This is a continuation of a previous WitCH. To provide teachers with some guidance on the new VCE Specialist Mathematics curriculum (Word, idiots, and see comments here), VCAA had posted two webinars, on proof by contradiction and proof by induction; we WitCHed it. VCAA has now added four new webinars, which can be viewed here, and with companion documents as indicated below.
Get to work.
Proof by induction – Transcript (Word, idiots) and Slides. (Previous comments and Critique at this WitCH.)
Proof by contradiction – Transcript (Word, idiots) and Slides. (Previous comments and Critique at this WitCH.)
Mathematical Investigations – Transcript (Word, idiots) and Slides. (27/02/23 Critique below.)
Introduction to Pseudocode – Transcript (Word, idiots) and Slides. (27/02/23 Critique below.)
Calculus – Transcript (Word, idiots) and Slides. (14/02/23 Critique below.)
Vectors – Transcript (Word, idiots) and Slides. (15/02/23 Critique below.)
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UPDATE: Calculus (14/02/23)
On we plow.
The calculus webinar is not nearly as bad as the proof webinars, but it is bad; some of the questions are very bad, and there are errors. And again, fundamentally, one must ask: what is the point? What is the purpose of a worse-than-a-million-other-videos “10 minute information bite”, which is actually 24 minutes, but still rushes through a bunch of poorly chosen examples?
The integration by parts section is kind of ok, but is very quick and not particularly clearly laid out. The single example worked through, Q11 of the sample exam 1 questions, requires two integration by parts, which a poor first choice.
The real problems begin with the discussion of surfaces of revolution. They begin with a “curve” y = f(x) rotated about the x-axis, and give the standard formula for the surface area, exactly as it appears on the exam formula sheet. Except the formula assumes that y ≥ 0. Now that wouldn’t be a problem, and we could accept the nonnegativity as understood, if it weren’t for the fact that VCAA examiners love to be nitpicking assholes, willing to dock students for the smallest, most meaningless infringement.
It immediately gets way worse. The first example discussed involves rotating such a “curve”, 
,
to form a solid of revolution.
The question is then,
Find the total surface area of the closed volume of this solid [emphasis in the original]
This is so dumb, and so nasty. And so dumb.
To begin, they can emphasise “closed” all they want, but “closed volume of this solid” doesn’t mean anything. Try “region”, perhaps? But OK, we know what they were trying to say: they want to include the area of the end-circles as well. Fine. Except it’s not close to fine. And it’s not correct.
First of all, why clutter a first surface area question by worrying about other bits? Who does that? Well, they do that, because they’re preparing kids for the nasty exams. The question is adapted from Q11 of the sample exam 1 questions. But there is a notable difference in the wording:
Find the surface area of this solid of revolution.
Well, at least the wording isn’t wrong. But the question is pointlessly sneaky, made sneakier by the decision to exclude the word “total”. Exactly so that the examiners can be nitpicking assholes. But, anyway, the question is stuffed.
Remember how the question began? They rotated a “curve” to form a solid of revolution. Except, rotating a 1-D curve does not create a 3-D solid: it only creates a 2-D surface. Which means there is no region at all to have end-circles to be included. If you want a 3-D region, then rotate the 2-D area under the curve. Nitwits.
After this wrong, nasty question, the webinar continues with a very bad, and arguably wrong, question. This second question, which is Q7 of the sample exam 1 questions, concerns the curve ![boldsymbol{y = sqrt[3]{x}}](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-0d444d3176e143c1be30262878310bec_l3.svg "Rendered by QuickLaTeX.com")
rotated around the y-axis, again falsely suggested to have given a “solid of revolution”. The question then asks,
Find the surface area of the part of this solid of revolution where x ∈ [0, 8].
Does this surface area include or not the two end-circles? Are you sure? What does “the part of this solid of revolution” mean? What would the surface area of that “part” be? Have fun in November, kiddies.
