See the Evil Mathologer and the Evil Marty, December 3

On Tuesday December 3, the Australian Mathematics Society will hold a free education afternoon at Monash University, Clayton, as part of their annual conference. The talk details are below, and full details are here (and the lecture theatre details are below). You aren’t required to register, but you can do so here (and it is appreciated if you do).

UPDATE The talks will take place in Lecture theatre G81 of the Learning & Teaching Building (the bus stop side of Clayton campus). There’s a map of Clayton campus here.

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1:30 Joanna Sikora: Advancing Women in Australian Mathematics: context, challenges and achievements

This talk reviews recent research undertaken by social scientists on women in mathematics. First, adopting a life-course perspective it summarises findings on the persisting gap in vocational interest in mathematics among adolescent boys and girls, including its potential to widen over time. Systematic differences between boys and girls in the choice of basic and advanced mathematics for ATAR (Australian Tertiary Admissions Rank) are discussed. Next, the consequences of these choices for tertiary education specialisations and availability of suitably qualified male and female graduates are considered.

Following this introduction, the talk summarizes research on underrepresentation of women in mathematics departments in Australia and across the world. The focus is on structural and institutional process which, over the course of individual careers, can amount to significant disadvantage even in the absence of overt discrimination. Topics discussed include cultural stereotypes that link perceptions of brilliance and academic talent with masculinity, gender differences in professional capital, i.e. peer esteem, accorded to male and female mathematicians, the gender gap in rates of publications and impact, documented bias in student evaluations and factors that enable success in establishing international collaborations. The talk concludes by summarizing the literature on practical steps that we can take to improve gender equity.

2:20 Julia Collins and Katherine Seaton: Knitting and Folding Mathematics

Mathematical thinking is not confined to mathematicians, but one place you may not expect to find it is in the world of crafts. Even the most maths-anxious knitters will display an astonishing familiarity with concepts from geometry, topology, number theory and coding, while modern origami artists are turning to mathematical algorithms to create models previously thought to be unfoldable. This talk will highlight a number of surprising connections between maths and craft, and will be followed by a hands-on session facilitated by Maths Craft Australia where people can create some mathematical craft for themselves. (Knitting/crochet needles and origami paper will be provided, but participants are also encouraged to bring their own! Knitting in the audience is strictly encouraged.)

2:45 Afternoon Tea

3:10 Marty Ross: How I teach, why the Mathologer is evil, and other indiscrete thoughts

In this shamelessly narcissistic talk I will reveal the One True Secret to teaching mathematics. Along the way I will explain why you can and should ignore STEM, calculators, Mathematica, iPads, the evil Mathologer, constructivism, growth mindset, SOLO, Bloom, flipping classrooms, centering children, lesson plans, skeleton notes, professional standards and professional development and many other modern absurdities.

3:35 David Treeby: How to Instil Mathematical Culture in Secondary Education

Over the past few decades, mathematicians have ceded the educational space to two groups: mathematics educators and technology companies. This has had a dire effect on what mathematics is taught and how it is taught. The result is a commodified brand of distorted mathematics. This talk will focus on how some well-resourced schools have resisted these changes, and how broad and equitable change will require the support of working mathematicians and their professional bodies.

4:00 Burkard Polster: Mathologer: explaining tricky maths on YouTube

In this session I’ll talk about my experience running the YouTube channel Mathologer and I’ll give you a sneak peek of the video that I am currently working on.

WitCH 29: Bad Roots

This one is double-barrelled. A strange multiple choice question appeared in the 2019 NHT Mathematical Methods Exam 2 (CAS). We had thought to let it pass, but a similar question appeared in last week’s Methods exam (no link yet, but the Study Design is here). So, here we go. First, the NHT question: The examination report indicates the correct answer, C, and provides a suggested solution:

\Large\color{blue} \boldsymbol{ g(x)=f^{-1}(x)=\frac{x^{\frac15}-b}{a},\ g'(x) = \frac{x^{-\frac45}}{5a},\ g'(1) = \frac1{5a}}

And, here’s last week’s question (with no examination report yet available):

Update (19/06/20)

As commenters have noted, it is very difficult to understand any purpose to these questions. They obviously suggest the inverse function theorem, testing the knowledge of and application of the formula g'(d) = 1/f'(c), where f(c) =d. The trouble is, the inverse function theorem is not part of the curriculum, appearing only implicitly as a dodgy version of the chain rule, and is typically only applied in Leibniz form.

