It’s difficult get a handle on VCAA these days, which seems to be suffering from Multiple Personality Disorder. There are some indications of significant and positive change but there also signs of businessing as usual. To be clear, the MPD is a dramatic improvement over VCAA’s previous decades of unceasing sociopathy and narcissism. But it also makes it difficult to comment on VCAA doings right now, since the mixed messages mean that we don’t know which VCAA is in control. Such is the case with VCAA’s recent exam redactions.

As John Kermond noted a few weeks ago, VCAA redacted parts of some questions on the 2022 VCE Methods and Specialist exams. Specifically, the five redactions directly addressed the five major errors that Burkard and I flagged as part of our critique of the exams, and which became the focus of the mathematicians’ open letter to the Minister of Education. That letter, combined with the very strong reporting on the 2023 exam screw ups, prompted the Minister to initiate the (genuinely) independent Bennett Review. The redactions come with the message,

This question has been redacted following the findings of the Independent Review into the VCAA’s Examination-Setting Policies, Processes and Procedures for the VCE.

My feelings at the time were mixed, and they still are. To be clear, the redactions are a monumental improvement over VCAA’s year of Deloitte-fuelled gaslighting. Not simply as a sign of generally lessening lunacy but so that VCAA’s specific wrongness did not spread further. Which happens. An idiotic and wrong 2021 Specialist question, which remains uncorrected and unredacted, made its way into two 2023 VCE textbooks, where it will presumably remain for the next decade. This is a routine occurrence. So, the 2022 redactions are a good thing. But not the best thing, not the most informative or courageous or honourable thing.

Nonetheless, I was going to leave the redactions thing be. But then, a couple days ago, commenter RF pointed out to me a VCAA Notice to Schools, on the redactions:

## 2022 Specialist Mathematics and Mathematical Methods examinations and external assessment reports

### Principals/Directors, VCE coordinators and VCE Mathematics teachers

In March 2024, the VCAA received the report of the independent review into its VCE exam-setting processes and procedures. This review was led by Dr John Bennett, who was previously the CEO of the NSW Board of Studies, now known as the New South Wales Education Standards Authority.

The report identified errors in the following 2022 Mathematics examinations:

• 2022 Specialist Mathematics, Examination 1, Section B, Question 3b

• 2022 Specialist Mathematics, Examination 2, Section A, Multiple Choice Questions 4 and 19, and Section B, Question 6f

• 2022 Mathematical Methods, Examination 2, Section B, Question 4eii.

In response, the VCAA has republished the above examination papers and external assessment reports with these questions redacted. These updates will ensure that schools and students have resources to support student learning with accurate information.

This Notice, although now a few weeks old, annoyed me, enough to now post on it.

VCAA has a ton on their plate right now, and I believe they are tackling things seriously and, at least in part, intelligently. By a mile, the main issue right now is revamping the exam setting procedure, as effectively as VCAA frantically can for 2024, and properly effectively for 2025 and beyond. I believe this is occurring. For this reason, I have erred on the side of giving VCAA some space. A secondary but important issue is for VCAA to correct their misinformation, and the 2022 redactions address this, at least in some manner, on that specific misinformation. But there are other important issues, for now and the future, on which VCAA’s conduct is mixed.

The Monster Issue for the future is for VCAA to deal with the current Study Design, which is a hideous, CAS-infested blurgeflug. So, VCAA having just set up a questionnaire, flagging an early review of the Study Design, is very good news. It is also a very good sign, because it indicates that VCAA is looking beyond the Bennett Review, beyond exam processes, to the central perverted nature of VCE mathematics. There is reason to be hopeful for the future. But the past is still with us and the past should be properly addressed, as an issue of honourable conduct and as a matter of misinformational importance.

It is not entirely obvious how VCAA should address past exams and reports that are in error. I would think that the QCAA approach, of keeping the exam as is while noting the question error and/or grading error, is the natural way to go. But I can also see some argument for redaction, to simply removing the mud. I’m not really convinced by that but I was willing to let the redactions slide. The VCAA Notice, however, is another matter.

At some point VCAA should properly confess to screwing up in 2022, and elsewhere. To be fair, the newly acting CEO has appeared to be genuinely contrite on radio in a manner in which the previous CEO was not. But this is not primarily about apology. It is about being on the record and being honest. Without question, some VCE students were cheated in 2022 and probably many were. This cheating may have amounted to only a mark or two, but as the QCAA example demonstrates, a single mark can change a life. At least it can when a curriculum authority acts with reactive integrity.

So what did VCAA decide about those 2022 students, and why? Yes, it was over a year later when the Bennett Report appeared, and perhaps nothing reasonable could then be done. That truth would stand even though it was over a year later solely because of VCAA’s year of denial of reality: we are where we are. But if so then say it. VCAA has to say something and, to the best of my knowledge, they never have. It is one thing to have a straight facts clean-up on the exams and reports. But for VCAA to issue such a bloodless Notice, without a word on those students affected in the sleazy background, is shameful.

Beyond treating the 2022 students with some respect, VCAA’s dealing honestly with the past events matters because it is a guide to the future. In sum, can VCAA be trusted going forward? Honestly, I don’t know. As I wrote, there are good signs. Prior to the Bennett Review, Burkard and I met with the acting CEO and I liked her. She seemed thoughtful and inquisitive, intent upon listening rather than defending VCAA. It was a hopeful sign. But other signs, such as the continued availability of the abysmal sample materials, and the dissembling performance of DoE apparatchiks at Senate Estimates, and now the airbrushing of 2022, are much less hopeful.

