This week I committed a heinous crime against my students. Yes, I lost my head entirely. My transgression came in the form of an exam problem. The results were not pretty.
The occasion of my sin was a unit on business math. I wrote a multi-part exercise on annuities. The assault on my students occurred in part (b), so let me set the scene of the crime by describing part (a). If you do not have a strong stomach, you may wish to stop reading at this point.
Okay, you were warned. Part (a) asked my students to compute the maturity value of a 15-year annuity: $80 deposited monthly in an account paying a 4.2% annual rate. If you do the math, which in this case merely means plugging the numbers into a nice formula and pressing the right buttons on your calculator, you discover that the annuity will result in about $20,000. Students did well on part (a).
Part (b) asked my students to compute the present value of the annuity in (a). That's the amount of money you would have to put into a regular compound-interest savings account today, in one lump sum, in order to have the same amount of money in the same fifteen years as the maturity value of the given annuity. Since you wouldn't be putting in any monthly payments, the present value must be a good-sized lump of money. We have a nice formula into which you can plug the pertinent amounts, after which the calculator does all the work. But then ... I struck!
I wrote part (b) without including the words present value. Instead, the problem merely said, “How much should he deposit in a savings account today at 4.2% (compounded monthly) so that he will have as much money in 15 years as the annuity in (a)?” Panic ensued. Hysteria! Tears!
I exaggerate. Slightly. Because I described the quantity I wanted them to compute instead of specifically giving them the name of the quantity, many students had no idea which formula they needed to plug into. They knew the name, but not the property. If I may paraphrase the response of some: “You didn't tell us what to compute.”
A few hearty souls resorted to first principles. They had the result in (a) and they had a formula for compound interest, so they plugged in the maturity value of the annuity into the future value of the compound interest account and solved for the initial deposit (the principal). Good for them. This approach was riskier than using the present value of an annuity formula because it required that they have the correct result in (a), but it was a valid calculation. (It was a little more work, too, than just using the specific present value formula.)
Other students desperately plugged into the same formula as in (a) and gave me the exact same amount again. Uh, right. Others used the same formula in (a) to figure out the monthly deposit required for an annuity and proudly told me the answer was $80 (or, more often, $79.99 because of rounding error). But that was for an annuity and was already given in part (a).
When I returned to the scene of the crime and gave the students back their exams, they were pleased by the overall results (the class average was high). Part (b) had been a disappointment for many, however. When the solutions went up on the board, several were stunned:
“Oh, is that all?”
“But you didn't say present value!”
“You could have told us!”
I replied: “But the text of the problem exactly describes what a present value is. You need to know what it is you're computing as well as how to compute it.”
Not everyone was mollified.
“Well, you could have given us a hint that you wanted present value!”
I replied: “Okay. Did you notice how I labeled the answer blank on part (b)?”
Saturday, November 04, 2006
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That's hilarious, but not unexpected. It is surprisingly hard to teach people to think.
P.S. Aren't they first principles or am I mixing them up with first principal?
Quite right on your P.S. Thanks! It is duly corrected.
I imagine this is the kind of mistake I might have made on a test... but only on a test.
Perhaps I'm falling into the standard student's denial of responsibility, but I do believe that I've been trained to not think during tests. Going into a kind of autopilot mode is a very old habit.
Imagine my surprise when my college classes required actual original thought--and not just in the regular coursework, but on the tests! Until then it seemed as though math classes and I had an understanding: thinking goes on during instruction and study, not tests. Tests, I must have thought, are for mindlessly chugging through numbers to prove mastery of some given algorithm.
Back when I taught college mathematics, I often commented that students don't have math anxiety. They have thinking anxiety.
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