Monday, November 24, 2008

Your calculator passed the exam!

Too bad you didn't

It may not arise too often in practice, but suppose you want to find the derivative of z = y2e4x with respect to s, where x = 2s − 3t and y = 7st2. It's a perfectly good problem for testing the differentiation skills of one's calculus students and their understanding of the chain rules for partial derivatives. As is often the case, there is a good way and a bad way to work out the correct answer. The bad way involves brute force substitution, replacing each occurrence of x and y and then differentiating with respect to s. Even in a problem as elementary as this one, finding ∂z/∂s by substitution and differentiation adds additional steps (like the product rule, for example). It is a daunting task to tackle the direct differentiation of z = (7st2)2 exp(4(2s − 3t)).

This could be avoided with a little thought, which reminds the attentive student about the chain rule. The individual derivatives in the chain rule formula are relatively easy to compute: ∂z/∂s = (∂z/∂x)(∂x/(∂s) + (∂z/∂y)(∂y/∂s). Most of my students remembered this rule and applied themselves to the necessary calculations. With a little care, one can obtain the correct answer in a few steps.

One of my students, however, decided to obtain the answer in one step. He typed the problem into his calculator, entering the slightly grotesque version that results from substituting for x and y in terms of s and t. He could never have been expected to compute the result successfully in the time available, but his TI-89 quickly spat out an answer that he merely transcribed on his paper.

He had done this before. He had merely written down some calculator results and I had given him only a handful of points for setting up the problems—but none for the answers. But now he was doing it again. Had he not learned his lesson? Perhaps he had learned a lesson other than the one I intended: He could either skip the problem because he could not do it, or he could give it to his calculator and hope to get a few points for the setup. Good thinking!

Too bad the points derived in that manner aren't enough to produce a passing grade.

I've long been concerned about the phenomenon of calculator dependency. I've seen its impact at all levels of mathematical instruction, but the example from my multivariate calculus class is particularly disturbing. The student who can't live without his TI-89 has somehow survived till calculus III. He doesn't know how to apply the basic rules of differentiation. How did he earn passing grades in I and II? Maybe we should give a math degree to his calculator.

But not to him.

16 comments:

Mozglubov said...

I'm surprised you actually allow TI-89 calculators in your exams. I go to the University of Toronto, and most of our mathematics exams are entirely without calculators (and, if calculators are allowed, they are usually not allowed to be programmable). Of course, non-programmable calculators are usually common in exams for applied mathematics-type courses like physics and some computer science courses, but it's fairly rare. Anyway, it just seems to be an interesting thing to note.

Zeno said...

That's a good point you raise, Mozglubov, and I have on occasion banned calculators from certain exams and quizzes. However, most of my colleagues allow them most of the time and our students are accustomed to them. Besides, I know a lot of the calculator's weaknesses and am willing to take advantage of them (like problems that use sec x, since TI-89's can't give answers with secants in them and students who try to work backward from calculator answers often get lost in secant problems).

The bottom line is that I am very hardnosed about requiring students to show their work in methodical step-by-step fashion. If they use their calculators to check their answers, fine by me. If they simply try to copy down their calculator answers, big trouble, as for the student in question in this post.

John Armstrong said...

Here's another vote against calculator use. I don't allow them in calculus classes, and I write my exams so that they aren't necessary.

unapologetic said...

Grr, that last post should have been me...

Zeno said...

The funny thing is, John, that I do generally write my exams so that calculators aren't necessary, but most students don't seem to realize it and use them anyway. I could ban them, but it seems superfluous to do so (and many students would swoon without their "security blankets").

intrinsicallyknotted said...

It's worse than that. I've had students who needed to use a calculator to compute 4 x 3. In our tutoring center, I'm starting to develop a reputation as the "calculator nazi" for refusing to let my students use their calculators for simple computations.

j said...

When I was in Calc III, I used to tutor other students in single-variable calc. I had students who couldn't do simple arithmetic without a calculator (i.e., can't differentiate 6x^7 because it would require multiplying 6 by 7). It probably doesn't help that the public school district in our town has adopted all this reform-math curricula that teaches elementary-school children to use calculators for everything. Mass innumeracy, yay!

I haven't been allowed to use a calculator in math class since Calc II. All the tests and quizzes have been no-calculator. And our homework assignments, if too difficult for pencil/paper, are difficult enough to require Mathematica. But I am still very attached to my TI-89, and I still cling to it for physics.

Sili said...

The really big calculators were just starting to get in vogue by the time I left 6th form and entered uni. I did try out one, but ended up returning it to the shop, since I couldn't figure out how to use it.