VCAA suffers from Multiple Personality Disorder, of course, because it is multiple personalities. But at least now there is seemingly some disagreement amongst the personalities, at least they’re not all on the same sociopathic page. We just have to hope that the personalities with some sense and some integrity win out. We’ll see.

I suspect one of those multiple personalities is the VCAA Legal Department, which will undoubtedly have thought long and hard about VCAA’s public response to the serious errors in the 2022 mathematics exams and examination reports (especially in light of VCAA’s numerous previous denials being totally discredited by the Bennett Review) and how best to spin it. I doubt that the response is about moral duty or ethical responsibility or doing the right thing. It is purely about managing legal risk.

One of the questions I have asked in writing to Education Minister Ben Carroll is:

What has the VCAA done to identify and compensate students who were demonstrably disadvantaged by these errors and the incorrect grading of those exams (disadvantage includes missing out on scholarships, Premier’s Awards, or a higher University offer)?

I believe it is reasonable and in the public interest to get an answer.

Thanks, John. I am sure you are correct, that VCAA Legal is a strong force for the airbrushing. In some sense that is reasonable: it is Legal’s job to err on the side of being denialist dickheads. But I had honest, obviously naïve hope that more considered VCAA personalities would have pushed back and over such denialism. I thought

someonewith clout within VCAA might have successfully argued,“These 2022 kids were dicked over because we stuffed up, and they remained dicked over simply because for an entire year we denied reality. The least we can now do is be straight about that and declare what, if anything or even if nothing, can and will be done about it.”

I’m genuinely disappointed and somewhat shocked that that has not occurred.

I think your question to the Minister, to which you won’t receive a reply, is good and reasonable. But I also think it is very likely at this late stage that nothing reasonable can be done to compensate any of the kids. Even earlier it could have been tricky, depending upon the consequences.

When the 2023 exam debacle began, Tony Guttmann wrote a letter to

The Age, which the know-nothing idiot editors declined to publish. It read, in part,“The VCAA claim that students are compensated for errors, but that is not always possible. I know of one student who failed to get the Premier’s Prize for being the top student because the official answer to the question was wrong. The student’s mark was corrected some months later, but no mention of the prize was made.”

You can argue whether something was “possible” after “some months”, but, in the 2022 case, after a year and a half, it is a lot more difficult to argue.

I also doubt, as armchair lawyer, that VCAA is in any legal jeopardy, pretty much no matter what they now say. In an official sense, VCAA seemingly followed properly formal processes to check, both internally and externally, the accuracy of the 2022 exam questions and their grading. Yes, this checking was obscenely inadequate, and it’s impossible to not suspect deliberately so. But still, if VCAA followed processes, I’d be very surprised if they could not now rely upon that, if they were now legally liable.

Minister Carroll has demonstrated integrity in the way he’s handled things. He shares credit for the good and significant reform happening with the VCAA’s processes. History will judge him favourably. When the day comes that he retires from public office, he leaves a legacy that he can be proud of. Minister Carroll has an opportunity to complete this legacy.

So I hope you’re wrong about getting no answer.

I’m afraid it doesn’t work like that in politics.

I remember I held out on signing an independent contract under Jeff (to my financial detriment). I thought that when Labor won in 99 (what a relief and surprise that was!), it would all be made good. Some make up pay for leave loading, but overall I was worse off – no justice there. (Though I won out big time over the longer term due to how Labor changed the pay ranges in response to Jeff’s nonsense).

Keep up the good work and remember the long game. Old habits take quite a while to change as does replacing personnel – it’s a war, not a single battle.

Thanks, JJ. I am aware that this is all very political. But part of that politics is VCAA having gotten absolutely whacked by Bennett. The political reality is that VCAA is now under scrutiny and will remain under scrutiny. In such conditions, it is in VCAA’s self interest to not be dickish about things.

This is a bit off topic, but having spent 2 hours not even getting through one page of the specialist maths textbook, I think we can safely say that maths (past year 10 or so) is basically impossible for 95% of the population. Saying that would save a huge amount of waste and heartbreak.

Thanks, Harry. It is off topic but that’s fine.

I get sick to death of cuddly inclusive types declaring that there is no such thing as a “maths person” (and thus that there’s no such thing as a not-maths person). However, I also believe that

manymore students would be capable of and interested in doing SM if they had had a decent education in the previous 12 years. It’d also help if the curriculum and the textbooks didn’t suck.So, although I have no idea of your background when attempting the SM text, I’d always be guessing it’s your background for the attempt rather than any innate inability.

I’m an adult trying to do specialist maths year 12. I have the textbook and a good teacher. I really think it’s safe to say about 5% max of people can do (year 11 or beyond) methods/specialist. It just can’t be done by the 95%.

I think we’d save ourselves and lot of time and aggro if we just accepted this basic fact.

Thanks again, Harry (except for the final sentence). You might be right about the 5% but I don’t know how recognising/declaring that will save us anything. Students select themselves out of SM anyway. Even if you are correct, my point still holds.

I think the problem is that we say year 11 and on maths is learnable when in fact it’s impossible for the 95%.

This sparks a whole industry of frustration and pointless effort.

Just curious Harry – when and what was your last year of maths education? Totally agree with Marty – lots more people could do it, but the system holds them back.

Did year 10 maths ok, then tried and failed Year 11 maths (A and B it was back then).

Then returned many years later as an adult started doing Year 11 maths methods and specialist, struggled through that, then trying to do Year 12 methods and specialist now, with a good tutor.

I can read/write and drive a car etc. My point is there are many people like me (in fact we are the majority) who just can’t do maths past year 10. It really is impossible.

