What If Not Everybody is Good at Maths?

Eugenia Cheng is one of the happy new faces of mathematics popularisation. She is adored by all. Well, almost all.

As a mathematical evangelist, Cheng wants everyone to join the Church, to fall in love with mathematics. To that end, Cheng has given a zillion talks, she writes a monthly column for the Wall Street Journal, and she has written a number of popular mathematics texts. Cheng has just released another book, Is Maths Real (reviewed here). Last week, to promote her book, Cheng had an article in The Guardian, What if nobody is bad at maths? It is not a good article.

Cheng begins in the familiar way, lamenting that people are so willing to say they are bad at maths, but that this attitude only develops later, after they are “scarred” from schooling:

But one thing I know is that when I help five- and six-year-olds with maths they typically scream with excitement, and only learn to fear it later.

Yeah, like when they have to learn to pay proper attention and do it.

The basic problem, in my view, is that in our haste to convey content – fractions, percentages, algorithms – we don’t pay enough attention to feelings.

Really? Maybe in England or America, but for primary kids in Australia the attention appears to be overwhelmingly on feelings, which is exactly the problem. Anyway, Cheng then tells us where it goes wrong:

I ask [her undergraduate art students] what they found so disagreeable, and there are clear recurring themes: memorisation, especially of times tables, timed tests, right-and-wrong answers, and being made to feel stupid for making mistakes.

Heaven forbid there be right-and-wrong answers. In maths. And again, always, with how cruel it is to make kids learn the multiplication tables. There is obviously a problem with unmotivated students, we may (and do) argue about and disagree on how to motivate them, but the suggestion that learning a few dozen patterned multiplications is arduous, or is optional, is plain nuts.

Cheng continues:

Often they felt alienated because they had searching questions – such as why does -(-1)=1; do numbers exist; is maths real – but they were told these were silly or irrelevant, and they should get back to their repetitive, algorithmic homework assignments.

We don’t believe it. We cannot remember ever meeting a maths teacher who would consider, let alone tell a student, that such questions are “silly” or “irrelevant”. A teacher may not offer much time to provide answers, and they may well suck at giving any decent answers, but for 99% of teachers that’s the worst it would get. Cheng is pretty much just making stuff up.

After, correctly, slamming Rishi Sunak for his idiotic plan to make maths compulsory to 18, Cheng goes back to fixing the problem. Which appears to be by treating kids forever as if they’re five years old:

When five-year-olds first encounter the subject, it’s as a creative, open-ended activity, involving play and exploration. They learn about numbers using colourful blocks that join up in different ways. They fit these shapes together and tell different stories with them. Just a year or two later, though, maths becomes a discipline with strict rules and a forbidding regime of right or wrong answers.

Yes, because that’s the way maths is. Eventually you have to get on with building the structure. Which comes with rules.

Instead we should try to maintain that sense of exploration and open-endedness, of trying out different approaches to a problem and seeing what works.

Really? Got an example?

What’s important about times tables, for example, is not the answers, but the different possible relationships between numbers.

Reverse mic drop.

Does one really have to point out how bad this is? Probably, not, but we’ll do it any.

First of all, of course the goddam answers are important. Secondly, yes, you want students to be familiar with, and always on the lookout for, relationships between numbers; that is, as it happens, one of the means by which the tables get solidified, and why it is simply not that arduous to learn the damn things. Thirdly, when you really get into the number relationships, you’ll do it with algebra, and there’s not a snowflake’s chance in hell of doing that with any success, or pleasure, if you don’t first know the damn tables by heart.

Cheng continues with the boringly familiar sales pitch, for the Real World and classroom gimmicks, “maintaining interest” in maths by making it less mathsy. She then takes a right-angle, and addresses the question posed by the title of her article:

The idea that anyone is naturally “bad” at maths is pernicious in several ways. It ignores the amount of work it takes to get good at it. And it does take work. But that work doesn’t need to be hard – it can be challenging, but with a sense of adventure and ultimately reward, rather than discouragement.

Well, no and yes. That work does kind of need to be hard. That’s the nature of work: it’s work. But, true, the work can be enjoyable and satisfying along the way. However, work aside, it doesn’t mean some people don’t just suck at maths. But apparently suggesting this as a possibility sends the wrong message:

The “bad” trope also provides people with an easy reason to give up, and the education system concurs by writing them off as fundamentally unsuited.

We’re anything but an apologist for “the education system” – ask ACARA – and we regard as unmitigated evil the premature writing off of kids who were never given a proper chance to enjoy and to excel at mathematics. But there is still reality to deal with.

As it happens, we don’t see that it matters to a teacher whether the student in front of them is “fundamentally unsuited” or not: the student is what they are, for whatever reason, and the teacher deals with what they’ve been given. But on the larger scale, truth matters.

Cheng closes with the obligatory “it’s not your fault” bit:

One thing is clear: if you think of yourself as belonging to the “bad at … ” camp, it’s not because you failed maths. It’s because maths failed you.

We really wish these popularising superstars would spend less time earning easy adoration and more time telling people the hard truth of what it takes to not fail at maths.

