The VCAA Draft and its Third Rail

We’ve looked a little more closely at VCAA’s Draft for the new mathematics VCE subjects. Yes, the time for feedback has ended, unless it hasn’t: the MAV are offering a Zoom session TODAY (Thursday 25/3) for members. God knows how or why. But in any case, it’ll be a while before VCAA cements the thing in place: plenty of time to ignore everyone’s suggestions.

The following are our thoughts on the Draft and Overview. It will be brief and disorganised, since there is no point in doing more; as we wrote, the content doesn’t matter as much as the fact that, whatever content, VCAA will undoubtedly screw it up. Still, there are some clear and depressing points to be made. We haven’t paid much specific attention to what is new nonsense, and what is the same old nonsense; nonsense is nonsense.

GENERAL POINTS ON THE DRAFT

• The draft looks like a primary school book report. Someone at VCAA really should learn .
• “Computational Thinking” is meaningless buzzery, and will be endemic, insidious and idiotic. It will poison everything. Every step of Methods and Specialist is subject to the scrutiny of Outcome 3:

“On completion of this unit the student should be able to apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in practical situations requiring problem-solving, modelling or investigative techniques or approaches.”

“Statistical functionalities of technology”. And, there’s way more:

“key elements of algorithm design: sequencing, decision-making, repetition, and representation including the use of pseudocode.”

“use computational thinking, algorithms, models and simulations to solve problems related to a given context”

“the role of developing algorithms and expressing these through pseudocode to help determine and understand mathematical ideas and results”

“the purpose and effect of sequencing, decision-making and repetition statements on relevant functionalities of technology, and their role in the design of algorithms and simulations”

“design and implement simulations and algorithms using appropriate functionalities of technology”

This will all be the same aimless, pseudo-exploratory, CAS-drenched garbage that currently screws VCE, but much, much worse. Anybody who signs off on this idiocy should hang their head in shame.

• CAS shit will now be worse than ever.
• There should be no CAS exam, at all.
• There should be no bound notes permitted in any exam.
• Don’t write “technology”. It is pompous and meaningless. If you mean “CAS” then write “CAS”.
• SACs have always been shit and will always be shit. The increased weight on them is insane.
• The statistics is the same pointless bullshit it always was.
• The presence of “proof” as a topic in Specialist highlights the anti-mathematical insanity of VCAA and ACARA curricula: proof has zero existence elsewhere. Much of what appears in the proof topic could naturally and engagingly and productively be taught at much lower levels. But of course, that would get in the way of VCAA’s constructivist fantasy, now with New and Improved Computational Thinking.

MATHEMATICAL METHODS

• Not including integration by substitution is still and will always be the most stupid aspect of Methods.
• Dilations must be understood expressed as both “parallel to an axis” and “from an axis”? But not in terms of the direction the damn points are moving? Cute.
• The definition of independent events is wrong.
• The demand that, for the composition , the range of must be a subset of the domain of is as pedantic and as pointless as ever.
• “literal equations” is the kind of blather that only a maths ed clown could think has value.
• The derivative of the inverse is still not in the syllabus, and everyone will still cheat and use it anyway.
• “trapezium rule” is gauche but, more importantly, what is the purpose of teaching such integral approximation here? Yes, one can imagine a reasonable purpose, but we’ll lay odds there is no such purpose here.

SPECIALIST MATHEMATICS

• The killing of mechanics is a crime.
• The inclusion of logic and proof and the discrete topics could be good. But it won’t be. It will be shallow and formulaic and algorithmised, and graded in a painfully pedantic manner. Just imagine, for example, how mathematical induction will be assessed on exams: “Students often wrote instead of . Students should be aware of the proper use of these variables.”
• There is no value here in “proof by contrapositive”, and it is confusing. Proof by contradiction suffices.
• They’re really including integration by parts? Incredible.
• The inclusion of cross products and plane equations makes some sense.

WitCH 58: Differently Abled

Like the previous post, this one comes from Maths Quest Mathematical Methods 11, and is most definitely a WitCH. It can also been seen as a “contrast and compare” with WitCH 15.

Subsection 13.2.5, below, is on “differentiability”. The earlier part of chapter 13 gives a potted, and not error-free, introduction to limits and continuity, and Chapter 12 covers the “first principles” (limit) computation of polynomial derivatives. We’ve included the relevant “worked example”, and the relevant exercises and answers.

WitCH 57: Tunnel Vision

The following is just a dumb exercise, and so is probably more of a PoSWW. It seems so lemmingly stupid, however, that it comes around full cycle to be a WitCH. It is an exercise from Maths Quest Mathematical Methods 11. The exercise appears in a pre-calculus, CAS-permitted chapter, Cubic Polynomials. The suggested answers are (a) , and (b) 81/32 km.

