A few days ago, the Centre of Independent Studies held a forum, *Ensuring Australia’s Maths Teaching Adds up to Success*. The stars of the show were Eddie Woo and Greg Ashman, and the video of the forum has now been posted.

Of course the main thing to note about this forum was the astonishing decision to not include any women. But, ignoring that disgraceful aspect, the forum is well worth watching. We have our thoughts (of course), but we’ll save them until after people have had a chance to watch and to comment.

### UPDATE (20/05/12)

Prompted by Sir Humphrey’s comment, here are two links to recent CIS publications, which are somewhat relevant to the discussion:

John Sweller – *Why Inquiry Based Approaches Harm Students’ Learning*

Glenn Fahey, Jordan O’Sullivan, Jared Bussell – *Failing to teach the teacher*

### UPDATE (20/05/22)

In the video, Greg Ashman discusses a pointless class activity based upon the game of Greedy Pig (at 21:30). In 2015, Burkard Polster and I whacked the maths300 version of this activity:

Burkard and Marty – *The Statistical Problem of Greedy Pig*

The primary link on CIS’s website (worth reading):

A secondary link embedded in the primary link (definitely worth reading):

A quick read through this yields:

“Report co-author, Glenn Fahey, blames an ideological bias among university-based teacher educators for failing to provide teachers a balanced view on how to teach mathematics effectively. He argues that this is a major factor explaining why teachers are underprepared for the classroom, and ultimately, why student outcomes are declining.”

Which also links to (most definitely worth reading):

I wholeheartedly agree with the assessment of ITE. I would also add that there are students being accepted into (M.Teach) ITE courses who are significantly and profoundly mathematically incompetent (with largely mathematically-irrelevant Bachelor degrees). However, because of the heavy focus (i.e. insistence) on inquiry-based learning (5E, E5 or whatever alphanumeric permutation is the latest constructivist fad), the cracks are sufficiently cavernous such that many wind-up ‘teaching’ mathematics in the classroom. If universities don’t teach teachers how to actually teach mathematics then how can they reasonably evaluate whether their students are able to teach the subject or not? From what I have seen, many M.Teach students are being ‘facilitated’ all the way to the classroom not knowing the subject, let alone how to actually teach it – and the quality of teaching shows.

Thanks, Sir H. I wasn’t thrilled with either report (while agreeing with everything they say). I’ve added links to the post.

In my M.Teach studies at Deakin, I don’t recall any emphasis by lecturers on inquiry-based learning or constructivism (whatever these terms mean) although I remember that one student mentioned constructivism in an on-line discussion forum; there was an assignment about various philosophies about how knowledge in constructed; this is where I became acquainted with the work of T. S. Kuhn which has served me well. Naturally most of the course was about teaching generally rather than teaching mathematics specifically; 4 of the 24 units dealt with teaching mathematics. As I have since learnt, there is much more to being a mathematics teacher than teaching mathematics! One only has to look at the selection criteria for jobs teaching in government schools in Victoria to realise this. My teaching method was double mathematics; I gathered that, of the 300 or so M. Teach. students in my cohort, only two of us chose double mathematics. Since graduating, I have often referred back to my studies in the M. Teach. often to find a reference. Still I did not expect the M.Teach. to give me all that I needed to get started; my first substantial appointment was made over the phone – “No pressure, but can you start tomorrow?”; my previous studies in mathematics are an important part of my background. I have just learnt this week that I have work until the end of the year for which I am grateful. Do I now “know” how to teach mathematics? I have some ideas, but much to learn.

If I may offer an opinion that may offend Math-Ed people… Universities being used to train teachers is not the correct model.

Universities are not there to produce “job ready” workers of the future, they are places of research, higher learning. Should all teachers have university degrees? I’m not going to answer that for all teachers, but in the case of Mathematics teachers: yes, absolutely.

So why is it an issue that Universities are training teachers? Mostly because Universities are very good at marketing themselves to schools saying “come study with us when you graduate” but not so good at asking “what skills do you want us to ensure teaching graduates achieve competency in/with?”

Look at pretty much any ITE course and the “subjects” these teachers are training for and then look at the advertised vacancies in schools nation-wide and it quickly becomes clear that the supply-side and demand-side of the scales are far from equal. Is this an issue? If we need Mathematics teachers each year to graduate to compensate for retirements and new positions created by new schools, population growth etc but a lot less than this number apply to University ITE courses to become teachers, is it OK to fill these places with (forgive me History teachers…) aspiring teachers from other key learning areas where supply at the moment exceeds demand?

I’ve been a mentor teacher for many years and, looking back, just over half the pre-service teachers I have mentored have stayed in the profession. Those that left were not always those with weaker academic backgrounds either.

So while it may be considered by some a good idea to parade Eddie Woo around the nation and have him say, “be like me – study to be a Mathematics teacher” (disclaimer: I’ve never actually heard Eddie say these exact words or even words to this effect, mostly because I don’t listen much), perhaps the issue is a lot deeper embedded.

End rant.

Eddy starts with an inquiry approach ( i think based on an old Maths300 lesson -“21”) thought it may cause a retort from Ashman or Sweller?

Excellent point. Although, to be fair, it is not clear how he uses the example in classes, or what message he intended by introducing the activity at the forum.

He clarified that, but much later on. Saying that his intent was to use it as a lead into Stage 4 Algebra, as an example of generalisation.

That said, his choice to do that without making the purpose of the task clear, immediately, is interesting (or less than perfect planning).

The game of 23 is related to Nim – maybe they are in effect the same. Players can realise who will be the winner well before the game ends; hence Eddie’s reference to 18. This is strategic thinking and, in my view, mathematical thinking. I inferred that Eddie was using the game to show the audience that there is a mathematical aspect of the game, beyond simple addition. His demonstration led to his main point that, for him, the mathematics that is important is mathematics that helps us to understand the world (in this case, the game).

I have used Nim in Year 8 classes, and in a public forum with parents and students. In the Year 8 classes (end of Year 8 to be precise), the purpose was to introduce the students to combinatorial games which is a modern branch of mathematics (but it is not in the Australian Curriculum). I spent two lessons on the game: in the first lesson we played the game repeatedly so that students got the hang of it. In the second lesson, I presented and explained a winning strategy developed by Bouton (Ann. Math. 1901/1902) and the students put it into practice.

