With a new year beginning, and with *Do the Maths* having struck a bit of a chord, it seems a good time to think about doing more than shooting stupid ACARA fish in ACARA’s stupid barrel. I’m open to suggestions for this blog, of what I might write about, *and* of what you might write about.

Let me know. The ideas don’t have to be detailed or clearly thought out, or even particularly plausible. Just anything that’s in the ballpark of this blog, and of which you’ve been thinking “I wish someone would write on this”. And, if you might be that someone, if you think you can contribute an interesting post, I’m happy to ~~steal your ideas and take all the fame~~ discuss your writing a guest post.

Any thoughts can be in a comment below, or you can contact me through the “Contact” tab above.

Well, firstly, good on you for day after day wading through the ACARA swamp – I think it comes under the headline, “I read this crap so you don’t have to!” And I’m glad it (you?) forced me to at least dip my toes into the dirty waters. Really, good on you Marty.

Some ideas:

1. Now you’ve identified the problems, how bout some solutions? (To put it in a cheesy way…) If you were Maths Education Tsar for Victoria, what would you change? Immediately? Quickly? Eventually?

2. Getting rid of CAS calculators – should we? how? In reality?

3. Did you ever tell the story of getting audited by the VIT mafia?

4. Long ago I quite liked this article: https://www.mathvalues.org/masterblog/2018/7/19/devlins-angle-post-1-sf48y (yep, that Devlin guy) and in particular this image: https://images.squarespace-cdn.com/content/v1/5aa6128850a54f0331abec9a/1539574862917-R0CR584KF169P82AND4N/deja.png?format=750w – how do people really do maths when presented with an interesting problem? Me, some combination of pen and paper, excel, python, desmos, physical models, KSEG (an old bit of easy fast geometry software).

5. Why have you never written the perfect (set of?) maths text(s) for high school? (I’m well aware that I haven’t.) How would you start?

6. A bit more on how your life as a blogger is going? Does the blog provide you income? More stress or more joy?

Thanks, AII, and RF and Terry. There’s good food for thought there. In brief reply:

1) I agree, my blog can be unrelentingly negative. So, posting some positive and constructive stuff would be worthwhile, and I’ve had thoughts along these lines, and am open to guest posts. I’ll write more in response to RF. But, I think it’s tricky.

I think there’s always a question with such debates/fights, does one propose solutions, suggestions for the way things should be done, or does one work politically, to establish power in order to implement such suggestions. Unfortunately, the educational malaise seems to me to be all about power. In particular, mathematics education is in the hands of ignorant, arrogant clowns, who could not recognise good mathematics teaching if it were shoved down their throats. As such, and although I appreciate there are a few soldiers on the ground doing great stuff, and I hope/plan to write about them, it’s for me not the main game. For me, the main game is pointing out naked emperors.

2. The answer to the first question is “Well, duh”. But note that this is a local, e.g. Victorian question. IB doesn’t have CAS. NSW doesn’t have CAS. So, again à la 1, it is a matter of politics and power, of taking Victorian education out of the hands of the techno-fetishists. Having said, perhaps there’s something I can write, and a story I can tell about The Debate That Never Happened. That could at least be entertaining, and might ruffle some feathers.

3. That’s just a story (well, two stories). But sure, I’ve added it/them to the shortlist.

4. Hmm. To quote Professor Smarts,

“My focus then, is on using math to solve real-world problems.”

I can understand how this is of interest to some people, but I have zero interest in it, and I think the focus on the “real-world” has been an absolute disaster for mathematics education. ACARA’s curriculum is overflowing with this swill.

Having said that, it might be interesting to write about the solving of

mathematicalproblems. That is much more the realm of Tony Gardiner: see, for example, this and this. But I can try to include more problems and discussion.5. I don’t think mathematicians writing school texts is a recipe for success. The ICE-EM texts, for example, although better than the current alternatives, cannot possibly be considered a success story.

But of course, decent and attentive mathematicians must be involved. There is work being done there (not by me) that I’d love to write about. It’s on the list, and with luck I’ll be writing some good news later this year.

6. Really? What might be the interest? Although, your second question is hilarious.

Thanks again for the suggestions.

I second points 1 to 3 above. The others are also of interest, but 1 to 3 (for different reasons) are what I would most look forward to reading.

In addition, a few ideas:

A – how to not be a crap teacher when given a crap curriculum. Where are the opportunities to do some decent work whilst all the time playing within the rules? (Ignoring said rules is the easy option – I want to know if it is possible to follow the rules but still do good; I’m guessing this would be a pre-VCE post however…)

B – stories (for inspiration) about people who actually managed to do good. I’m thinking Fitzpatrick & Galbraith level here, not Mike Deakin and/or Russel Love (sure, they were also brilliant, but as far as I know neither of them taught at a high school).

C – there seem to be a few early career teachers who comment here on occasion. Something to help them not lose hope – think “Adam ruins everything” style where at the end of the episode he shows how there is still hope for change.

A: In my experience there are always opportunities to insert interesting ideas, here and there, that are not mentioned in the curriculum, at all levels. A well-qualified teacher will spot them and exploit them.

