Tootering Your Own Horn

Eddie Woo is reportedly concerned about private tutoring. His warning comes courtesy of SMH‘s education editor, Jordan Baker, in an article entitled ‘Be very, very careful’: Experts raise warning on private tutoring. The article begins,

Maths teachers including high-profile mathematician Eddie Woo have sounded an alarm on private tutoring, warning that bad tutors could be “fatal” to students’ future in the subject.

Eddie said it, so it must be true. And, Baker quotes another expert, the chief executive of the Australian Tutoring Association, Mohan Dhall:

I am absolutely dismayed at the lack of creativity and lack of real-world applicability most tutors bring to maths …The main problem stems from this idea that they focus on the outcome – ‘this is what students need to know’, rather than ‘this is what kids need to learn to be interested and engage’.

Finally, Baker quotes expert Katherin Cartwright, a lecturer in mathematics education at The University of Sydney. Cartwright, according to Baker, is concerned that poor tutoring could lead to a lack of confidence:

If it becomes about skill and drill and speed, and it becomes an anxious, emotional issue for students, then they are not going to like it, and they will not want to take it further.

Yep, of course. The most important consideration when framing an education is to be sure to never make a student anxious or emotional. Poor, fragile little petals that they are.

Baker’s fear-mongering is nonsense. Almost every line of her article is contentious and a number contain flat out falsehoods. Beginning with the title. Woo and Dhall and Cartwright are “experts” on the issues of tutoring? According to whom? Based on what? Perhaps they are experts, but Baker provides no evidence.

OK, we could concede Baker’s point that Eddie is a mathematician. Except that he isn’t and we don’t. Not that it matters here, since most mathematicians are unlikely to know much about the role of tutoring in Australian education. But the false and pointless puffery exemplifies Baker’s unjustified appeals to authority.

What of the declared concerns of Baker’s “experts”? Cartwright is supposedly worried about “skill and drill and speed”. This in contrast to school, according to Baker:

Most schools no longer emphasise speed and rote learning when teaching maths, and now focus on students’ understanding of key concepts as part of a concerted effort to improve engagement in maths across the system.

This hilarious half-truth undercuts the whole thrust of Baker’s article. It is true that many schools, particularly primary schools, have drunk the educational Kool-Aid and have turned their maths lessons into constructivist swamplands. But that just means the main and massive job of competent Year 7 maths teachers is to undo the damage inflicted by snake-oilers, and to instil in their students, much too late, an appreciation of the importance of memory and skill and efficient technique. Such technique is critical for formal success in school mathematics and, which is sadly different, for the learning of mathematics. Baker seems entirely unaware, for example, that, for better or worse, Year 12 mathematics is first and foremost a speed test, a succession of sprints.

As for Dhall, does he really expects tutors to be more offering of “creativity” and “real-world applicability”? Dhall seems blissfully unaware that most “real-world” applications that students must suffer through are pedagogically worthless, and are either trivial or infinitely tedious. Dhall seems unaware that some subjects have warped “applicability” into a surrealist nightmare.

And Eddie? What worries Eddie? Not much, as it happens, but too much. Eddie’s quoted comments come from a NSW podcast, which appears to have been the genesis of Baker’s piece; stenographic fluffing is of course the standard for modern reportage, the cheap and easy alternative to proper investigation and considered reflection.

Eddie’s podcast is a happy public chat about teaching mathematics. Eddie is demonstrably a great teacher and he is very engaging. He says a number of smart things, the half-hour podcast only being offensive for its inoffensiveness; Eddie, or his interviewer, was seemingly too scared to venture into a deep public discussion of mathematics and the sense of it. The result is that, except for the occasional genuflection to “pattern”, Eddie may as well have been talking about turtle farming as teaching mathematics.

