## Saturday, January 28, 2012

### I've half a mind

When it's early in the semester, I tend to cut my students a little more slack. Of course, I expect them to pay attention when I explain why I take off points for some calculations that manage to produce correct answers. For example, how many minutes does it take you to travel 12 miles at 18 miles per hour? Here's what one student told me:

Yeah. Well, I'm really not happy with that. Sorry, but 12/18 is simply not equal to 40. Equality is supposed to be a transitive property, folks! Of course, this could be redeemed with the appropriate use of unit conversion:

This I like. Careful use of units is a powerful way to keep one's calculations in order and to make sense of the results. Full marks! But then you get the woefully calculator-dependent student who presents this travesty:

Heck, you can keep your puny old leap-seconds! My students can conjure up a dozen seconds out of the thin air of feckless rounding. This is a particular gripe of mine. You actually need to grab for a calculator to compute two-thirds of sixty? Good grief!

Thoughtless calculations like these were sprinkled throughout the early semester quizzes and exams. But the pièce de résistance came in a different problem. One that had nothing to do with rounding. I gave my students (gave them, mind you) some volume formulas. All of the most popular shapes were there: cone, cylinder, sphere, box (ahem! Sorry. I mean rectangular parallelepiped, of course). The formulas were actually written out on the assignment sheet. I then asked my students to use the formulas to compute the volumes of some specified shapes. One of the shapes was a hemisphere.

Sure enough, several students decided the formula for a sphere was the best match they could make, computed the result, and ended up with an answer that was two times too big. Arrggh! Naturally, I took off points for that mistake. One of my students waxed indignant when he got his paper back and issued a two-part complaint: (a) I had not given them the formula for the volume of a hemisphere and (b) I had not done an example in class where we had to divide a result by 2 to get the correct answer.

I offered a plea of “no contest” to both charges. They were irrelevant. I patiently explained: “I have higher expectations of my students than merely plugging mindlessly into formulas. I want my students to think about what they're doing. This is not just a plug-in and grind class. Sorry.”

But not very.

The Ridger, FCD said...

I patiently explained: “I have higher expectations of my students than merely plugging mindlessly into formulas. I want my students to think about what they're doing. This is not just a plug-in and grind class. Sorry.”

Yes, but did you tell them that on the first day, so they had a chance to drop this abysmally difficult and unfair class?

Zeno said...

Oh my yes, approximately twice a day for two weeks, if I recall correctly. (Maybe more often than that. I tend to repeat key ideas.)

Kathie said...

"Yes, but did you tell them that on the first day, so they had a chance to drop this abysmally difficult and unfair class?"

But did you put it in the course syllabus??? (The war stories my husband could tell...)

Erik said...

Since you failed to provide the equation for the hemisphere, they would have to fall back to doing a triple integral over x, y, and z to compute the volume of an arbitrary shape. Unfortunately, it's still not solvable because you failed to specify which half of the sphere the hemisphere was taken from, and thus the appropriate intervals for the integrals are unknown.

Zeno said...

Kathie: Yeah, the syllabus has language about reasoning ("recognize and explain" what they're doing!).

Erik: You are correct: I really need to give my algebra students more information if I expect them to spontaneously generate the tools of multivariate calculus (with extra points for reinventing the notation!).

RichS said...

I'm not so sure the person who wrote 2/3 = 0.67 used their calculator. I can't think of a calculator that would display only two digits after the decimal point, unless you specifically ask it to (which seems unlikely).

Perhaps this student has been trained by his or her calculator to think of common fractions in terms of decimal approximations. (0.67*40 = 40.2 was surely done by calculator though.)