I think the 5% who can do this sort of maths really really really just dont get how difficult it is for the 95%. Hence all the pointless initiatives to ‘sell’ advanced maths to kids, talk about careers etc, only to see that point when the students are confronted by pages full of incomprehensible equations nobody can make sense of.

And the nonsense of ‘maths anxiety’ whereas the reality is that its just impossible to do for the 95%. The real ‘anxiety’ is when you work out its flatly impossible to do.

Hi Harry – my question was really about whether you had done times tables/mental arithmetic in primary school (without which maths is so much harder).

You’re right that it is really hard, but it can be done. It’s much easier at school than later if you have a good teacher (what you are doing is very hard – though can be done with a good tutor – though it will take much longer than doing it each day at school with a good teacher).

We get how hard maths can be – I spend my whole time encouraging students and working out ways to best teach and scaffold the concepts so they get it. I don’t know how many could do the advanced maths, but suspect with the right help, it’s a very very much bigger number than 10% – the system is set up to make it harder for them these days. My weakest students are working really well and getting it (although it’s not Year 11 and 12). A very small minority of people genuinely have this thing called Dyscalculia (like dyslexia) but even they can learn more maths with the right support.

We are all different and have strengths and weaknesses – maths comes relatively easy to me – in PE I was always one of the ones hoping I wouldn’t be picked last for teams, but with good coaching, I could be OK. There are many many other things I’m not much good at, but I know that with application I could do OK (though never be really good but good enough).

Keep going – you are showing what so many these days seem to lack – determination and application. I suspect you may find you disprove yourself over time (sounds like you’ve done Year 11 and if you can do that, then you can do Year 12). It can be done (with lots of blood, sweat and tears) – but a good teacher will make it so much easier.

Been studying Yr 12 maths (used to be maths A and B, now called methods and specialist) for 24 years as an adult. Had excellent paid tutors , best textbooks etc.

It can’t be done. If you aren’t in that 10% or so who ‘get’ abstract mathematics, then no matter what you throw at it, you will never be able to absorb it.

The stories about declining enrolments in more advanced mathematics, are, in fact, good news stories, as people are simply rationally working out that it is impossible, and if they do enrol in these units, they will come up against the 10% or so who can do these units, and they will fail or get 51% or so.

I agree with the criticisms on this blog about ‘Maths anxiety’ not being a thing- the real anxiety is being confronted with a page full of what looks like noodles and realizing that its just impossible to understand.

I really dont think the 10% of people who can actually do advanced abstract mathematics really have ever understood the 90% of people who cannot.

Absolute nonsense.

Actually, I really don’t think the 90% of people who can’t actually do advanced abstract mathematics really have ever understood the 10% of people who can.

(Not that I believe any of those figures, which have been plucked out of thin air to support a baseless assertion and are unlikely to bear any resemblance to reality).

Calc 1 is equivalent to Spesh 3/4, so try picking up a good college Calc 1 textbook instead, it will be much better written.

Whats calc 1, a university course?

Even if the book is better written, it will still be impossible.

Harry, there’s a possibility that you don’t know as much about this as indicated by your blanket declarations.

Ok, maybe it isn’t 5% can do, 95% can’t do. Maybe its 10%/90%.

But the fact is that post year 10 mathematics is a leap into an abstract world that most people can’t do. And I think we have to be more honest about that fact.

And I think the 5 or 10 percent who can do it, just dont understand that its flatly impossible for the others.

You’re still missing my point, seconded and thirded by JJ and Joe.

And fourthed by me, who simply hasn’t been able to be bothered weighing in to a very old very tired discussion.

I don’t think Calc 1/2/3 are much more abstract than year 10 maths. They are definitely more complicated subjects in terms of the content required and the amount of work required to solve problems, but I remember it still being mostly calculations with numbers, instead of writing proofs or dealing with abstract objects.

Is calc 1/2/3 a university course or something?

Yeah

So is it standard across Australian universities, is it just literally called ‘Calculate 1’ or something like that?

Different universities call the courses different names, but afaik Calculus 1/2/3 are pretty standard courses in terms of subject content. Obviously it will vary a bit depending on the university.

Ok but I’ll bet Calculus 1,2,3 is still impossible.

I have the Jacaranda textbooks for Year 12 Methods and Year 12 Special Maths. My tutor agrees with me that they are well written books. The problem is the subject is impossible to understand.

The books suck.

I think they are pretty good. What totally derails me is when they ‘jump steps’ in their algebra explanations, ie doing three changes to an equation at once. They assume knowledge, which is very confusing when you’re learning something for the first time.

I’m not sure why you think you’re in a position to be a good judge of the quality of the books.

Hi Marty. This has been an interesting off topic development of your post and definitely worth following up. The VCE Maths textbooks currently in use are basically ‘cook books’ full of recipes. Compare Cambridge and Jacaranda to Porter’s Further Elementary Analysis or Parsonson’s Pure Mathematics or Matthew’s Calculus etc. I think it’s safe to say that CAS calculators are the main cause in the reduction of actual mathematical content. Textbooks now have to explain how to actually use them! What if there were more than just the two major VCE CAS calculators currently in use? Furthermore, Online Maths posts sometimes show tricks which can be used with 1 calculator but not the other. Hardly a level playing field is it?

Re: What if there were more than just the two major VCE CAS calculators currently in use?

Like if, say, Mathematica was currently in use …?

Yes, it is absurd that the textbooks explain how to use a CAS calculator. But not even close to the total absurdity of commercial trial exams whose solutions for Exam 2 consist of 75% CAS calculator screenshots (usually a single brand) and 25% of actual mathematics (the reason for this is that teachers allegedly want it because the don’t feel confident using CAS calculator ‘tricks’).

