Calculators, I (The Technical Post)

Gentle readers: This post contains math. If you are a Not-Math-Dweeb you may not find it at all interesting and may want to just skip to "Calculators, II" above. If you are (like me) a Math-Dweeb of some flavor, you may find this interesting. I've been tinkering with these two posts since last semester when the issue came up and finally decided to just post.

I've was on a committee last semester (yes, more meetings) to design a calculus test. We'd run into some difficulty on asking about using a linearization to estimate a function's value. On a previous test, we had a question similar to this:

Use a linearization to estimate the value of ln(1.1).

I had a student who responded with something like this:

L(x) = ln(1.1) + (1/1.1)(1.1-x) (approximately equals) 0.0953 + (1/1.1)(1.1-x),
so ln(1.1) (approximately equals) 0.0953 + 0 = 0.0953.

(I've probably been somewhat more clear than the student has, but you get the idea.) How do I grade this? The student has in fact used a linearization to estimate the value of ln(1.1)--they just chose to use the linearization of ln(x) at the value x = 1.1.

In the narrow view, the student has answered the question correctly and deserves full credit. (Heck, they even know how to linearize a function, which is essentially what the question is asking.) In the broad view however, they are completely clueless as to why we might use a linearization. In this sense, they have no idea what is going on in the problem, and should probably get very little credit. (I don't remember anymore what I eventually decided when I graded this.)

Of course, they were only able to do this because they had a calculator that could give a decimal approximation to ln(1.1). This makes writing tests in a calculator age all the more challenging: How do you write a question like this in a way so as to circumvent such an answer?

There are ways to get around it, but most are unsatisfactory. Here's some of the initial options we discussed:

1. Tell them where to linearize the function. (Downside: They don't have to make an intelligent decision about where would be a good place to linearize, which is part of what we would like to test.)
2. Give them information about f(1) and f '(1) for some unknown function, and ask them to estimate this. (Same downside as #1.)
3. Tell them to make the linearization at a point where ln(x) is a whole number. (Downside: This still seems to give away a lot, really.)
4. Tell them they have to do this based entirely on values for ln(x) which they already know without use of a calculator. (Downside: Someone who is a Wise Ass (but who may or may not be Tall) will say they just "happened to know" that ln(1.1) is about .0953.)

Not very good choices, really. But I came up with two new ones while I was starting this post, which I like:

1. Give them a table of values for an unknown function f and f ', and ask them to use a (wisely chosen) linearization to estimate the value of f at some intermediate point.
2. Have them come up with the linearization for a function based at two different points, then ask them to choose an appropriate one of the two to use for estimating the desired value.

See, and you thought writing these posts wasn't accomplishing anything for me? (We ultimately went with the second choice.)

So I've solved the intermediate problem, and now I have a deeper problem to work with: Why do I care if they can do without a calculator what the calculator can do for them?

The classical problems in most calculus texts covering linearization tend to be problems like estimating the values of ln(1.1) or the square root of 10. While it's relatively cool that some simple calculus techniques and a back-of-the-envelope calculation can give you such estimates, this is not really what most people do. There are certainly many reasons to use linearization other than these kinds of estimates. I discuss with my class the linearization of the motion of a pendulum, for example (although they are not ready to understand the details of the differential equation). There's a lot of physics based around simple linearizations like sin(x) (approx equals) x or ln(x+1) (approx equals) x-1. Also, it's the beginnings of the theory for how the calculators and computers know the value of ln(x) or sin(x). (Admittedly, we are talking baby steps here, but you have to start somewhere.)

I think there might be value in asking the students to come up with and use a linearization in the simple way we do, even if their calculator can find the value more quickly and accurately. It's a good check to see if they understand both how to find a linearization, and what it means--namely, that it is a simplification of a complicated function. So maybe it's an unrealistic problem which is simultaneously a good test question.