64 Replies to “What If Not Everybody is Good at Maths?”

  1. I’m forever grateful to my grade 6 teacher Mr Goertze who on discovering we were all shit at math had us spend 10 minutes a day for several weeks to get proficient in tables to 12×12.
    The worst part about the tables are bad crowd is it is easy to make them easy. Jump math had a great pdf on an ideal sequence.

    I’d ask the anti tables folks if memorizing a sequence of 26 letters is also a torture we shouldn’t put kids through as it might expose a bias against math.

    1. Thanks, Stan. Yes, this way or that, the tables just ain’t that hard. It is the stupidest possible hill for these let’s-just-have-fun advocates to die on.

  2. To me, there is a great similarity between learning mathematics and learning music. At the beginning stages, the introduction is a gentle introduction to material designed to appeal to the learner and invite some experimentation. However, at some point, it is clear that progress is not possible without the guidance of a competent and sensitive teacher and the discipline of learning the rudiments and the technique. While this may be tedious for the student, the payoff is much greater efficiency, the ability to progress more rapidly, the internalization of musical structure and the satisfaction of being able to perform more serious music. I am sure the process is similar if you want to be an effective tennis player or carpenter. What is needed in the syllabus are more interesting exercises and examples, and not the mindless pap that too many texts have to offer. For example, ask the pupils to use each of the digits from 1 to 8 exactly once to create two numbers whose difference is as small as possible; I have tried this and the class interactions are quite interesting.

    1. Thanks, Ed. I had thought to make that music analogy, given Cheng is an accomplished musician, but decided not to clutter the post. The digits problem you mention is excellent. I’ve just recently been doing versions of it with primary kids, as it appears in Tony Gardiner’s excellent Mathsteasers books.

  3. There is a teaching issue with learning multiplication tables. If a student misses out on learning tables in primary school, then it seems to be a perpetual problem. What does the Year 9 teacher do when one of the students in the class admits to not knowing multiplication tables? Similarly what does the Year 10 teacher do when one of the students in the class admits to not knowing fractions? Some gaps in knowledge can be remedied quickly; others, not so quickly. Suggestions welcome.

    1. Step 1 is to scream “Damn Jo Boaler!” at the top of your voice. I’m not sure about Step 2.

      1. Terry’s point, whilst valid, raises the question of why the Year 7 or 8 teacher did not realise this issue…

        But I digress.

        Cheng is clearly a highly educated person. She writes well (in the sense that her books are easy to read) and has a high level of Mathematics education.

        Something makes me wonder therefore if she is in a position to tell me why students who are not as literate and have never been good with numbers are hating the subject.

        Unfortunately, it is exam/reports season and so this thought-bubble needs to deflate now.

        1. Thanks, RF, although I feel you’re Dorothy Dixering yourself.

          Of course, as I know you know, Year 7 and Year 8 teachers *do* realise that plenty of their kids learned bugger all arithmetic at primary school. The question is, what do they then do about it? If the school is good, they will, systematically, do something along the line of Stan’s Mr Goertze. They will begin Year 7 by declaring that no students need look at any of the godawful statistics and some other dispensable swill assigned for that year, and the school will use the time saved to begin the year with a proper Arithmetic Boot Camp, which is then reinforced throughout the year. If the school is bad, what’s a teacher going to do? They can tell the kid and the parents that the kid needs to catch up, they can give the kid the sheets or the programs to do so, and 99% of the time the kid/parents will agree gratefully and enthusiastically, they’ll get on with it industriously for two or three days, and then they’ll give up. Then, a couple years later, Terry comes along …

          I don’t think Cheng writes well, or even close.

          1. Indeed, it was not a genuine question.

            As to whether Cheng writes well; maybe it depends on your definition of “well”.

            I read “How to bake pi” many years ago and found it was easy to read and the mathematics contained within was (as far as I could tell) correct. Sure, I still don’t get what “category theory” and suspect I never will.

            Did I learn anything from the book? Not really. Was it a good book? Debatable. I’m not sure that makes it bad writing though.

            As to the times-tables issue though – there are many, many issues at work. You left out the bit where the student gets themselves a tutor who “did methods in year 11 and is now studying commerce” and then comes back to school saying “my tutor says…” which may or may not a true account of the conversation. And let us not forget the students who have done K**** for many years and think they know a lot more than they do…

            The idea of an arithmetic/algebra “boot camp” is a nice one though. I might try to borrow that.

            1. Sometimes the tutor is a disaster. Sometimes the tutor is Marty and, with all due humility, knows a hell of a lot more than the teacher.

              1. More often than not the tutor is a con artist. “I’ll get your child ahead” they say. And then the child doesnt pay attention in class because they already ‘know’ the work because their tutor taught it to them. (It goes without saying that the child is not ahead at all and actually understands jackshit). And dont get me started on the con artists who run tutorial with 25-30 students, each student paying 30-40 dollars to sit there for an hour and work through a worksheet. Marty, I would say that often the tutor is worse than a disaster, the tutor actually causes damage. And yet these con artists keep doing brisk business every year. More teachers need to tell parents that the tutor they pay good money to is a con artist causing academic harm to their child. There are certain ethnic groups that these con artists particularly exploit.

                1. Hi, A. I honestly have no idea the proportion of tutors that are bad. What makes you believe the “more often than not”?