It turns out, according to the webinar, that they didn’t want the two end-circles, just the surface of revolution. So why the hell talk about a solid at all? Why not simply stick to rotating a curve to give a surface, which is what has actually occurred? God works in wondrous ways, and VCAA works in wondrously idiotic ways.
After this, it’s on to rotating parametrised curves, going through Q9 of the sample exam 1 questions. This material is better, but/and a couple points should be noted. First, the surface area formula includes absolute values, although the exam formula sheet does not; God knows why. Secondly, the problem worked through only refers to “the surface area of revolution”, with no reference to any “solid”, and the same is true of the similar Q8 of the sample exam 1 questions; God knows why not.
It smells as if someone at VCAA imagines there is a difference between the function and parametric settings, the first creating a solid and the second not; which is false. Rotating a 1-D curve sweeps out a 2-D surface, however the curve is defined.
Finally, the webinar works through a logistic equation question, Q10 of the sample exam 1 questions. The question is badly wrong: (b) is plain wrong, and the webinar indicates that (c) reads differently than intended. See this WitCH. Specifically, (c) asks for a “differential equation” to be solved, but what is intended is that an initial value problem be solved; these are not the same thing. Note also on (b), the webinar “solution” nowhere uses the initial population. But if the initial population is too high then the population will decay to the limiting value. It follows that the webinar “solution” is wrong, even on its own terms.
The webinar presentation of the logistic equation is also poor, which is minor compared to the problems indicated above, but is annoying. The lack of any graphs to illustrate the behaviour is silly, and may have helped lead VCAA into error. It is also gauche to solve of the DE by first flipping to give an equation for dt/dP. It is not wrong, but it is gauche.
That’s it. It is bad. Not as bad as the appalling proof webinars, but it is bad.
UPDATE: Vectors (15/02/23)
Once again, we have to ask what the point is to a “ten minute information bite”, which is 20 minutes, but still rushes way too fast, and with way too little care, through a bunch of new material? The main problem with the vectors webinar is that there is no purpose to it, but there are specific issues as well. So, once again, here we go.
The webinar begins with a discussion of the vector product,
also known as vector cross product
Well, no. It’s typically know as the vector product or the cross product. But no biggie. The biggie is that the fundamental properties of the vector product are stated without a single mention that any of these properties might be proved, much less how one might prove them; the right hand rule also receives not a single mention.
There is then an extraordinarily confused section on parallel, orthogonal and independent vectors:
“Facts”.
This slide implies that the zero vector is permitted to be “parallel”, but suggests the zero vector cannot be “orthogonal”. (See posts on this issue, here and here and here.) But then there is the voiceover:
When vectors A cross B equal null vector implies parallelity is similar to vectors A dot B equal zero implies orthogonality. Note that parallel vectors could be the two vectors either have the same direction or have the exact opposite direction from each other, that is they are not linearly independent or if either one has a zero length.
We have tried, hard, to figure out what this is supposed to mean. We have failed.
It is then onto planes, with the fundamental equation introduced in the form 
. No clue is given to the origin of this equation. (UPDATE 16/02/23: We should have also noted that the webinar refers to this equation as “the vector equation of a plane”, which it kind of is, but which directly contradicts the formula sheet using the exact same phrase to refer to a parametrised equation for a plane.)
Having then gone on to give the standard Cartesian form, the webinar goes through the first example. Which is wrong. The question is a garbled version of Q12 of the sample exam 1 questions (or, more likely, Q12 is a corrected version the webinar question). The question asks for “the” cartesian equation of a plane perpendicular to two given vectors. Clearly what was intended was the two vectors be perpendicular to the normal of the plane. So, just a clunk, but a clunk that should not appear in such a webinar. The definite article is also wrong, since a cartesian equation for a plane can always be scaled, but this is a minor irritation in comparison to the webinar’s other flaws.