In the first lesson, students asked good questions and formulated sensible hypotheses without any prompting. I have seen some remarkable things in the second lesson after presenting the winning strategy. I defined the heaps of counters, and my Year 8 students were able to tell me who can win with best play simply by just looking at the initial set-up and a little calculation without making a single move!

The public presentation to students and parents was part of an information session for prospective students and their parents. The purpose of using Nim in this event was to demonstrate to parents and prospective students that at my school there were opportunities for capable students to be challenged. Anyway, we just played … but again, like Eddie Woo’s experience … students, who were Year 6 students from neighbouring schools, could work out who would win well before the end of the game. Even a senior non-mathematics teacher dropped in, played the game with me, and liked it so much he took the documentation home to play Nim with his children. We spread the good news.

thanks Terry, yes it was NIM I meant to refer to not Maths300. Your description is excellent. I agree doing these sorts of inquiry activities does promote engagement, enjoyment and deeper thinking. I was just noting that Eddie started with this Inquiry approach & I guess unbeknownst to him Greg’s PhD is against this approach. It was interesting a day or so after the presentation Greg provided another 2 studies and argued against Eddie’s approach. Also, with Sweller in the audience, Eddie was probably unaware Sweller has said there is NO place for inquiry based learning in the classroom. I personally believe there is a place & much of the evidence Sweller provides has a lot of definition issues – e.g., Mayer (2004) Should There Be a Three-Strikes Rule Against Pure Discovery Learning?

Thanks, George. Can you point to where “Sweller has said there is NO place for inquiry based learning in the classroom”?

it was in this TES article – https://www.tes.com/magazine/archive/tes-talks-john-sweller

but maybe the reporter misunderstood? I’m guessing Sweller probably meant ‘pure’ inquiry, which is part of the problem of definitions in Educational research i referred to earlier.

Thanks, George. I read the article briefly, which seems to have been written by Sweller. But the lead-in to the article, which, seemingly, is what got up your nose, appears to have been written by someone else:

The godfather of cognitive load theory writes that teachers need to recognise the working memory limits of their students and that this dictates that problem-solving, inquiry-based learning and other common pedagogies have no place in the classroom.I’m more than happy for you to take a whack at Sweller, and maybe Greg, who occasionally reads this blog, will whack back. But I think you should be whacking the content of the article, not the lead-in.

I’ll make one further comment. You state that “much of the evidence Sweller provides has a lot of definition issues”. Greg touches on this point at 24:00.

i’m sure he will take a wack back, but I think the reporter’s summary of Sweller’s points in the lead is a fair & reasonable interpretation.

Re Mayer 2004.

Ashman & Sweller’s 2019 paper says-

Across multiple domains ranging from mathematics to reading comprehension, researchers have repeatedly demonstrated that fully guided forms of instruction are more effective for novice learners than unguided or partially guided forms of instruction (see Mayer 2004; Kirschner et al. 2006).

Mayer 2004 is clear on pure discovery but not the hybrid forms of discovery like that used by Eddie – the conclusion was – guided discovery is better than pure discovery.

I’ve started on Kirschner et al who also cite Mayer 2004, so it is a key ref. They also cite 3 other refs, 1 is a book which I costs over $100, 1 is a small RCT the other a conference paper which i can’t access.

Well, Greg will probably what read you’ve written (which is difficult: this thing called formatting is a good thing …). But, he’s a busy guy.

I’ll take a look at your references later although this is not my fight. But I think you’re taking a weird tack here. You don’t like the message of the lead-in, but you don’t address the article by Sweller, which you raised, to which the lead-in supposedly leads.

If you want someone to look at Sweller, maybe choose the one source that best captures whatever it is that annoys you, and I/Greg/anyone can focus on that, if we so choose.

ok I just pointed out Eddie used a strategy significantly criticised by Sweller and Ashman in a forum focused on improving maths teaching. I did write a few things to Sweller, not surprisingly he did not reply. I read Greg’s stuff, most of i agree with, some of it I don’t, when i don’t, i try to read the refs supporting his views, but, Greg’s web page needs a paid subscription to comment. Anyway enough said, we all have more important things to do.

OK, up to you. If you have a specific criticism of a specific part of a specific reference by or about Sweller, you’re welcome to state it. Otherwise, I’ll assume we’re done.

thanks Marty all done, thought i spend the arvo watching a good saints win

The statement that “fully guided forms of instruction are more effective for novice learners than unguided or partially guided forms of instruction” appears to me to be obvious.

I have never seen an example of a teacher who uses “pure discovery” as their teaching method.

Which of course proves that the Discovery fetish is no cause for concern.

I meant that no teacher I know pays the idea any attention. But maybe I don’t move in the right(?) circles.

That lead-in looks like a wedge.

Nim is a hell of a lot more complicated than 23.

That was interesting to watch. I liked that Eddie Woo focused on the question, ‘What is the mathematics that matters?’ (to teach in high school mathematics) because that is a question I wonder about a lot. But then his answer was limited perhaps by time and the need to sell mathematics itself to an audience (and so I just got an inkling: the kind of reasoning used in mathematics, as exemplified by the 23 game). That seemed like just the start of an answer. I would have liked to have had more on that.

I cringed a little at the focus on initial teacher education, because teachers adapt throughout their careers to meet the extensive demands put upon them, and match the cultural attitudes in schools about what is important. I think that if people want teachers to be different, then they need to scrutinise these demands and how it shapes what teachers focus on throughout their careers. I’m only a year and a half past finishing my MTeach and already the training I received through it seems insignificant.

Thanks, wst. I think Eddie’s answer was limited by his limitations. Why did you cringe at the focus on ITE? They* were, rightly, trashing it as useless. Isn’t that the point?

*) Fahey and Ashman. Eddie, of course, just wants to be loved.

I think blaming Initial Teacher Education means finding fault with beginning teachers. People often say if only they (we) were better or more prepared in some way, then that would solve most of the problems we have. And I don’t believe that. (What training is going to be enough to succeed in schools that don’t value the subject, with a study design that focuses on calculators? What do they expect of us?)

It also just clear to me that teachers learn throughout their careers and adapt, and that adds up to way more than their initial education anyway.

What?! I don’t accept your first sentence at all. Ashman wrote very recently in this very point.

Oh, sorry. I should read that then. I don’t understand though. If the problem is that beginning teachers lack skills, isn’t that a fault with beginning teachers? They didn’t cause the fault, but it is a fault.