VCAA’s Draft Feedback Due TODAY

We’ve been remiss in not writing further on VCAA’s draft for the new mathematics VCE subjects. It’s just, for reasons we’ll explain briefly here and flesh out elsewhere, we’ve struggled to face up to this new nonsense.

But, feedback is due TODAY (midnight? – see links below), and we really oughta say something. So, here are our brief thoughts and then, after that, why we believe none of it really matters:

• “Computational thinking and algorithms” is pure snake oil.  Inevitably, it will be nothing but wafer-thin twaddle for the training of data monkeys.
• The increased weight on these meaningless, revolting SACs is insidious.
• If we read it correctly, more weight will be placed on the non-CAS Methods/Specialist exams; it is not remotely close to enough, but it is good.
• Statistics was and is and will always be an insane topic to emphasise in school.
• Foundation Mathematics: Who Cares?
• General Mathematics: Who Cares?
• Mathematical Methods: same old swill.
• The deletion of mechanics from Specialist Mathematics is criminal, but the topic had already been so bled to meaningless that it hardly matters.
• In principle, the inclusion in SM of “logic” and “proof and number” and “combinatorics” is a good thing. We’ll see.
• Similarly, in principle the making of SM12 presumed knowledge for SM34 is good; in practice, it is almost certainly bad. Currently, a good teacher at a good school will take the freedom in SM12 to go to town, to show their students some genuine mathematics and real mathematical thought. In the future, that will be close to impossible, and SM12 will likely become as predictable and as dull as MM12 (and MM34 and SM34).

And now, why doesn’t any of it matter? Because, fundamentally it doesn’t matter what you teach, it matters how you teach. What matters is the manner in which you approach your subject and your students, and none of that will change in other than a microscopic manner. Nothing in VCAA has changed, nothing in the general culture of Victorian education had changed. So, why the Hell would twiddling a few dials on utterly insane subjects assessed in an utterly insane manner make any meaningful difference?

Everything VCAA touches, they will turn to shit. That will continue to be true until there is a fundamental cultural shift, in VCAA and generally.

I hate this place.

• The current (pre-COVID) study design (pdf) is here.
• The draft for the new study design (word) is here.
• The key changes overview (work) is here.
• The link for feedback (until March 9, 2021) is here.

WitCH 52: Lines of Attack

Yes, we have tons of overdue homework for this blog, and we will start hacking into it. Really. But we’ll also try to keep the new posts ticking along.

The following, long WitCH comes from the Cambridge text Mathematical Methods 3 & 4 (including an exercise solution from the online version of the text).

UPDATE (07/02/21)

Commenter John Friend has noted a related question from the 2011 Methods Exam 1. We’ve added that question below, along with the discussion from the assessment report.

MitPY 11: Asymptotes and Wolfram Alpha

This MitPY comes from frequent commenter, John Friend:

Dear colleagues,

I figured this was as good place as any to ask for help. I’m writing a small test on rational functions. One of my questions asks students to consider the function where and to find the values of for which the function intersects its oblique asymptote.

The oblique asymptote is so they must first solve

… (1)

for . The solution is and there are no restrictions along the way to getting this solution that I can see. So obviously .

It can also be seen that if then equation (1) becomes which has no solution. So obviously .

When I solve equation (1) using Wolfram Alpha the result is also . But here’s where I’m puzzled:

Wolfram Alpha gives the obvious restriction but also the restriction .

emerges naturally (and uniquely) from this second restriction and I really like that this happens as a natural part of the solution process. BUT ….

I cannot see where this second restriction comes from in the process of solving equation (1)! Can anyone see what I cannot?

Thanks.

WitCH 51: Logging the Possibilities

The question below is from the second 2020 Methods exam (not online), discussed here. You may wish to brush up on your modal logic before attempting the question.

MitPY 10: Square Roots

This MitPY comes from a student, Jay:

I have a question relating to polynomial equations. For context I have just finished Y11 during which I completed Further 3&4, Methods 1&2 and Specialist 1&2.

This year during my maths methods class we covered the square root graph, however I was confused as to why it only showed the positive solutions. When I asked about it I was told it was because the radical symbol meant only the positive solution.

However since then I have learnt that the graph of also only shows the positive solution of the square root, while shows both. I am quite confused by why they aren’t the same. The only reason that I could think of is that it would mean would be the same as , and while the points (-2,-4) and (2,-4) fit the latter they clearly don’t fit former.

Could you please explain why these aren’t the same?