It is then on to lines, and the angles between planes and lines. The vector equation of a line is introduced without a hint of its origin, and with the no hint of the connection between the line and the parametrisation of a bug travelling along the line. The subsequent three examples, Q14 and Q15 of the sample exam 1 questions (and Q16 is also mentioned), are fine. There are some clumsy “given by”s, and some improperly definite articles, which we’ll address when we get around to hammering the sample questions (as an update, on this post).
Overall, the vector webinar is by far the most coherent and useful of the four content webinars. There is clunkiness, and the introduction of new concepts and properties as Commandments From God is shameful, but there is nothing nearly so painful or so wrong as exists in the other webinars.
UPDATE: Mathematical Investigations (27/02/23)
We learned three things:
(1) The “Mathematical Investigations” are going to be yet another mathematically trivial horror show;
(2) VCAA has mastered the art of the content-free webinar;
(3) VCAA is in desperate need of an audiovisual expert. Or a novice. Or a kid. Get a 10 year old kid. They would do a better job with the sound.
UPDATE: Pseudocode (27/02/23)
We really appreciated the minute of silence at the beginning of the webinar. It gave the listener a fair chance of changing their mind before wasting seven minutes of their life. Other than that, all we can think of, and all we can be bothered with, is to point out this (for slide 6), and this (for slide 8, and generally).
Introduction to pseudocode:
This whole section gives an overbearing emphasis on the pseudocode and the importance of reserved words, which is bizarre. What really confuses me is that nothing presented in these subjects suffices any introduction to algorithms, it’s just a few uses of say `for` loops, or something similar. Kind of insulting to the beautiful field of algorithms and data structures where you’re solving problems with nontrivial techniques, which belongs more in algorithmics (well, in theory).
> Reimann sums
Misspelt.
Calculus:
> Provided the integral of v du/dx exists
Um… Am I being unreasonable here, but is there a better way to have a discussion about elementary antiderivatives? The “integral does not exist” is what I’d say for an integral that diverges, but I’m not sure what the presenter suggests by that.
That is indeed a weird moment of pseudo-rigour. There’s worse.
Franz, is this from the video or the slides? Not defending the graphic, but bare slides can make builds look worse than they are.
I watched the video of integration by parts. What could a VCE teacher or student learn from this video that could not be learned, just as easily, from a text book?
Probably nothing Terry – but the reason a teacher or student would watch these is to try to get a sense of what VCAA wants to see written as working in an exam.
Which makes any errors people find quite significant!
There’s (at least) a minor error and a huge error. Plus plenty else to consider.
I’ve spent many hours “considering” these “resources”. Spotting errors is important, of course, but trying to decipher what VCAA wants from students is my main game and damn it is tough!
Indeed. There is a very bad error, but I think there are more important reasons to criticise these videos.
OK – I’ll start with a minor gripe from the Vectors video.
The presentation keeps talking about THE normal to the plane. There is no one normal to a plane, there are infinitely many vectors normal to a plane. We only need one of them, but this doesn’t mean the normal is unique. Nor does it need to be.
You are correct, and you are warm.
This is what I imagine a typical conversation to be at VCAA:
Person A: [Government/higher up/media/private school] says we need to teach kids [X]!
Person B: [X] is too difficult to teach properly so let’s teach them [Y := oversimplification of X|analogy for X|application of X] instead.
Person A: Good idea! But it’s [too informal|too difficult] to write questions about that examiners can mark in a few minutes!
Person B: Let’s just add a bunch of expectations for answers to [Y] that seem like they’d make sense. We can then make a rubric based on that, which examiners can use!
Person A: Great! Do you know any experts in the field of [X]?
Person B: [I|Person C] completed a subject in [tangential field to X] during undergrad! [I|Person C] should be perfect for the job!
I don’t see how the introduction to pseudocode they gave could exist without a conversation similar to this.
Very funny (in the black humour sense), and I know nothing to contradict it.
