Little Steps for Little Minds

Here’s a quick but telling nugget of awfulness from Victoria’s 2017 VCE maths exams. Q9 of the first (non-calculator) Methods Exam is concerned with the function

    \[\boldsymbol {f(x) = \sqrt{x}(1-x)\,.}\]

In Part (b) of the question students are asked to show that the gradient of the tangent to the graph of f” equals \boldsymbol{ \frac{1-3x}{2\sqrt{x}} } .

A normal human being would simply have asked for the derivative of f, but not much can go wrong, right? Expanding and differentiating, we have

    \[\boldsymbol {f'(x) = \frac{1}{2\sqrt{x}} - \frac32\sqrt{x}=\frac{1-3x}{2\sqrt{x}}\,.}\]

Easy, and done.

So, how is it that 65% of Methods students scored 0 on this contrived but routine 1-point question? Did they choke on “the gradient of the tangent to the graph of f” and go on to hunt for a question written in English?

The Examiners’ Report pinpoints the issue, noting that the exam question required a step-by-step demonstration …. And, [w]hen answering ‘show that’ questions, students should include all steps to demonstrate exactly what was done (emphasis added). So the Report implies, for example, that our calculation above would have scored 0 because we didn’t explicitly include the step of obtaining a common denominator.

Jesus H. Christ.

Any suggestion that our calculation is an insufficient answer for a student in a senior maths class is pedagogical and mathematical lunacy. This is obvious, even ignoring the fact that Methods questions way too often are flawed and/or require the most fantastic of logical leaps. And, of course, the instruction that “all steps” be included is both meaningless and utterly mad, and the solution in the Examiners’ Report does nothing of the sort. (Exercise: Try to include all steps in the computation and simplification of f’.)

This is just one 1-point question, but such infantilising nonsense is endemic in Methods. The subject is saturated with pointlessly prissy language and infuriating, nano-step nitpicking, none of which bears the remotest resemblance to real mathematical thought or expression.

What is the message of such garbage? For the vast majority of students, who naively presume that an educational authority would have some expertise in education, the message is that mathematics is nothing but soulless bookkeeping, which should be avoided at all costs. For anyone who knows mathematics, however, the message is that Victorian maths education is in the clutches of a heartless and entirely clueless antimathematical institution.

5 Replies to “Little Steps for Little Minds”

  1. Correct as always Marty.

    For years as an IB Mathematics examiner, I remember thinking that we had it easier than the VCE paper-markers. For a start we were paid in US dollars which for a while back in the 2000s was a really nice exchange rate… but more importantly, the MARKING SCHEME was released TO ALL schools rather than just a summary of what a correct answer looks like.

    One of the best features of such a marking scheme was the annotations provided to guide the award of marks. In a question like this one, such an annotation would have been “…evidence of the product rule (seen anywhere) 1 method mark…”

    For what it is worth I suspect a lot of students had run out of time by Q9 on this paper or (also quite likely) thought that for 1 mark, the answer you gave above was perfectly fine.

    If VCAA would release proper marking schemes we might all be able to stop taking our medication for the headaches provided each year trying to work out which version of the truth is going to be marked correct this year…

    …don’t see it happening.

    1. Thanks, Number 8. True, if the examiners were competent and had a decent command of mathematics (and English) then they might be able to put together a better grading scheme, and that couldn’t hurt.

      I don’t believe, however, that this would solve the fundamental problem. There is simply no good way to grade a “Prove this triviality” question. Smart students will inevitably be penalised for their perfectly reasonable “Well, duh!” responses. And, the examiners are unwilling to test anything of any substantial size or depth.

      1. I never said the question itself was any good… just suggesting how questions of this structure (with more appropriate mathematical content) could be assessed.

        A 1-step “show that” does not make any sense to me.

    1. Thanks very much, George. Nicole speaks very well and, yes, she captures beautifully the ritualism, and thus the meaninglessness, of current VCE assessment. An important confusion, however, is that Nicole, and many others, see this as a battle between traditional education and some freer, exploratory form, which Nicole would prefer. But current VCE is not traditional education. It is nothing but the putrid remains, the decaying carcass of a once strong and proud and respected educational system. Only a handful of teachers are aware of that, and to Nicole it is probably unimaginable.

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