For each x-value (domain) there is only one y-value (range) that is associated with it. Which I'm good with; that's a perfectly solid explanation of what makes something a function.

What floored me, though, is that only partial credit was given for answering by saying that the table represents "y = x² - 1". For instance, one scorer commented,

While the ordered pairs given in the table do satisfy the equation y = x ² - 1, this is not the only such function, since, for example, y = (x ⁶ /10) - (3x ⁵ /10) - (x ⁴ /2) + (3x ³ /2) + (7x ² /5) - (6x /5) - 1 also satisfies the relationship in the table. The response provides no reason to support that this is a functional relationship.

Wow. Well. The scorer has done a fine job of showing off an ability to devise a sextic equation for six data points; good for them. But where I come from, we've got something called "proof by construction", and if someone asks "Is it the case that X?", one way to answer it is to provide a concrete example of X. If you're asked "does this table represent a function?" and you respond by giving a function that covers the table, you've pretty much answered the question.

Now, if I put this on an exam—and I conceivably could, because part of what I cover when I teach semantics is the concept of a function—I'd probably make it a five-point question and dock the student a point for only providing the quadratic equation, because I'd be testing very specifically on material I covered about the domain-range relationship. But that's 80% credit; whereas for the NAEP exam, the possibilities for this question are "correct/partial/incorrect" (other questions, incidentally, have more gradations), and the sidebar just notes the percentage of students who were correct, with "partial" not included.

Again, I wouldn't give full credit to an answer like that, but I think the NAEP doesn't give nearly enough. Or, perhaps, is writing questions that encourage perfectly adequate answers that it doesn't count as correct.

Comments

[mrmorse.livejournal.com]

To me, the issue is not that it's possible to write an equation that fits all the points and which is a function. It's that it's possible to write an equation which fits the points which is not a function. The existence of a particular equation is a useful fact about the points, but it's not an answer to the question asked. Unless you state that one value of x only gives one value of y, you haven't shown that it's a function.

You can't claim that y=x^2-1 is a function unless you can say why x=y^2-1 is not.

I agree, however, that the scorer also missed the point.

[tahnan]

It's that it's possible to write an equation which fits the points which is not a function.

True. But this is a little tricky—because unless a math class goes into subtleties, I think it's understood that something of the form y = ...x... is a function from x to y. Now, that's not entirely true (most things that get taught in pre-college math classes are probably not entirely true, since they gloss over complications), since you could write y = (0 or 2 when x = 0, 1 or 3 when x = 1...) or y = ±√x (or really, ±x, or anything with a ± that isn't followed by a 0 since that indicates "either of two values"). You could; but how often does anyone? How often does anyone, in high school classes?

That is to say: you're right that being able to write down an equation for something doesn't guarantee that it's a function. But if the convention that you've been taught for years of algebra classes is that y = ...x... is how you write functions, I think it's dicey to take off too many points for someone who writes that down to illustrate a function.

[rhu]

I agree that y=x^2-1 is a bad answer, because the question clearly specifies that "the table above shows all the ordered pairs (x,y)" (emphasis added).

Specifically, this is a discrete function whose domain includes six specific values; since each one maps to a single value in the codomain, it is indeed a function. That's the only criterion that matters.

To construct a formula that coincides for those points is interesting, but not actually an answer to the question that was asked.

[tahnan]

Well, sure. Though I have to admit that when I saw the question—which is why I put the answer behind a cut tag—my own thought process was: "Well, yes, it's a function, because no value of y is repeated. In fact, let's see, ah, given the symmetric values, it's a parabola centered on the y axis. Oh—though if this table represents the complete set of points, there's no real reason to extrapolate. Well, anyway, function." In other words, my own well-informed thought process went "right answer; partial answer; rejection of partial answer", which is one reason it's hard for me to fault students who come up with the second step of my thought process without going on to the third.

And like I said, of course the only criterion that matters is the domain/range fact. That's true for determining whether anything is a function. On the other hand, there's also not really anything wrong, I don't think, for providing a way of getting from x to y, even if y is only defined for six values of x. (If a student had written the unlikely answer of y = x² - 1 for x in {-2,1,0,1,2,3}, would it still be as wrong?)

The point I made in response to Matt above is that to provide a formula that coincides for those points is, if y = ...x... is how you write functions, actual evidence that what you have is a function. That "if" was left implicit, but I think it is implicit in most high-school classes.