Keeping up with the Joneses

In math, we say that 45 > 42 because 45 represents a larger quantity than 42. So, it seems fair to say that if Peter gets 42 points on a test and Claire gets 45 points on the same test then Claire did better than Peter. Right?

Hmmm... Suppose there were 50 points possible on my test. Suppose there are 10 questions and each is worth 5 points. Then perhaps Claire got 9 problems completely correct and 1 problem completely incorrect, and Peter got 1 point off of 8 problems. So, who did better? Is one large error better than several smaller errors? What seemed so clear now becomes murky.

In the "safe" world of adding up points earned and dividing by the points possible, Claire gets an A- (90%) and Peter gets a B (84%). But, if the above were true, it seems challenging to me to try to decide who did "better".

Keeping with the above scenario, Peter would have two "perfect" problems and 8 "imperfect" problems. What if Peter got full points on the problem on which Claire earned zero points? So, he understands that idea way better than Claire. That would also mean that Claire understands 8 questions better than Peter, but only in a small way on each.

Suppose, instead, the test were out of 214 points? How much of a difference is 3 points?

Comparing students to students is the real issue here. We, as teachers, do this all the time. The SAT does it. Colleges do it. Students do it. Why? To see who is better, or worse, or the same as us? How does that help my learning?

Let us keep Peter with his 84% and Claire with her 90%. What do these scores tell us about how they are doing overall? Not much. Those scores provide us only some measurement of how they did on one individual test. Perhaps Peter has never scored as high before. Perhaps Claire has never scored as low before. Now what? Does that diminish the value of Claire's score? Does that augment Peter's? Might we now say that Peter has, in a way, done "better" than Claire?

No, I go for Option Q2. With such scores on a test, do not be concerned with such questions as "Who did better on this test: Peter or Claire?" Rather, ask the question "What does Peter's performance on this test tell us about how Peter is doing at learning the ideas being tested?" Ask the same question of Claire. But don't bother with trying to make sense of comparing the two.