B: There are, and always have been, mathematics teachers who do a good job teaching students. Unsung heroes. Maybe there are many readers of this blog who can trace their interest in mathematics back to a teacher at school. There is, of course, the perennial question of what “good” means in this context. Perhaps only students know who the good teachers are, and why – but they may not agree with each other.

Thanks Terry. In response:

A – examples please. The more the better. Stuff that is accessible to most levels of ability, but perhaps skewed towards the stronger end while still being within the curriculum.

B – names/stories please. The more specific the better.

A – Thanks to Marty, examples abound in this blog. Prove Pythagoras’s theorem and its converse. Emphasise that by definition. Give students experience in problem solving using Problemo or problems from the Australian Mathematics Competition. Use problems from Tony Gardiner’s books alpha, beta, gamma. Show students how to calculate the date of Easter for any year – an intro to algebra (attached). Give students an essay to write as a formal assessment.

B – I won’t name names. But I reckon that if you look around at the other teachers in the school, not necessarily mathematics teachers, you will find good teachers. Talk to them. Watch them. Learn from them. Maybe invite them to your class. Listen to the advice of your students. And, in the end, as someone once advised me, be yourself.

BTW, it is fashionable in some schools to have large open classrooms that might hold 50 or more students in a class. A class is taught by two or three teachers to keep the average in check. I can’t believe that this system is effective, but it does have the advantage that you can see other teachers at work. (Architects have a lot to answer for.)

2009-Vinculum-Easter

Examples may abound, but it is still exhausting finding them for a specific lesson to be given tomorrow. It should not be the responsibility of a teacher to be hunting blogs or whatnot. Where the hell is ACARA? Where is AMSI? Where is MAV? Where is AAMT? Where is AMT? Where is VCAA? Where is anybody?

Thanks Terry – I am very much a novice when it comes to hunting out these books of Gems, but Gardiner and a few others I’ve read about on this very blog have been brilliant reads.

As for watching other teachers… I actually chose to do this in the years BC (before Covid) and sat in on two classes of the same subject, same year level, different teachers. It was quite amazing to see how two VERY different styles were still highly effective. Unfortunately, transferring said skills to my own Mathematics classroom has been… slow progress to say the least.

The “team teaching” idea works for some, but a lot of teachers are against it for some reason. I’ve tried it a lot and with a second teacher who is keen on the idea, it does work and is successful. As you say though, architecture can often be a challenge… two classes of 15 becoming one class of 30, not so bad. Two classes of 25… now there is a “real world” mathematics optimisation problem!

RF: I too have seen team teaching in practice, some good and some not so good. The good ones are where the teachers work as a team. It’s almost theatrical, as if it has been rehearsed. There is certainly an art to it.

I have ordered

* Felix Klein, Elementary mathematics from an advanced standpoint

* Euler, Elements of algebra

through bookdepository. Sometimes you can find pdf files online.

During 2022, I made some use of past papers on mathematics competitions from Australia and the UK.

Thanks, Terry. I agree. But A is exhausting. You’ve got 200-odd years of experience to draw from: others do not.

Thanks, RF.

On A, I agree that it is an excellent topic, but I don’t think I’m the person to write on it. There are some people I would love to write on this, but they are very busy and very modest.

On B, I talked to Peter Galbraith for an hour about the writing of F & G, about those times, and his having played football with a member of the 1966 Premiers. I had intended to record (with Peter’s permission), but alas pushed the wrong buttons.

I would love such stories, but I don’t know if I have the power to get them.

On C, if teachers want some Panglossian hope, they can always seek out Smiling Eddie at WooTube.

More seriously, I agree, as with #1 of AII’s suggestions, that some constructive content would be worthwhile. I’ll try …

So I would like to see more discussion about the poor quality of teacher education at uni. The quality of primary mathematics teaching is particularly worrying. I think this is related to to raising the profile of Mathematics (NOT STEM/STEAM) in general.

My area of study is the use of manipulatives at high school level so that is something I would like to discuss too.

Vicky, I know that you did not want to hear about STEM, but … your comment about STEM/STEAM reminded me of something that bugs me. There is a lot of talk about getting more female students into STEM course/careers. In fact we are very successful in this regard. University courses in medicine, dentistry, physiotherapy, psychology, vet science are dominated by women. Of course those arguing for more women in STEM do not mean these professions. Are they not science? My theory is that already many women are entering STEM professions – it’s just that they are choosing the ones that pay well.

One of our Year 10 students – a very capable student – decided to take her work experience in a primary school. When she reported back at an assembly, she said that is was very helpful – she had decided that teaching was not for her. She went on to say to the teachers present, “I don’t know how you put up with it.”

I confess that I do not know anything about the use of manipulatives at high school – but some of my students could give lessons on this.

Thanks, Vicky. Of course “poor quality” doesn’t begin to describe it, but teacher training is strategically very difficult to attack. The thing is, although there are systemic reasons for the poverty, any proper analysis must be done subject-by-subject and lecturer-by-lecturer. That information is difficult for me to obtain, and also obviously any critique could become personal. It is easier to attack “ACARA” and “VCAA” than it is, Professor Twit. But, if people email materials from education subjects, I’m happy to consider it.