Eddie’s comments on tutoring are a very minor part of the podcast, a response in the final question time. This is Eddie’s response in full:

When I think about external tuition – again just like before this is a really complex question – there is tuition and then there is ‘tuition’. There is some which is enormously helpful to individual students to come in at a point of need and say “you have got gaps in your knowledge, I can identify that and then help you with those and then you can get back on the horse and off you go, fantastic”. There are other kinds of tuition which are frankly just pumping out an industrial model of education which parents who are very well intentioned and feel like they cannot do anything else, it is like “at least I can throw money at the problem and at least they are spending more time on maths hopefully that will help”. Maybe it does and maybe it is making your child hate maths because they are doing it until 9pm at night after a whole day? That to me is heartbreaking.

I think that students need to be very, very careful and parents need to be very, very careful about how they experience mathematics. Because yes the time is a worthwhile investment, it is a practical subject, but if you are just churning through, often tragically learning things which actually are just machine processes. I have students come to me and they say “I can differentiate, I am really good at that, I am only fifteen years old”. You don’t need to know what differentiation is, but they come to me with this ability to turn a handle on this algorithm this set of steps. Just like me; I don’t know how to bake, but I can follow a recipe. I have no idea what baking powder does or why 180 degrees Celsius is important but I can follow steps. That is okay for a cake because you can still eat it at the end, but that is fatal for mathematics because you don’t know why you are doing any of the things that you are doing. If that is what you are, you are not a mathematician, you are a machine and that is not what we want our children to become. We have to be careful.

Eddie says plenty right here, touching on various forms of and issues with tutoring, and school teaching. The issues do not get fleshed out, but that is the nature of Q & A.

Eddie also gets things smugly wrong. Sure, some tutoring might be characterised as “industrial”. But more so than schools? How can mass education not be industrial? This isn’t necessarily bad: mostly, it just is. Unless, of course, little Tarquin’s parents have the time and the money to arrange for individual or small-group lessons with an, um, tutor.

All the concerns Baker and her experts raise about tutoring apply as much or more so to school education and, as a matter of business necessity, are largely a reflection of school education. And, how do tutors and tutoring companies deal with this? Some well, some poorly. But mostly with industry, which is not a dirty word, and with good and honest intent.

Baker notes the underlying issue, seemingly without even realising it:

However, Australian students’ performance in maths has either stalled or declined on all major indicators over that period, and academics have raised concerns about students arriving at university without the maths skills they need.

Why do parents employ tutors? Having enjoyed and suffered forty years of tutoring, in pretty much all its forms, we can give the obvious answer: there’s a zillion different, individual reasons. Some, as Eddie suggests, are looking for a little damage control, the filling of gaps and a little polishing. Some, as Eddie suggests, think of mathematics, falsely, as a syntactic game, and are looking for lessons in playing that dangerously meaningless game. Some believe, correctly or otherwise, that their teacher/school is responsible for little Johnny’s struggling. Some are trying to get darling Diana into law school. Some are hothousing precious little Perry so he/she can get a scholarship into Polo Grammar or Mildred’s College for Christian Ladies.

But, underlying it all, there is one obvious, central reason why parents employ tutors: parents are unsatisfied with the education their child receives at school.

Why are parents unsatisfied? Are they right to be? Of course, it depends. But, whatever the individual analyses, the massive growth of the tutoring industry indicates a major disconnect, and either a major failing in schools’ performance or a major blindness in parents’ expectations, or both.

That would be a much more worthwhile issue for Baker, and everyone, to consider.

Feynman on Modernity

We plan to have more posts on VCAA’s ridiculous curriculum review. Unfortunately.

Now, however, we’ll take a semi-break with three related posts. The nonsensical nature of VCAA’s review stems largely from its cloaking of all discussion in a slavish devotion to “modernity”, from the self-fulfilling prediction of the inevitability of “technology”, and from the presumption that teachers will genuflect to black box authority. We’ll have a post on each of these corrupting influences.