                  1. Based on my sample of students over the years, most tutors are con artists. More often than not when I talk to a student (for reasons such as not paying attention in class, working in class on stuff not related to what Im teaching, poor result in a test etc) I discover after some probing that the student has a tutor and the tutor has said that they can get the student ahead. Very rarely do I discover a student who is being tutored in parallel with what is being taught in my classroom. In my experience, good tutors – who work with a student on their weaknesses and try to consolidate and reinforce what the student is being taught at school – are very rare.
                    And dont get me started on students who ask me how to do an inappropriate question (that is, a question well ahead of what Im teaching or taught at the time). When I probe where the question has come from they nearly always say their tutor gave it to them. And then look puzzled when I say Ill be damned if Im gonna do the tutors work for them.

                    1. If the kid is properly solid on the standard material then some advancing is fine. Not obviously the best choice, rather than going deeper, but fine. Too often, however, the kid is doing advanced stuff without the proper grounding in the earlier stuff, and so the “advanced” work either a mess or too thoughtlessly mechanical, or both.

                    2. X, when your learning how to sketch graphs of quadratics:

                      Advancing is when the tutor says theyll get you ahead by ‘teaching’ you how to sketch graphs of cubics.

                      Going deeper (extending) is when the tutor teaches you how to algebraically investigate the effect of a parameter in the equation of the quadratic eg What do the graphs of y = x^2 + kx + 2 look like for different values values of k.

                      “I’ll get your child ahead” is typically lazy and exploitive tutoring that does much harm and very little good. It is snake oil that many parents and students continually fall for and buy. I have not met the student who did/would not greatly benefit from tutoring that probes for and addresses weaknesses, reinforces and consolidates what has been taught at school, and extends (or goes deeper) what has already been taught at school. Such ethical tutoring requires patience, skill, experience, knowledge and preparation. Con artists do not offer this.

                    3. My issue on this front, as a tutor, has been parents demanding that their kids “get ahead”. Selective school admissions culture, and such.

                  2. Admiral Rickover, in the 1950s, thought that enrichment was not as valuable as acceleration, for top few percent students. Just felt it was better use of their time and that the wonder of new material/tools was better than going in depth on earlier stuff in the standard curriculum.

                    Not saying he was an oracle, but he wrote some interesting things on education during the 50s. And was very bright himself. And founded the first (and best) power-producing nuclear program.


                    (off topic…but at least it’s not a YT video…interesting find when I googled his education writing…he had a book…borrowed it from the library once.)

                    P.s. I think tutoring can be a great addition. Would you similarly say that extra coaching in sports or music, over what the school provides, has no value? That other Anon is just jealous.

                    1. Your argument’s kinda weird. I don’t think the other Anonymous (why can’t you guys use names …) was talking about the “top few percent”.

                    2. It wasn’t an argument. More of an isolated point. A nuance. A flourish.

                      You know me by my claw, mean Marty. 😉

                    3. I would estimate that at least 40% of students think (or their parents think) that theyre in the top 8% of the state.
                      I had a Parent-Teacher interview once where the mother insisted despite overwhelming evidence to the contrary that her child could and should get a study score of at least 40 for Maths Methods. I asked her whether she seriously thought her child was in the top 8% of students in the state. Without batting an eyelid she said yes.
                      PS – Jealousy has nothing to do with it, frustration with con artists has everything to do with it . I speak from experience and base my statements on evidence. To listen to many parents youd think virtually every child was gifted and should be advanced. Reality checks are never welcome. Con artist tutors make the job of teaching so much harder than it should be (again, I am speaking from experience).

  4. Easy target, but famous target, so get in there!

    Still an easy target though. Like with the ACARA curriculum, it might be more interesting to find sensible points that Cheng has argued for.

    The book is incredibly bad. I couldn’t handle more than a few pages.

    1. Really? Cheng is an easy target? If I’d known that I wouldn’t have bothered. But I’m not sure you’re correct.

      Also, why do you think Cheng’s book is bad? (I have heard one reason and I suspect another, but I know nothing of much substance.)

      1. It almost parodies its own mission by being completely over-the-top in inclusivity. The book is confused, is it amount mathematics being “real” or is it about the author, or is it about social justice enforced using math?

        If you wish to indulge in masochism, you could read some….

  5. I asked my maths students what they found so disagreeable [about art], and there are clear recurring themes: you never really had to learn anything, and even when you did, you weren’t really rewarded for it because your marks largely came down to the teacher’s subjective opinion, and you were made to feel stupid simply because you and the teacher had different taste.

  6. When we ask children to learn maths that they are not ready to learn, and they do not experience success, it is discouraging for them and they begin to think of themselves as ‘not a maths person’. The Australian Curriculum tells us what year 9 students are to learn, but it is difficult to teach index laws to students who do not have a good handle on multiplication.
    The problem is expecting students who have gaps to learn what they can’t yet learn, and not having the time to go back far enough to build on the foundations that they do have.
    I think Gonski was on the right track suggesting that we expect students to gain a year’s worth of knowledge in a year, rather than to get to a particular level.
    Eventually we stream students: not everyone does Specialist Maths in Year 12, so why does the curriculum expect all students to learn the same things in year 10 maths?
    Learning the maths that you are ready to learn is actually enjoyable. Why can’t we celebrate the success of those who learn and allow those who are not there yet to have the time they need to learn instead of them getting a fail on that topic and moved on to the next topic.