As indicated by the solution in the first examination report, the intent seems to have been for students to have explicitly computed the inverses, although probably with their idiot machines. (The second examination report has now appeared, but is silent on the intended method.) Moreover, as JF noted below, the algebra in the first question makes the IFT approach somewhat fiddly. But, what is the point of pushing a method that is generally cumbersome, and often impossible, to apply?

To add to the nonsense, below is a sample solution for the first question, provided by VCAA to students undertaking the Mathematica version of Methods. So, the VCAA has suggested two approaches, one which is generally ridiculous and another which is outside the curriculum. That makes it all as clear as dumb mud.

WitCH 28: Tone Deaf

We haven’t yet had a chance to go through the 2019 VCE exams, but this question was flagged to me independently by two colleagues: let’s call them Dr. Death and Simon the Likeable. It’s from Mathematical Methods Exam 2 (CAS). (No link yet.)

UPDATE (05/07/20)

Even ignoring the stuff-ups, this question is ugly and pointless; the pseudo-applied framing is ugly and pointless; the CASification is ugly and pointless; the back-to-front integral is ugly and pointless; the matrix equation is ugly and pointless; the transformation is really ugly and really pointless. Part (f) is the pinnacle of ugliness and pointlessness, but the entire question is swill, from beginning to end.

And then there’s Part (e). “This question was not answered well” the examiners solemnly intone. Gee, really? Do you think your question being completely stuffed might have had something to do with it? Do you think maybe having a transformation of x when there’s not an x in sight may have been just a tad confusing? Do you think that the transformation then resulting in a function of t was maybe not the smartest move? Do you think writing an integral backwards was perhaps just a little too cute? Do you think possibly referring to the area of, rather than to the value of, an integral was slightly clunky? And, most importantly, do you think perhaps asking a question for which there is an infinite and impenetrable jungle of answers may have been an exercise in canyon-sized incompetence?

But, sure, those troublesome students didn’t answer your question well.

Part (e) was intended to have students find a transformation of the function f that effectively switches the behaviour on the intervals [0,4] and [4,6] to the intervals [2,6] and [0,2].  Ignoring the fact that the intended question was asked in an absurdly opaque manner, and ignoring the fact that no motivation for the intended question was either provided or is imaginable, the question asked was entirely different, and was ridiculous.

Writing the transformation out,

    \[\boldsymbol{\left\{\aligned &X = ax + c \\ &Y = by + d, \right.\endaligned}\]

we then have

    \[\boldsymbol{\left\{\aligned &x = \frac{X - c}{a} \\ & y = \frac{Y - d}{b}. \right.\endaligned}\]

So, the function y = f(t) y = f(x) can be written

    \[\boldsymbol{\dfrac{Y - d}{b} = f\!\left(\dfrac{X - c}{a}\right).}\]

Solving for Y, that means our transformed function Y = g(X) can be written

    \[\boldsymbol{g(X) = b\, f\!\left(\dfrac{X - c}{a}\right) + d.}\]

Well, this is our function g unless a = 0, in which case g doesn’t exist. Whatever. Back to the swill.

Using the result from Part (d), we have Part (e) asking for a, b, c and d such that

    \[\boldsymbol{\int\limits_2^0 + \int\limits_2^6 \ \left[ b\, f\! \left(\dfrac{X - c}{a}\right) + d\right]  \, {\rm d}X \ = \ \dfrac{15}{\pi}.}\]

What then are the solutions to this equation? The examination report lists a couple of families and then blithely remarks “There are other solutions”. Really? Then why didn’t you list them, you clowns? 

We’ll tell you why. Because the complete solution to this monster is a God Almighty multi-infinite mess. As a starting idea, pick any three of the variables, say a and b and c, to be whatever you want, and then try to adjust the fourth variable, d, to solve the equation. We’ll offer a prize for anyone who can give a complete solution. 

This question is as good an example as there can be of the pointlessness, the ugliness and the monumental klutziness of VCAA’s swamp mathematics.

WitCH 27: Uncomposed

Ah, so much crap …

Tons of nonsense to post on, and the Evil Mathologer is breathing down our neck. We’ll have (at least) three posts on last week’s Mathematical Methods exams. This one is by no means the worst to come, but it fits in with our previous WitCH, so let’s quickly get it going. It is from Exam 1. (No link yet, but the Study Design is here.)