I’m happy to consider you writing on manipulatives for this blog, or you can be more specific about what/how you might want the discussion to me. Feel free to email me to discuss further.

Yes Marty

Is it possible to convince students and teachers that performance in VCE is not so important? If a student wants to do a university STEM course then it just a matter of convincing some admissions officer to take them. As a former such officer I know institutions are under pressure to accept almost anyone. (Those applicants who I thought had no chance of succeeding at Victoria and I knocked back would often find a place elsewhere.) So for a secondary school student the best long-term strategy is to prepare themselves for what is required in tertiary courses. Current ACARA courses fail this. I see the need for a site that can give guidance on that. And a collaboration between teachers experienced in schools and those in universities or TAFEs – a hefty part of the readership of this site – could start the discussion on what is needed.

Let me suggest what aspects I would push in such a discussion.

My career was mainly involved with teaching students of engineering. For this I view mathematics as a language required to understand other subjects. Becoming fluent in a language requires extensive practice, a shallow pass through a topic is not enough. For example, as well as the general principles of coordinate geometry, I would like a student to recognise immediately that represents an ellipse. And I find students actually enjoy repeatedly returning to a topic from different angles. I would ban formula sheets and cheat sheets in tests to promote this fluency. But in the distance I hear the Huns at the gates shouting “stop rote learning”.

More generally, I like when teaching to relate each topic to a wider cultural context. For example I was recently teaching negative numbers for the first time. Long puzzled by our silly notation for such things, I was led to looking at the introduction of double entry accounting and how (maybe) that gave us the modern notation. Relax Marty, not looking to some fantasy about the achievements of our first nation peoples.

Tom, you’re hired. I’ve already suggested to you, over a coffee, that you might write something for this blog.

My memory is that we discussed the possibility of a separate site to provide a more positive critique. I actually booked a WordPress site. But then I became busy saving the world which I thought should take priority. Sorry. [Strange: Hugh Hunt, the Deputy Director of the Centre for Climate Repair at Cambridge tracked me down via this site.]

I’m happy to contribute but will not as much as I hoped.

Here is a suggestion. We pick a topic and investigate it thoroughly. Should it be taught and if so to what extent and which parts, and to what age students? How does it relate to other subjects, or to the world at large? Are there later topics that depend on this one? Does it require complete understanding or should we be content to use it as a functional black box? (eg the formula for solving quadratics) Reports from people who have taught the topic, what works and what fails? Should it be examined and if so how? The person who picks the topic gets to make an initial submission and to summarise the contributions at the end. We concentrate on one topic at a time; the alternative is chaos. So the convenor (Marty) selects the next topic.

OK, sure, I’m happy to give it a go, launch a new series. Except, I’m not sure about your final sentence. I’m happy for anyone to choose the next topic, including the first topic. (If more than one topic arises, I’m then happy to decide if need be.)

So, what might be the first topic?

Polynomials.

You can go to great lengths and depths but it is also a topic with early entry points if you consider solving linear equations to be a form of polynomial algebra.

That strikes me as an excellent choice. Let me have a think of how to frame a post for the discussion, and we’ll get going.

This will be interesting. I have never thought of polynomials as a topic in its own right. Just as something that pops up everywhere.

Marty. Can we run these discussions under a different name? I would like some separation from BADMathematics! (Have we talked about this before? My memory is failing.)

Sure. If you run it. I’d then flag the new posts here as well.

I was thinking “polynomials” as an idea, not as a topic. The distinction is a bit abstract so allow me to elaborate on what I was thinking…

Polynomials are a subset of the algebra universe and polynomial algebra has a different set of “rules” to say “numbers” when students first discover the idea(s).

Polynomials can be added, subtracted, multiplied, divided and can have factors, just like numbers, but in order to fully appreciate this, you first have to be very clear on what those operations actually mean.

The topic “polynomials” is a pretty big one in Methods Units 1 to 4 and is largely (from what I have seen) presented as a grab-bag of “skills” with little to know consideration as to why the “rules” of polynomial algebra actually exist. The overlap with graphs of functions seems to be more-or-less assumed rather than carefully constructed and the examination questions at Units 3&4 seem to test button-pushing and algebraic manipulation more than anything else. Throw in some really basic calculus (OK, in Specialist it gets a bit more interesting, especially with the inclusion of complex numbers) and there is half the exam paper some years.

So… a discussion about what is actually important, what SHOULD be examined and HOW it could be assessed I think could be a really interesting pursuit.

I think we agree. “Polynomials” could cover a ton of ideas and smaller topics over various year levels (and uni level). But it’s up to tom how it’s presented.

Tom,

Many students already realise that ATAR is not especially important. “A large proportion (around 60 per cent) of domestic undergraduate university offers are reported as non-ATAR/non- Year 12” (Pilcher, S. & Torii, K. 2018).

On the other hand, when I talk to high school students, they have ambitions that require a high ATAR, but they know so little about the university system.

Student: “I want to be a physiotherapist for an AFL team.” Me: “What ATAR is required to get into a physiotherapy course?” Student: “75?” Me: “Look it up.” Student (some time later): “Maybe I won’t be a physiotherapist.”