    1. An excellent question and one that I believe has no simple answer.

      Defining “a year’s worth of knowledge” is in and of itself a challenge and that is before we even get onto how best to measure growth…

      Grouping students based on age is easy, so schools do it. Grouping on ability has been suggested a lot by a lot of people. Will it ever happen? I’m not sure.

      1. My school grouped based on ability (ie. streamed) this year. It was taken as an opportunity by some to cherry-pick their classes and offload the most challenging students to the least experienced. The school lost two teachers in the first week with many more looking to leave.

        1. Thanks, Anonymouskouri. Do you think that’s an argument against streaming, or an argument against bad management?

          1. Does it depend on why the streaming occurs? Is it to benefit students or to benefit teachers?

            Streaming mostly seems to be a “regrouping” of classes. It might look nice on paper, but I’m not sure this form of ability-based-classes does much good, overall.

            1. At some point you stream, don’t you? Even if it’s not called that. Is there any school that doesn’t “stream” at Year 10?

              1. If you are going to teach a totally different curriculum, then yes, perhaps streaming works (VCE could be considered “streamed” in a sense).

                So yes, any school that offers a VCE Mathematics class to Year 10 streams students.

                What I think may be problematic is creating ability streams in a year level but then teaching the same material to all (and doing the same assessments). In the end that means some teachers have an easier time than others. Is this bad management? Perhaps.

        2. “It was taken as an opportunity by some to cherry-pick their classes and offload the most challenging students to the least experienced.”
          Leadership in any school that allows this to happen should be sacked on the spot. (But unfortunately it is often the ‘leaders’ who teach one or two classes who do the cherry-picking).
          The “least experienced” were probably told some weaselly crap like “It will be good experience for you to teach these students”. We are currently in a climate where teachers who have this done to them can give leadership the finger and move on to a school with a better culture.

    2. Thanks MW, although I’m not quite sure what you’re arguing. ACARA’s Year 10 is already absurdly weak, and absurd, and is way insufficient for a kid aiming for Specialist, and just barely, maybe, sufficient for Methods. If a kid has significant gaps when they reach Year 10 then they’re obviously screwed, and they’ve probably been screwed by Year 7 or Year 8, if not earlier. Schools already know this, and typically provide a Fake Year 10 for such students.

      If you’re gonna advocate policy change, why not tackle directly the absurdity of so many students having such large gaps that go unaddressed for years? Primary schools do not attempt to teach arithmetic properly, and they do not demand mastery of the little they do teach.

      In short, the problem is obvious, the solution is obvious, but no one with clout both has a clue and gives a shit.

  7. Marty I enjoyed your initial post on Eugenia, and the reference to Britain’s Rishi Sunak. Attached is another source of enjoyment, from that country’s Telegraph newspaper earlier this year about a teacher at Winchester College—the privileged school that Rishi attended and where he learnt his mathematics.


    1. Thanks very much, Ian. Tony Gardiner’s the guy to weigh in, if he’s around. But here are my thoughts.

      First of all, I’d disagree a smidge with “This is the first stupid thing Sunak has said …”. Sunak is (the continuance of) a disaster.

      I’d also disagree with most students needing less maths. This is intended to be elitism by ability, but in fact it’s elitism by wealth. There are zillions of kids who never get a decent chance to prepare for and compete on level terms in maths comps.

      I know nothing about SMP, but there are also seems to be confusion between setting a strong curriculum and an (arguably overly) abstract curriculum. You can have one without the other.

  8. Being opposed to right and wrong answers has perplexed me since I was in primary school. Knowing you have the right answer is great! These days I go so far as to say that nobody is actually opposed to right and wrong answers. Being so would imply fundamentally being opposed to learning anything – people’s names, how much to pay for a beer at the pub, what Essendon won by over the weekend. That is – being opposed to “right and wrong” answers is merely being opposed to answers.

    A lack of quality mathematics teachers at a primary (and early secondary) school level is a fast track to making maths boring. Good maths teachers (as I suspect most people here had) make the hard work enjoyable. I learned my times tables to be better at them than the kid doing year 8 maths in year 3.

    My year 7 teach emphasised the importance of the method. If we did not set out our work perfectly we had to repeat the question regardless of our correct answer. If you got an answer wrong you did the entire question again. The whole class learned very quickly both that it was faster to doing things properly the first time and that if you didn’t know how to do something you should make an effort to learn it, whether by asking for help, reviewing worked examples etc.

    Anybody who says they don’t like exact answers in mathematics should be asked about the last time they purchased … anything.

    1. Thanks, Wilba. There’s also something weirdly blind about Cheng’s claim. Yeah, kids may be get frustrated with getting wrong answers, but they *love* getting right answers. They love the very idea that there is a right answer.

  9. I’m an interloper, an English teacher, but with a Philosophy and Philosophy of Maths background, and my experience has been that some students will say that they like ‘right answer’ subjects, and others the opposite. This division is pretty much the core of the ‘arts v science’ split in personality, no?