Also, sorry for the grammatical error above: “It is also just clear to me…”

Well, I’d call it a shortcoming, not a fault. But *of course* beginning teachers are gonna have shortcomings. Who could possibly think otherwise, and who would possibly blame the teachers, except in individual screwball cases?

I read Ashman’s blog post and now realise it is possible to criticise ITE without finding fault with beginning teachers. You could, for example say that it is a waste of time, which I think is implied by:

In most cases, I would rather be able to take students straight from a science, maths or arts undergraduate degree and train them in the mechanics of teaching on the job, than have them go through the process of two years of initial teacher education.I agree that of course beginning teachers will have shortcomings. That’s why I find it troubling that people think that the key way to improve educational performance of students is to change ITE, like when Tudge announced the ITE review:

“The review of initial teacher education courses is the most critical element towards lifting [school] standards.”That does feel like blaming the teachers to me.In the article by Fahey, Sullivan, and Bussle, they only say it mildly:

At least partly, underprepared early career teachers contributes toward underperformance of Australian students, especially in mathematics.I will add though: at my uni, they were pretty transparent about all our assessments being based on making sure we achieved the AITSL standards. In the Fahey et al article, they criticise the emphasis on technology. But using technology for teaching is part of two standards, so they had to assess us on it and explicitly required us to explain how we used it. If they want to change teacher education, I think they could achieve that by changing the AITSL standards.

(Sorry for rambling on a lot. I’m a bit unwell today.)

Your First paragraph: Yes.

Your second paragraph: Yes and no and no.

Yes, the Tudge line, is of course ridiculous, and I hammered it here.

But no, that does not amount to blaming teachers, either generally or blaming new teachers in particular. That doesn’t mean teachers can’t be criticised, but such criticism can be, and I think generally is, quite separate from the criticism of ITE.

And no, even if reforming ITE is not the key to the fundamental fix that some might wish, it is not irrelevant either. It is relevant if only for making would-be teachers endure two years of pointless, tortuous bullshit. And it is more relevant than that. ITE is a lunatic asylum, and that has to be addressed.

Your third paragraph: FSB is a pretty silly paper, and I can’t be bothered attempting to decipher that quote in order to critique it.

Your fourth paragraph: a huge yes. Your last sentence is correct, and critical. The AITSL standards are way worse than pointless; they are pure poison.

And that really brings up the main point, of why a focus on ITE is not the key. It is not that ITE is insane, it is that

absolutely everythingin education is insane. They are all out of their minds, and they pervert everything, and everyone, they touch.If I may offer an opinion:

SOME of the AITSL standards MIGHT NOT be pointless if they were better phrased. I’m thinking of 1.2 and 2.1 here:

1.2: understand HOW students learn. Yep, good idea.

2.1: basically KNOW YOUR CONTENT. Yep, good idea.

If I had to pick a third to include it would be 1.5 – differentiate (don’t get me started on the choice of phrasing…) to make the content accessible.

Some are poison, especially 3.4, ALL of 6 and most of 7 made all the worse by the “PD” gravy train and the VIT.

ITE is not the singular issue here though. These standards would still be an issue even if ITE were over-hauled.

AITSL standards are worse than pure poison …

You’re lucky you don’t have to formulate a Professional Development Plan each year that requires you to explicitly link your goals to the AITSL standards. Someone pass me the hemlock laced beverage already.

(Actually, I don’t bother wasting my time on that rubbish. I simply write down random numbers for appearances sake. But younger teachers full of vim and vigour …)

RF, you’re missing the point. I address this at 43:00 of the Hell video.

I remember the video well. Especially the “cheese question”.

AITSL standards to me say two things:

1. Someone with some authority somewhere thinks there is an issue with education in this country.

2. That same someone (and “someone” could refer to a large group or department) has NO IDEA how to fix the issue.

And JF, I do have to fill in *that column* on my annual PD plan. Every year I write 1.2 and 2.1 and hope no one notices… some school leaders are too busy to care much.

Yes, but again you’re not acknowledging the fundamental poisonous of the AITSL standards: if someone is asked to pay attention to thirty standards then they will pay proper attention to none. That means it is utterly irrelevant that a couple of the AITSL standards are correct and important. It sums to nothing, and a nothing about which naive young teachers will tie themselves in knots.

@RF: Great(?) minds think alike (although I think you should spend a little more time and choose random numbers).

“hope no one notices… some school leaders are too busy to care much.”

I think you can depend on that. The fact is that any decent goal is going to connect to a couple of those standards – it’s totally dumb to force teachers to waste precious time doing this.

@Marty: “It sums to nothing, and a nothing about which naive young teachers will tie themselves in knots.”

I totally agree. And I’ve seen plenty of older and experienced teachers also “tie themselves in knots”, to the point of a couple breaking down in tears.

THIS is one of the many reasons teachers are leaving teaching. And the word gets out to all the coodabeen teachers ….

Any beginner – be they engineer, doctor, lawyer, or teacher – will have some shortcomings. That’s why there’s this thing called experience. But I think ITE actually creates many of the shortcomings that beginning teachers have.

In what way(s)?

If I had to guess:

1. Over simplifying the idea of what a “school” actually does.

2. Giving the impression that as a graduate they will be able to fix everything if they only remember what (Insert name of whichever education research person is in vogue at the moment) says to do.

3. Providing no training whatsoever in how to un-jam a photocopier.

Yes, that sums it up. And 3) is never to be underestimated!

Eddie had two minutes of clarity from about the 38 minute mark to the 40 minute mark.

The key point in that 2 minutes being,

“That is what the teaching of mine that is resonating with people, actually, focusses on.”

The what being, “the things that are clear, that make sense of the different pieces of what … picky-picky detail”. Picky-picky detail being a term mentioned by Greg, that relates to direct instruction (see https://gregashman.wordpress.com/2016/05/22/a-day-at-researched-melbourne/).

Nailing the picky-picky detail, in classrooms across the country, is part of the way, a large part of the way, that “Australian schools can become a world leader in mathematical education.”

If that was to happen, Eddie’s videos may not get so many views. But I am sure he would not mind.

Also in that 2 minute window Eddie said,

“it is silly to try and divorce these things, and we end up in all kinds of problems when we do.”

The it, being 21st century skills and picky-picky detail. Thoughtful and experienced teachers have known this for many moons. They also know that choosing A over B results in many more problems than choosing B over A.

Compare the level of positivity oozed by the term 21st century skills vs that oozed by picky-picky detail.