Student: “I want to be an oncologist.” Me: “What does it require to be an oncologist?” Student :”Do a course in oncology.” Me: “Check it.” Student: “I did – I saw it on the internet.” Me: “Check it again.”

Student: “I like history and politics.” Me: “You could do an Arts degree.” Student: “But I can’t even draw.”

Thank you for the invitation Marty. Your blog for some time now has indeed (as you make quite clear) focused largely on the new ACARA Maths curriculum, its shortcomings and associated matters.

The curriculum is but one element of the whole story. And while it’s a necessary condition of successful maths learning and teaching, it’s not of course the only one. In Bad Maths, over the long term, you’ve made reference to all sorts of other matters, many of which are totally suppressed in general discourse, or referred to but little.

An example is the public/private school paradigm that exists in this country, educational apartheid as described by some. The divide is stark in our own state of Victoria, and the consequences are huge. And it’s not something we’re able to do much about!

There are two big picture matters I’d like to write about, and to hear comments from you.

1. Inside schools we still labour under an industrial model that despite superficial changes operates as it always has. Units/groups are created, students to a teacher, and off they go, for the duration, to manage in whatever ways are possible, some good, some bad, others somewhere in between. Some imaginings and some modern implementations are all possible; we do have the capacity to change things. Teachers need the support of their peers; we all function better when we have mentors. Not in the staff room, not at staff meetings, but alongside when working with students, to share and expand knowledge and experience, to grow continuously.

2. The assessment regimes we all labour under. Consider senior secondary level VCE Maths. Teachers of final year 12 Unit 3 and Unit 4 subjects are required to expend enormous amounts of effort to create and administer course-work tasks that accord to strict guidelines. They “must be a part of the regular teaching and learning program and must not unduly add to the workload associated with that program.” Teachers and students both know this is patently untrue. The intention is that students be given credit in their final subject score for work done through the year. Do we know that this indeed works for their benefit? And at what cost to the focus that might otherwise be given to formative assessment activities along the way?

Ian, you’ve raised some great points. The School Assessed Coursework (SAC) system, certainly in mathematics, is long overdue for a scathing and damning review.

A few decades ago there was a system of internal assessment called Common Assessment Tasks (CAT). Some readers might recall this diabolical system. There were many bad things about it and not much good. Bad things included:

1) Authentication of work was a joke.

2) The workload involved marking the CATS was huge.

3) The mental and physical pressure placed on students was enormous. Many students got little to no sleep over a 1-2 week period (since the CATs could be taken home and worked on).

4) Students with tutors were greatly advantaged over students without tutors.

The VCAA (actually it was called the Board of Studies – BOS – back then) eventually – dragged kicking and screaming – abolished the CAT system. And replaced it with the SAC system. Ostensibly SACs removed all of the bad things whilst keeping the ‘good’. It did so (but not completely) for 1) and, to a lesser extent, 3). However, 2) was made much worse (and would be much, MUCH worse if teachers strictly followed the VCAA guidelines). This is despite VCAA saying that it would ease teacher workload. As an aside, maybe there’s a shortage of qualified VCE mathematics teachers because many of those teachers don’t want the huge SAC workload …

The (good) intent of the CAT system was to provide assessment that was different to exams. It did, in a poorly conceived way and at a great cost to teachers and students. The SAC system was intended to also provide assessment different to exams. It does not. It is ludicrous to think otherwise. VCAA refuses to admit this, hence we have the pedantic and officious auditing of SACs in a ridiculous attempt to force square pegs into a round holes.

As far as I know, VCAA had made no attempt to review the efficacy of the SAC system. Under the new Study Design, SACs have been given even greater weighting. I have seen no evidence to support any logical rationale for doing this. Someone’s ignorant little thought bubble, no doubt. It would be interesting if VCAA had the guts to survey teachers on SACs. Marty, does your blog have the capacity to run a simple poll with a couple of simple questions to be responded to?

I might to try and find some time to write a critique of the SAC system, perhaps collaboratively.

Ah the CATs… Methods CATs were bad enough, but those Specialist CATs in some years…

There was one advantage (in theory) and that was consistency. Until of course, one could go to a newsagent in (say) Box Hill and purchase a set of solutions… I never did either as a student or teacher, but none of the cheating was a secret.

SACs theoretically improved things a bit, but in practice we all know this is not the case. I know for sure that when there is only one class for a particular subject (usually Specialist in many schools) finding a teacher to volunteer, knowing they will have to write all the SACs themselves (made all the worse by the “audit” system and don’t get me started on MAV allowing private tutors to attend the SAC workshops…) becomes a rather difficult task.

Some school leaders get this. Many don’t.

VCAA’s latest study design insists that all SACs must now be tech-active does not help the situation either.

Yeah, ain’t nostalgia wonderful.

Indeed, the VCAA is very good at arguing the ‘in theory’ merits of all it does. But not so good in accepting all the ‘in reality’ demerits of what it does.

Yes, it wouldn’t be too hard for the MAV to prevent private tutors attending its SAC presentations, and I’m sure attendances are high enough that this wouldn’t cause a financial impediment.