    My son has been complaining about maths recently for this reason (he loves the visual arts) and I have tried and failed to appeal to the beauty of formal Maths in an effort to bring him round (he’s only 10, so plenty of time). Interestingly, he’s pretty good at Maths (at least NAPLAN says he is) suggesting being turned off for this sort of reason is not just a question of being no good at it. I have often heard the argument that motivation and engagement are bound up with competence, but I am yet to be convinced – my son is certainly a counter example, as are the majority of 15 year old boys I have taught who are 100% motivated to become football stars, or have sex. Competence seems neither a necessary nor sufficient condition of motivation.

    I have a lot of sympathy with the idea that at least some memorisation and formal drilling is essential for achieving advanced mathematical capacity. Given the woeful state of maths capacity of most school leavers, though, would it be such a bad thing to accept a lowering of the already dismal level, in return for them not hating it so much?

    Or, to put it another way – which is worst: 60% of people leaving school bad at maths and hating it, or them leaving school a bit worse at maths but at least open to future learning? A lot of schooling policy comes down to trade-offs, and this may be one that we are getting wrong.

    Also, it’s interesting that, whether you take the ‘touchy-feely-experiment more’ route, or the ‘throw-the-textbook-out-the-window-and-make-them-learn-their-timestables’ route, everyone seems to agree that the one-size-fits-all curriculum, composed of expected levels and progressions, isn’t really doing anyone any favours.

    1. Hi, Joe. Here are brief responses to some of your more specific claims/suggestions. I’m happy to discuss any or all.

      1) I’m not sure about the arts v science split you suggest. I’ll think about it. But I don’t think it negates my claim that (basically) all kids want to know the rules, if there are rules to be known. Kids, particularly younger kids, loathe the fluffying of mathematics, the business of making up your own method and meaning, and so forth. (Fluffying is not the same as giving kids a multi-step exercise/problem, where there are many possible, and perhaps actual, approaches. The steps, however chosen and ordered, are still based upon rules.)

      2) NAPLAN is not mathematics, or much of anything. It is not even called mathematics. It tells you next to nothing.

      3) I have no idea what mathematics your son has seen, either through you or the school. (Most ten year olds have seen essentially nothing of worth, and are not prepared for proper mathematics even if they were given the opportunity.) But, if your son has the grounding to try genuine problems (for his age/level), and if he has tried such problems but it doesn’t appeal to him, why should you care?

      4) Sex and football are hardly compelling counterexamples to the idea that motivation and competence are related.

      5) The lowering to the “already dismal level” to which you refer was enacted for exactly the kind of reasons you’re giving to lower it even further. How low are you willing to go? For what purpose? Just so the kids can come out saying they don’t hate maths, even if that is at the cost of not having seen any maths?

      6) The kids will not be “open to future learning”, because they will have been taught that “learning” can be achieved without ever having to sit still and pay attention to an authority.

      7) Your ‘throw-the-textbook-out-the-window-and-make-them-learn-their-timestables’ line makes no sense. One wants both the tables and the textbooks. Yes, the textbooks are appalling. But you want decent textbooks. In primary school as well.

      8) The “one-size-fits-all curriculum” has issues, but the issues pale in comparison to the issues with “do what you damn well please at whatever damn rate you wish, and we’ll never test you or bother you”. The clear majority of kids would be well-served by a solid, coherent curriculum that is delivered with the expectation of mastering certain material at certain times. With the clear majority substantially catered for, teachers might have some proper time to cater to the non-majority. As it is, teachers are already trying to fit the curriculum to N sizes, for way too large a value of N. It is absurd, it is failing, and it drives teachers nuts.

  10. Thanks for taking the time over my comments Marty.

    I don’t think your points about whether or not NAPLAN, or contemporary primary school curricula, are ‘real maths’ are relevant to my main argument, which is simply that some kids at least seem to neither like nor be motivated by subjects in which they are expected to get right answers (we could call what my son does ‘schmaths’ if it helps). I’d also hazard the claim that some kids don’t even respond to ‘rules’ very well (BTW – do you really endorse SRK’s views of Art expressed above, that in Art lessons, for example, you ‘never really learn anything’? Not sure how much either of you were being serious there?).

    I apologise for the failed joke about sex, but would like to know why you think my comparisons are so obviously wrong. People can be motivated and engaged in activities that they are no good at, was my point: perhaps there are ways to motivate people similarly about maths, to make the hard yakka of memorisation and practice more palatable?

    Coming back to ‘schmaths’, I’m confused by your definitions. On the one hand, you say that my son has not met any real mathematics because most 10 year olds have seen ‘nothing of worth’. So how then, can any kid have had any experience of maths ‘fluffed’? Fluffed or not, you’re still saying they’re being exposed to maths of some sort, right? This might seem like being a bit precious about semantics, but I think it’s important to be clear about exactly what you mean, especially if you’re going to make pretty radical claims like ‘NAPLAN has nothing to do with maths’. Maybe you have some other blog posts you can point me to which explain what you think maths actually is, and how, if at all, it could ever be taught at primary school?

    I am not especially well informed about the progressive educators who so clearly get your goat, but I am not sure they would feel it fair to say of them that they reject altogether the notion of ‘rules’. Maybe all they’re saying is that, much as with any other words or syntax rules, a grasp of the rules governing mathematical language is best acquired through a process of structured interaction, in which each student is enabled to make the necessary connections with their existing vocabulary at their own pace and in their own order of priority. I’m not sure I’d agree even with that myself, and would probably be more sympathetic to your position, but we have to argue with the best construal of our opponent’s viewpoint that we can.