Well, VCE exam 2s generally are at least 75% CAS so teachers who want practice exams like that are right in their judgement.

I completely disagree. I think it’s absurd and reprehensible to replace mathematical working with CAS screenshots in solutions. And I think teachers wanting this because they’re not confident using calculator ‘tricks’ to answer questions (code for “I don’t want to do the maths, I want to push buttons”) is a disgrace.

With real mathematicians now involved in writing the VCE maths exams, I hope most exam questions will require mathematical acumen rather than CAS calculator ‘tricks’, no doubt much to the chagrin of the button pushers.

Play nice.

Whether we like it or not, the truth is that at least 75% of exam 2 questions are trivial as long as you know the right CAS commands, so a teacher who wants to maximise their students’ study scores should spend a lot of time teaching how to use the CAS.

Exam 2 is more of a CAS exam than a math exam, and I doubt that the addition of a couple of mathematicians to the writing team will be able to change that immediately, CAS is just too ingrained in VCE mathematics.

I won’t enter this argument unless required, to separate you two, but I’ll make a few obvious points:

1) CAS poisons everything. Thanks, Dave and thanks, Kaye. What you’ve done to the State is immeasurable.

2) The CAS instructions, whatever their merit or necessity, make the VCE textbooks the most disgusting, butt ugly, unreadable garbage ever published.

3) The most important question you guys are discussing is whether the introduction of mathematicians to writing and vetting VCE exams will make a significant difference to the level of CAS poisoning. We can’t really predict, but I would take a middle ground between Joe and John.

There is no question that Joe is substantially correct, that CAS is ingrained, and turning around this Titanic will require a change in the SD and the deprogramming of thousands of gullible teachers who have been indoctrinated into this insane cult. That will take time.

One the other hand, John is not wrong. Any and every decent mathematician who looks at the standard CAS swill cannot do so without screaming “What the fuck is this?” 2024 is a transition year, and I would imagine no significant change, except substantially fewer errors and non sequiturs. But from 2025, when mathematicians should be properly on board, I do not believe they will sign off on the traditional VCE-CAS idiocy. They are not going to give up their incredibly precious time to simply ignore bullshit.

What would be the impact on Victoria if Specialist Mathematics were dropped from VCE ?

And just not replaced?

I don’t really understand why it’s an add on to methods rather than just a higher stream.

But I wouldn’t want to get rid of it entirely. Higher maths kids are more than ready for complex numbers and differential equations before first year uni. If nothing else, it would set back physics and maths degrees a semester or so.

Any pretence that STEM mattered would be blown to pieces – the symbolism would be huge – not to mention the notion that the dumbing down of education was in full swing would be very very obvious. Decades of students managed it without major issues – can’t see why it should be dropped. Why not go back to the old Pure and Applied?

While we’re at it, Terry, we should also drop English Literature, Physics, Chemistry, Psychology … Come to think of it, we should drop every subject in VCE. Students should leave school at Yr 10 and work or get an apprenticeship. Those who want to go to university should sit common university entrance exams, which they would prepare for by studying for perhaps another 2 years. There should be many institutes that provide the opportunity for such study because there will probably be many students wanting to do this. The universities could outsource the running and grading of those exams to a single body. There would be a wide variety of subjects students could sit exams on, ranging from sciences and mathematics to humanities. They would be subjects that students could build upon once at university, and would be offered at various levels, from elementary to advanced, from theoretical to practical. Then everyone could be catered for.

(Terry, I’m sure you’ve heard of the thin edge of the wedge).

I just wanted to clarify, I am in favor of this site, I like how it calls out bureaucracy in education (I have experienced a lot of bureaucracy, not really in education or mathematics as I dont work in those areas).

I like it when this site exposes problems in the exams and flawed maths reasoning, as it needs to be exposed, but have to be honest and say that most of the time, I can’t really spot the mistake the exam paper/problem.

I DO think there is a big blind spot by the 10% who can actually do abstract maths to the 90% who simply can’t, and this causes a lot of wasted effort, costs and personal frustration out there. This is why there’s all this nonsense about maths anxiety and making the subject more approachable etc etc, having more talk about careers in science etc etc, when in fact when you are confronted by the reality abstract mathematics you are confronted with a subject impossible to understand.

I think we need to lose our social taboo on flatly coming out and saying that advanced (post yr 10) mathematics is impossible for about 90% of the population.

One day I’d love to write a guest post about all this if the site agreed to it.

harry, I appreciate that you seem to be on the same basic page as this blog. I also have a high tolerance for disagreement, and wish there were more. I detest groupthink. But you keep making these grand and unsubstantiated claims, and keep ignoring what others write. You don’t know as much as you think, and the grandstanding character of your comments don’t contribute much.

Ok so lets focus on what we can agree on:

(NB when I say ‘Maths’ here I mean ‘Year 11 and onward Maths Methods and Specialist’)

– Maths is hard and requires effort and discipline and should be taught rigorously.

– Maths anxiety is not a thing.

– Some people have an innate ability for maths

– There is a lot of silliness and bureaucracy in the education regulators and a lack of rigor in exams.

And what we disagree on:

– The innate maths people are about 5-10% of the population

– That people without innate maths ability really really struggle with it, to the point that its just not viable to do and will destroy their ATARs if they try

– (maybe you do agree with this) a lot of the ‘hucksters’ who surround maths education are really exploiting the fact that maths is basically impossible for the people without innate ability, but there’s a cultural taboo on just bluntly saying that, so we constantly try to ‘dumb down’ the subject, or build it up by saying how you can be a famous scientist if you learn maths etc etc (which is great till you get to the incomprehensible formulas that just look like noodles).