VCAA has always insisted that SACs be CAS-active. It doesn’t even let you split the SAC into a CAS-free section. And don’t get me started on its dumb-ass and unexplained requirement that SACs must have no writing space for students and marks are not to be shown. (According to an auditor, doing this gives too much help to a student on how much work they need to show for a question. Yep, water finds its own level). VCAA flatly refuses to give a written reason, instead it refers to ambiguous, vague, non-existent statements given in an almost-impossible to find document.

The tide has turned – school leaders who don’t get this may well find themselves without staff in the coming years.

In the meantime, you have idiot education bureaucrats scratching their heads (and getting splinters) wondering why there’s a shortage of teachers for subjects such as Specialist Maths and why student enrolments in Specialist Maths are plummeting (a drop of 32% since 2005). None of them could see an elephant in the room if it trampled them. Their solution is to hold lots of summits, have lots of reviews and ask the clowns that caused the disaster what should be done.

There was a time…

When 3 hours to do 9 questions was the standard for a Mathematics exam.

Those 9 questions usually amounted to a sum total of less than 30 sentences as well and could easily fit on half an A4 page (well, maybe a full page for the Applied Mathematics paper)

When the school Mathematics curriculum (beyond “leaving certificate” which is not part of this discussion) prioritised depth of skill.

When school textbooks had hard covers and would not break a toe when dropped on one. The books themselves were also not republished with new covers every three years to keep the publishers’ profit margins strong.

I suspect the 70-plus year old teachers I had in the 90s were some of the last to remember those times… but geez those teachers were good.

Yeah, a no-nonsense curriculum taught by no-nonsense teachers with a no-nonsense end-of-year exam. A lot changes (little for the good) in less than 40 years.

It would be great if all those old Pure Maths and Applied Maths exams were made available. But I don’t think VCAA will ever do this because comparisons to the ‘modern’ exams could then be readily made. (In fact, I doubt VCAA would even have copies).

They are all in the State Library of Victoria from memory.

I have the 1966 papers somewhere (found them in a pile of books I saved from the dumpster)

Vinculum occasionally publish papers from 100 years ago as well, so someone must have a set (HINT, HINT, HINT for anyone reading this who is willing to share – the copyright will have run out a long time ago…)

RF, and Everyone, the copyright status is unclear. This is being investigated. Please stay tuned, and please don’t post old exams without first checking with The Blog Tsar.

The attached may be of interest. I can’t imagine that there are any copyright problems here – according to p. 2, this is meant to be shared; Marty can overrule me if he thinks fit.

Overruled. email me about it.

It also doesn’t belong on this post. Patience, and stay on topic.

I did those exams. I was taught by those teachers.

Thank you John, it’s always a pleasure to see reference to the historical context, and your description of Common Assessment Tasks and an earlier Board of Studies.

We may in 2023 have missed an opportunity to do things better, and to improve the current system, that which replaced the CATs.

In May 2019 MAV hosted a forum, in a late afternoon session at Brunswick Secondary College, as Stage 1 of consultation to seek feedback on the structures being proposed for VCE Mathematics. Three key background papers were provided, and three curriculum structures A, B and C.

Dr David Leigh-Lancaster, the then VCAA Mathematics Curriculum Manager, conducted the session. He too talked about the historical context, and the need to think now about one or two cycles ahead. It was implied, surely, that it was important to consider the views of teachers. He did tell us that it would be good to find out what students think. To my knowledge nothing of that kind ever happened.

To embark now on preparation to teach the new content (certainly in Specialist Maths) is as always seriously compromised by the overriding matter of school-based assessment. To this writer and teacher, it wrings the very life and joy from the task.

Indeed, VCAA talks (and talks and talks) the talk but never walks the walk.

The net result of all the talk was Foundation Maths. An opportunity to do things (much, much better) was definitely lost. As for school-based assessment, my advice is to do what the accountants of all big business do.

Of possible background interest: I hear that Dr DLL (btw the PhD is in education not mathematics) finished at ACARA at the end of 2021 and is now a private education consultant.

Thanks, ian. I’m happy for you to draft something on either or both. Just email to discuss at any time.

Yes Marty I will start work on some content.

Hi!

I’m new to your blog, so apologies for not knowing all the content you have on here. I stumbled on it while looking for research on the “big ideas” in mathematics and giggled when reading familiar sentiments on the topic.

I haven’t read all the comments above, but I would love some of your insight on the following crappy things about maths:

1) what are your opinions of maths textbooks? Specifically, the textbook itself, what it presents, the questions they ask, the theory they offer, and the digital platforms that accompany them (e.g. Cambridge, person, Mathspace, maths pathways, education perfect, Edrolo). Are there any that are less crappy than the others?

2) speaking of the digital space… in your opinion, what should the purpose of the digital mathematics space be? What should it look like, sound like, evoke, and be, if it should be at all? (youtube, TikTok, Geogebra, Mathigon). And how involved should the digital space creep into teaching?