How sad it is that the critical stuff, that is unquestionably a prerequisite for the other critical stuff, has been demonised to a point that it is spoken of (even by advocates of it) as picky-picky detail, that has to “come along for the ride” (Eddie).

How sad and how ridiculous. Turning that around, alone, would go a long way towards Australian schools becoming a world leader in mathematical education. Whether or not it will stop the stagnation/declining numbers is debatable.

The decline has been bemoaned for decades. Decades in which the focus of mathematics teaching was picky-picky detail and decades in which it was not. So, …

Thanks, wst, and a few comments in reply.

0) I don’t see “picky picky details” as a pejorative, and I’m not sure it was intended as such. Sure, it doesn’t glitter with magic fairy dust, like “21st Century Skills” and “Critical and Creative Thinking”, but I think the focus should be on the dishonesty of these huckster phrases, not pondering how one might cloak proper learning with pointless buzzery.

1) I don’t think Eddie’s “two minutes of clarity” are all that clear, or helpful, or correct. The need to “maintain unity with” A and B is not telling me how this is to be done. As is, it’s just another motherhood statement, at which Eddie excels.

2) It’s not totally clear to me why Eddie’s (popular) videos are popular, or that a new national/worldwide focus on picky picky details would make them less popular. The

contentof Eddie’s videos mainly seems to be adequate – sometimes good – explicit teaching. Eddie does details. But of course plenty of other people do details, too. And, at times, Eddie gets the details wrong. No one gives a stuff. Including Eddie.3) Obviously a large part of Eddie’s charm is his sunny personality. Myself, I just want to slap him. In the CIS video, he was absolutely excruciating. In general, I think he’s a fraud.

That’s harsh, Marty. Eddie Woo is a maths teaching phenomenon. (So we’re told. Repeatedly).

Yes, he’s a phenomenon. So was the tulip mania.

0) I did not think p-pd was being used pejoratively. But the term conjures an unnecessarily ugly image. The fact it is used, in a non-pejorative sense, says something.

1) He did, albeit briefly, hint at how to maintain unit, using his example (23) as a lead into what would likely be the explicit teaching of Stage 4 Algebra. To be fair, that forum was not ideal for a description of such.

(not wst 🙂)

0) Maybe, but I get tired of tiptoeing around wording, unless we scare the horses. If people are going to learn maths then they are going to have do the hard yards, to attend to the details. This is front and centre, and I think it is self-defeating to pretend otherwise.

1) The game of 23 as a “lead in” to what? For whom? A year three or year four student, maybe. And then what? What year(s) does Eddie teach?

It’s at best small beer.

0) Agreed. Acting like it is not necessary is why we are where we are. That led to the re-defining of, or at least multiple meanings for, mathematics.

1) The next forum, maybe. 🙂

I first went to a workshop for Mathematics teachers back in the early nineties on using games in Mathematics teaching. I didn’t think much about it for many years but when I was given a highly talented year 8 class, I decided to use the game “First to 50” , choosing a number from 1 to 6. My intentions were multiple; primarily to get students to enjoy playing a game involving numbers. But first I made them play against a 6 sided die, to make sure they would win. Secondly, like Eddie, I soon got them to realise that getting to the ‘magic number 43’ would ensure they would win. A knock out tournament resulted in an incomplete analysis of the game but we went on to discuss other ‘magic numbers’ in this game which then led on to developing a winning strategy. We also then discussed designing alternative games where you would or would not want to start first etc. There was a lot of very useful mental arithmetic going on; repeated subtraction, multiplication, modulo arithmetic etc; the importance of which was so clearly stated in the “Essence of Mathematics’. Marty, I think your ‘small beer’ comment was out of line actually.

Huh. If my “small beer” slap was out of line, what was my calling Eddie a fraud?

Thanks, Rob, and you may be correct. Maybe I undersold the activity.

I’ll admit, once again, that Public Eddie gets up my nose. He occupies, and wastes, a rare public space for maths ed discussion. Eddie’s cloyingly chirpy, I-just-told-another-joke willingness to say almost nothing with an edge, nothing of substance or import really pisses me off. He’ll never risk his balls.

And, whatever the merits of the 23 game, I thought it was pointless in the context of that forum; Eddie said too much and too little, for it to make any sense. So, maybe all that made me go too hard on the actual game. I try to be fair, but maybe I wasn’t.

But, maybe I was fair. I still have large doubts about the game. Let me ponder, and I’ll respond later.

Rob, I’ve thought more about this but don’t have much more to add. I don’t think “small beer” was out of line, but “wrong beer” is probably more accurate. Think of all that is wrong with maths ed at the moment. Is “We don’t have enough games at in maths classes” high up on your list?

In the main, I think the discussion that storyteller and wst and Terry and Glen have been having covers things well, and I agree mostly and strongly with Glen. I don’t see such games as much more than gimmicks, and typically time-consuming gimmicks. As for the 23/N game itself, sure the game encapsulates the idea of division and remainder that you want to capture. But really, how long do you wanna go with that? How much is it really worth?

I also have to note, you suggested the 23/N game worked well with a “highly talented year 8 class”. I’ll take your word for it. But isn’t year 8 for learning the power of algebra? What we were these kids doing in year 4 or year 5, when they were supposedly learning about division and remainder? And, if you’re gonna do division and remainder in year 8, wouldn’t the Euclidean algorithm be a more valuable use of time?

I interpreted Eddie’s use of the game 23 as a way to engage people in the audience – whoever they may have been – rather than a sample of one of his lessons; he clearly posed his main question “What is the mathematics that matters?”; he used the game to demonstrate that mathematical questions arise in even simple games; he concluded that, for him, the mathematics that matters is the mathematics that helps us to understand the world more clearly. It was obvious to me that he ran out of time, but I could see where he was going.

See https://nrich.maths.org/1272

Hi Terry. I followed your link but it didn’t shed any light on what EWoo was trying to accomplish.

His question makes my skin crawl. “The mathematics that matters”? All of it bloody well matters! But who cares? Aren’t we supposed to be talking about teaching?

Sorry, I have a bad habit of reacting viscerally to Woo.

Thanks Glen, The link was about the game 23. That’s all.

I interpreted the presentation by EW as giving us a guide to what mathematics should be taught in schools, and the presentation by GA as giving us a guide to how we might teach our students about that mathematics.

I wonder: Were the speakers given some direction as to what they should talk about? Who was in the audience? I guess that they would have been given some advice on these matters: this is normal practice.