After 24 years of spending a fortune as an adult on books and teachers and everything else, I’m calling it: maths is impossible if you dont have innate ability. And I know many many many many other people just like me (who accepted it in high school and sensibly moved on to other things).

That’s better. I’ll reply tomorrow, indicating where I agree and disagree. But I’ll say up front, I agree with most of this but the 5-10% is meaningless, because being mathsy is not a yes-no characteristic. And however it might be solid in your mind, you’re simply making the stat up, with no evidence for it whatsoever.

Ok, great. And just maybe two more points that we might disagree on:

– The people who have innate ability for maths dont really get how hard it is for the non-innate people (a bit like the way people who can walk dont understand people in wheelchairs)

– The comment ‘and I never needed to use algebra in my life’ is, in fact, profoundly true for most people. In my maths learning days I used to scoff at this comment as being ignorant and wrong, till I looked at my own life and realized how right it was.

Harry, I strongly disagree with this “innate ability” stuff. Most humans are capable of most things at non-elite levels, if they make the choice to put time in (understanding you can’t put time into everything). Completing a high school course is not like being a top tier athlete or researcher etc where there are only a few spots and it is highly competitive. But maths is a large cumulative body of knowledge and skills that takes time to accumulate.

Look at the course Learning how to Learn, it is a bit corny, but decent. And the main presenter is Barbara Oakley who was a self-proclaimed non-maths person until she needed to learn as an adult.

Nah, harry is correct. It’s just a question of whether it matters.

This blog post got very derailed. But I’ll continue down the path a bit more

I got stuck on this idea. I still don’t feel that SM should be unobtainable for the majority of the high school population, but it’s just a matter of prior experiences and choices. I tried to see if there was any literature on this, but it looked messy! Even given there are some known genetic factors to intelligence and that IQ is a fairly stable measure of aspects of intelligence, mapping that to what is required for top level high school maths is not clear.

The few teachers (and John K in the comment below) I talked to about this all thought that most students should be able to reach specialist maths – provided they had been properly supported and chose to put the required effort in. But the couple of non teachers I talked to (including an ex-school psychologist) thought that it would be only maybe 20-30% of students or less.

This difference might be part of teacher training and culture concerning the effects of teacher expectations… but even that seems oversold. There is so much noise and selective/agenda driven citations.

[Aside: Boaler When You Believe In Your Students They Do Better

is the most painful, very selective citations – but even she slips and implies the affect is only “significant for students of color”.]

This review https://doi.org/10.1207/s15327957pspr0902_3 seems to summarise things pretty well. The abstract gives a good summary of a long dive into the research following the ‘famous’ 1968 Pygmalion paper of Rosenthal and Jacobson.

Thanks, Simon. It’s an interesting question, in which I have little interest.

As a teacher/lecturer, you are dealt the hand you are dealt: you do your best for the students you have been given, whatever their abilities and for whatever the reasons their abilities are at the level they are. If you get the kid in Year 10, you cannot wave a magic wand to undo the damage of the previous ten years.

More broadly, kids are doing less and less hard mathematics because mathematics education has long ago lost the plot. The issues of innateness and stigmatised social groups and whatnot are trivial in comparison.

Thanks Marty, Yes – you play the hand you’re dealt with the students. But you also have been dealt a hand in other games that can affect larger change, which is part of why you write this blog.

I only went to the “teacher expectations” thing as I was struck by the different opinions in my small n data set – not because I thought it is the largest effect in overall maths education.

Thanks, Simon. I’ll reply to your latest comment as a fresh comment, below.

OK, harry, I’ll reply in substance now.

ON “WHAT WE CAN AGREE ON”

I agree in spirit but I have important objections to your “we agree” list. First. a minor nitpick.

A couple times now you have alluded to the importance of “rigor”, but either you’re using the wrong word or you’re misunderstanding the nature of school mathematics. Even good school mathematics is not rigorous in anything like a proper mathematical sense. One simply has to cheat. Probably the word that best captures what good school mathematics is, is “coherent”. The stuff hangs together in a natural manner, which involves an unclear and largely unstated combination of logic and intuition and blind faith.

More importantly, you treat “innate ability” as a yes or no attribute. That’s just silly.

I have other objections, to your first point, but they’re better covered in the second part.

ON “WHAT WE DISAGREE ON”

You declare that “The innate maths people are about 5-10% of the population”.

Again, even if one believes that mathematics ability is innate to some degree, and I do, it is silly to claim it’s on-off, like being blind or sighted.

But let’s take “innate ability” to mean “sufficient innate ability to succeed at proper senior school mathematics”. Even then, and whatever “proper senior school mathematics” means – which you do not seem to understand – you’re pulling that 5-10% out of your ass. You have no damn clue what the percentage is and you come across as a clown each time you pronounce that you do.

But that’s not my fundamental objection. My fundamental objection is that you are so blinkered by “innate ability” that you don’t even see the other factors, contributing massively to students choosing to not do higher VCE mathematics, and why the ones who choose to do so do poorly.

I totally agree with you that there is a huge industry, populated on the one hand by maths ed grifters, and on the other hand by lovey lovey feel good academic twats, trying to convince anyone and everyone that “we’re all maths people” or whatever. It is obscene.

But to be able to succeed at maths, at whatever level, takes a hell of a lot more than “innate ability”.