3) How can you not be a crap maths teacher? Who should maths teachers be looking at for this? For example, in Craig Barton’s book, “how I wish I taught maths”, what do you rate on a scale from ‘Eh, not so crappy’ to ‘absolute dog shit’?

4) are there any non-crappy math curriculums out there in the world??? I commented earlier about the Texas standards (which aren’t perfect), but I would love your analysis of Texas or any other standards out there in the world and how they compare to the Aussies.

5) what are the top 5ish research papers/blogs/books that people involved with maths education should read when making/producing/involving themselves with maths education? Especially for the people who will be influencing the young people. I’m also thinking of the people publishing maths content for the masses, creating curricula, education policymakers, and the teachers who are in the thick of it.

6) Standardised test. Are they ever a good indicator of maths capability? For example, standardised tests in Texas public schools are high-stakes and affect your pay as a teacher, how much money a school will get from the gov’t, and affects who is hired, fired and promoted as a teacher, principal, and superintendent. Too many failures can also result in a school shutting down. On the other hand, if students “fail” the test (thanks to George W.), they don’t get ‘left behind’. Students move onto the next grade but have other consequences, such as extra maths classes, fewer extracurricular classes (bye-bye art and music classes), and before and after school maths tutoring.

Hi texas, and welcome. What this blog desperately needs is a front page, so that people can find their way around. It’s on the list …

You have some great questions, and I don’t have many great answers. But it would be good to post on them, as questions for commenters to then contribute ideas. I’ll do that soon, but in brief:

1) In Australia, the texts and platforms are pretty much all garbage. There are just a few exceptions of which I’m aware. But your question is very good: what are the accessible resources, from anywhere?

2) I cannot see any use for digital resources, but of course I’m a Luddite. Still, I’m happy to post the question, even if I’ll be arguing for the Negative.

3) An important but somewhat amorphous question. It is not of so much interest to me, and in fact the question frustrates the hell out of me: it is maddening to have individual teachers knock themselves silly trying to make up for institutional failure. I simply want to go to war. Nonetheless, we are where we are, and teachers now have hungry students now.

4) There are obviously better (Singapore) and worse (Australia) curricula, but it would be worthwhile discussing why: why does this instruction work and that one fail?

5) Similar to (1), I have few suggestions, but others may have more.

6) This one seems of less interest. What is required is a culture of learning: with that, a proper appreciation of testing follows naturally; without that, who cares?

I’ll do some posts on your some of your questions soon.

I taught in the IB space for a fair chunk of my career – the course is good (I think it was better pre-2010ish) and the curriculum documents are easy to understand.

IB textbooks are a mixed bag though. When I was teaching the course in a school with leaders that really cared… we had a library of books the teachers would use because different books did different topics well. Finding books suitable for non-English first language students was also a priority. Unfortunately, these books are probably long out of print.

In Australia, the NSW exams seem to be the most… non-crap in the sense that they are not pretending to be something they are not; the questions are clear but challenging (and freely available online). The IB papers you have to pay for (about $3USD per paper) but they are nice to work through and remember a time when things were different…

As to how to be a non-crap teacher… when you work it out, let me know!

Re: 5).

I would highly recommend Greg Ashman ‘Filling the Pail’ for inclusion in the top 5.

Given the paucity of detail in the Study Design, I think a blog post that archives VCAA policy, provides clarifications, precedents, definitions and elaborations etc. (based on what is said in Examination Reports, VCAA documents etc (*)) would be very useful. Especially since VCAA sometimes suffers from memory loss.

Example of a precedent:

Using the second derivative test to establish the nature of a stationary point is acceptable:

Mathematical Methods 2021 Examination 1 Report Q8(b) (p9).

Example of a clarification:

In a ‘show that’ question, students are required to provide a detailed progression to the given answer:

Mathematical Methods 2021 Examination 1 Report Advice to Students p2.

Example of an ‘elaboration’ (or definition):

Diagonal (aka slant, oblique) asymptotes:

A linear asymptote that is not vertical or parallel to the axes.

A line is a diagonal asymptote of the function as if

.

Similarly when : .

If then the line is a horizontal asymptote.

Special case: If is a rational function and where (such a form might be obtained using polynomial division), then is a diagonal asymptote of .

IF is a diagonal asymptote of as then

1) .

2) .

3) .

Similar equations hold IF is a diagonal asymptote of as .

An elaboration/definition could also be written for (VCAA’s definition of) a point of inflection.

Such a post would have a similar format to MELting Pot: The Methods Error List.

There would be a blog post for Methods and a separate post for Specialist.

I realise that this suggestion might be deemed too unwieldy to be practical. Or it might not be considered to be in the spirit of these blog posts. But I think it would be an excellent resource (not the least reason being to keep the bastards honest). Anyway, it’s just a suggestion.

* Personal correspondence regarding VCAA policy would not be acceptable.

Thanks, JF. It’s an interesting suggestion. There’s no question that VCAA’s shifting, semi-visible policies are frustrating. But I think, as you consider, the idea may be too unwieldy. The good thing about the error lists is that what goes into them is (relatively) clear cut: an error is an error. I’m open to the idea of other “list” posts, but unless the criteria for inclusion were very tight, I think it’d be painful to maintain, and of limited use.