    Finally, is it fair to say that you feel things used to be done very much better? When, exactly? Is there clear data suggesting that the collective maths capacity in Australia has dramatically declined? I know PISA suggests a clear decline, but TIMSS the opposite – though perhaps you feel the same way about them as NAPLAN?


    1. Thanks, Joe. Happy to engage, but again it’ll take me time to get to back to you. Very briefly:

      *) I didn’t say your son hadn’t met any good maths. I said I had no idea what he had seen.

      *) I know you weren’t arguing generally on the basis on NAPLAN. What you wrote was that NAPLAN says your son is pretty good at maths. My point is that NAPLAN is the flimsiest of evidence for that (without denying, of course, the possibility that your son is indeed good at maths).

      *) PISA is also garbage: see here. TIMSS is most definitely not garbage, but the reports about Australia’s “success” in TIMSS are highly flawed: see here.

      *) You sex joke didn’t fail. I liked it, and I don’t think it was wrong, just inadequate. I once was invited to give a talk, which was to include my response to prompt questions, one of which was “The five most important things to learn in school are …”. My response was “Girls, girls, girls, girls and girls”, accompanied by a photo of a stunning Audrey Hepburn.

      More, on the substance, later.

    2. Hi again, Joe. A little more in response.

      a) You wrote,

      “some kids at least seem to neither like nor be motivated by subjects in which they are expected to get right answers (we could call what my son does ‘schmaths’ if it helps)”.

      Kids like to play, and kids like to know what is expected of them. What they do not like, cannot like, is to be permitted to play for years, and then have to magically fulfil some precise expectations.

      That’s exactly what happens. Primary school is full of schmaths, as you put it, with no mandatory methods, no clear expectation of mastery, and no testing whatsoever. Then, eventually the kids gotta get on with fractions and algebra, and plenty of them are utterly screwed. They don’t have the mastered techniques for the precision required, and they don’t even know what they don’t know. The upshot is that most then get watered down fractions and algebra, which at that stage they still understandably detest, and it all cascades into insufficient preparation for all that follows.

      b) You wrote

      “I’d also hazard the claim that some kids don’t even respond to ‘rules’ very well”

      What can you possibly be arguing? Young kids certainly like to *know* the rules, which is what I wrote. If kids don’t learn to appreciate rules, the importance and power of discipline, then they simply haven’t been properly taught.

      c) I think SRK had a good point, that there is not the same structure to school art as there is with maths, or even with schmaths, but I don’t particularly want to get into a debate about art lessons. I’ll make a general point, however, that *all* school disciplines are losing the structure they have. You want to tell me that this is not true of English?

      d) Yes, you are playing semantic games with whether most kids meet real maths. *Of course* kids will see some arithmetic, for example. Big fat hairy deal. The real question is whether they develop any proper sense of and facility with arithmetic, and have the appropriate facts and techniques at their fingertips. Which only comes with proper practice.

      e) I did not write ‘NAPLAN has nothing to do with maths’: I wrote ‘NAPLAN is not mathematics’. NAPLAN has a test in Numeracy, not mathematics. Look at some NAPLAN questions and get back to me. Don’t forget to bring your calculator.

      f) I think you know what maths is well enough, or better. What I don’t think you are aware of is how weak the curriculum is, and how poorly even this weak curriculum is generally taught, particularly in primary schools. I can point you to posts, and I’ll link a couple, but I think to do so is missing the point.

      I get sick to death of talking about it, but we always have to begin with the multiplication tables. If kids are to develop any proper facility with fractions and algebra then they *must* know their tables, to 12, by heart. Yes, of course there is much more, but the tables are non-negotiable and a no-brainer.

      BUT BUT BUT, the overwhelming majority of primary teachers do not ensure that that their students know the tables. Many of these primary teachers, and probably still a majority, do not regard students knowing the tables as even particularly desirable. Given this, what can possibly be the point of my detailing what these teachers might properly do?

      For posts, you can start here and here.

      g) I’m in no mood to provide the “best construal of our opponent’s viewpoint that we can”. See remark (f).

      h) Yes, things used to be better. Are there stats to indicate that the general level of mathematics gotten worse? I don’t know: you’d have to argue all manner of things, such as retention rates. But why haven’t they gotten demonstrably *better*? The classes are much smaller now. There is much more money now. There is more training of teachers now. There is way more maths ed theory now. There is a whole massive industry now. So where are the gains? And what is being offered to the top end of kids, at least in Victoria is just garbage, way way worse than it used to be.

      1. Hi Marty

        Not sure if you will be still looking at this entry, but thanks so much again for taking the time with my ideas. I have been having a look around your website over the last week and very much enjoying reading some of your other material, getting a much better idea of what you think in the process. I was also really grateful to follow up a link to some TIMSS raw results you provided, and I was frankly a little shocked by the gap between Australia and the top achievers (I had been assuming that the scaled score was a bit of a ruse to exaggerate differences and get media attention, but the opposite seems true). I had been looking for similar raw scores for the PIRLS reading a few weeks before, with no luck, so it was good to see that they might be out there somewhere.

