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.

There’s Madness in the Methods

Yes, we’ve used that title before, but it’s a damn good title. And there is so much madness in Mathematical Methods to cover. And not only Methods. Victoria’s VCE exams are coming to an end, the maths exams are done, and there is all manner of new and astonishing nonsense to consider. This year, the Victorian Curriculum and Assessment Authority have outdone themselves.

Over the next week we’ll put up a series of posts on significant errors in the 2017 Methods, Specialist Maths and Further Maths exams, including in the mid-year Northern Hemisphere examsBy “significant error” we mean more than just a pointless exercise in button-pushing, or tone-deaf wording, or idiotic pseudomodelling, or aimless pedantry, all of which is endemic in VCE maths exams. A “significant error” in an exam question refers to a fundamental mathematical flaw with the phrasing, or with the intended answer, or with the (presumed or stated) method that students were supposed to use. Not all the errors that we shall discuss are large, but they are all definite errors, they are errors that would have (or at least should have) misled some students, and none of these errors should have occurred. (It is courtesy of diligent (and very annoyed) maths teachers that I learned of most of these questions.) Once we’ve documented the errors, we’ll post on the reasons that the errors are so prevalent, on the pedagogical and administrative climate that permits and encourages them.

Our first post concerns Exam 1 of Mathematical Methods. In the final question, Question 9, students consider the function \boldsymbol{ f(x) =\sqrt{x}(1-x)} on the closed interval [0,1], pictured below. In part (b), students are required to show that, on the open interval (0,1), “the gradient of the tangent to the graph of f” is (1-3x)/(2\sqrt{x}). A clumsy combination of calculation and interpretation, but ok. The problem comes when students then have to consider tangents to the graph.

In part (c), students take the angle θ in the picture to be 45 degrees. The pictured tangents then have slopes 1 and -1, and the students are required to find the equations of these two tangents. And therein lies the problem: it turns out that the “derivative”  of f is equal to -1 at the endpoint x = 1. However, though the natural domain of the function \sqrt{x}(1-x)} is [0,∞), the students are explicitly told that the domain of f is [0,1].

This is obvious and unmitigated madness.

Before we hammer the madness, however, let’s clarify the underlying mathematics.

Does the derivative/tangent of a suitably nice function exist at an endpoint? It depends upon who you ask. If the “derivative” is to exist then the standard “first principles” definition must be modified to be a one-sided limit. So, for our function f above, we would define

    \[f'(1) = \lim_{h\to0^-}\frac{f(1+h) - f(1)}{h}\,.\]

This is clearly not too difficult to do, and with this definition we find that f'(1) = -1, as implied by the Exam question. (Note that since f naturally extends to the right of =1, the actual limit computation can be circumvented.) However, and this is the fundamental point, not everyone does this.

At the university level it is common, though far from universal, to permit differentiability at the endpoints. (The corresponding definition of continuity on a closed interval is essentially universal, at least after first year.) At the school level, however, the waters are much muddier. The VCE curriculum and the most popular and most respected Methods textbook appear to be completely silent on the issue. (This textbook also totally garbles the related issue of derivatives of piecewise defined (“hybrid”) functions.) We suspect that the vast majority of Methods teachers are similarly silent, and that the minority of teachers who do raise the issue would not in general permit differentiability at an endpoint.

In summary, it is perfectly acceptable to permit derivatives/tangents to graphs at their endpoints, and it is perfectly acceptable to proscribe them. It is also perfectly acceptable, at least at the school level, to avoid the issue entirely, as is done in the VCE curriculum, by most teachers and, in particular, in part (b) of the Exam question above.

What is blatantly unacceptable is for the VCAA examiners to spring a completely gratuitous endpoint derivative on students when the issue has never been raised. And what is pure and unadulterated madness is to spring an endpoint derivative after carefully and explicitly avoiding it on the immediately previous part of the question.

The Victorian Curriculum and Assessment Authority has a long tradition of scoring own goals. The question above, however, is spectacular. Here, the VCAA is like a goalkeeper grasping the ball firmly in both hands, taking careful aim, and flinging the ball into his own net.