I have been invited to give a key-note presentation to young doctors on the use of statistics in research, and the organisers have given me details of the composition and size of the audience, the length of time of my presentation, what the audience wants to hear from me, and what the other key-note speaker will be talking about. I suppose that EW and GA were given a similar briefing by CIS.

Thanks Terry. How is what Eddie spoke about a guide? All math matters.

Also, there is this thing called a curriculum.

Maybe you’re right though, if Ewoo thought he was talking to a room of politicians, it’s probably in the ballpark.

They always speak to a room of politicians, even when none are in the room. That’s because what they say always gets back to the politicians. So they make sure the gravy train always hears what it wants to hear.

Speaking of which, has anyone heard any politician make any comment on the recommendations made in the ITE report. Maybe the new Federal Education Minister – who seems decent – will make some comment. Although I’m guessing it’s old news from an old government and what’s needed will be a new report … That report will get buried with all the other past reports, because it contains recommendations that would actually cost money.

And commenting on Victoria, I’m amazed that the media are giving the Victorian Education malad-Minister such an easy ride. We can thank the total disaster that is the public health system for that. Victorian education is in total crisis (as is the health system) and we have malad-Ministers – and a Premier – acting solely in self-interested political survival.

I don’t think I’ve recovered yet from the last Woo video I endured.

Well I watched it. I’m made very uncomfortable by it. I hope I don’t sound at times like Woo when I present. I found it a very disconcerting experience. Woo is slippery. Greg wasn’t taken to task. I didn’t like their foci.

Can’t really fault Greg seriously. Woo is too easy to fault but he wasn’t always wrong! At least in my opinion.

I doubt very much you sound like Woo. And yes, Woo is slippery, meaningless except in his meaninglessness. Fahey is a smug, manipulative nitwit, a classic CIS clown, and the forum was always going to be weird and silly. But I thought Ashman was great: serious and pointed and accurate, and non-distractingly funny.

Fahey and Ashman agree on one point. Pre-service teachers should learn about cognitive load theory.

If cognitive load theory means, as it has been explained to me, that each person can deal with only a finite, but relatively small, number of new ideas at a time, then I would say that this is pure common sense. There is a limit on how many balls we can juggle at once – and clearly this varies from person to person. (One for me!)

I used to teach gymnastics to boys; if a boy can’t execute a cartwheel properly there are usually several problems with his style. But you don’t correct them all at once. One at a time will do.

The difficulty with solving problems in applied mathematics that often the context is too complicated; there are too many variables to consider, and realistically often the mathematics is too advanced. In school mathematics, we need a simple context that captures the essence of what we are trying to teach the students about mathematics.

Thanks for sharing the youtube video; it was interesting to watch.

Let me offer some comments on using games in a mathematics lesson based on my short experience in teaching secondary mathematics.

First, games are often enjoyable activities.

Second, sometimes a game can be used to help students to understand mathematical ideas. When playing the game, students should be aware of the link with the relevant mathematical idea.

Finally, a game can inspire students to ask interesting mathematical questions. Is it an advantage to go first? What if we changed the rules of the game?

Fourth, a game is typically a cheap sleight of hand, where everybody is pretending the students are learning when in fact they’re playing a game.

Maybe that is often the case. On the other hand, the teacher can plan the lesson so that this is not the case. The objective is to learn something about mathematics; the game is simply an enjoyable means to this end.

You accidentally gave the whole game away (so to speak). Thinking of a game as “a means to [an] end’ is exactly the problem. The game is never going to get you to the end. It might spark the beginning, but little more. And, sure, a teacher

mightplan a lesson with a valuable inclusion of a game. But the overwhelming majority do not.We are living in a time of diminishing attention span, diminishing discipline, diminishing value of teaching and learning. Everything has to be “enjoyable”. We are living in an era of playing games. The last thing students need in a maths class is another bloody game.

You are correct on one point: I want my games to be cheap; they should not involve any expensive equipment. I also want my game to get to an end fairly quickly; we have only 70 minutes for the entire lesson. Choosing a game that fits the purpose is a time consuming activity. However, sometimes students get so hooked on the game, that they want to play it over and over again even in other lessons. “Can we have that game again please sir?” “What, you want more?”

I am correct on all points.

Terry I am certain that you are a great teacher to your students. But loving a game, especially when faced with a dreary day at school, is simply that: loving a game.

When my son asks for another slice of home-cooked pizza, I know it isn’t for the sneaky cherry tomato I hid under the pepperoni.

Thanks for the compliment, but I am not a great teacher; enthusiastic maybe but not great. I have a lot to learn.

You are a great teacher.

Would that our political debates were as civilised as the interview with Eddie Woo and Greg Ashman.

Nonsense.

I assume that the speakers were told who is likely to be in the audience so that they could pitch their presentations accordingly. Does anyone know who was in the audience?

BTW, I note that there were no mathematicians on the panel. Can you imagine a panel discussion about how we should teach science in our schools with no scientist involved? Or the same question about history.

Yes, and that’s a really big problem. It’s related, I think, to the fact that most people are glad to proclaim they’re no good at mathematics, but perish the thought if they said they were no good at reading …

Greg Ashman blogged on this really well: https://fillingthepail.substack.com/p/we-dont-need-no-education?token=eyJ1c2VyX2lkIjo4MzA4Njg3NywicG9zdF9pZCI6NTYzOTg3NDMsIl8iOiJEa2wwRSIsImlhdCI6MTY1Mzc5MzUyNCwiZXhwIjoxNjUzNzk3MTI0LCJpc3MiOiJwdWItMjIwNjMwIiwic3ViIjoicG9zdC1yZWFjdGlvbiJ9.NezXkHcnmNewF0RzQbpx-ZeDLq5pbj9rT6Gha3oXaQU&s=r

I find this a fascinating point. I am frequently annoyed by panels about education that have no teachers on them. But you are right, if we want to fix maths education, mathematicians must be part of that discussion.

(1) Should primary school teachers also have a role in the debate? I suspect that teaching in a primary school may be quite different from teaching in a secondary school. Is there a place for specialist mathematics teachers in a primary school?

(2) I agree with Greg’s point that problems in the “real” world are complicated. One of the advantages of applying mathematics to games is that there is no hidden information.

(3) On the other hand, I have had used open-ended questions such as “How much does it cost to keep a dog?” (due to Peter Sullivan). Students realise that they need to make assumptions. The hard part is developing the model: Which variables do I include? And almost all students can contribute to the discussion at this point. This is part and parcel of applied mathematics.

