This post will take the form of Betrayal, with a sequence of five stories going backwards in time.
Last year, I was asked by an acquaintance, let’s call him Rob, to take a look at the draft of a mathematics article he was writing. Rob’s article was in rough form but it was interesting, a nice application of trigonometry and calculus, suitable and good reading for a strong senior school student. One line, however, grabbed my attention. Having wound up with a vicious trig integral, Rob confidently proclaimed,
“This is definitely a case for CAS”.
The integral in question was of the form
with b ≥ 0 a real parameter. The parameter made the integral undeniably vicious, perhaps “solvable” in terms of gamma functions and the like via complex techniques, but not doable in any manner that would be clarifying for the intended reader. All that mattered in the context, however, was to determine the sign of I for b positive. The sign of I, however however, could not be read off by an easy inspection. Since the interest was in perturbing from b = 0, however however however, all that really mattered was the sign of I for b positive and close to 0. And that provided the non-CAS “in” to solving the problem.
The point was to think of the integral as a function of b. (Of course F still also depended upon x, and was integrated with respect to x, but that could be left implicit.) Now, was simple enough that it was immediate that I(0) = 0. Moreover, by “differentiating under the integral sign”,* we had
and in particular .** Then, F'(0) was simple enough to easily conclude I'(0) > 0. Since also I(0) = 0, it followed that I(b) > 0 for small positive b, the problem was solved, and the CAS went back in its case.
*) The technique is slightly beyond high school, but so was Rob’s article.
**) Yeah, yeah, those should be partial derivatives inside the integrals. But if a student is happy to integrate with respect to x while keeping b constant, they can just as happily differentiate with respect to b while keeping x constant.
About five years ago, I was asked to tutor a Year 12 Specialist Mathematics student, let’s call him James. It was pretty silly, since James was smart and studious, and attended a very good private school, let’s call it Scotch College. James needed no help whatsoever. But, James’s mother insisted, and so we kept up the meetings. Given James needed no help with Specialist, we typically talked about mathematics instead. It was fun.
During the year, James showed me a number of his Specialist SACs and SAC-preps. Scotch’s stuff was good. The SACs et al were based on natural scenarios, with well-constructed (and error-free) questions, and with an emphasis on algebraic rather than numerical framing. Nonetheless, and inevitably, the SACs also contained plenty of questions that were intended to be done with CAS.
Typical was an assignment problem that required maximising a quantity something like
with a, b, c and d suitable and fixed (e.g all positive and c > d). I can’t quite remember the form, but I can remember the message. James and, it seems, every one of his classmates saw the messy function, contemplated the messy derivative to come, noted the 2 Marks or whatever on offer, and decided it was a case for CAS.
This happened repeatedly throughout the year. James would show me some function he had needed to manipulate or maximise or whatever, and if there was a hint of hard algebra or calculus, he would have reached for the CAS. The notion that a calculation might be easily handled or avoided with a little thought or a delicate touch was too rarely considered. Specialist discouraged such notions, even at Scotch.
In 2009, the Mathologer and I wrote a (not very good) opinion piece for The Age, decrying the state of Australian maths ed. Our piece included a two-sentence slap at CAS, which spurred a vocal CAS fan, let’s call him Victor, to email us. Victor was friendly, agreeing with most that we wrote, but Victor also claimed that research showed that the use of CAS strengthens algebraic skills, and that CAS allows the students to explore and to focus on understanding, and so on.
I replied to Victor, focussing on his claim about the research on CAS and algebraic skills. I indicated that I would be interested to read such research, and that if he suggested “one or two” references then I would read them. I made it explicit that I did not want ten references thrown at me. Victor somehow interpreted this as an onerous restriction to not provide more than ten references, but restrained himself and sent six. I read none.
In 2008, a teacher friend, let’s call him Fred, emailed a pro-CAS professor, let’s call her August. Fred asked August for the strongest academic references demonstrating the pedagogical value of CAS. This prompted August to email one of her minions, noting that “We do need a good answer to this question”. August’s email, however, was sent to Fred by mistake. Fred then quickly received a second email from August, indicating that it would “take a bit of time” to gather the references. If August and her minion ever compiled their “good answer”, they never got back to Fred to let him know.
Around 2005, soon after I started pondering maths education, someone handed me a paper with the title The Case for CAS. That was the first time I was touched by this stuff, and I remember my reaction: “Uh oh”.