*When I’m with them I’m confused*

*Out of focus and bemused …*

That’s pretty much it, except for the “bemused” bit. But our cheap-joke title is also asking a genuine question: what does ACARA think is the essence of “problem-solving”? How do you solve a problem like ACARA (does)?

The answer is a classic bait and switch; the bait is Singapore, and the switch is to “inquiry-based learning“. Here is ACARA CEO, David de Carvalho, in a recent report (Murdoch, paywalled):

*However, Mr de Carvalho said problem solving was at the core of the curriculum in Singapore, whose students consistently topped the global education rankings, …*

There is plenty more, similar bait in ACARA’s comparative study of Australia and Singapore (discussed here and here). So, let’s take a closer look at the bait, at the problem-solving “core” of Singapore’s mathematics education.

A “problem” in mathematics can mean many different things. In particular, a problem can be absolutely routine, what would normally be referred to as an “exercise”, and is there for the practice of basic skills. But not all exercises are routine. An exercise may require more care in setting up, or involve nastier numbers entailing trickier computation, or more subtle manipulations of the equation(s). It is still an “exercise”, in the sense that it is there primarily there to test and to practise specifically chosen skills, but it can be a *hard exercise*. It may stretch the student, but within clearly defined parameters, with the required facts and skills clearly understood.

At some point such hard exercises would more naturally be called “problems”. If they’re sufficiently difficult you might call them “hard problems”. But none of that changes the essential nature of these exercises/problems, that they’re there for the testing and practicing of clearly defined facts and skills. *And*, in that way, these problems *presume* some prior mastery of those facts and skills. The harder the problem, the greater the mastery presumed.

This is the way to understand “problem-solving” in Singapore’s mathematics education. We have more to learn, but everything we have found so far points to exactly what one would expect: in Singapore, “problem-solving” largely amounts to the serious practicing of hard, up to *very* hard, exercises, based upon a prior mastery of fundamental facts and skills.

It is easy to get a sense of this simply by searching for “Singapore test papers”. This is one such site, and this is a Primary 6 test paper from that site. Not all the questions are hard, but they get plenty hard. Some of that difficulty is in the material being more advanced — Primary 6 students do a decent amount on rate and ratio problems, including some algebra — but that’s not the only reason. There is plenty harder, and the reader is encouraged to hunt, but here is a quick, telling example from the Primary 6 paper:

*Which of the following fractions is nearest to 2/3?*

*1) 3/4 2) 5/6 3) 7/9 4) 1/3*

*That’s* a Singapore maths problem. Just a fraction comparison question, but a *hard *fraction comparison question. You can’t possibly do the question quickly without being light on your fraction toes.

That’s the bait, Singapore’s problem-solving. And now, the switch: what does ACARA mean by problem-solving?

It is abundantly clear that ACARA’s notion of a “problem” is not remotely like Singapore’s focussed and difficult exercises. ACARA’s “problem-solving” is of a *much* more open-ended and exploratory nature. It is inquiry-based learning, with the little kids being intrepid little Lewises and Clarks. This is immediately clear from De Carvalho’s conscious decision to highlight a ridiculous “why”-hunting exercise, with the kids supposedly discovering Pythagoras for themselves.

It is also abundantly clear from ACARA’s documentation. Front and centre in the draft mathematics curriculum is the diagram below. It is one of the silliest, over-egged pieces of nonsense we’ve ever seen:

This craps smells very much of CCR. Whatever its origin, notice that at the bottom of the pretty blue list of “Mathematical approaches” is “problem-solving and inquiry”. This is then explained:

*Problem-solving and inquiry – skills and processes that require thinking and working mathematically to understand the situation, plan, choose an approach, formulate, apply the relevant mathematics, selecting appropriate and efficient computation strategies, consider results and communicate findings and reasoning; Problem-solving and inquiry approaches that involve thinking and working mathematically include experimenting, investigating, modelling and computational thinking.*

Ugh! But let’s go on.

