*now*matters more than what you didn't in October. Students need an accurate picture of what they're studying, and "Test #4" doesn't provide that.

Great. But here's something about SBG that's been bugging me for a while.

There's something wrong here, but what is it?

*The kid showed that she knows all the triangley stuff, but dropped the ball on the square root side of things. She gets a 5/5 on Finding Sides of Right Triangles, but gets a 2/5 on Understanding Square Roots.**The kid got a question about right triangles wrong, so she gets a 3/5 on Finding Sides of Right Triangles.*

Neither of these ideas is quite right. Knowing how to find the square root of 1 is not an all-or-nothing affair. Understanding isn't binary. Rather, understanding comes in degrees, and if a piece of knowledge is weakly understood then it's especially likely to falter when under pressure.

If you aren't super-comfortable working with right triangles, trying to solve a right triangle problem will be mentally taxing, and when you engage in mentally taxing behavior, you mess things up. But you don't mess up the things that are rock-solid. I doubt that I'll mess up single-digit addition when working on a Calculus problem. Rather, when you're using up mental resources it's the infirm and tentative knowledge that falls apart.

It's the sort of thing that we see all the time on mathmistakes.org.

This student said something silly, but it's artificial to attribute this to either his understanding of solving quadratic equations or his understanding of what the equation symbol means. It's both.

Would you ask this student to reassess on

*Doing Arithmetic with Negative Numbers*or*Finding Equations Given 2 Points*? Neither? Both?
There's a larger point here. The idea that you can create a quality assessment that targets an individual skill is a myth. Take the slope question above. You could make the numbers easier so that the arithmetic probably wouldn't be a problem. For instance, you could use (0, 4) and (2, 10). But this is far too easy -- understanding means being able to apply a skill to a difficult context. So you toss in more difficult numbers, but then you're no longer purely assessing a kid's ability to find a line that passes through two points.

I don't know what this means for SBG or reassessing, and I hope that (in addition to challenging the premise of my post) we hash this out in the comments. Maybe this is an argument for fewer, but more substantive standards, like "Doing Stuff With Lines." I'm not sure, though.