Let’s suppose we have a year 10 kid, contemplating senior mathematics. Before even beginning, he needs, sufficiently solidly:

a) The desire, the willingness to work hard enough;

b) The mathematical background;

c) The mathematical ability.

What “sufficiently solidly” means depends upon the demands of the subjects. And, the solidity of one aspect can compensate for the porousness of another. And, a person who denies that there is innate mathematical ability would presumably fold it into a) and b). It doesn’t matter. However you work it, a kid coming into senior mathematics, if he’s not Terry Tao, needs a hell of a lot more than ability.

And this is what really pisses me off about your “some kids are not up to it” shtick. Sure,

plentyof kids are not up to it. But the overwhelming reason they’re not up to it is not any lack of innate ability. It’s that they were never given the chance. They have never been taught enough mathematics solidly enough, NOR have they been taught the importance of desire, so of course they’re gonna blow off senior mathematics, or be blown away if they don’t.I am livid with the fact that the vast majority of Australian kids are never given a shot. So I don’t want to hear about their lack of innate ability. Even if true, it is not remotely the critical truth.

Ok thanks for the response.

Yes totally agree that everyone should be given the chance at maths.

But also think that many many people (me included) just do not ‘get it’, cannot get it, will never get it. Even after throwing massive resources and time at it.

(I’m probably in a bad mood because I’ve just tried to re-read and re-do the specialist vectors chapter, and its totally incomprehensible, as its always been. How do you multiply vectors, which are arrows??!).

And your response does really indicate that people do need the ‘mathematical ability’. And that people without it really really struggle.

Agh! My response indicates nothing of the sort. Plus your book fucking sucks.

Ok but you said the attitude ‘we’re all maths people’ put out by the hucksters is wrong. And thats exactly my point. I got taken in by the hucksters as well. Many many people are NOT maths people (me included), and never will be.

And you ‘get’ vectors, and I dont get them (along with virtually everything else in the specialist subject). And that proves my point- you are an innate maths person and I am not (and I reckon I’m in the majority).

I made a lot of points. You are obsessing on one. You don’t listen.

ok you’re saying that many people fail higher mathematics because of factors not linked to innate ability.

So sure, innate ability is not a sufficient condition to do higher maths but in my opinion it is a necessary one.

I think that is probably true. But even if true I think that, at this time, after decades of maths ed perversion, it is a very very secondary truth. At least as concerns VCE mathematics.

Right so what I’m saying is that if you dont have a certain level of innate ability at maths, then you’re just not going to be able to ‘get’ maths, no matter what you do. We are NOT all maths people. And we disagree how many people are innate/not innate, I believe its 10/90%. And I also DO believe it is a very clear cut thing, like being blind or sighted.

Also what is wrong with the Jacaranda books? My only experience (apart from finding say 5 mistakes in the first 330 pages of Spec Maths Yr 12 which is not too bad a rate) is that they cause me problems when they ‘jump’ steps and assume knowledge, but otherwise they seem ok (all my teachers also said they were ok) but you are saying they are too basic or too advanced…?

FFS.

1) If you believe innate maths ability is a yes or no thing then you are utterly deluded.

2) Once again you pull that idiotic percentage out of your ass. Fine. You’re allowed to believe what you believe. But you have absolutely no damn clue, not a shred of support for the proposition. None. And given you have absolutely no damn clue, why should anyone else care?

3) Primarily, the Jacaranda books suck because they are written for a piss poor,

incoherentcurriculum. It is logically impossible to write a good textbook for VCE mathematics. In ways Jacaranda makes some decent attempts and in other ways they do not. But the attempts are doomed to substantial failure because they are trying to present topics, as guided by VCAA, that do not make much intrinsic sense.Ok but you agree that there are people who just dont have innate mathematical ability, and just can’t do maths (as I’ve defined it). And in fact by trying to do it they are setting themselves up for a lifetime of failure.

Yes, the delineation between Specialist and Methods seems to be all over the place with topics recurring in both.

Have you met someone who lacks innate maths ability? Have you really understood the world from their point of view?

I believe that my skills for empathy are far superior to your skills for listening.

Ok you said this:

What “sufficiently solidly” means depends upon the demands of the subjects. And, the solidity of one aspect can compensate for the porousness of another. And, a person who denies that there is innate mathematical ability would presumably fold it into a) and b). It doesn’t matter. However you work it, a kid coming into senior mathematics, if he’s not Terry Tao, needs a hell of a lot more than ability.

But when you just can’t get it, where the concept of maths are just too abstract too hard to understand, then the solidity of the other aspects just wont compensate.

Which you’ve now said five thousand fucking times.

Ok I give up then.

Everyones a ‘maths person’ and anyone can do it, and lets just put a glamorous picture of a scientist on the cover and that’ll make up for how incomprehensible it is for many many people, innately.

Oh and you can ‘multiply’ an arrow by another arrow (?)

Yes, that’s exactly what I’m saying. Thank you for summarising me so pithily and faithfully.

Ha ha ha, I found your post hilarious. Frankly, you are too simple minded to understand anything that is fairly abstract. And you and your tutor are sort of delusional to think that after failing maths at year 10 level several years ago, you can just simply walk in a Y11 Spec and MM class and continue your struggle from there. Or maybe your tutor just inputs that idea into your head to get your dime??? May I suggest you to relearn year 10, or year 9, or year 8 or maybe primary???

As per CAS in MM2, I am in all sort of conflicts with it. I am from IB background and since moved here, I had to invest tremendously in CAS use and spending majority of class time on it as I do not want to disadvantage my students. I teach both Specialist and MM and always favour Specialist I wish they could remove CAS from Specialist.