ACARA is explicitly linking “problem-solving” to inquiry based learning, but it is worse than that. This problem-solving is more than an approach to the curriculum, it *is* the curriculum. From ACARA’s What Has Changed and Why:

* The content descriptions and the achievement standards in the consultation version now explicitly include the critical processes of mathematical reasoning and problem-solving from the proficiency strands. This results in a mathematics curriculum that supports deeper conceptual understanding to make mathematical learning more meaningful, applicable and transferable to students. *[emphasis added]

That is, “problem-solving”, meaning inquiry-based learning, is now to be part of the *content* of the Australian Curriculum. De Carvalho can claim that “ACARA is not making any recommendations about pedagogical approaches”, but his claim is clearly, ridiculously false. And here is the falsehood in the flesh. Here is just one of a zillion such *content* descriptions, this one from Year 2 Number:

*model situations (including money transactions) and solve problems involving multiplication and division, representing the situation as repeated addition, equal groups and arrays. Use a range of efficient strategies to find a solution. Explain the results in terms of the situation*

This is garbage and, with absolute certainty, it is not Singapore.

What is Singapore doing while Australia is playing these idiot inquiry games? The students are learning their damn multiplication tables, so they can go on and do *Singapore* problems. Problems worth doing. Problems that the vast majority of Australian students haven’t a hope of being able to do.

It is a blatant and insidious lie to claim that ACARA’s problem-solving push in mathematics is even remotely like Singapore. And it is a hugely damaging lie. Inquiry-based learning is a disaster; it is already here in Australia and it is already disastrous. As we have written elsewhere, the poor kids aren’t Lewis and Clark; they’re Burke and Wills. They don’t have a chance in hell of getting anything solid, of retaining anything from these aimless treks.

Does one need a proof that inquiry-based learning is a disaster? No. It is obvious on its face, to anyone with any decent understanding of what mathematics is and how children learn. But, for anyone who needs a proof that dumb is dumb, Greg Ashman has written an excellent post on ACARA’s Singapore nonsense and the evidence for the failure of inquiry-based learning.

ACARA are bait-and-switch swindlers and swill merchants, and they should be disbanded. *That’s* how to solve a problem like ACARA.

That solution just seems like pie-in-the-sky dreaming. Unfortunately. Not that I have anything better to add.

The fractions question is nice. I like it how when you calculate the difference, the algebra is quite simple, and then it turns out to just be knowledge of being smaller than when . My son (currently in third grade) is doing these in his class right now. They are much easier than that, but I tend to give him lists of problems at home. I’m pretty certain he can do that one, but I want to check. He’ll get a kick out of it being a problem in Singapore for Y6.

The problem is not that he is doing no good problems at school. The problem is that there aren’t enough of them. With the current curriculum, there is so much baggage on each of these exercises that the focus is not on cementing skills, and anyway, with time constraints it becomes impossible.

With this new curriculum, it will only get worse. What a shit situation.

Glen, what is this whole blog other than pie-in-the-sky dreaming? No one with any power is listening, and it is far from clear that the people with any power have any power. It is a fundamental and endemic cultural decline.

Yes, the fraction question is nice. It’s nowhere near the most difficult question on that paper, but it requires a sense and facility that is simply not part of the AC, and is rarely taught in schools. It is also, ironically, a decent “inquiry” question, in that there are at least two fundamentally different but reasonable ways to do it. Not lost-in-the-jungle ACARA inquiry, but inquiry nonetheless. HOWEVER, that inquiry, the whole question, should come *after* you have the basic mechanics of common factors/multiples and equal fractions. It would be appalling as a question for learning the basic facts and skills of fractions.

I forgot to reply to this one.

My son really had fun with the question. I told him to expect it to not be easy — it was for older students, and also I started an egg timer. He had paper and a pen. He did it in about 2 minutes. He drew a few pictures of dots and squares, I didn’t ask him why but I guess that was his working out.

He did get the right answer. He wrote next to each option the distance from that option to . I could see that his first guess was wrong, but he changed it. He got such a kick out of the right answer. I rarely see him so happy about doing an exercise. Thanks for the question.

Of course he then asked everyone he could the same question (especially adults) for the next few days.

Impressive. Year 3? I assume the dots was effectively a common denominator thing: 2/3 OF a suitable number, etc. I wouldn’t try on my Year 4 yet. We’re just doing fractions now.

Yes he’s in year 3. To be fair, his teacher has him helping the other students in the class with their fractions when they do “maths time”, and it has been like that for the last three weeks. So he is quite comfortable with them.

I’ll have to ask him what the dots and squares mean this afternoon. I wonder if that was a strategy he was taught in class that I don’t know about. (I hope it is not like the pie-charts mentioned elsewhere…. but I suspect not, since he did get the right answer.)

Yes you were right Marty — they were representing ratios and partially masquerading for common denominators. He will grow out of it… I think it isn’t too harmful.