Now that I work at a school (helping the janitor) I see that the maths course actually instructs people in how to use the TI89 and make them available to the students. I invigilated an exam this Summer and apparently (at least some of) the questions are designed to test the students' understanding of the calculator (not that all of them *understand*). I don't think that's necessarily bad. It's a tool and it's good to learn the use of tools.

That said - by the time I entered uni the calc teacher had already caught on to this issue of some people having graphing calculators and some not. He simply reformulated the questions to include the answers. So instead being asked to find the derivative or integral of something, we were simply asked to *show* that it was such and such.

Of course I've forgotten it all these ten years later, but I'm actually reading my old books now to exercise my brain a bit. Just finished series in order to have a look at complex analysis. Can barely remember any of it, and I know that I'll have to try my hand at the problems if I'mn ever to relearn it. But seing as how I'm reading in bed, that's not gonna happen anytime soon.

William said...

I always hated the "show your work" thing, when I'd already done the work in my head. Ah well.

Of course I feel that no post on calculator dependency would be complete without a link to "The Feeling of Power". :)

David said...

I teach maths and I use my TI-89 to set up problems :-) On the spot, on the fly you can start with a nice answer and work backwards to a question.

Students, OTOH, are banned from using a TI-89 at the exam (UK 'A' Level). But for learning purposes I'd encourage its use. One of my students generated some fascinating curves playing with the Polar coordinates graph sketching function.

Don't Panic said...

Ah, 6 times 7, my old nemesis. We meet again...

Seriously, it was literally until I reached the age of 42 that 6*7 stumped me, at least as something I should know by rote. I could work it out by knowing that 6*6=36 add 6, means 2 carry the 10, add to 30, makes 42. Not all that convenient in the middle of some other complex calculation. Of course this might make me look, to someone wanting a quick answer, like I was numerically illiterate. But by that age I'd long ago earned a PhD in Physics, so I don't think that's the case.

We all have our quirks. I quickly reach for the calculator where others might do the arithmetic in their head. But by the time I do so the problem has been reduced to something sensible and I'm not just plugging "stuff" into one of these new-fangled gadgets.

Geez, now I've googled the TI-89. Makes my trusty ol' HP15C look like an abacus.

Zeno said...

My own trusty HP-15C travels with me everywhere. It has its own special pocket in my briefcase and is ever at my side. How could I live without it?

Even better: When one of my students says, "Oh, Dr. Z, I forgot my calculator. Can I borrow yours?", they take one look at my little HP, make the warding-off-evil plus sign, and leave me alone.

Alan said...

I remember calculus I when it was necessary for curve sketching to know which was larger, the square root of 5 or Pi^2/3. we weren't allowed even nonscientific calculators, that curve was pretty screwed up looking in the end. bah!
I must admit, I'm still an undergrad but I"m hopeless with a ti-89, i can work it out by hand before i can figure out how to clear the damn screen. I do like to use a calculator to protect from the evil 2x3=5 that turns up so often in mental calculation. I don't really like graphing calculators, I"m an engineer, the specific shape of a graph rarely matters and when it does, only matlab will do. eat my shorts TI.

Anonymous said...

Without a ti-89, i would have never passed calc 1 or any of the higher calcs. I would not be graduating w/ an econ degree either. Thanks ti-89!

Most of my exams were not if you could figure out the problem, but you have to figure out each problem in a set limited amount of time. The people that failed usually knew what they were doing, but were slow. A ti-89 solve function has saved tons of time, and for nasty chain rules and log's differentiation has made it one second calc.

unapologetic said...

With a TI-89, Calc 1 students don't have to understand a thing about the material. You can speed right past algebrizing the calculus and just stick the formula into a program that does the differentiation or integration for you.

Sure, it saves tons of time you'd otherwise have to spend learning what the calculus means. I want my students to spend that time, so I don't allow their use on my exams.

btw: captcha is "fordeher". Why not fordehim? sexist blog software...

Anonymous said...

Most of my math classes so far have had very little focus on understanding what your doing, and much more on just knowing the formula and spitting out what you put in it. In this situation I don't see a single thing wrong with using a TI-89, because you haven't learned anything except maybe how to get the most out of your calculator. Of what value to me is it to know how to work with the quadriatic formula by hand if I don't know what I've actually done?

Maybe I've just had horrible math classes up till this point, but if you don't teach the application of the math, what value is doing it without a calculator? I'd love to take a calculus class where we're taught about the theory behind the use of this math and what it can accomplish, and where doing the actual problems is a relatively smaller role compared to the theory. Or maybe I just wasn't paying enough attention and it was a footnote at the start of the chapter about quadriatic or differentiation or whatever.