        I won’t take any more of your time, but did want to quickly describe some personal context. I have been yo-yoing between progressive and non-progressive educational philosophies for my whole career (usually I decide that whatever the current bosses are telling me must be wrong). My last but one employer was one of Australia’s highest performing schools, and it had a strongly anti-progressivist viewpoint, with direct instruction, phonics and more phonics, and ‘knowledge rich’ curriculum being the order of the day. I arrived thinking I was finally going to learn how to teach, and left feeling that the most important thing I’d learnt was how to play the system. In maths, repeated NAPLAN practices in exam conditions, and some wonderful drilling and explicit instruction techniques, won superlative results in NAPLAN and VCE Further Maths (but not so much in methods or specialist), In English it was worse: NAPLAN creative writing prep was teaching kids how to rip off a Nobel Prize winner’s short story – and as I was leaving VCE English results were going through the roof as we found we could just get kids to memorise a teacher’s writing and then reproduce it in the exam. The story being told, meanwhile, was that our excellent results were because we bravely eschewed progressive and trendy fads. And the library was kicked out of its custom- built building because, you know, encouraging kids to find stuff out on their own is for hippies, right?

        Of course, that these more old fashioned approaches can be misused in this way does not invalidate them, but it does go some way to explaining why I am currently bouncing along in the direction I am.

        Best wishes, and thanks once again – I intend to keep an eye on the blog perhaps contribute again


        1. Hi again, Joe, and thanks very much for providing some of your background. It explains your quite reasonable suspicion of guys like me.

          It is always good to be queried and challenged. One of the problems with a blog like this is that the most of the readers, and even more so the commenters, will genuinely agree with me. Of course I could encourage others by pretending to be more open to “progressive” approaches, but I’m not so I won’t.

          Of course I share your concern about schools playing the game. I think, however, the problem is much less with the playing than it is with the game. If schools were concerned with acing TIMSS, for example, I’d be a hell of a lot happier. And, when I did the Victorian HSC in the 70s there was absolutely no concern other than to play the game. But, you couldn’t possibly play the HSC game other than by learning the (very good) subjects very very well. Now, in VCE, learning the subject, at least in maths, is of little help for the game, and probably in sum makes things worse.

          In brief, if the test is good then you can do a hell of a lot worse than teaching to the test.

  11. I do think there are “math types” and “verbal types”. It was a late realization that I had in high school. Since, I’m sort of good at both (was in GT classes in both, etc.) But I did see kids who were killing advanced history and English (reading extra books, adult insights) in late high school and then NOT on the calculus track. I do think there’s a general trend of intelligence making everything easier. But you do start to see kids who are Harvard level writers and not math strong. And NO, I don’t think it comes from their teachers being too fluffy or not fluffy enough. Or not showing them the beauty of math. Or showing it too much. It’s just something about them.*

    Similar story…I remember telling a faculty member that I thought all kids in a Ph.D. chem program should take both semesters of quantum chemistry (non-physical types only had to take the first one). I talked to the head of department about it, with another student with me. He was an organic type. And in his 40s. And still scarred from dealing with the damned pitchforks (psi). He said, truthfully that he had never needed quantum. The look of remembered pain in his face made me question my “good for everyone” assumptions. A few years later, his research group made hundreds of millions of dollars from some natural synthesis of a vitamin (or a drug or hormone or something). And they did it with traditional Snape in the dungeon wet chemistry skills. Not Arithmancy (I mean computational chemistry).

    P.s. The director of Rushmore said that the hero would “be one of those arty types who would never really do well in math”. But at least we’ll always have Paris…I mean the hardest geometry problem in the world (standard calc application problem).

    *This is not to say they can’t learn basic skills. The human organism has a lot of adaptability.

  12. Perhaps this has been raised elsewhere on this blog (if so – link?) but NAPLAN, TIMSS etc are, in essence, tests. If the test sucks, the data will most likely suck.

    Assessment, from where I sit as a high school Mathematics teacher, really drives what students learn. If a teacher says “this won’t be on the test” then it is not revised, not learned, forgotten (take your pick). So if the test sucks…

    On the matter of the plethora of “research” in Maths-Ed… there doesn’t seem to be much of a clear consensus. School leaders and curriculum-setters can therefore, to some extent, pick-and-choose the research that supports their view. Don’t like assessments? Don’t think tables need to be rote-learned? There will be research out there to support this view.

    Of course, if the data sucks… which brings us somewhat full circle.

  13. This is a (reluctant) response to Ian and Marty, and the general thread. It is a minefield – so apologies for the length.

    Cheng is clearly a talented person. And Mackinnon was a most unusual, and curiously effective, maths teacher. But their various *public* contributions are primitive and dangerous.
    They identify a vague “problem” – which is fine. But they jump too quickly to a silly “(re)solution” that will appeal to just one particular faction – when what is needed is to recognise: that most of us do not really understand what constitutes mathematical ability and achievement; that there are different aspects to be taken into account; that there is wide experience out there; that our own experience and competence may be limited; etc.. In other words, instead of chewing it over in a balanced way within the community, they offer their own trite commentary.
    The Enlightenment is dead; long live Twitter (and let’s all put our trust in superficial “popularisers”).