Our first such post is on a quote by Richard Feynman. For another project, and as an antidote to VCAA poison, we’ve been reading The Character of Physical Law, Feynman’s brilliant public lectures on physical truth and its discovery. Videos of the lectures are easy to find, and the first lecture is embedded above. Feynman’s purpose in the lectures is to talk very generally about laws in physics, but in order to ground the discussion he devotes his first lecture to just one specific law. Feynman begins this lecture by discussing his possibly surprising choice:

Now I’ve chosen for my special example of physical law to tell you about the theory of gravitation, the phenomena of gravity. Why I chose gravity, I don’t know. Whatever I chose you would’ve asked the same question. Actually it was one of the first great laws to be discovered and it has an interesting history. You might say ‘Yes, but then it’s old hat. I would like to hear something about more modern science’. More recent perhaps, but not more modern. Modern science is exactly in the same tradition as the discoveries of the law of gravitation. It is only more recent discoveries that we would be talking about. And so I do not feel at all bad about telling you of the law of gravitation, because in describing its history and the methods, the character of its discovery and its quality, I am talking about modern science. Completely modern.

Newer does not mean more modern. Moreover, there can be compelling arguments for focussing upon the old rather than the new. Feynman was perfectly aware of those arguments, of course. Notwithstanding his humorous claim of ignorance, Feynman knew exactly why he chose the law of gravitation.

This could, but will not, lead us into a discussion of VCE physics. It suffices to point out the irony that the clumsy attempts to modernise this subject have shifted it towards the medieval. But the conflation of “recent” with “modern” is of course endemic in modern recent education. We shall just point out one specific effect of this disease on VCE mathematics.

Once upon a time, Victoria had a beautiful Year 12 subject called Applied Mathematics. One learned this subject from properly trained teachers and from a beautiful textbook, written by the legendary J. B. “Bernie” Fitzpatrick and the deserves-to-be-legendary Peter Galbraith. Perhaps we’ll devote some future posts on Applied and its Pure companion. It is enough to note that simply throwing out VCE’s Methods and Specialist in their entirety and replacing them with dusty old Pure and Applied would result in a vastly superior, and more modern, curriculum.

Here, we just want to mention one extended topic in that curriculum: dynamics. As it was once taught, dynamics was a deep and incredibly rich topic, a strong and natural reinforcement of calculus and trigonometry and vector algebra, and a stunning demonstration of their power. Such dynamics is “old”, however, and is thus a ready-made target for modernising zealots. And so, over the years this beautiful, coherent and cohering topic has been cut and carved and trivialised, so that in VCE’s Specialist all that remains are a few disconnected, meat-free bones.

But, whatever is bad the VCAA can strive to make worse. It is clear that, failing the unlikely event that the current curriculum structure is kept, VCAA’s review will result in dynamics disappearing from VCE mathematics entirely. Forever.

Welcome to the Dark Ages.

VCAA Puts the “Con” into Consultation

As we have written, the Victorian Curriculum and Assessment Authority is “reviewing” Victoria’s senior secondary maths, which amounts to the VCAA attempting to ram through a vague and tendentious computer-based curriculum, presented with no evidence of its benefit apart from change for the sake of change. Readers can and should respond to the VCAA’s manipulative questionnaire before May 10. In this post we shall point out the farcical nature of VCAA’s “consultation”, as evidenced by VCAA’s overview and questionnaire.

The overview begins by framing VCAA’s review with the following question:

What could a senior secondary mathematics curriculum for a liberal democratic society in a developed country for 2020–2030 look like?

This is peculiar framing, since it is difficult to imagine how a society being “liberal” or “democratic” or otherwise has any bearing on the suitability of a mathematics curriculum. Why would a good curriculum for China not also be good for Victoria?

One could easily write off this framing as just jingoistic puffery; neither word reappears in VCAA’s overview. It is, however, more insidious than that. The framing is, except for the odd omission of the word “suitable”, identical to the title of the Wolfram-CBM paper promoting “computer-based mathematics” in general and Wolfram-CBM in particular. This paper is the heavy propaganda gun VCAA has procured in furtherance of its struggle to liberate us all from the horrors of mathematical calculation. Though the Wolfram-CBM paper never states it explicitly, this makes clear the purpose of the framing:

“[L]iberal” and “democratic” and “developed” amounts to “rich enough to assume, demand and forever more have us beholden to the omnipresence of computers”.

The VCAA overview continues by noting the VCAA’s previous review in 2013-2014 and then notes the preliminary work undertaken in 2018 as part of the current review:

… the VCAA convened an expert panel to make recommendations in preparation for broad consultation in 2019.

