Henry Pollak, in his essay, “What Is Mathematical Modeling?”

Probably 40 years ago, I was an invited guest at a national summer conference whose purpose was to grade the AP Examinations in Calculus. When I arrived, I found myself in the middle of a debate occasioned by the need to evaluate a particular student’s solution of a problem. The problem was to find the volume of a particular solid which was inside a unit three-dimensional cube. The student had set up the relevant integrals correctly, but had made a computational error at the end and came up with an answer in the millions. (He multiplied instead of dividing by some power of 10.) The two sides of the debate had very different ideas about how to allocate the ten possible points. Side 1 argued, “He set everything up correctly, he knew what he was doing, he made a silly numerical error, let’s take off a point.” Side 2 argued, “He must have been sound asleep! How can a solid inside a unit cube have a volume in the millions?! It shows no judgment at all. Let’s give him a point.”

What a fantastic dilemma.

Pollak argues that the student’s error would merit a larger deduction in an applied context than in a pure context. In a real-world context, being wrong by a factor of one million means cities drown, atoms obliterate each other, and species go extinct. In a pure math context, that same error is a more trivial matter of miscomputation.

The trouble is that, to the math teachers in the room, a unit cube *is* a real-world object. They can hold a one-centimeter unit cube in their hands and, more importantly, they can hold it in their minds.

The AP graders aren’t arguing about grading. They’re trying to decide *what is real*.

What a fantastic dilemma.

**Featured Comment**

Whenever I wanted to give students the most amount of partial credit, my coop teacher would ask me the poignant question, “What exactly are you assessing?” I found this was a great question to continue asking myself. So, in the example you gave, are you assessing students’ ability to perform mathematical functions correctly or are you assessing their ability to connect those math functions to the real world?

Benjamin Dickman alerts us that Pollak’s piece is online, free, along with a number of his modeling tasks.