Re 3.

The sweet spot is somewhere in between the extremes. Both sides know that, but seem to rarely acknowledge it, perhaps fearing loss of ground.

The sweet spot is not easy to implement in a systemic manner, unlike the extremes.

Nonsense.

🙂

No no no, if you want to teach mathematical modelling, abstraction, mathematical framing, that’s its own thing. It is something additional to whatever other mathematics you are trying to teach.

In 2018 I created a subject to teach specifically abstraction, among other things. It is delivered in the first year of university and is for math students who will have already completed high school mathematics and feel good about math. It is a notoriously difficult subject. Abstraction is important but it just seems to make the life of students in primary or high school harder.

Maybe it could be taught in an enrichment class.

And we find ourselves at the impasse where you define school mathematics to be … and they define school mathematics to be …

Do we? School mathematics is what is in the school mathematics curriculum. Curricula are different in different places, but essentially you are learning how numbers and functions work in school.

You can teach modelling and abstraction after having mastery of the mathematical objects that you will end up with. So you *could* teach something along these lines that involves arithmetic after learning arithmetic, which *can* be done in school. But this is distinct from learning about arithmetic. The arithmetic should come before the modelling and abstraction with arithmetic.

Yes, we do.

Yes, we do find ourselves at an impasse. But it is not about “we define” and “they define”. It is about “we are correct” and “they have the power”.

Don’t pretend this is about negotiation and sweet spots. It is about sanity and leadership by the insane.

Maybe you could elaborate. Do you disagree with how I defined school mathematics? Or something else? Your comment is far too clever for me.

Glen,

My “clever” comment was simply answering your question (Do we?).

Re school mathematics.

School mathematics is what actually happens in the classroom.

School mathematics is extremely variable, due to, in part, the tsunami of advice/directives about how to teach (or not teach) what is written in the curriculum; a tsunami that has pummelled teachers relentlessly over the last 30 years.

I agree that mathematical modelling is complicated if the situation in complicated, or the required mathematics is complicated – and not for the feinthearted. I have used the question “How much does it cost to keep a dog?” with VCAL students. I regard this question as a question on applied mathematics – but most VCAL-Numeracy exercises are examples of applied mathematics.

On (2), the advantage of applying mathematics to mathematics is, not only does it not have the complications of the real world, it also does not have the distraction of games.

Here are some comments on “Ashman’s first principle of educational psychology: Students tend to learn the things you teach them, and don’t tend to learn the things you don’t teach them.”

I realise that this principle is framed in the context of classroom teaching, and it contains the qualifying word “tend”, but it does deal with learning.

AI has offered us new insights into learning. AlphaZero is a computer program that plays chess. Given only the rules of chess, the program quickly learned how to play at a very high level without any further instruction. Of course, AlphaZero is not a human being.

Srinivasa Ramanujan was a famous Indian mathematician. Although he was largely self-taught, his discoveries astounded the world’s leading mathematicians. Of course, Ramanujan was an exceptional human being.

However, at a more mundane level, “Teach Yourself” books have been in print for more than 75 years. The idea underlying this series of books is that even ordinary human beings can learn from a book without direct instruction. And many of us mere mortals often learn stuff from the internet.

There are many ways to learn.

Personally, I think students tend to learn the things you

assess, so long as you’re clear about what you’re going to assess.ah … the voice of experience!

So … do you tell students exactly what you’re going to asses? Or do you say that everything taught is examinable? And where does that leave students sitting external exams – in the case of VCE, what’s assessed is what’s stated in the Study Design.

Students are generally lazy and always looking for short-cuts – telling them what you’re going to assess guarantees that they will not bother with learning anything outside of that. I don’t think students deserve to be told anything more than “If it’s on the course, it’s potentially examinable”.

Hi John.

To be fair, I’m just a relief teacher. So this is theoretical for me. I think my answer is not really – I usually don’t know enough about what’s going to happen to do that. If I could, I probably would. I’m lazy so I don’t judge people for that.

However, I’m including all kinds of assessment here – informal in-class stuff as well. Recently Terry Mills showed an assignment he got students to do in class, and said how he thought of it as an opportunity for students to learn through completing it. I liked that.

I think on the whole my (highly imperfect) approach to teaching follows two steps:

1) create situations where students seek my help to learn or understand something.

2) try to help them.

I find Step 2 is relatively straightforward compared to Step 1, which is multifaceted and difficult. Assessment is part of Step 1. It signals what we value. Students pay attention to it, and seem to remember tests more vividly than lessons. The focus of assessments is communicated over a long time, sometimes accidentally.

Also, I read this idea about assessment being important in a book a long time ago. I just remembered what it was called: Learning to Teach in Higher Education by Paul Ramsden. He argued for the importance of assessment from the student’s perspective. I found it very compelling at the time. However, perhaps it applies more at the university level.

I reviewed this book by Ramsden once. I too enjoyed it.

I have seen examples where assessment is not so important for all high-school students. First, there are students who just don’t care; they don’t even want to be at school. Second, I have met many students who do not care about an assessment if it is not included as part of their final grade; e.g. a mid-year examination. I’d venture to say that many students don’t care about NAPLAN.

Students remember tests for the wrong reason. They focus on marks, rather than feedback.

And tests (usually) don’t signal everything that we value. It’s impossible to write such an assessment. VCAA mathematics Exam 1’s would be 3 hours long if they signalled everything that was valued. They are simply a snapshot of what is valued.

I remember a year when there was nothing about inverse functions on the Maths Methods exams. Most teachers were swearing and cursing that they shouldn’t have bothered teaching inverse functions.

The moment I tell a class that partial fractions will only be assessed for the non-repeated linear factor case is the moment most students in the class will deliberately choose not to bother with repeated linear factors or irreducible quadratic. The moment I tell a class that a question of type X from topic Y will be on the test is the moment that suddenly all most students care about is question X from topic Y. Nothing else from topic Y matters anymore. Students don’t care about learning, they only care about the mark. You only have to see students (and parents) being (trying to be) ‘strategic’ in getting the best ATAR. Students (and politicians and so-called ‘experts’) do not see learning as the preservation of a culture.

Education in Australia is a joke. It’s not about education, despite the mewling and bleating of politicians and education ‘experts’. Education in Australia (and particularly Victoria) is a plaything for so-called (self-proclaimed) education experts and politicians to indulge in social engineering.