By the by (or maybe not…), but I’d wager that most VCE General / Further mathematics students would struggle with that Singapore Grade 6 test.

SRK, that’s an unnecessarily narrow target.

Great idea! I’ll ask my students tomorrow to do just that! And I’ll let you know how they go. Anyone care to make a prediction on the results?

Will you impose a time limit?

Assuming a 5 minute time limit, I predict 1/3 of your students (rounded down to the nearest whole number) will succeed. My prediction is higher than you might expect, because I’m assuming 1/4 of them will simply guess the correct answer.

The funniest part of your experiment will be the looks on the faces of your students when they’re told calculators are not allowed.

I doubt Methods Unit 1/2 (or even Methods Unit 3/4) students will do too much better.

JF: Would a time limit matter? It seems to me that many assessments have time limits that do not allow students to show what they can do. Rather they show what students can do in a time limit that is quite restrictive. For practical reasons there have to be some time limits, but do they have to be so restrictive?

A time limit always matters. I have no doubt that if 2 hours was given for this question, the success rate would be closer to 80-90%. On the other hand, I stand by my prediction if the time limit is 5 minutes:

In this case we’re simply haggling over the amount of time.

I’d be impressed with a success rate of 80%-90% even with a two hour time limit.

Time limit: 10 minutes.

VCE Further Maths Class = 0% success rate.

VCE General Maths Class = 0% success rate.

Their attitude was good, their approach to the problem was shocking.

Will try my colleague’s VCE Methods 1+2 and 3+4 class, hopefully tomorrow.

He suggested that, maybe, one student might get it: “Might”.

What a fascinating, yet depressing, exercise.

The seeds sown of a “progressive” education bearing fruit?

To quote Dirty Harry: “That’s a hell of a price to pay for being stylish.”

Will keep you posted.

Given the 0% I assume you weren’t permitting them to guess, you were asking for a method/solution?

On the bright side:

Time limit: 3 minutes.

Class of some of the brightest Yr 11’s = 90% success rate.

Good attitude (embraced the challenge), multiple approaches that all reduced to using the lowest common denominator of 36.

The class enjoyed being compared to an average Singaporean grade 6 student.

To quote Frankenstein: “It’s alive! It’s alive!”

Yes, I insisted on a method that would arrive at a solution beyond mere guesswork. Nobody guessed, they all had a crack – and all approached it by drawing pie charts and comparing the size of the pieces to try to determine which piece was closest in size to 2/3 (I kept copies of all their working). Over 10 years of “doing maths” and pie-charts were the universal approach..? WTF? After failing to recall a method to solve the problem, one Year 11 student begged me to be allowed to used her calculator – and she still couldn’t get the correct answer! When I guided them through the solution, they were stunned. The Year 11’s were captivated and wanted (!) to do more maths. I asked them what they were up to (not my class, not my kids – I was filling-in for an absent teacher) in General Maths and they said “linear equations, something to do with substitution”. As it turned out, they were studying simultaneous linear equations and how to solve them using substitution or elimination. They had no idea. I slowly went through a couple of examples for them and you couldn’t have asked for a better, more attentive bunch of kids. They got the hang of it pretty quickly, and nailed a couple of examples themselves. They wanted to learn – no, they wanted to be taught! – but hadn’t been, not really. They had been subjected to enquiry-based constructivism (or whatever the terminology is for this particular flavour of b.s.) and the result has been, and continues to be, a disaster. It makes me sad – and frikkin angry. The kids deserve a lot better than the crap that they are subjected to, and the powers that be wonder why they are disengaged and why there are serious and persistent behaviour problems. GIGO = Garbage In, Garbage Out.

SH, my experience is fairly limited, but my sense is that inquiry-based teaching is extremely rare at VCE level. (Of course, there are teachers who “teach” by telling students to read the worked examples in the textbook, and then doing the exercises, but I don’t think that is what is meant by “inquiry-based teaching”).

I would be curious to hear from other, more experienced teachers, if my impression is correct.

SRK, I am sure that this is not what is meant by inquiry based learning. I suspect is that this is the way in for investigative projects in mathematics at VCE level. While I am not opposed to this, one issue is that many teachers have not had any experience themselves in investigative projects .

Rare @ VCE level? Sure. Rare < VCE Level? Not a chance, which is where the damage is done.

That is frankly shocking.

