Hand Me That Tool, Will Ya?

To tell you the truth, I don't remember a whole lot about elementary school. I remember the faces of a few of my teachers. I have pretty clear memories of Sister Mary Vincent, my fifth grade teacher, mostly because she was about 110 years old and a variety of creative ways of expressing her displeasure, including one memorable instance in which she picked one of my classmates up by the hair, dragged him to the blackboard, and slammed his head into the board five or six times to put the fear of God into him.

One thing I do remember is that I was absent for a week or so during third or fourth grade, I think it was, when the class covered the process for deriving a square root. I remember coming back to school and having the teacher tell me I'd have to make it up on my own following the instructions in the text. I gave it a shot, and never did figure it out. It remained a conspicuous hole in my mathematics background up until the first hand-held calculators showed up in the early 70's, at which point deriving a square root became something any moron with fingers could do. Not that I can ever recall ever having the need; the derivation of square roots has turned out to be about as useless a math skill for the average person to possess as one can well imagine.

Reason I bring this up, yesterday I got to sit in on a joint meeting of the middle school and high school math departments, one of the first such meetings that has occurred at my school in many a moon. Todd and Will, the respective department chairs, had asked the teachers to read in advance a packet of articles relating to math pedagogy, and had arranged to have the 30 or so teachers divide up into groups, one group for each of the topics that the articles had covered. My group turned out to be the group discussing calculators, and it wound up being a very interesting discussion. The original focus of the discussion had to do with the reality that while middle school students are at this point in time allowed, even encouraged, to use calculators for routine arithmetic operations, students in at least some of the grade nine classes are forbidden to do so.

I haven't spent a lot of time (any time, actually) researching the arguments of the pro- and anti-calculator factions in education or tracking the history of the debate, which has of course been around ever since calculators became ubiquitous and cheap. But I get the gist of the conflict, and got a quick review of the core positions as the discussion unfolded. The anticalcs argue that use of the calculator gives kids an excuse never to learn the basic computational skills without which one will never truly understand, much less master, mathematical thinking. The procalcs argue that there are many very bright kids who are good mathematical thinkers who for organic or developmental reasons struggle with arithmetical operations, so that use of the calculator actually frees them up to go further and deeper at the conceptual level than they ever would be able to if they were forced to do the (likely to be faulty) arithmetic on their own. Out of the instance of the freshman program, larger questions emerged. What is the logic of allowing, or forbidding, calculators in the first place? Should there perhaps be a consistent schoolwide (K-12) policy or position statement that says when and why we are going to ask, or forbid, the students to use calculators? What are the mathematical non-negotiables? Are there some operational procedures — the derivation of square roots, for example —that the invention of the calculator has made irrelevant or unnecessary for students to learn? Is long division one of those operations? Multiplication? Addition? Where do you draw the line?

From there we bumped up to the next larger frame of reference: laptops. We are well on our way to the implementation of a One-to-One laptop program in grades 4-12 (as of this moment, only grades 11 and 12 remain technologically unenhanced.) Laptops have not only multifunction calcuators of various kinds wired in, they have very sophisticated mathematical modelling programs as well, like the one that one of our physics teachers used on the opening chapel ceremony, which was built around a theme of "hearts beating as one," to give a scientific demonstration of a drum head doing a complex series of simultaneous vibrations. The concept came across very clearly, despite the fact that there was perhaps no one in the audience, with the possible exception of the teacher himself, who could have done the math necessary to produce the demonstration onscreen. So when kids walk into the math room, do we tell them to check their laptops at the door? Do we define certain types of activities or lessons or assessments as being laptop-friendly or laptop-hostile? Bottom line, is telling math students they can't use calculators any more or less reasonable than telling English students they can't use word processors? Spellcheckers? Dictionaries? When is a tool a tool, and when is a tool a crutch? Answers may vary.