Really? On whose authority does this anonymous panel consist of experts? Expert in what? How was this “expert panel” chosen, and by whom? Were there any potential or actual conflicts of interest on the “expert panel” that were or should have been disclosed? How or how not was this “expert panel” directed to conduct its review? Were there any dissenters on this “expert panel”?

The only thing clear in all this is the opacity.

The overview provides no evidence that VCAA’s “expert panel” consists of appropriately qualified or sufficiently varied or sufficiently independent persons, nor that these persons were selected in an objective manner, nor that these persons were able to and encouraged to conduct the VCAA review in an objective manner. 

Indeed, any claim to breadth, independence or expertise is undermined by the constrained formulation of the questionnaire, the poverty of and the bias in the proposed curriculum structures and the overt slanting of the overview towards one particular structure. Which brings us to the issue of consultation:

There is no value in “broad consultation” if discussion has already been constrained to the consideration of three extremely poor options.

But, “consult” the VCAA will:

The VCAA will consult with key stakeholders and interested parties to ensure that feedback is gained from organisations, groups and individuals.

Well, great. The writer of this blog is a keenly interested stakeholder, and an individual well known to the VCAA. Should we be waiting by the phone? Probably not, but it hardly matters:

The VCAA has provided no indication that the consultation with “key stakeholders” and “interested parties” will be conducted in a manner to encourage full and proper critique. There is very good reason to doubt that any feedback thus gained will be evaluated in a fair or objective manner.

The overview then outlines three “key background papers” (links here). Then:

… stakeholders are invited to consider and respond to the consultation questionnaire for each structure.

Simply, this is false. Question 1 of VCAA’s questionnaire asks

Which of the proposed structures would you prefer to be implemented for VCE Mathematics?

Questions 2-8 then refer to, and only to, “this structure”. It is only in the final, catch-all Question 9 that a respondent is requested to provide “additional comments or feedback with respect to these structures”. Nowhere is it possible to record in a proper, voting, manner that one wishes to rank the Wolfram-CBM Structure C last, and preferably lower. Nowhere is there a dedicated question to indicate what is bad about a bad structure.

The VCAA questionnaire explicitly funnels respondents away from stating which structures the respondents believe are inferior, and why.

The good news is that the manipulativeness of the questionnaire probably doesn’t matter, since the responses will be presumably just be considered by another VCAA “expert panel”.

The VCAA overview gives no indication how the responses to the questionnaire will be considered and provides no commitment that the responses will be made public.

The VCAA overview goes on to provides outlines of the three structures being considered, which we’ll write upon in future posts. We’ll just comment here that, whereas Structures A and (to a lesser extent) B are laid out in some reasonable detail, Structure C looks to be the work of Chauncey Gardiner:

What is written about Structure C in the VCAA overview could mean anything and thus means nothing. 

True, for a “detailed overview” the reader is directed to the Wolfram-CBM paper. That, however, only makes matters worse:

A 28-page sales pitch that promotes particular software and particular commercial links is much more and much less than a clear, factual and dispassionate curriculum structure, and such a pitch has absolutely no place in what VCAA describes as a “blue-sky” review. By giving prominence to such material, the VCAA fails to treat the three proposed structures in anything close to a comparable or fair manner. 

If there were any doubt, the overview ends with the overt promotion of Structure C:

The distinctive proposal … contain[s] aspects which the Expert Panel found valuable … There was support for these aspects, indeed, many of the invited paper respondents [to the 2018 paper] independently included elements of them in their considerations, within more familiar structures and models.

Nothing like putting your thumb on the scales.

It is entirely inappropriate for a VCAA overview purportedly encouraging consultation to campaign for a particular structure. A respondent having “included elements” of an extreme proposal is a country mile short of supporting that proposal lock, stock and barrel. In any case, the cherry-picked opinions of unknown respondents selected in an unknown manner have zero value. 

Though woefully short of good administrative practice, we still might let some of the above slide if we had trust in the VCAA. But, we do not. Nothing in VCAA’s recent history or current process gives us any reason to do so. We can also see no reason why trust should be required. We can see no reason why the process lacks the fundamental transparency essential for such a radical review.