Passed year 10. Failed Year 11.

And yes, there are a lot of people like me who just dont get abstract maths.

And you’re post (although a bit nasty) is quite honest- its just impossible to get or understand. This is the right attitude!

Passing a Yr 10 maths exam means nothing. I have taught students from literally a hundred different schools and I can say this for a fact. It means nothing.

It might have been a bare pass, it might have been a pass on an exam written in such a way that everyone passes (so that the boat doesn’t get rocked by student and parent complaints) And these days, students at many schools are not even doing Yr 10 exams or even tests because, apparently, it’s too stressful for them post-Covid (and no doubt it makes life as a teacher much easier). There are many reasons why a pass on a Yr 10 maths exam means nothing.

Can you do the following?

1) Rationalise the denominator of .

2) Re-write by completing the square.

3) Solve in three different ways.

4) Find the values of k such that has no real solutions.

This is (in most decent schools) Yr 10 level stuff. For me, being able to answer these questions is a litmus test for whether a student will struggle in VCE Maths Methods (let alone Specialist Maths) or not. Of course there are other skills a student needs to have competence with, but I find the above is a quick way of determining where a student is at and what remedial work might be required.

And it’s not just about being able to execute one skill in the context of a question about that skill, it’s about being able to identify and execute several skills in a context where you do not know what skills are needed.

Learning maths is like learning carpentry, or learning how to play a musical instrument etc. And at some stage, we hit the wall where our ability fails despite all of our efforts. But I do not believe that wall is during secondary school.

If you cannot do the above questions, then you need to go back and (re-)learn the maths upon which the answers are based.

Victoria’s comment that “Or maybe your tutor just inputs that idea into your head to get your dime???” is not far-fetched. I am aware of many tutors who do exactly this.

In conclusion, maybe you need to lower your goals and go back a few years to Yr 9 or Yr 10 maths. (Re-)Establish the foundations and build yourself up to where you can experience success at the VCE level.

I’ve lost track of whether or not you have already said this but if you have, please remind me: Why have you returned to VCE maths study after several years out of school?

John, this does not help.

Also, should point (3) have an on the end?

And not even a spoiler alert?

(Yes, very careless of me. Lucky the proof reader is not asleep at the wheel).

Hi Marty, I wanted to thank you for continuing this interesting and I think important digression; namely the discussion with Harry. If Harry is struggling to comprehend the concept of ‘multiplying’ two vectors, I can only suggest that he gets a new tutor. Furthermore, the arbitrary 5% or 10% figure for innate ability is indeed nothing short of theatrical. In my experience, any kid who has a knowledgeable and passionate teacher has a real potential to do well in maths provided that they have the basic numbers skills firmly at their fingertips. Another point; the SPEC exam questions are often easier than MM questions since the topics are only covered in a superficial way and are often more predictable. I also agree with John Kermond that MM paper 2 is more like a CAS exam not a Maths exam. Why not simply make a better and longer MM Paper 1; no CAS and no notes? Like the good old days.

Ok, this is what I’m talking about.

I can understand adding two vectors- one starts where the other finishes.

But I just can’t get my head around how you can multiply two vectors- one arrow is wrapped into another arrow(?).

And there are many other topics in Spec maths which are just too abstract as well.

And there are many many other people like me out there.

You guys get it – and thats great. I’m not being sarcastic. But you need to understand how difficult it is for many people.

Vectors are not arrows.

Ok but they’re often shown as that.

See, its easy for you to understand multiplying vectors, its hard for many others.

And numbers are shown as squiggles on a page. How do I multiply the squiggle for 3 by the squiggle for 9?? It doesn’t make any sense!

Don’t pile on. harry is not expressing things well and is way overconfident in his opinions, but he’s just a guy. What’s Jacaranda’s excuse, for Christ’s sake?

Dont mind at all. For many of us, maths is really just a bunch of squiggles. See, you guys get it (great-well done) but most of us dont.

Now you’re whining again, and I feel like backing Ron. Whatever your abilities, innate or learned, the main problem here is that your textbook sucks.

I assume that by “multiplying vectors” you mean the dot product. Number multiplication and the dot product are different things; it’s lazy to just call it “multiplying vectors” without further explanation, and if a textbook or a new tutor didn’t make it explicitly clear that they are very different concepts then get a new textbook or a new tutor.

Ok I meant the dot product, the book and the teacher did explain it.

And its still utterly incomprehensible. Again, you guys get it. I’m a bit jealous but good on you. But many many people do not and will never get it.

I don’t believe I couldn’t teach you the dot product in five minutes. I’m with Joe.

You could teach the rule, sure, a x b x cos(of the angle between a and b).

But the problem is understanding why this is so.

Many people just cant ‘accept’ a rule. Something I’ve noticed- people good at maths just accept a rule but dont try to understand it deeper. Its the people that try to deeply understand it that hit the rocks and get frustrated.

That “rule” is not the dot product of two vectors.

ok I thought this:

“The dot product of two vectors is equal to the product of the magnitudes of the two vectors, and the cosine of the angle between them”

But thats wrong….

Yes, your book sucks.

Well I’d better write a letter to Jacaranda then!

I had a look at the Jacaranda textbook. Yes, that’s really how it defines the dot product.

The real definition of the dot product is as such: . You can generalise this to n-dimensional vectors by just adding new terms, e.g.

You can derive the geometric “definition” from this if you really want to, using trigonometry.

And I rest my case. Thanks, Joe.