If you want students to learn, the last thing you should do is tell them what you’re going to assess. It’s best to say that everything covered in class and set for homework is examinable. Then they can’t be ‘strategic’ in what they decide to ‘learn’.

Every time I write a SAC I tell students that the SAC will be different to any past SAC and there is no point memorising how to do a question from some past SAC. But many persist in living in a fantasy world, and then complain that the SAC was too hard because the questions were different to the ones they had ‘learnt’ (euphemism for memorised like a trained parrot or copied into their bound reference, ready to be regurgitated onto the page). Their bound references consist of every past SAC with the solutions. Students are lazy, they constantly want short-cuts to ‘learning’. They don’t want to learn, they just want to get a score. And teachers are the clowns that have to entertain them.

Re: bound reference: The bound volume is helpful if it is the student’s own work – with some guidance from the teacher. “Learning through writing” is the mantra. (BTW, I think that Victoria is the only state that allows such a resource in mathematics examinations.)

My students are not particularly good at taking notes. I did a small literature review of the topic last year, and I found a particularly wide-ranging survey paper on note taking – but almost no mention was made of mathematics. I wrote to the author who told me that, in the literature, note taking in mathematics is hardly ever discussed.

This is a tricky point. Once students get to the point where they are able to take ownership of their mathematical knowledge and truly construct it, note taking takes on a very powerful meaning. I remember when this transition occurred for me and from that point onward mathematics went from being sometimes difficult to something that was very explicit, either I had built the knowledge for myself, in which case I could do everything or I had not, in which case my knowledge and skills were spotty.

That’s the eventual upside but early on, what is good note taking in math? I certainly agree that students should take notes, but I have no idea what good practice at that level looks like. A really interesting problem.

There used to be a ‘reason’ for the ‘Bound Reference’. The ‘reason’ has been forgotten as time passed. Maths teachers who began their careers 20 years ago would never even have known the ‘reason’. So the reason gets lost in time and becomes “Because that’s what we’ve always done.” Legacy thinking.

I wish I had a dollar for every time I’ve heard a line like that in response to my question “Why do you do things this way?”

Funnily enough, the ‘reason’ became relevant once more when the Great Mathematica Experiment began.

Here is a short history lesson:

Pre-80’s, no notes were allowed. Then, up to 2005, two double-sided A4 sheets of pre-written notes were allowed in the ‘Calculator-enabled’ exam. From 2006 onwards a ‘Bound Reference’ was then allowed. This happened because of the Great CAS-Calculator Experiment, began in the early 2000’s in Maths Methods. The CAS-Calculator was allowed in Maths Methods exams but not in the Specialist Maths exams. Specialist Maths only allowed the Graphics Calculator. But then it was decided that from 2006 the CAS-calculator be allowed in Specialist Maths. Because it was dumb for students in the Great Experiment to be using a CAS-calculator in Methods and a Graphics calculator in Specialist. Among other things, it meant those students required two separate and expensive calculators. But the memory of the CAS-Calculator was large enough to store a book, and students who used it in the Specialist Maths exams would have an unfair advantage compared to students who only had the graphics calculator. There were other relative advantages of course (which apparently were neutralised by using appropriate exam questions – a debate for another day) but it was decided that this particular advantage would be neutralised by allowing all students to bring in a bound reference to all exams. They could have as many notes as they wanted. Notes in a CAS-calculator were rendered irrelevant.

And of course, like a weed, the CAS-calculator infested and infected the Mathematics curriculum to such an extent that it wasn’t long before every student was using one. But of course, the ‘Bound Reference’ remained, even though its raison d’être did not. The clowns who pontificated and decided upon these things either didn’t realise this or wanted to maintain the status quo because everyone, especially students and teachers, were happy with this. I suspect the latter. And those on the gravy train – noses in the trough – of selling notes were particularly happy. Students are inexperienced and can’t be blamed for not seeing the writing on the wall. Teachers don’t have this excuse, they are – mostly – just dumb and apathetic.

Now we have Mathematica and students are allowed to access Mathematica notebooks in the exams. The relative advantage these students have is massive, and would include access to notes if Bound References weren’t a thing. So the raison d’être for a Bound Reference returns. Although the clowns who pontificate(d) and decide(d) upon these things would hardly have realised this.

And on the topic of ‘Bound References’, can anyone give a sane, rational reason why a Formula Sheet is provided for Exam 2 when students can bring in a bound reference??? More legacy thinking.

Peter Liljedahl’s book Building Thinking Classrooms in Mathematics has a chapter about note taking. I think it’s interesting to read, although it’s within the context of a lot of proposed changes in how we teach mathematics. A lot of the challenge is to convince students that the notes are meant for their own benefit, rather that just to please the teacher.

He tried prompting students to write notes as a reminder to their “future forgetful selves” and then had them return to them three weeks later.

According to him, the younger students (in Years 4, 5, and 6) are better at taking notes independently than those in Years 7, 8, and 9: “in general, … by Grade 8, students have become encultured into a more mindless form of note taking, and they have lost the ability to decide for themselves what to write down.”

So, early high schoolers apparently need a lot of support to write notes: perhaps graphic organizers, guiding questions, reminders about what the notes are for, etc.

Or, and hear me out, they could have a decent, succinct and to-the-point textbook.

That would certainly mitigate the need for note-taking, although not remove it. It would mean that the teacher could give a different style of notes.

Although I suspect there’s a very common type of teacher who would simply hold the book in one hand and write its contents on the board with the other, for the class to happily write down.

I tell my students that what I ‘write on the board’ is the minimum of what they should be writing down, they should be supplementing my ‘board notes’ with things I say that I don’t write down that they consider useful or important.

But most students will not take notes. They prefer to let the lesson wash over them like waves on a beach. I wish I had a dollar for every student who has asked me how to do a question from the set work, I say that I did a similar one as an example in class the other day, I ask to see their notes from that lesson in order to refer them to the example I did, and they have nothing. Nada. Bupkis. And then they wonder why I get angry and tell them to find someone who DID take an accurate set of notes.

Hi Terry. You are right, of course, that students will learn things on their own as well. And yes there are many ways to learn.

It isn’t about classifying and appealing to all of those different varieties of ways. It is about learning in the main. The core thrust of what you are doing. That is what needs to be clear. When you’re playing games — whether they be actual games or other colourful tasks — this core thrust is unclear. It often ends up just using up class time.

