Don’t Accept No Subsitutions

As a byproduct of the recent differential equations WitCH, I promised to write something on integration by substitution. I’ve been pondering that, and I will write it, soon. But there’s also a little VCAA story about substitution I’ve been thinking about posting for a long time, and it seems natural to do that first. In order to tell the VCAA story, I’ll also give a little history to frame the story. Because you young ‘un readers of this blog need some history.

Once upon a time, senior Victorian Mathematics subjects made sense. There was, for the serious HSC mathematics students, Pure Mathematics and Applied Mathematics. (Less serious mathematics students took General Mathematics, which was still pretty serious.) With two serious and sensibly framed subjects, topics could be covered once, properly and in depth. So, for example, all of the techniques of integration were covered in Pure Mathematics, and all of the combinatorics was covered in Applied Mathematics. The combinatorics example points out the permeability of the Pure-Applied border; combinatorics was chosen to be part of Applied Mathematics because the decent coverage of binomial and hypergeometric and Poisson probability made it natural to do so. The important point, however, was that although topics from each subject would depend upon the other, individual topics did not straddle the border. Then, things changed.

Fifteen years ago, after I had returned to Victoria and started pondering school mathematics as a grown up, I had no sense of how much things had changed. I knew HSC had become VCE, and that Pure and Applied had been replaced, eventually, by Mathematical Methods and Specialist Mathematics, but I didn’t understand what these changes meant in practice. I learned.

The first thing I learned was that Methods was treated as a stand-alone subject and was a weaker subject. This was self-evidently absurd, and damaging. Having both strong students and not-so-strong students taking the same Methods subject meant that topics covered in Methods could not be treated in the appropriate depth for Specialist students.

The second thing I learned was that the two mathematics subjects then on offer were, and still are and more so, effectively meaningless. That meaninglessness was of course suggested by the any-name-will-do titles, but I learned that the subjects offered even less than the titles advertised: there was nothing particularly special about Specialist, and there was nothing remotely methodical about Methods. I learned that both subjects were a mish-mosh of thin, ill-defined and ill-fitting topics, and that these topics were routinely splayed across the two subjects. Which brings us, finally, to integration by substitution.

Mathematical Methods contains a few but very far from all of the techniques of integration. And one of the techniques it does not contain is integration by substitution. This is obviously absurd since, almost immediately, students will be integrating by substitution, if only in an ad hoc manner. And, Methods’ preclusion of substitution as a taught technique means that the technique will always be done in an ad hoc manner. Thus, for example, we have the following formulas in a chapter summary from the best Methods textbook:

And, we have the following on the Methods exam formula sheet:

The ugliness, the forgettability, the plain absurdity of this beggars belief. It is so absurd that many Methods teachers will teach substitution in at least a quick, cookbook manner so that their students will have at least a little resilience. And as well as being absurd, this stuff routinely screws up Specialist students, who get the message, for instance, that 1/THING = log(THING) + c, give or take a constant factor chosen somewhat at random.

Fifteen years ago, I saw this no-substitution madness and I talked to teachers, who shrugged their c’est la vie shoulders. I still had no idea how this nonsense had originated, but I decided gently (for me) to try to do something about it. I was friendly then* with a teacher who was in the good books with VCAA. This teacher was scheduled to be on a VCAA committee, which was to tinker with the VCE mathematics subjects. So, I told the teacher that the one obvious, important and simple change was to move substitution from Specialist to Methods. The teacher seemed to agree with me, and agreed to make this suggestion.

The teacher then reported back: the teacher had made the suggestion and the other people on the committee had laughed. They had laughed because they knew the suggestion had come from me. And they laughed because they had no intention of accepting the suggestion. And they did not.

And that’s when I first got some sense that VCAA was arrogant and as stubborn as a mule, and that the absolute majority of teachers in the inner sanctum of VCAA were pig-ignorant. And hence, eventually, this blog.

*) Not now.

11 Replies to “Don’t Accept No Subsitutions”

  1. Dubious title (using the double negative … only the Rolling Stones are allowed to do that), nice anecdote and very nice pictures of possibly (*) the best secondary school textbooks we are likely to ever see (then again, textbooks are only as good as the syllabus they cover, so perhaps one day …)

    * Lucas and James is a strong challenger.

    Having the substitution technique in Maths Methods makes a lot of sense. But let me be more specific – having the technique of \displaystyle linear substitution in Maths Methods makes a lot of sense. I know it gets taught by some teachers (myself included) in Methods, if only to answer questions like

    If \displaystyle \int_0^a f(x) dx = a find \displaystyle \int_0^{5a} f\left( \frac{x}{5}\right) dx.

    The VCAA method for solving such questions (see Examination Reports) uses dilations and areas and I tend to find this confusing. I think it’s much simpler to make a linear substitution. However, it does open up the can of worms of the concept of the ‘dummy variable’ and this can take a while to explain.
    (Personally, I prefer constructing a constant function f(x) = k that satisfies \displaystyle \int_0^a k dx = a and then use it to calculate the answer. But it’s amazing how many students don’t understand that if f(x) = k then \displaystyle f(x/5) = k).
    I’ve always said that such questions give an unfair advantage to Specialist Maths students.

    There have been questions on past commercial Maths Methods Exam 1 papers that required explicitly knowing the formula \displaystyle \int \frac{f'(x)}{f(x)} dx = \log_e (f(x)) + c. I have strongly objected to these questions for a number of reasons, not the least being that it’s outside the scope of the study design. Nevertheless, apparently it’s a reasonable question to ask, although no-one could ever convincingly tell me why (and one like it has never appeared on a VCAA exam). I actually wouldn’t mind if this was part of the Maths Methods study design.

    I whole-heartedly agree with your implication that Maths Methods and Specialist Maths mesh together like a glove and a sock (or should that be water and oil?)

    1. PS – the 20 min window for editing currently seems to not be available. No biggie, I’ll let readers figure out where the extra “\displaystyle = a” is.

      1. Thanks. Yes, it’s working fine now. But not for my first comment (but I think I know why).

        And I meant to say \displaystyle f\left( \frac{x}{5} \right) = k.

        1. I made one correction. I have no idea what this second correction is intended to be, but your “personally” parenthetical makes no sense. Also, assuming Anonymous = Back in Black, I can make that correction.

          1. Thanks. Please do.

            The “Personally” parenthetical is a method I like to use: Given \displaystyle \int_0^a f(x) dx = a, there will be a constant function \displaystyle f(x) = k for which it’s true. So figure out what that constant function is and then use it in \displaystyle \int_0^{5a} f\left( \frac{x}{5}\right) dx.

            1. Too much work. Tell me precisely what you want the parenthetical statement to be, and I’ll insert it.

    1. I’d have to think about it. Now, of course, the list is very, very long.

      But note that, in schoolground fashion, but it’s important, VCAA pissed me off first. VCAA didn’t ignore me in the above story because I had pissed them off. There was nothing personal about it. They ignored me because they ignored everybody.

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