Oh, that’s nothing. Some of my Year 12 FM students can’t tell me (or anyone else for that matter) what 25% of 40 is off the top of their heads (i.e. without reaching for a calculator). A couple can’t even tell me what 25% of 100 is (without reaching for…). One of my physics students converted 50mm to 0.5m (in order to determine the potential energy stored in a compressed spring) and, when I asked him to look closely at his conversion, began to explain to me that, “Um, 50mm is 0.5m, innit?” to which I replied “Multiply both by 2” and he dutifully did, getting 100mm = 1m, thereby sensing that something was a bit “off” but couldn’t quite put his finger on the source of the error. Needless to say, rectification of his basic understanding of units and their conversion became the focus on the remainder of the lesson. Now ain’t that even more shocking? (Proactive apologies to Marty for the digression into physics but it is a good example, n’est ce pas?). 🙂

Physics with numbers is permitted. Posts on VCE’s Aristotelian physics will be deleted.

One of my friends had a growth in his ear. He went to a specialist in Melbourne to have it checked. When the results came back, he was terrified because the measurement of the growth was in cm instead of mm. I gather this is not an uncommon error in the world of cancer.

I’m shocked you’re shocked.

I’ve clearly come to the right place! This blog is an island of sanity. Hat’s off to you, kind sir (and to those like-minded souls who get stranded here from time to time).

I’m just trying to keep myself sane. Anybody else’s sanity is an unintended byproduct. But, you’re welcome.

Thanks, SH. I’m not surprised at your results. Note that the question is worth 2/45 on a 1 hour paper (if I counted correctly), which is ballpark what JF permitted.

One thing we don’t know is the percentage of Singapore students who got the question correct, or the percentage of general Singapore Primary 6 students who would know how to do the question quickly. Undoubtedly it’s way higher than Australia Year 6, but we don’t know how much way higher.

I explained to both the Year 11 and Year 12 classes that it was a question in a test for Grade 6 Singaporean students and the result was stunned silence. One student (the girl who failed to get the correct answer using her calculator) said that she now felt that she was dumb. Naturally, I had to counsel her (and her classmates) that they are not at all dumb (they are not, they’re actually quite bright kids). On the bright side, the Year 11’s desperately want me as their Further Maths teacher next year. On the dark side, Further Maths is a vestibule of manure, as is Methods, which I don’t (yet) teach (it’s coming, I can smell it). What has become of high school mathematics..? Surely this has not happened purely by chance. “ACARA does not play dice with the curriculum”. Or does it?

Question: Anyone got copies of the Vic HSC Pure & Applied maths books from the mid-80’s? If my memory serves me, one was a light brown colour and the other a mid-blue colour. No idea of publisher. I’d be fascinated to compare “now and then”. Sadly, mine have been lost in space-time.

Please Do Not Feed or Annoy the Animals.

Fitzpatrick and Galbraith: Yellow and Green, then Red and Green, dating from the early 70s, until Labor hacks killed Victorian maths. They are excellent and as rare as hen’s teeth. I worked my butt off trying to get permission to digitise them and distribute them (for free). Peter Galbraith was flattered and agreed immediately. Bernie Fitzpatrick’s son seemed puzzled and disinterested, and finally said no.

You’re referring to the classic Lucas and James textbooks Sixth Form Applied Math (brown cover) and Sixth Form Pure Math (Blue cover). Brilliant textbooks from a brilliant era in Victorian mathematics (up to the mid-80’s). Published by Lloyd O’Neil (https://abiawards.com.au/lloyd-oneil-award/)

Impossible to find nowadays – because of course most libraries (and schools) have chucked them out. According to trove: https://trove.nla.gov.au/work/18926421?keyword=Sixth%20Form%20Pure%20Math

Fahrenheit 451 comes to mind …

PS – You are far too kind, Sir H. I would have told those kids they were not dumb, they were simply victims of “enquiry-based constructivism (or whatever the terminology is for this particular flavour of b.s.) and the result has been, and continues to be, a disaster”. This sort of bullshit teaching does not deserve protection.

Ah, the other books. I didn’t like L & J as much. I thought F & G were always clearer. But yes, really solid, really good books, and I really wish I had a set.

Thanks JF. Hardly too kind, just careful what I post. Re: “I would have told those kids…” = LOL, I did precisely that! They might need further counselling as PTSD might have set in. Yes, they are hapless victims and, yes, they are now aware of it. Once the PTSD wears off, they’ll be angry. I suspect that it’ll be around the time when the VCE results are posted. Sadly, this sort of bullshit teaching is being taught to the naive and idealistic new crop of young teachers (i.e. mid to low 20’s) by the various academic ‘experts’ who swallow and regurgitate this scheissdreck by the shovel-load.