In all my classes, I make the learning intention clear, and give students clear criteria by which they can judge their success.

Do the students understand that the success criteria is for success?

I’m not convinced that students pay much attention to learning intentions and success criteria even though I give them some thought; e.g. In this lesson, we will have one important objective: to continue to get to grips with fractions. You will be successful in the lesson if you solve most of the exercises and learn at least one thing about fractions. (Year 8)

That’s great, I believe you. I’m not sure how that responds to my comment though. I can tell students that I’m going to teach them algebra, but if we then play games, then what I’m actually teaching them is the game. Maybe with a bit of algebra on top. But in the main, it’s the game.

In Terry’s defense, sometimes a game is the best way to convince certain students (and most schools have a fair few) that what they are about to learn is useful/important.

Debriefing the game is essential and can take a lot of preparation. In some topics (probability especially) this can be pretty straight-forward (not always easy though). In other topics it can take a lot more time and effort.

Glen’s comment does remain correct in every sense though – if we teach students how to calculate their expected win/loss in a game of crown & anchor, are we teaching probability or a specific example? It possibly depends on the student.

Yes, a game can be a useful gimmick. But it is a gimmick. And, if it is a time-consuming gimmick then it is a bad gimmick.

Gosh I’m coming off a bit negative. I’m not an old grouch who teaches like a drill sergeant.

I love to use gimmicks in class. They are great. I wear stupid t shirts with dumb math jokes. In breaks students like to guess what the joke means. That’s a game.

I make jokes about pictures I draw on the board.

I write bits of the lesson from the future on the board, which I will write around later. Students like to guess what it is out of context.

I make a deliberate (written) spelling or grammar mistake each lesson, the student that finds it gets a gold star and the adoration of their peers.

I use gimmicks when teaching new concepts. Like turning off and on lights in an array to show kids binary.

These things I have done and yes it gets them talking. But I just think that these “games” are super quick, improve the mood, and are basically harmless fun. The vast majority of our time is on doing math. (And showing each other our math.)

I want to ask a question and this forum seems like the quickest way to reach an opinionated-yet-educated audience so here goes:

If I was (say) teaching Year 10 Probability (so they know SOME set theory and maybe a bit of conditional probability but very little about expectation and variance/standard deviation and have likely never seen a binomial distribution.

Suppose now I tell the class I run a casino and have invented a new gambling game. I demonstrate the game and have students play it (with me as the house, of course). Then I set them a task of telling me if the game is fair or not, AND if it is not fair, who has the advantage and how much?

Is this “playing a game” or is this some type of formative assessment?

I have my own opinions (as do colleagues, and we disagree frequently) and “I” may actually be a colleague, but the story works better in first person…

Does every student do the task?

Does every student submit their answers to the task?

What’s the time limit on submitting answers.

Are the questions written down and handed out?

At what stage of the probability teaching sequence is the task given?

Depending on the answers to the above, my opinion is that you’re role playing a question that has several parts to it and that it’s a formative assessment, not playing a game.

Thanks JF and Terry – I’m not completely across the details but it sounds like I owe this one a chance.

It looks like students do the activity and then there is a debrief. It seems to come right at the start of the teaching sequence – one of those “hooks” I keep hearing about.

RF, if it comes at the start of the teaching sequence then it’s a game. And that’s just DUMB. It deserves NO chance.

It’s just plain dumb at the start of the teaching sequence to set students this task.

I think it would be VERY suitable AFTER the appropriate ideas have been explicitly taught.

As they say in comedy, timing is

.

.

.

.

.

.

.

.

everything.

I’d ask: What is the purpose of the game?

BTW, a useful reference on formative assessment is Black, Paul; Harrison, Christine; Lee, Clara; Marshall, Bethan and William, Dylan (2003). Assessment for Learning- putting it into practice. Open University Press.

As a demonstrated worked problem that illustrates whatever you are teaching, it sounds alright.

But as the later comments reveal, if you give students this to do before they know how, it does not sound like a good idea to me.

However I guess it does relate to this whole “let them fail, then teach them how to succeed” idea. I think this can be useful, but you do need to teach them *something* beforehand.

All fair points. Thanks.

Now, how to convince colleagues to alter their teaching sequences… a discussion for a different forum.

You might also ask, what advantage does the game have over simply explicitly teaching students about the mathematics involved?

The game might illustrate a mathematical idea in a way such that some students will better understand the idea as a result. “Oh – that’s what you were driving at!” But if the game is complicated, the complications can hide the key mathematical idea.

Although I am interested in game theory, I agree with Marty’s sentiment that, too often, a game is the first point of call for teaching students. And not only in mathematics – I see it in other disciplines too.

Complications can arise if the game involves students using computers. Often I see activities being used in class that require the students to do something on their computer. Almost always, some students do not have a computer, or their computer is not charged, or their computer has crashed, or they left their computer in the locker. (Can I go and get my computer please?)

I also have a view about using casinos as the context for mathematics problems that some may regard as puritanical.

More on Eddie Woo:

What did you learn? Apart from the fact that Eddie can prove 0! equals 1.

That was not in the clip. It describes his work in NSW with teachers and students in ordinary schools.

Yes it was.

Sorry; I missed that bit the first time, but saw it the second time around. I would guess that the selection of that bit was the responsibility of the producers. The main the point of the clip was what can be done with promoting mathematics in a school that has no obvious advantages.

And my main points are that: (a) Well, duh; (b) Eddie ain’t God.

I agree with you. However the clip suggests that much more can be done to promote mathematics in schools building on the capabilities of the teachers in the schools. This may be obvious, but it is good to hear it being said.

There are many good mathematics teachers in all the schools where I have worked. How do we harness these skills and enthusiasms collectively to improve interest in mathematics?

Get out of their way, and get VCAA out of their way, and get VIT out of their way, and get ACARA out of their way, and get the army of maths ed twats at university out of their way.

Just go away and let teachers teach. The good teachers will then be able to do great things, and the others could not possibly do worse than they’re doing now.

So, when am I gonna appear on 7:30?

You could volunteer to be on the box.

You’re a funny man, Terry.

Problem solved in the time it took you to type your comment, Marty.

It is the obvious solution but will never happen because there are too many snouts in the trough on the gravy train profiting from NOT getting out of the way …

Sorry; I missed that bit the first time, but saw it the second time around. The main the point was what can be done with promoting mathematics in a school that has no obvious advantages.