VCAA is either (a) unable to see that they are consistently employing people not up to the job, or (b) they see it but are unwilling to do anything about it, or (c) they see it and they want to do something about it but, for whatever crazy bureaucratic reason, they are incapable of doing something about. I can think of no fourth option.
If I was a gambling man, I would bet 2 units on option A and 1 unit on option C.
Option B just seems unlikely.
Gripe #2: quote from the Vectors transcript: “The fact that the cross product is really the determinant of a three by three matrix, …”
I would argue that the matrix determinant is a convenient method for finding the cross-product but it is not at all the definition.
No mention of the Right-Hand-Rule. Interesting.
It’s ok to think of the determinant as the definition of the cross product. Little else in that part of the presentation is ok. Including no mention of RHR.
Just a note to (non)commenters. The purpose of WitCHes is for you guys to do the work, which you usually do much of, and well. But, for whatever reason, and I’m guessing it’s going nuts with the new school year, there’s been little contributed to this WitCH. So, and given how topical this WitCH is, I’ll do it myself.
In the next day or so, I’ll update the WitCH. If you have questions or comments, now is the time to ask/make them.
(I haven’t put much thought into this, so please forgive…) I felt that the narrator in the Vectors video was a bit confused about the concept of linear dependence of a set of vectors.
I was always of the belief that any set of vectors that contained a zero vector was automatically a linearly dependent set since the zero vector is parallel to all other vectors by convention.
I’m not sure why (or if) this has to be the case, except for internal consistency, but it did seem as though the narrator and/or creator of the slides did not think it was an important enough concept to be clear about.
Most textbooks I have consulted say the same thing, and I do have memories of VCAA examiners reports confirming this but haven’t checked them all as yet (often it comes up as a MC Question, so it can be difficult to know exactly what they think.)
Thanks, RF. Yes, that part is a mess. It’s difficult to even decipher what was intended. Part of my homework for updating the WitCH …
So… my main criticism of these videos is that, if a student were to watch them, they would likely emerge more confused about important concepts than if they had read a (quality) textbook on the matter.
These videos are presented as though they are exemplars (authorised by VCAA nonetheless!) which is so many levels of wrong.
Are they for students or for teachers? I had assumed they were for teachers, and they’re bad enough for that purpose. For students, …
A friendly correspondent has just pointed out to me that VCAA has (finally) posted sample exams for Methods and Specialist. I haven’t yet had a chance to look at them.
From the SM Paper 2, just a query… the paper says “miner” bird. I always thought it was “mynah” or “myna” bird.
It’s VCE. Maybe they mean a canary in a coal mine?
Seeing Q2a on the sample Specialist Exam 1 leaves me with very little hope that the examiners know what they’re doing.
And, again, uh-oh. Thanks, Michael, and everyone. Just trying to get a difficult blog post out of my hair, and then I’ll look at this stuff.
You are all working way too hard. 2(a) is utterly, insanely, meaningless, and obviously so. Nothing further need be, or really can be, said. I wish I hadn’t looked …
That is quite incredible, and without official worked solutions I really have no idea what would be expected working to reject n0=1 (if we infer minimum n0). Full working for Q2b first?
Hi Tungsten – the inequality is not true for n=2,3,4. So, if their intent is to use induction to prove the inequality for all n >= n0, that n0 must be at least 5. But the wording is odd…
I agree, but in order to accept n0=5 then we need to prove for all n>=5 (Q2b) surely? So for 2a if testing say n=2 is good to reject n0=1 then how do we justify not rejecting n0=5 is my point.
I looked at both sample examinations for MM, briefly. It seems to me that both examinations assess students’ knowledge about the same material.
For example, compare Q7 on MM Exam 1 (Sample) and Q4 on MM Exam 2 (Sample). In terms of probability, these questions appear to be essentially assessing students’ knowledge of the same point.
Do we need two examinations in mathematics subjects?
As long as there is CAS, yes. If there is no CAS, then no.