Update (15/06/20)

The examination report (and exam) is out, so it’s time to wade into this swamp. Before doing so, we’ll note the number of students who sank; according to the examination report, the average score on this question was 0.14 + 0.09 + 0.14 ≈ 0.4 marks out of 4. Justified or not, students had absolutely no clue what to do. Now, into the swamp.

The main wrongness is in Part (b), but we’ll begin at the beginning: the very first sentence of Part (a) is a mess. Who on Earth writes

“The function f: R \to R, f(x)  is a polynomial function …”?

It’s like writing

“The Prime Minister Scott Morrison of Australia, Scott Morrison is a crap Prime Minister”.

Yes, you may properly want to emphasise that Scott Morrison is the Prime Minister of Australia, and he is crap, but that’s not the way to do it. This is nitpicking, of course, but there are two reasons to do so. The first reason is there is no reason not to: why forgive the gratuitously muddled wording of the very first sentence of an exam question? From these guys? Forget it. The second reason is that the only possible excuse for this ridiculous wording is to emphasise that the domain of f is all of R, which turns out to be entirely pointless.

Now, to Part (a) proper. This may come as a surprise to the VCAA overlords, but functions do not have “rules”, at least not unique ones.  The functions f(x) = -4x^2\left(x^2 - 1\right) and h(x) = 4x^2-4x^4, for example, are the exact same function. Yes, this is annoying, but we’re sorry, that’s the, um, rule. Again this is nitpicking and, again, we have no sympathy for the overlords. If they insist that a function should be regarded as a suitable set of ordered pairs then they have to live with that choice. Yes, eventually ordered pairs are the precise and useful way to define functions, but in school it’s pretty much just a pedantic pain in the ass.

To be fair, we’re not convinced that the clumsiness in the wording of Part (a) contributed significantly to students doing poorly. That is presumably much more do to with the corruption of students’ arithmetic and algebraic skills, the inevitable consequence of VCAA and ACARA calculatoring the curriculum to death.

On to Part (b), where, having found f(x) = -4x^2\left(x^2 - 1\right) or whatever, we’re told that g is “a function with the same rule as f”. This is ridiculous and meaningless. It is ridiculous because we never did anything with f in the first place, and so it would have been a hell of lot clearer to have simply begun the damn question with g on some unknown domain E. It is meaningless because we cannot determine anything about the domain E from the information provided. The point is, in VCE the composition \log(g(x)) is either defined (if the range g(E) is wholly contained in the positive reals), or it isn’t (otherwise). End of story.  Which means that in VCE the concept of “maximal domain” makes no sense for a composition. Which means Part (b) makes no sense whatsoever. Yes, this is annoying, but we’re sorry, that’s the, um, rule.

Finally, to Part (c). Taking (b) as intended rather than written, Part (c) is ok, just some who-really-cares domain trickery.

In summary, the question is attempting and failing to test little more than a pedantic attention to boring detail, a test that the examiners themselves are demonstrably incapable of passing.

WitCH 26: Imminent Domain

The following WitCH is pretty old, but it came up in a tutorial yesterday, so what the Hell. (It’s also a good warm-up for another WitCH, to appear in the next day or so.) It comes from the 2011 Mathematical Methods Exam 1:

For part (a), the Examination Report indicates that f(g)(x) =([x+2][x+8]), leading to c = 2 and d = 8, or vice versa. The Report indicates that three quarters of students scored 2/2, “However, many [students] did not state a value for c and d”.

For Part (b), the Report indicates that 84% of students scored 0/2. After indicating the intended answer, (-∞,-8) U (-2,∞) (-∞,-8] U [-2,∞) or R\backslash(-8,-2), the Report goes on to comment:

“This question was very poorly done. Common incorrect responses included [-3,3] (the domain of  f(x); x ≥ -2 (as the ‘intersection’ of  x ≥ -8 with x ≥ -2); or x ≥ -8 (as the ‘union’ of x ≥ -8 with x ≥ -2). Those who attempted to use the properties of composite functions tended to get confused. Students needed to look for a domain that would make the square root function work.”

The Report does not indicate how students got “confused”, although the composition of functions is briefly discussed in the Study Design (page 72).