Secret Methods Business: Exam 2 Discussion

This is our post for teachers and students to discuss Methods Exam 2 (not online). There are also posts for Methods Exam 1, Specialist Exam 1 and Specialist Exam 2.

UPDATE (09/09/21) The examination report is here (a Word document, because VCAA is stupid). Corresponding updates, including the noting of blatant errors (MCQ20, and see here, and Q5), are included with the associated question, in green.

UPDATE (31/12/20) The exam is now online.

UPDATE (21/11/20) A link to a parent complaining about the Methods Exam 2 on 774 is here.

UPDATE (24/11/20 – Corrected) A link to VCAA apparently pleading guilty to a CAS screw-up (from 2010) is here. (Sorry, my goof to not check the link, and thanks to Worm and John Friend.)

UPDATE (05/12/2020)

We’ve now gone through the multiple choice component of the exam, and we’ve read the comments below. In general the questions seemed pretty standard and ok, with way too much CAS and other predictable irritants. A few questions were a bit weird, generally to good effect, although one struck us as off-the-planet weird.

Here are our question-by-question thoughts:

MCQ1. A trivial composition of functions question.

MCQ2. A simple remainder theorem question.

(09/09/21) 56% got this correct, which ain’t great.

MCQ3. A simple antidifferentiation question, although the 2x under the root sign will probably trick more than a few students.

(09/09/21) 86%, so students (or, more accurately, their CASes) weren’t tricked.

MCQ4. A routine trig question made ridiculous in the standard manner. Why the hell write the solutions to other than in the form ?

MCQ5. A trivial asymptotes question.

MCQ6. A standard and easy graph of the derivative question.

MCQ7. A nice chain rule question. It’s easy, but we’re guessing plenty of students will screw it up.

MCQ8. A routine and routinely depressing binomial CAS question.

(09/09/21) 50%, for God knows what reason.

MCQ9. A routine transformation of an integral question. Pretty easy with John Friend’s gaming of the question, or anyway, but these questions seem to cause problems.

(09/09/21) 35%, for Christ’s sake. The examination report contains pointless gymnastics, and presumably students tried this and fell off the beam. (The function is squished by a factor of 2, so the area is halved. Done.)

MCQ10. An unusual but OK logarithms question. It’s easy, but the non-standardness will probably confuse a number of students.

MCQ11. A standard Z distribution question.

MCQ12. A pretty easy but nice trigonometry and clock hands question.

MCQ13. The mandatory idiotic matrix transformation question, made especially idiotic by the eccentric form of the answers.

(09/09/21) 25%, obviously a direct consequence of VCAA’s idiotically cute form of answer.

MCQ14. Another standard Z distribution question: do we really need two of these? This one has a strangely large number of decimal places in the answers, the last of which appears to be incorrect.

“Do not round too early …”

It seems their idea is actually “do not round at all”.

MCQ15. A nice average value of a function question. It can be done very quickly by first raising and then lowering the function by units.

(09/09/21) 36%, presumably because everyone attempted the thoughtless and painfully slow approach indicated in the examination report.

MCQ16. A routine max-min question, which would be nice in a CAS-free world.

(09/09/21) 53%, for God knows what reason. The question is simple to do in one’s head.

MCQ17. A really weird max-min question. The problem is to find the maximum vertical intercept of a tangent to . It is trivial if one uses the convexity, but that is far from trivial to think of. Presumably some Stupid CAS Trick will also work.

(09/09/21) 42%, which is not surprising. The examination report gives absolutely no clue why the maximising tangent should be at x = 0.

MCQ18. A somewhat tangly range of a function question. A reasonable question, and not hard if you’re guided by a graph, but we suspect students won’t do the question that well.

MCQ19. A peculiar and not very good “probability function” question. In principle the question is trivial, but it’s made difficult by the weirdness, which outweighs the minor point of the question.

(09/09/21) 15%, which is worse than throwing darts. Although the darts would be better saved to throw at the writers of this stupid question.

MCQ20. All we can think is the writers dropped some acid. See here.

(09/09/21) 18%. Now, what did we do with those darts? As follows from the discussion here, the suggested solution in the examination report is fundamentally invalid. One simply cannot conclude that a = 2π in the manner indicated, which means that the question is at minimum a nightmare, and is best described as wrong. Either the report writers do not know what they are writing about, or they are consciously lying to avoid admitting the question is screwed. As to which of the two it is, we dunno. Maybe throw a dart.

UPDATE (06/12/2020)

And, we’re finally done, thank God. We’ve gone through Section B of the exam and read the comments below, and we’re ready to add our thoughts.

This update will be pretty brief. Section B of Methods Exam 2 is typically the Elephant Man of VCE mathematics, and this year is no exception. The questions are long and painful and aimless and ridiculous and CAS-drenched, just as they always are. There’s not much point in saying anything but “No”.