Will look into the textbooks that you mentioned. My old mate from high school might have both of them (he’s a bit of a hoarder). Will check with him. 🙂

I do teach some math teachers in training, and I do try and teach them well. I figure you’re referring to education academics rather than mathematicians. Some of them are not so bad. More than this, though, is true. At some universities, for example the one I work for, mathematics teachers in training are at least partially taught mathematics and mathematics education by mathematicians. So you’re including them (and me) in your last sentence.

Glen, here is a question that Greg Ashman asked on Twitter a couple days ago:

Who are the education academics in Australia who specialise in mathematics teaching and who advocate for explicit teaching, times tables etc.?So, a couple questions:

1) Do any of your “not so bad” maths education academics get a passing grade under Ashman’s criteria?

2) Do any of you “mathematicians in maths ed” get to see to-be primary teachers?

Personally, and not counting low-level cannon fodder, I have met exactly one Australian maths education academic who I thought was doing good rather than serious harm.

“Cannon Fodder”. LOL, that was/is going to be the name of the book that I will be writing re: my experience of transitioning into teaching via an M.Teach course. I’m contemplating other titles, but any assistance would be greatly appreciated.

I’d make a slight change to the title:

“Canon Fodder”

Excellent adjustment.

For Q2, there is much less contact. Just one subject that we teach as far as I know compared to a whole major (around ten subjects). That’s normally justified by the idea that primary teachers teach “everything” and so the course isn’t focused on mathematics.

The fact that we have contact at all (also the more extensive contact for high-school teachers) is due to one of the not-so-bad maths ed academics, who happens to be the course coordinator. I’d have to ask him explicitly re: times tables.

About Greg’s question, I think it might be a bit poorly worded. Because I’d say most would “advocate for explicit teaching” but some would spend say 10% of maths time on that, whereas others might spend 90% of time on it. So it’s more about how they plan to achieve mastery of foundational skills. If they say by explicit instruction, you’ve got your answer. If they say by something else…. then you’ve got your answer.

Thanks, Glen. it sounds like your to-be teachers to at least some extent get contact with mathematicians, which is better than most places, but the real damage is in primary schools. This is also not remotely about what your course coordinator believes, it is about the clear messages the to-be teachers receive. You’d need to convince me that the message overall, what they come away with, isn’t pretty standard nonsense. This still holds, and I question whether more than a handful of your to-be teachers believe it.

Re Glen’s question, it was a quick Twitter thing, to people who know him and his research. They understood the question, and the *crickets* indicated the answer.

Clarification: referring to those who run post-grad teacher education courses e.g. M.Teach (Secondary), which I am currently experiencing in all of its academic vaingloriousness. Not sure what happens in Bachelor level maths teacher education courses as I’m not familiar with them, and my lack of familiarity is only exceeded by my lack of interest. But M.Teach (my pathway into teaching)? Ay yay yay (slaps head).

During my MTeach (Secondary) – which I completed through Deakin only in 2019 – there were times when I wondered what was the point of a particular idea, only to find out that it was useful in my placement and teachers with whom I worked had never considered the idea. I felt like I was ahead as a result of the course.

Without your stating the idea, I don’t believe you.

Marty, I will offer one of several instances. Experienced teachers will realise that I am new to secondary teaching.

Anyway, during the MTeach, we were ask to find out where our students were born. “What is the value of this in teaching mathematics?”, I asked myself. But I did it anyway. Several of my Year 9 students came from Kerala in India. So I devoted one class to the attached exercise. I think it went well.

One teacher told me that many of the Indian students take lessons out-of-school in speaking Indian. I thought that I was ahead.

lessons-year9

Clearly those two years weren’t wasted.

It feels like they’ve tried to copy the IB curriculum circles, but with gross colours.

And gross meaninglessness.

Yes, it looks like a gross pizza that’s well past it’s eat-by date. You’d throw it in the garbage bin if you saw it in the fridge.

The LSD Special, with anchovies.

I looked through the Sg tests.

Off the top of my head, I’d say that they are a good assessment of a student’s numeracy. (For me, numeracy=applied mathematics.)

Based on my experience in a classroom today, I’d say that many Year 9 Australian students could not cope.

How little you must think of applied math :(.