The other day the NY Post screamed in horror that in New York State students receive partial credit on math questions.

But aside from multiple choice, true/false and fill-in-the-blank, most math teachers routinely award partial credit. What gives?

The Post is mostly wrong. Not a big surprise. There’s an interesting discussion involving at least one Bronx math teacher in the comments section of a post at Gotham Schools.

But the Post’s shock is not completely a manifestation of their special blend of ignorance and hyperbole. Something is wrong with the questions they cite, and that something is 2. 2 points. Quite simply, where partial credit is to be awarded, it is necessary to divide up points appropriately. And “2” leaves not enough room to do it well.

For example, I’m looking at the sample exam for Integrated Algebra II/Trig (my kids sit for the real thing next week).

There are eight free-response (ie, partial credit) 2-point questions. Here’s a few examples:

Find to the nearest minute the angle whose measure is 3.45 radians.

Forget for the moment that radians are customarily multiples of π. Let’s look at the work. Convert radians to degrees by multiplying by 180/π. Now, if the kid uses 3.14 for π, he gets 197.77. But if he uses the π button on the calculator, he gets 197.67.

In New York State π = what it says on the calculator, and π ≠ 3.14. That’s important to know in life, because it might be on the regents.

So the poor schlump actually has to occasionally do calculations, is more familiar with the 2 decimal place approximation of π, and correctly converts 197.77 to degrees and minutes. That’s 197 degrees and .77 left over, times 60 (60 minutes in a degree), 46 minutes. New York State wants 197.67 which is 197 degrees and 40 minutes.

Another gets 197.77, and just leaves it.

Another kid has been taught to trust the calculator, gets 197.67, and writes 197 degrees, 67 minutes.

Another kid sees the number is not in terms of π, and understandably misreads the problem as a conversion of degrees to radians, and multiplies by π/180, obtaining 23π/1200.

Another says: 6.3 radians is a bit more than a circle, so 3.45/6.3 is about 55%, times 360 is 198.

And another converts 3.54 radians to degrees and minutes.

And another converts to 197 degrees, 40 minutes, and 14 seconds.

And another multiplies by 360/π, gets the wrong answer, but accurately converts from 395.34 to 395 degrees 20 minutes.

All of these are worth something. All show some skill, some understanding, but some sort of conceptual error, some sort of computational error, or both. But are all of these responses worth the same thing? To New York State, yes. Each of those is a one point response. But math teachers would far rather have a 4 point question, where some of these are worth 1, some worth 2, some worth 3… Partial credit only works if it can be awarded sensibly.

Another example, also from the sampler:

$\frac{4x}{x-3} = 2 + \frac{12}{x-3}$

First, how do we solve it? Some variation possible here

1. Note that x ≠ 3 (otherwise those denominators would be 0, big no no).

$\frac{(4x)(x-3)}{x-3} = 2(x-3) + \frac{12(x-3)}{x-3}$  (multiply through by x-3)

$4x = 2(x-3) + 12$ reduce each term

$4x = 2x - 6 + 12$ distribute, and solve for x, obtaining 3. Note that we started by saying x≠3, so we conclude “No solution” or just write {} or ∅, the empty set.

or

2. Find a common denominator, throw away the common denominator, and check.

$\frac{4x}{x-3} = \frac{2(x-3)}{x-3} + \frac{12}{x-3}$
$4x = 2(x-3) + 12$  (and we’ve seen it from here. Need to check the answer, 3, and when we do, we will discover the 0 in the denominator)

3. Add fractions and cross multiply

$\frac{4x}{x-3} = \frac{2(x-3)}{x-3} + \frac{12}{x-3}$
$\frac{4x}{x-3} = \frac{2x-6+12}{x-3}$
$\frac{4x}{x-3} = \frac{2x+6}{x-3}$
$4x^2 - 12x = 2x^2 - 18$
$2x^2 - 12x + 18 = 0$
$2(x-3)^2 = 0$, and so we discover that x = 3, then check and discover it doesn’t.

In any event, how many steps are involved? How many places to lose a point? But this is a two point question. And not all of the mistakes are equal. Is a sign error equivalent to forgetting to restrict the domain? I don’t think so. But the regents do. What if a kid got to the next to last line in the last example, and stopped. Partial credit? Sure. But the same as the kid who finished and didn’t check.  And the kid with an early arithmetic error that led to a solution of -3. Had he restricted the domain, he would have stopped without a check. How many points off? ONE. One point off. They are all 1-point errors, because with a two-point question, there is no other kind of deduction possible.

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So why does New York State use two-point free-response questions? Idk. But I do know that “The Field” – teachers, have complained bitterly, and been ignored. Especially at the high school level, where questions involve multiple steps, having higher point values allows teachers to assign appropriate partial credit.

1. June 10, 2010 am30 9:19 am 9:19 am

I love your list of all the ways a student who at least somewhat understands the material could go wrong.

For comparison, on the Massachusetts high stakes test, short response answers like these don’t ask for work and don’t give partial credit, but there are also longer, multi-part free response questions, that are scored on a 1-4 rubric.

• June 14, 2010 am30 7:10 am 7:10 am

It’s interesting to watch how kids goof, and predict how others will goof. It’s challenging, right, because we have to imagine having an incomplete grasp of material that we know well. It’s not good enough to correct mistakes. It’s really worth trying to understand what the underlying misunderstanding was… and then, when you have a collection of misunderstandings you’ve encountered, you can reapply them to other problems and predict where and how kids will go off.

At least that’s what I try to do.

And then, on days when I’m clever, I can adjust my teaching to engage the misunderstanding and deal with it before it leads to mistakes.

Same sort of thinking leads to my new hot question: “What’s a mistake you think (another) (a younger) (a weaker) student might make on this problem?”

In any event, I’d prefer little questions with little point values to be MC. Once a question gets bigger, with more points, then partial credit makes sense. Frankly, even 4 is a low number for that.