Here are our question-by-question thoughts:

Q1. What could be a nice question about the region trapped between two functions becomes pointless CAS shit. Finding “the minimum value of the graph of ” is pretty weird wording. The sequence of transformations asked for in (d) is not unique, which is OK, as long as the graders recognise this. (Textbooks seem to typically get this wrong.)

Q2. Yet another fucking trig-shaped river. The subscripts are unnecessary and irritating.

Q3. Ridiculous modelling of delivery companies, with clumsy wording throughout. Jesus, at least give the companies names, so we don’t have to read “rival transport company” ten times. And, yet again with the independence:

“Assume that whether each delivery is on time or earlier is
independent of other deliveries.”

Q4. Aimless trapping of area between a function and line segments.

Q5. The most (only) interesting question, concerning tangents of , but massively glitchy and poorly worded, and it’s still CAS shit. The use of subscripts is needless and irritating. More Fantasyland computation, calculating in part (a), and then considering the existence of in part (b). According to the commenters, part (d)(ii) screws up on a Casio. Part (e) could win the Bulwer-Lytton contest:

“Find the values of for which the graphs of and ,
where exists, are parallel and where

We have no clue what was intended for part (g), a 1-marker asking students to “find” which values of result in having a tangent at some with -intercept at . We can’t even see the Magritte for this one; is it just intended for students to guess? Part (h) is a needless transformation question, needlessly in matrix form, which is really the perfect way to end.

(09/09/21) This could be a horror movie: Revenge of the 1-Pointers.

Part (c) was clearly intended to be an easy 1-pointer: just note that a (tangent) line without an x-intercept is horizontal. But, somehow 77% of students stuffed it up. So, how? The examination report sermonises, thusly:

“The concept of the ‘nature of a tangent line’ was not obvious for many students.”

This suggests that the report writers don’t understand the concept of a concept. It also indicates that the report writers didn’t read their own exam. Q5(c) reads as follows:

“State the nature of the graph of ga when b does not exist. [emphasis added]”

Three sub-questions ago, ga is defined to the be the tangent to a function, and b is defined to be the x-intercept of this tangent (even though it may not exist). So, what was obviously not obvious to the students was the meaning of a vaguely worded question framed in poor notation. God, these people are dumb. And sanctimonious. And dumb.

Part (g), another 1-pointer, is concerned with the function p(x) = x3 + wx. The question asks, badly, for which values of w is there a positive t such that the tangent to p at x = t will have x-intercept at -t. 3% of students were smart enough (or lucky enough) to get the right answer, and 97% of students were very smart enough to go “Fuck this for a joke”, and skip it.

The solution in the examination report is lazily incomprehensible, but the idea was to just do the work: equate p'(t) with the rise/run slope of the tangent and see for which w there is a positive t that solves the equation. It turns out that the equation simplifies to w + 5t2 = 0, and so as long as w < 0 there will be a solution. It also turns out that specifying t positive is entirely irrelevant. Which is what they do.

It is worth noting the question can also be nutted out qualitatively, if one knows what graphs like y = x3 + x and y = x3 – x look like. If w ≥ 0 then it it easy see the tangent to p(x) = x3 + wx will always hit the x-axis before crossing the y-axis, so no chance of giving a solution for the exam question. If w < 0 then look at tangents at points t between the two turning points. At a turning point the slope of the tangent is horizontal. Then as the slope goes negative the x-intercept of the tangent comes in from ∞ (or -∞), until eventually the x-intercept is 0. Somewhere along the way, there has to be a t that gives a solution for the exam question.

Part (h), another 1-pointer, is the final question on the exam, and is a stuff up. This time, 98% of students were very smart enough to skip the damn thing. As well, as mystery student PURJ pointed out to us, the examination report is also stuffed. The question again concerns the function p. It had been noted earlier that the tangents to p at t and -t are always parallel. Then Part (h) asks, in idiotic matrix notation, how p can not be translated or dilated to so that the new function still has this parallel tangent property. Yep, the question is framed in a negative manner, and the examiners whine about it:

“The key word in this part is restrictions.”

Nope. The key word in this part is “idiotic”. (And, “ungrammatical”.) In any case, the examination report indicates that the key to solving the question is that the transformed function must still be odd. This, as PURJ pointed out, is wrong. A vertical translation is just fine but the resulting function will not be odd. The correct characterisation is that the derivative of the transformed function must be even. In sum, this means one can transform as one wishes, except for horizontal translations. Which equates to the report’s matrix answer, without any noting of the “odd”, but not odd, contradiction.