I’ve updated my VCAA SD post with a bunch of links to new VCAA materials for the new SD.
If I can summarise the HITs discussion, it would seem that HITs is a cut above standard maths ed: standard maths ed is either trivial or false; HITs is trivial and false.
Is it supposed to be asking for the cartesian equation of the plane PARALLEL to a and b in Example 1 for the vectors? They say perpendicular, but n is orthogonal to a and b, and is also perpendicular to the plane…? I’m new to this content myself so I don’t know for sure but that’s a pretty bad mistake to include. Mostly though my least favourite thing in the vectors one is the complete lack of derivation for any of the formulas (how hard would it have been to write that n dot a = n dot r because r-a is along the plane hence n dot (r-a) = 0? for example.)
Thanks, ans. Bulls-eye and bulls-eye and bulls-eye. For clarity, I regarded the error you spotted as a “minor” error: it’s probably a (bad) mis-wording rather than a mathematical error. There is a very large error than no one has yet commented on (although John Friend noted it to me off-blog).
hm okay… is it to do with the angles between vectors and planes? they’re always talking about finding ‘the’ angle, although there’d be an acute and an obtuse; and then if the dot product between the direction vector and the normal vector you read from the cartesian equation is negative, you’d end up with a negative angle after using their 90 – theta rule… I’ll keep looking though
Not that video. Although there are other errors, or at least incomprehensibles, in the vectors video.
I don’t think this is The Issue either, but in looking at the surface area part (also new to me), it uses the absolute value of g(t) in the formula (which makes sense; VCAA go absolutely berserk if you give an area as negative) but it’s not included in the non-parametric forms meaning you’d get negative values for something like ‘find the surface area of -sqrt(1-x^2) rotated around the x axis for -1 <= x <= 1, and just plain wrong values if the function crossed the x-axis at some point.
Their convention that 'surface area' does not mean 'total surface area' seems a bit awful. Do universities use this as well?
Update: they’re really just going to use ‘maximum’ to describe the supremum of the logistic graph all year aren’t they? I suppose it’ll never matter, since they’ll never divorce the logistic function from some kind of application. Then if anyone points out that for a specific question (eg. with the bacteria) they’re referring to a supremum and not a maximum, they’ll say it is equal to 48000 bacteria because given the context you should round up. How do they miss the point of mathematics so badly…
And that would be the major error. (But is “supremum” correct either?)
aps, any time you want to take over the blog, I’m happy to hand you the keys.
thank you 🙂
I was starting to think that the major error was the entire Mathematical Investigations video…
Is supremum correct? You would know much, much better than me, and I’m invested now. Is this issue that a supremum is specific to a set and as such can’t be a property of a function?
No, I’m fine the with the supremum of a function. The question is, does the function in question have a supremum? (It definitely doesn’t have a maximum, so VCAA screwed up. That’s decided.)
‘The supremum is the least upper bound of a set S, defined as a quantity M such that no member of the set exceeds M, but if epsilon is any positive quantity, however small, there is a member that exceeds M-epsilon’ -from Wolfram Mathworld. Wouldn’t that mean that since lim_{t\to\infty} P = 48000 and P<48000 for all t in R, that 48000 is the supremum?
Ah, damn. You’re correct. I hadn’t notice the initial condition. My point was to be that if the population begins too large, then it will decay towards the limiting value.
So, we’re just stuck (on that slide) with the error you noted: supremum ≠ maximum.
I forgot to reply to your comment about the absolute value. You are absolutely correct, and VCAA needs to clarify this.
I am fine with not having absolute values in any of the formulas, with the understanding that you chuck in an absolute value or a negative when needed. So, the formula sheet and the non-parametric presentation in the slides is fine by me. But, chucking in the absolute values in the parametric part of the slides then suggests that this might be some nonsense about which VCAA will obsess. You can’t have it both ways.
The entire thing is a complete mess. VCAA is incompetent.