    We here all realise that *Maths is more important than most people think*. (I recommend Ian Stewart’s recent book: “What’s the use?”.)
    So one would like everyone to gain a sufficient grounding in order to achieve a degree of autonomy as adults.

    England is unusual in pretending to make everyone do the same maths until age 16, and then allowing 80% to simply *stop*. In reality, this simply means that central government (or society) *dodges the need to make awkward distinctions* and leaves it to students and schools to sort out the resulting mess.

    The “mess” arises because we *refuse* to embrace the more reasonable, and achievable, goal of giving (almost) everyone a decent common grounding *up to age N*, after which students may follow suitable distinctive pathways (differing: in curriculum; in difficulty; in the vocational/academic balance; in institution; in target-qualifications; etc.).
    We (in England) know that “not everyone finds maths equally appealing or manageable”; but we find it uncomfortable to admit this and to discuss the most appropriate response (choosing a value of “N”; devising related provision; and then making it work).

    Given the centrality of maths in the modern world, the essential first response would appear to be
    (a) to sort out how to design the first seven or so years of school so that everyone gains as solid a grounding as possible,
    (b) to work out how to teach (and how to *train* teachers to teach) this solid foundation in a way that supports whatever follows.

    Having spent 50+ years trying to understand other systems, I have the impression that a decent answer to these basic questions is available (from the far East; from Hungary/Poland/USSR pre-1990; etc.). The approach is humane; it is initially slow – so that everyone grasps the basic principles – after which everyone is routinely moved on to the general or abstract; it is both child-centred and maths-centred; and it insists from the outset on linking symbols-and-reality (e.g. through word-problems).
    This requires initial, and ongoing, training in the *craft* of maths teaching. And it requires that teachers be given a detailed map (= carefully crafted textbook), and the ability to read that map (so that they can become masters of their craft over time: understanding 1-10; addition/decomposition; internalising number bonds and multiplication tables; place-value; arithmetic; measures; and fractions/ratio; etc.).

    At some point, as students proceed through our extended “education and training” systems, there has to be a parting of the ways! This may be at 18 (on leaving high school), or at 16 (as in England), or at 14/15 (as in parts of Europe), or earlier (as in Singapore). Modern democracies seem to have difficulty facing this – for it is indeed a delicate issue.
    But dodging it is not in students’ interests; and it makes teaching a much less attractive profession.

    Dodging this issue also encourages factionalism in trying to protect serious mathematics and potential mathematicians (on the broadest sense). Failed systems tend to grasp at straws in seeing mathematical ability as “given”, and hence to see those with this “ability” as a small group that can be “identified” and protected (irrespective of whatever hare-brained schemes may be adopted for the great unwashed). This is dangerous nonsense: we have no idea (at least, no idea that commands a consensus) how to “identify” mathematical ability *in advance*. Such ability emerges in part in response to the diet provided and the inner development of the student. So rather than imagining, and then trying to protect, some “chosen elite”, the only defensible strategy is to provide for a suitable “large” number (initially *for all – up to age “N”*; then for a decreasing percentage), knowing that such an approach provides opportunity for a much larger number to emerge.

    I once sat in on some Nick Mackinnon lessons. His old school selects (at 13/14) more strongly than any other school I know (on ability and pedigree); and they devised an approach that I have seen nowhere else. They also made it work – after a fashion. (The problem is that their students start way ahead, and they almost all go on to do very well in some field; so it is hard to distinguish those who *chose* to go in a different direction, and those who were in fact *killed off* mathematically.)

    What one should certainly *never* do is to assume that those with uniquely privileged school backgrounds (I believe Cheng was at Roedean) have got a clue when it comes to analysing the wider scene in which the rest of us live, breathe and work.

    In particular, Mackinnon’s suggestion (at the end of his article) *utterly distorts* the philosophy which was meant to underpin UKMT (which was NOT to “identify” an elite, but to *provide* rich material for *a very large group in ordinary schools*, so that they might then have the chance as they pass through adolescence of deciding whether to take this curious subject more seriously).
    [Mackinnon’s proposal was tried in Singapore in the 1990s (in the form of the “Gifted Education Project”), where an elite were identified and given a special diet. It generated much debate, and was changed in the 00s – so that everyone in the academic stream (students and teachers) had potential access to the richer material. (I can explain why if asked.)]

  14. Yep, maths is hard. It’s not like there’s a ‘phobia’ and if you get over that, you can do it, it’s just impossibly hard all the time.

  15. Thanks very much, Tony (particularly given your reluctance to enter the minefield).

    To isolate your key point, the importance:

    (a) to sort out how to design the first seven or so years of school so that everyone gains as solid a grounding as possible,

    (b) to work out how to teach (and how to *train* teachers to teach) this solid foundation in a way that supports whatever follows.

    In regard to (a), is the design really all that hard?

    On a previous post, you mentioned the MNP books and your own Mathsteasers companions, both of which I’m using in my current attempts to infiltrate my daughters’ primary school. The Mathsteasers are great books, and although I’m not as thrilled with MNP, they seem solid and coherent.

    I know textbooks cannot capture the whole of the subject, and certainly not all of how to teach it (and then there’s your (b)(ii): teaching the teachers …). But isn’t something like MNP-MT a clear forward step? It seems to me *way* better than anything else I’ve heard about in Australia.

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