Hi Harry

How I would teach this is to say, well you learned multiplication with numbers, but as we get into more complex maths, we start to broaden the idea of multiplication. (actually that ties into matrices etc etc). So you give the student the idea that multiplication with numbers is a starting point for a simple concept that we then generalise (which is actually the basis for more complex maths).

And so as you say it makes no sense to multiply arrows (Marty – I know but this is scaffolding), so we define something which is useful in other areas and has some features of number multiplication and on we go. Hope this makes sense.

Hi, JJ. I’m not sure why you tagged me. Should I be objecting to something?

It was that I was calling vectors arrows

Ah, I see. I’m fine that.

I do not think it’s helpful to consider the dot product as “vector multiplication” pedagogically, I think that the concepts should be fully separated when first introduced, and maybe some similarities should be pointed out later. Calling it “multiplication” leads to confusion, as the dot product is completely different from typical multiplication. The dot product of two vectors isn’t even a vector.

Nitpick. It’s not a million miles from the madness of introducing the thing geometrically rather than algebraically.

Harry – This video with the slightly clickbait title Why can't you multiply vectors? by Freya Holmer gives a good presentation of a different approach [connected to geometric algebra] to thinking about vector products. It might get you away from that textbook thinking that the dot product is defined as a.b = |a||b| cosθ and closer to what multiplying vectors can mean, although it’s not the only way to think about the idea…

Freya started as a game designer and she’s done some good maths videos recently (I really like her ones on splines and Bezier curves), and she was motivated to dive into the maths from needing to build tools for making games.

As said by people in this thread, vectors are not arrows, but to me this feels a bit like mathematicians who insist teachers should not say multiplication is repeated addition. Multiplication starts as repeated addition and vectors start as 2D and 3D unanchored directed line segments, aka arrows. Generalisations come later.

A good analogy, and I stand semi-corrected.

(Sorry about the delay in your comment appearing. The number of links meant I had to manually approve it.)

Thanks! (and sorry I made extra work for you)

Oorah!

Marty:

One group of people you dislike are the maths ‘hucksters’ – people who try to say ‘everyone is a maths person’ and try to dumb down the curriculum, or treat maths education as just a giant PR exercise, where if we just put some sugar over it and some glamorous photos of people in labcoats everyone will study it and all will be well. They are the ones who mislabel maths frustration as maths anxiety, and say its an easily treatable condition. I dislike these people as well; I was taken in by them for many years.

But why are there so many of these people? Why aren’t there English hucksters or History hucksters to the same extent there are so many scammers in Maths ed?

For I believe two reasons:

1. Maths is an inherently difficult subject on a different level of difficulty to most other subjects.

2. We have a social taboo on openly admitting that people can’t do or struggle with certain subjects, and have to say that everyone is wonderful.

And by not admitting the above two points we are setting a lot of people up for failure and frustration, and encouraging a lot of trendy nonsense and bureaucracy in maths education.

Surely we can agree on this?

What we can agree upon is you repeatedly working the discussion to be on your hobby horse being rude and boring.

Ok. I actually thought it was a pretty good post, better than the others (which might have been a little bit overly sweeping). I’ll go quiet now.

It was. But you had already well used up your lives.

Hi, Simon. Thanks for going down the rabbit holes about innateness and whatnot. Of course I trust Boaler as far as I can throw her (and I’m all for attempting it). But the same is really true for any social science research, including the Jussim-Harber paper to which you refer. I don’t accept any such “conclusions” unless I have good reason, by either reading enough of the paper myself or, perhaps, by the paper being vouched for by someone I trust. (I don’t know the “famous” pygmalion paper: should I?) I’ve now grabbed Jussin-Harber and I may give it a look.

But I still wonder how much it really matters, at least to us, even us as crusaders for change. How would either acceptance or rejection of Jussin-Harber’s conclusions, for example, change at the moment what one might advocate for?

I didn’t know about the Pygmalion paper either – but apparently it has been very influential in the teacher expectations thread of research and associated teacher education ideas. And yes, social science research is a mixed bag containing a wide range of modes that are not always accepted in the harder sciences, let alone in mathematics!

And how much it really matters? I don’t know, a little.

There are a couple of entangled ideas in our subthread of conversation: What range of levels are students “intrinsically” capable of reaching? What are reasonable teacher expectations and the effect of those beliefs and their communication.

From a teaching perspective:

Communicating high expectations of students is a low-cost and, according to some, fairly high-return teacher move. It feels right that you should hold high but reasonable expectations of all students – not let them fail for lack of trying or belief in their abilities. Although, the summary of Jussin-Harber puts possibly a more realistic spin on it. And I do wonder how Boaler in particular squares it with the lowering of curriculum standards. And there would be a cost to some students in communicating they should be able to do something that they can not…

From a crusader for change perspective:

The idea of intrinsic ability probably matters. Should a multi-year curriculum sequence be developed assuming all students can be supported to reach the highest levels, or with the understanding that not all can make it and there should be different streams that cap out at different levels of abstraction. Given that, what are the appropriate knowledge and skills that could/should be emphasised in different streams. And at what year levels is it better to start streaming vs differentiating in the same class?

Thanks, Simon.

With the “teaching perspective”, I don’t see that the question of innateness has any bearing whatsoever. I agree, that one wants to encourage students to achieve what will be just or not quite within their reach, always with an eye to the individual student and how they will react to small success and small failures. But this approach is the same, whether the student got where there are by good/bad teaching and study, or whether they got there by good/bad genes.

With the “crusader perspective”, I can imagine a world where the issue of innateness may be relevant. Maybe it matters in Singapore. I don’t think it is practically relevant in Australia: innateness is swamped by other forces.