In summary, the VCAA review is unprofessional and the consultation process a sham. The review should be discarded. Plans can then be made for a new review, to be conducted in the professional and transparent manner that Victoria has every right to expect.

Reviewing the VCAA Review – Open Discussion

The VCAA is currently conducting a “review” of VCE mathematics. We’ve made our opinion clear, and we plan to post further in some detail. (We’ll update this post with links when and as seems appropriate.) We would also appreciate, however, as much input as possible from readers of (especially critics of) this blog.

This post is to permit and to encourage as much discussion as possible about the various structures the VCAA is considering. People are free to comment generally (but carefully) about the VCAA and the review process, but the intention here is to consider the details of the proposed structures and the arguments for and against them. We’re interested in anything and everything people have to say. Except for specific questions addressed to us, we’ll be pretty much hands-off in the comments section. The relevant links are

Please, go to it.

The Wolfram at the Door

(Note added 20/4: A VCAA questionnaire open until May 10 is discussed at the end of this post. Anyone is permitted to respond to this questionnaire, and anyone who cares about mathematics education should do so. It would be appreciated if those who have responded to the questionnaire indicate so in the comments below.)

Victoria’s math education is so awful and aimless that it’s easy to imagine it couldn’t get much worse. The VCAA, however, is in the process of proving otherwise. It begins, and it will almost certainly end, with Conrad Wolfram.

We’ve long hoped to write about Wolfram, the slick salesman for Big Brother‘s Church. Conrad Wolfram is the most visible and most powerful proponent of computer-based maths education; his Trumpian sales pitch can be viewed here and here. Wolfram is the kind of ideologue who can talk for an hour about mathematics and the teaching of mathematics without a single use of the word “proof”. And, this ideologue is the current poster boy for the computer zealots at the VCAA.

The VCAA is currently conducting a “review” of VCE mathematics, and is inviting “consultation”. There is an anonymous overview of the “review”, and responses to a questionnaire can be submitted until May 10. (Below, we give some advice on responding to this questionnaire. Update 25/4: Here is a post on the overview and the questionnaire.) There is also a new slanted (and anonymous) background paper, a 2017 slanted (and anonymous) background paper, a 2014 slanted (and anonymous) background paper, and some propaganda by Wolfram-CBM.

In the next few weeks we will try to forego shooting Cambridge fish in the barrel (after a few final shots …), and to give some overview and critique of the VCAA overview and the slanted (and anonymous) background papers. (We hope some readers will assist us in this.) Here, we’ll summarise the VCAA’s proposals.

The VCAA has stated that it is considering three possible structures for a new VCE mathematics study design:

  • Structure A.1 – the same warmed over swill currently offered;
  • Structure A.2 – tweaking the warmed over swill currently offered;
  • Structure B – compactifying the warmed over swill currently offered, making room for “options”;
  • Structure C – A “problem-centred computer-based mathematics incorporating data science”.

What a wealth of choice.

There is way, way too much to write about all this, but here’s the summary:

1. Structure C amounts to an untested and unscripted revolution that would almost certainly be a disaster.

2. The VCAA are Hell-bent on Structure C, and their consultation process is a sham. 

So, what can we all do about it? Pretty much bugger all. The VCAA doesn’t give a stuff what people think, and so it’s up to the mathematical heavy hitters to hit heavily. Perhaps, for example, AMSI will stop whining about unqualified teachers and other second order trivia, and will confront these mathematical and cultural vandals.

But, the one thing we all can do and we all should do is fill in the VCAA’s questionnaire. The questionnaire is calculatedly handcuffing but there are two ways to attempt to circumvent VCAA’s push-polling. One approach is to choose Structure C in Q1 as the “prefer[red]” option, and then to use the subsequent questions to critique Structure C. (Update 25/4: this was obviously a poor strategy, since the VCAA could simply count the response to Q1 as a vote for Structure C.) The second approach is to write pretty much anything until the catch-all Q9, and then go to town. (20/4 addition: It would be appreciated if those who have responded to the questionnaire indicate so below with a comment.)