I can’t resist returning another VCAA policy to prominence:

From the 2007 Exam 2 Report Q4 part (c)(i):

“When the inverse function is asked for, the domain must be given. Students will be penalised in the future if the domain is left out. If only the rule for the inverse function is asked for, the domain does not have to be given.”

By the way, notice the wording:

“… rule the inverse function …”

rather than the idiotic ‘… rule the inverse function …’ that has been used in the exams for the last few years.

Some good examples JF. But I must disagree with your consequence 2) of the definition of a diagonal asymptote. The limit of as gives us no value for the limit of . Consider for example . I guess you are correct if we stick to rational functions.

Hi Tom.

I must disagree with your disagreement. For your example:

, say, has the oblique asymptote as :

.

And we note that , (using a CAS or ‘by hand’ and the sandwich theorem) as expected.

2) is correct for any function that has a diagonal asymptote (including rational functions, of course).

Note: 1), 2) and 3) are conditional on a diagonal asymptote existing. (And I should point out that the converse is NOT true).

The e (ha ha) for an elaboration on diagonal asymptotes is that it enables the existence (or lack thereof) of a diagonal asymptote to be (and the equation calculated when it exists) for function, not just rational functions. This also provides a context for a good application of a CAS.

I think it’s a disgrace that the study design implicitly wants specialist maths students to simply re-write a rational function (using long division) and then blindly state the linear term as the diagonal asymptote without any understanding whatsoever. (The of why this works is trivial once the definition is available). This is exactly the sort of crap that is encouraged in Maths Methods in the context of horizontal asymptotes.

John I agree with your comments about the study design. The issue I raised is a technical matter that is best discussed face-to-face. But we need to be scrupulously rigorous on this site. So here goes.

The derivative of here is . Here the first term has oscillations that increase without limit and the remaining terms approach 2. In case you are not convinced, I attach graphs of (i) the original function and (ii) .

The general principal is that differentiation is not a continuous operation. I designed this function as an extreme example.

[First image missing. Please see follow up.]

Hi Tom.

There is a diagonal asymptote as x approaches infinity.

The oscillations you refer to are occurring as x approaches infinity, and I agree that there is no diagonal asymptote as x approaches infinity.

Oops I uploaded both images but it is only showing the second. And it seems the edit function does not allow me to change the uploads. So here is the other image.

Have just seen your latest post John. Here there is an oblique asymptote in the sense that as . But the curve gets increasingly crinkly. So one may decide not to call such a thing an asymptote. In that case your theorem is correct 🙂

Hi Tom.

I previously agreed that there is no diagonal asymptote as x approaches infinity. I was wrong. y = 2x + 1 is a diagonal asymptote as x approaches infinity since

.

The limit is trivially proved using the sandwich theorem since .

Re: "But the curve gets increasingly crinkly." I don't see that this matters. What matters is that the limit is zero. y = 2x + 1 is a diagonal asymptote, not withstanding the ‘crinkly’ approach of the function.

But I agree that 2) fails in this case. You are clearly correct since the limit for 2) does not exist. I was careless. I should have added the proviso that 2) gives the gradient of the diagonal only when the limit exists.

So I think that if a diagonal asymptote y = mx + c exists, then 2) will either exist (and give the value of m) or will not exist (in the sense of 'infinite oscillations'), in which case 1) must be used. I wonder whether there's a case where the diagonal asymptote exists but the limit in 1) does not exist …?

There is also a diagonal asymptote as x approaches infinity: y = 2x + 1. I proved this in my earlier post (January 15, 11:42 PM):

.

Again, the required limit is trivially proved using the sandwich theorem since

.

There is no trouble using either 1) or 2) to get m = 2.

PS – It is probably worth reminding readers of the following blog post here: https://mathematicalcrap.com/2019/02/13/witch-8-oblique-reasoning/

(Hard to believe that was nearly 4 years ago!)

In fact, it can be proved that if exists then

.

That’s the condition I should have attached to 2).

Hi again John

Ok L’Hopital eh?

Let’s run this to earth; we are close to full agreement. I don’t think that we need the derivative to have ‘infinite oscillations’. I chose such a function just to make a strong point. But I could have chosen where oscillates between 0 and 4?

Hi Tom.

Yes, this example also has a diagonal asymptote .

The limits in 2) do not exist but the limits in 1) and 3) exist.

So if you suspected had a diagonal asymptote , you’d calculate m and c using 1) and 3).

And then you’d prove that was indeed a diagonal asymptote by calculating .

Re: L’Hospital’s Rule.

There is a version of l’Hospital’s Rule that I don’t think is met very often at the undergraduate level. See page 109 (of Rudin’s Principles of Mathematics) below:

Click to access rudin.pdf

Proof of the previous theorem I posted is trivial using this version of l’Hospital’s rule.

But here is an example that does NOT have an oblique asymptote (superficially the behaviour as x –> positive infinity might seem similar to your example above):

.

It is simple to use the definition and prove there is no diagonal asymptote. This is something I think Specialist Maths students ought to be able to do.

