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Sunday, February 26, 2006

Graphing aids

There are several blogs that I read regularly. One of my favorites is Learning Curves, by the pseudonymous math professor Rudbeckia Hirta. She goes to some lengths to hide her real name, as well as the the name of the university somewhere-in-the-South at which she is head of the Mathematics Department. Not that she’s unwilling to leave some clues for the knowledgeable: she has posted photos of herself, a map of her neighborhood, a shot of the Math Department’s building, and even pix of her house. So it’s clear that anyone who knows her or is associated with her university will know her secret identity. Apparently she’s protecting herself from strangers, not that I’m nosy enough to try to figure out who she is or where she teaches.

In any case, Ms. Hirta recently wrote about the TI graphing calculator and Apple’s Grapher software:
It’s no secret that I’m not a big fan of the graphing calculator.... Now that I have found Grapher in the Utilities folder inside my Applications folder (that, among other things, makes very pretty graphs that can be output as .pdfs), my limited needs for the graphing calculator have waned somewhat.
This post got me thinking about calculators and graphing. It’s the second recommendation that I’ve received recently for Grapher, so I really have to check it out. One user describes it as “very useful and frustrating,” which is not particularly encouraging. The documentation is daunting, but clearly there isn’t enough of it. Sigh.

Oh, well. I’m going to have to explore it anyway, since it promises to be a useful tool for me and for other math teachers. It does 2-D and 3-D color graphs, animations, polars, parametrics, and so forth — but I don’t really have any overall sense of its capabilities.

And what about the TI graphing calculator? This device has become ubiquitous in American high schools, and quite common in colleges. It clearly does a lot of good things for students, but it just as clearly has done a lot of harm. Let’s look at four examples of what Texas Instruments claims as its principal virtues [numbers added]:
  1. A high school math teacher who works with students at a school geared toward the arts, finds that TI technology helps him overcome the disinclination toward math exhibited by many of the students, while it also enhances his own professional growth.

  2. An Arkansas teacher with a long history of implementing technology in the classroom uses TI handhelds and related peripherals to introduce his students to the latest analytical and problem-solving techniques in math and physics

  3. At-risk students receive a boost through programs that use TI graphing handhelds to reinforce basic mathematical concepts and prepare them for Algebra 1.

  4. TI-83 Plus and TI-89 handhelds help Pre-Calculus and Calculus students at North Shore Senior High School visualize concepts and achieve higher passage rates on the AP Calculus exam.
I’ve deliberately selected a variety of success stories. Of course this is advertising by TI, but that doesn’t mean that we should dismiss the stories out of hand. Note the four different themes here: #1 claims that the calculator increases motivation for many students, #2 says that it gives access to the newest and (perhaps) best techniques of problem-solving, #3 points out advantages for at-risk pre-algebra students whose skills are not up to par, and #4 addresses the opposite extreme, students preparing for AP exams in calculus.

As I said, we shouldn’t dismiss these stories out of hand. Far from it: even without following the links to read the details of these four stories, I don’t doubt that they provide significant arguments for the benefits of using the graphing calculator. It can clearly be a great help for a wide variety of students in a wide variety of courses.

Then why does its use distress so many math professors, including but certainly not limited to Rudbeckia Hirta? Does it also distress high-school math teachers?

Indeed it does. While the graphing calculator can definitely provide motivation and assistance, it can also foster inappropriate dependence. For example, if a student can’t multiply a number by ten or by negative one without whipping out his calculator, he is demonstrating an unfortunate lack of number sense. Now there are probably a few people who would never succeed in multiplying by ten without a calculator, but surely most students can develop that much number sense before high school. The calculator is hindering their learning, not helping it.

Insofar as the calculator is a substitute for skills that ought to be learned, it’s a harmful tool. Of course we’re going to have disagreements about what those skills are. Should a high-school student be able to divide one number by another, using just pencil and paper? Most of us would say yes, but certainly not all of us. Most of us would say that it’s OK to use the graphing calculator to perform long division most of the time — after all, we do it ourselves — but there still need to be enough contexts in which we ask kids to do it without a calculator. On the other hand, most of us would say that there is no longer a need for a student to learn to perform square roots by the pencil-and-paper algorithm, and most of us have forgotten the details of that algorithm oursleves. (On the third hand, we would want any high-school student to be able to determine whether 8.25 or 8.75 is closer to the square root of 70 — without a calculator, of course.)

These ideas led me to divide most tests into two sections, one allowing calculators and one forbidding them. But I don’t do that in every class, having been burned a few years ago by a parent who threatened to sue the school system if we didn’t allow her son to use a calculator on every math problem. More on that in a later post.

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