We shall have much more to write, and hopefully sooner rather than later. As always, readers are free to and encouraged to comment, but see also this post, devoted to general discussion.

The Crap Aussie Curriculum Competition

The Evil Mathologer is out of town and the Evil Teacher is behind on sending us our summer homework. So, we have time for some thumping and we’ll begin with the Crap Australian Curriculum Competition. (Readers are free to decide whether it’s the curriculum or the competition that is crap.) The competition is simple:

Find the single worst line in the Australian Mathematics Curriculum.

You can choose from either the K-10 Curriculum or the Senior Curriculum, and your line can be from the elaborations or the “general capabilities” or the “cross-curriculum priorities” or the glossary, anywhere. You can also refer to other parts of the Curriculum to indicate the awfulness of your chosen line, as long as the awfulness is specific. (“Worst line” does not equate to “worst aspect”, and of course the many sins of omission cannot be easily addressed.)

The (obviously subjective) “winner” will receive a signed copy of the Dingo book, pictured above. Prizes of the Evil Mathologer’s QED will also be awarded as the judges see fit.

Happy crap-hunting.

Obtuse Triangles

Whatever the merits of undertaking a line by line critique of the Australian Curriculum, it would take a long time, it would be boring and it would probably overshadow the large, systemic problems. (Also, no one in power would take any notice, though that has never really slowed us down.) Still, the details should not be ignored, and we’ll consider here one of the gems of Homer Simpson cluelessness.

In 2010, Burkard Polster and I wrote an Age newspaper column about a draft of the Australian Curriculum. We focused on one line of the draft, an “elaboration” of Pythagoras’s Theorem:

recognising that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds

Though much can be said about this line, the most important thing to say is that it is wrong. Seven years later, the line is still in the Australian Curriculum, essentially unaltered, and it is still wrong.

OK, perhaps the line isn’t wrong. Depending upon one’s reading, it could instead be meaningless. Or trivial. But that’s it: wrong and meaningless and trivial are the only options.

The weird grammar and punctuation is standard for the Australian Curriculum. It takes a special lack of effort, however, to produce phrases such as “right-angled triangle calculations” and “generate results”. Any student who offered up such vague nonsense in an essay would know to expect big red strokes and a lousy grade. Still, we can take a guess at the intended meaning.

Pythagoras’s Theorem can naturally be introduced with 3-4-5 triangles and the like, with integer sidelengths. How does one then obtain irrational numbers? Well, “triangle calculations” on the triangle below can definitely “generate” irrational “results”:

Yeah, yeah, \pi is not a “surd”.  But of course we can replace each \pi by √7 or 1/7 or whatever, and get sidelengths of any type we want. These are hardly “triangle calculations”, however, and it makes the elaboration utterly trivial: fractions “generate” fractions, and irrationals “generate” irrationals. Well, um, wow.

We assume that the point of the elaboration is that if two sides of a right-angled triangle are integral then the third side “generated” need not be. So, the Curriculum writers presumably had in mind 1-1-√2 triangles and the like, where integers unavoidably lead us into the world of irrationals. Fair enough. But how, then, can we similarly obtain the promised (non-integral) fractional sidelengths? The answer is that we cannot.

It is of course notable that two sides of a right-angled triangle can be integral with the third side irrational. It is also notable, however, that two integral sides cannot result in the third side being a non-integral fraction. This is not difficult to prove, and makes a nice little exercise; the reader is invited to give a proof in the comments. The reader may also wish to forward their proof to ACARA, the producers of the Australian Curriculum.

How does such nonsense make it into a national curriculum? How does it then remain there, effectively unaltered, for seven years? True, our 2010 column wasn’t on the front of the New York Times. But still, in seven years did no one at ACARA ever get word of our criticism? Did no one else ever question the elaboration to anyone at ACARA?

But perhaps ACARA did become aware of our or others’ criticism, reread the elaboration, and decided “Yep, it’s just what we want”. It’s a depressing thought, but this seems as likely an explanation as any.