There has been an insidious de-emphasising of limits in the VCE curriculum over the last couple of decades, and the new Study Design continues this. I don’t know how students are expected to a lot of what is stated in the new Study Design without limits.

Joining the party very late. As this blog has an appreciable number of non-maths topics, a suggestion for one may not be out of place. I recently thought again about a footnote (labelled “Disclosure”) to https://mathematicalcrap.com/2022/03/06/keith-devlin-makes-an-idiot-of-himself/#more-15873 in which Marty refers a piece https://www.maa.org/external_archive/devlin/devlin_06_07.html by sometime-subject of his blog, Keith Devlin. In that piece, Devlin makes mention of letters to The Age in response to an early piece by Marty in the same newspaper, and seems to sum them up by saying that “History, physics, social studies come to mind immediately [for teaching logical thinking].” The order here may just be alphabetical but I would like to know what Marty and others have to offer on the question of history as venture for mathematicians. This question seems to have some relevance also as mathematics, with its formal language traits, has appeal to people not necessarily strong in the natural sciences (a weakness I share). In the preceding I would want to exclude the study of the history of mathematics, although the line may be drawn differently — this would not seem to be the main point anyway.

Hi Christian. Happy to think about anything, but can you clarify? I don’t know what “history as a path of mathematics” means.

Hi Marty, first, I am sorry for having changed my wording before the final version – I recall that I changed “history as a path of mathematics” to the above “history as venture for mathematicians”. None of which is very clear at all, which is why your question is entirely warranted. I admit this was (and maybe remains) a reflection of my still not very clear perception on what to ask here. Let me try as follows to fill this gap:

Keith Devlin, the letter writers to the Age, and (hopefully) others seem to think that history is a subject that can be addressed by means of logic (which I think is also a bit easier than with social sciences that he also mentions). For otherwise, how can you learn logic from it, as he and the letter writers posit? So that means that, if a mathematician doesn’t have any ideas for, say, his or her next research paper, and is thinking about doing something more immediately connectable to outside world, then history may be a choice. I, for one, can attest to history having been one of my earliest interests in early youth or even childhood; although there may be some reason for this due to the harrowing nature of the history of my country, Germany.

In the ideal case, historical writing may provide not only a solace for the bad stuff that is going on in the mathematical profession, or in the teaching of mathematics as is somewhat more relevant to your blog; it may also be a source of fresh thinking within maths as one sees the logic arise through careful connection of these and those historical facts, or at least finding the fallacies in others’ arguments. But maybe this is a bit far-fetched.

I am only aware of a tiny handful of mathematicians’ writings on history (minus history of mathematics as I noted) – to give a list of this kind may also make an interesting post.

Thanks, Christian. I’m still a little confused, although I sort of get it.

I’ll leave aside the issues with Devlin’s sniping at me. Suffice to say that my article was clumsy and that Devlin is a self-promoting asshole, who may be fine with the logic of history but is totally ignorant of the logic of world conflict.

History is fascinating and, I think, a really hard gig. In mathematics, for example, we teach the cleaned up version of ideas that have been developed over thousands of years. (Which is the fundamental reason why inquiry “learning” sucks balls.) To read about the clumsy (with hindsight) way in which such ideas were teased out and employed is astonishing and impressive and frightening. When Burkard and I were writing our Mathsmasters column, I was always looking for history-based topics, anniversaries of more obscure mathematicians and the like, and I read a *lot* of history. It was great, and it was often really hard work just to understand even the secondary sources. It really brought home to me the incredible hard work of guys like Smith and Cajori and Neugebauer.

As for ideas for this blog, I have a few history-based items on my to-do list, and I’m more than happy if someone wants to contribute a historyish post. Separately, what might be good is a post for history of mathematics resources, if only so people would shut up about The Crest of the Goddam Peacock. Now on my to-do list.

Since I was referring to Keith Devlin in my first post in this thread, and since you mention him as well, I should make it clear that my above ‘contemplations’, so to speak, of history as maths-related blog material was not based on the utterances of Devlin. At the most, I would say that your mention of the events back then, with Devlin’s sniping, got me to think about the relation of history to maths, given also some other ties that I had seen earlier. Which is about the only credit I would be willing to give Devlin on this.

I definitely agree that history is a hard gig. Views on, say, the role of driving events are oftentimes still in flux, and maybe will often remain so for a long time to come, whereas in maths, things are relatively (or even very) stable once they have matured. (One example for what may count as a “cleaned up version” in your sense, although not over the space of thousands of years, that comes to my mind is the nowadays decreased role of the determinant in linear algebra and its replacement with linear independence considerations. I should acknowledge that I read words to this effect somewhere.) I would still think that the orderly way that mathematics uses to arrange thoughts is of some help in making this gig a little bit less hard. At least, to put it in the negative, I believe that a confused account of some historical matter would be painstaking to read in particular for many people used to (or able to appreciate) good mathematical writing.

Thank you for your mention of references and I look forward to your foreshadowed posts.

Just a note that the issue of a historical approach to teaching maths is continued in the Teaching Maths site:

Thanks, tom. I’ll also do a separate post on it. (Sorry for the delay: not working on all cylinders right now.)