Thanks. Yeah, I can imagine ‘write the integral that gives the surface area…’ questions with parametric equations where students are penalised for not including the absolute value (if it’s not needed, they’ll penalise them anyway, unless the student somehow proves its not needed, which no student would ever do, because it would be a waste of time). And the inconsistency is definitely a mess.
At least one thing is consistent: the sample Specialist exam 1 also refers to the ‘maximum’ of a logistic graph. However, this is probably because they just copied all the webinar questions to make the samples.
I assumed the reverse, that the webinar copied from the exam. But it doesn’t matter: wrong is wrong. And there is more bad on the exam.
Oh good point, I hadn’t even thought of that. But, are we stuck on the erroneous slide? The next one has Michael MacNeill’s contact details… any idea how that would go down?
Yes, I have a pretty good idea how that would go down …
What about a question form, eg. ‘I intend to tutor Specialist Maths, as such it’s important that I’m clear on VCAA’s definitions, such as that of a maximum. In your calculus information bite it came across that the maximum of a function can be outside the range of the function. As this conflicts with my current understanding of the definition as influenced by [links to reliable sources; will Wolfram Mathworld do?], I’m wondering if you could clarify VCAA’s definition so I can in turn pass it on to students who might be confused? Thank you!’ Or something along those lines. Maybe throw in a line about how I know mathematical precision is important to the VCAA…?
And I’m sure he’ll just state that yes, a maximum can be outside the range or that they only use the term loosely and then add that it’ll be evident from the context that’s provided what the answer is supposed to be and then maybe add that VCAA doesn’t want to confuse students with technicalities (as if!!!). Maybe he’ll see the intended insult straight away and simply not respond. But idk, maybe it’ll get him to devote just a second of thought to the fact that some of VCAA’s definitions are entirely against the mathematical consensus, and maybe that’ll be worthwhile?
aps, of course what you suggest is perfectly reasonable. But I think it’s fair to say that, particularly since (but not only because of) the media coverage of the 2022 exam errors, VCAA is feeling a bit under siege. (I am not totally irrelevant to this.) And, VCAA is defensive at the best of times.
I am always of the opinion that people should be holding authorities to account, and in regard to VCAA, teachers-students don’t complain nearly enough. So, I’m not for a minute suggesting you don’t write to VCAA (in the polite manner you’ve suggested). Just keep in mind that any response (or silence) may be coloured by the circumstances.
The other point is, VCAA is of course already aware that their new materials contain problems. They may not yet have cottoned on to the supremum error, but they must know that teachers are grumpy. I don’t know much about such teacher-VCAA interactions, but they must have been occurring.
I’ve sent the email now. I’ll let you know if anything interesting happens with it.
Thanks, aps. You should at least email me before commenting with any such details. Although there is no legal obligation (of which I’m aware) to keep any such communication confidential, you (and everyone) should still be respectful of the presumed privacy of any such conversations.
Yes, that sounds reasonable.
The sample Specialist exams are bad. I have zero time, but I’ll try to update bit by bit, on all the materials and slide shows and whatever, as quickly as possible.
I’ve updated the post with comments on the calculus webinar. It is bad, and there are a number of things people missed. (The logistic equation WitCH will be updated soon.)
I’ve updated the post with comments on the vectors webinar.
I’ve added a small update to the vector webinar critique.
I’ve updated the post with (what we’ll call) a critique of the Mathematical Investigations webinar.
And, I’ve updated the post with a pseudo-critique of the Pseudocode pseudo-webinar.
Unless someone raises something worth an update here, all future updates on VCAA’s “New Stuff” materials will be on the VCAA’s New Stuff Post.
I can’t find the pseudo-critique, did it get lost in the aether or am I blind?
Ah, just think of it as a Felliniesque update …
Thanks, Juliet. I must have pushed the wrong button. It’s there now (for what it’s worth, which is not a lot).