Agustin Torres | All-or-nothing grading doesn't reflect learning

Suppose you have a bucket with infinite space.

If this is starting to sound like one of those out-of-touch-with-reality math problems, it is, but please bear with me.

When there is one minute left until noon, add 10 balls to the bucket, labeled one through 10, and then remove the ball labeled "one." When there are 30 seconds left until noon, add another 10 balls to the bucket, labeled 11 through 20, and remove ball with label two. Keep doing this - each time the time interval reduces by half, add 10 balls, and remove the one that has the smallest number. At noon, how many balls are there in the bucket?

Here's a hint: The answer is either zero or infinity.

Could you confidently choose an answer within five minutes? Neither could I.

But if the above were a question in a multiple-choice mathematics midterm, then you are either right (in which case you get some points) or wrong (in which case you get no points, or maybe even lose points, depending on the test). Your thought process doesn't matter.

At Penn, that seems to be the case with some of the intro-level calculus courses, namely Math 104, 114 and 240, which frequently use multiple-choice questions for both midterm and final examinations.

But what's wrong with multiple choice? Isn't it an efficient and fair way to measure academic ability? Indeed, multiple choice allows for fast and easy grading, but it has implications about the way a subject is taught.

Before making any judgments, it's convenient to distinguish between methods of assessment.

For instance, a multiple-choice test may or may not award partial credit. In the latter case, if you select the right answer, you get full points - wrong answer, no points. But if partial credit is given, your work is taken into account, and points are awarded in accordance with the progress made toward the solution.

Partial-credit multiple choice is a reasonable assessment method. If you select the right answer, then there's no question about it - you get full credit. However, if you don't, then you're still rewarded for your understanding and ability. The problem lies in multiple choice without partial credit.

In many cases, solutions to the problems depend on mechanical procedures that are prone to errors. Chances are you may get a sign wrong, forcing you to choose the "stupid mistake" answer.

In these cases, one silly error invalidates the entire problem, even when you understood the principles necessary to solve it. Not accepting partial credit sends a terrible message to students: Penn only cares about the final result, not how you get there.

And, in fact, the impact on students' grades between assessment methods can be dramatic.

In the spring of 2006, the common final exam for Math 114 was all multiple-choice. College Dean Dennis DeTurck, who was the instructor for my section, graded the exams using the partial credit method. Had there been no partial credit, my final score would have been 24 points lower than it was with partial credit - more than one full letter-grade higher.

While you may argue that in the end, it doesn't matter, since partial credit would skew the entire class' grading curve, this conclusion is not necessarily the case. In the mentioned final examination, the average with no partial credit was 14.1 out of 18. With partial credit, it was 147 out of 180. These results mean that all the students whose score increased by more than six points would have scored higher with partial credit.

So, then, why use no-partial credit multiple choice? Maybe it's because nobody wants to waste time deciphering the mathematical squiggles of a class of more than 50 students.

"If you are grading over 50 exams per person sheer fatigue degrades the process," Bob Powers, temporary undergraduate chairman of the Math department, said in an e-mail. Powers, however, does grant partial-credit in his classes.

Maybe there's a reason why a lot of students don't enjoy calculus classes at Penn. Ultimately, mathematics is about definitions, theorems and proofs. It's about solving challenging problems with elegant notation. Math problems should be open-ended and interesting, encouraging creativity, rather than following a plug-and-chug recipe.

And just in case you're wondering, the answer to that initial problem is zero. Don't worry about learning how to solve it, though - that won't get you any points.

Agustin Torres is an Engineering sophomore from Monterrey, Mexico. His e-mail address is torres@dailypennsylvanian.com. The Monday Burrito appears on Mondays.

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