Eddie Woo’s Mental Connections

Eddie Woo has been annoying for a long time. Eddie knows much less than he realises and his smiling inanities, which are invariably swallowed whole, are a continual distraction from real issues and real solutions. But he’s gotten worse. Eddie Professor of Practice Woo has graduated from being a distraction and an annoyance to being an active menace.

Yesterday, Eddie and a couple of his University of Sydney maths ed colleagues had an article appear in The Conversation:

‘Why would they change maths?’ How your child’s maths education might be very different from yours

Greg Ashman has already had a good whack at the article but it warrants another, much harder whack. Greg summarises the article as “flawed”, which is true, but which fails to capture the magnitude of the article’s badness in the way that “hole” fails to capture the magnitude of the Grand Canyon. The article, cowritten by the Guru of Australian Mathematics Education, Eddie Professor of Practice Woo, is simply appalling.

The article begins with the very funny scene from Incredibles 2, with Mr. Incredible struggling with his kid’s maths homework:

The humorous disconnect underlying the scene is the basis for the article, and the authors first confirm, sort of, Mr. Incredible’s point, that “math is math”:

Pythagoras’ Theorem is as accurate today as it was when it was discovered millennia ago …

And just exactly how accurate might that be, fellas?

The critical insight, of course, isn’t that Pythagoras is as “accurate” as it ever was, but that it is as true as it ever was. It is the rock solid, eternal truth of mathematics that makes it so powerful and that must underly its teaching. There is something almost Freudian about the authors beginning with such a gaffe, suggesting that they really don’t get it. It makes one already question if there is value in reading on. There isn’t, but we will.

The humour in the Incredibles scene is that although maths hasn’t changed, maths education has:

… teachers today also teach maths very differently from when parents were at school. … The teaching and learning of maths has undergone a transformation in past 30 years [sic].

30 years past, you say? What a coincidence. Just the other day we were pondering the world of thirty years ago, when poor kids had to suffer educational horrors such as solving problems in maths competitions.

Yes, maths is taught differently now, and of course the writers are setting the stage for arguing how much better it is now. And of course they are out of their minds.

Their argument begins with a section titled,

Mental connections not procedures

Dear God. Even if they don’t quite mean it, why would they write it? Freud knows why: because they really do mean it.

The section begins:

In the past there has been a focus on teaching students procedures, such as times tables …

Um, times tables are not procedures, they’re related and naturally organised facts. But we get it: you hate the times tables. And the canary truth is there, as always: if you do not recognise that kids gotta learn their tables, up to 12, by heart, then you have you no business being anywhere near the education of these kids.

The authors also name a couple of procedures, “how to work out the circumference of a circle” and how to “solve an equation”, whatever that is supposed to mean. Let’s assume, for the sake of fantasy, that they don’t really want to kill the teaching of such procedures. What then, is the new “focus”? Well,

We now appreciate the importance of forming mental connections between concepts.

This is the strawiest straw man that ever left Strawtown. The implicit claim that maths ed in the past was not concerned with connecting concepts is simply absurd. But maybe maths ed guys are more concerned now:

For example, when students understand the connection between similar triangles and trigonometry they understand the definition of trigonometric ratios at a much deeper level.

Sure. And where is the evidence that trig is now taught and, more importantly, now learned, with any stronger connection and, more importantly, with any greater tangible benefit, than it ever was? None is provided.

The entire section is fantastic gaslighting, all with the flavour of, and with vacuous snowjob references for, the 21st century skills nonsense. There is no clear argument, much less any evidence for an argument.

The next section of the article is no section at all. It is titled,

Problem solving and reasoning

There follows no reference whatsoever to problem solving. The section merely notes that

the shift in mathematics education is reflected in key mathematical proficiencies in the Australian school curriculum

Yes, and the Australian mathematics curriculum is appalling, including that its proficiencies completely undermine the proper, foundational role of facts and procedures. It is unclear why the authors didn’t follow up on “problem solving”, but in preemptive response to the absent idiocy, see here and here and here and here and here.

The next section of the article is, unfortunately, really there. It is titled,

Encouraging inventiveness

And how are we to do that? Well,

[Teachers] are deliberately showing students different and multiple ways to represent mathematics problems to give students the space to develop understanding. This also give [sic] them an opportunity to reason, model and engage in mathematical thinking.

No. What it does is give the students an opportunity to get lost in a jungle of half-learned methods, none of which are ever properly practised or learned in any internalised, functional manner. For example,

your child might bring home problems to solve using the area model for multiplication, which looks quite different to a traditional method. For example, we teach how 8×27 can be modelled in parts – 8×20 and 8×7.

Boiling it down, the “area model for multiplication” referred to here is the shrivelled, deformed heart of this appalling paper, and it demonstrates just how clueless are the authors. It is thus worth considering this example in some detail.

To begin, of course most methods of computing 8 x 27 will involve breaking 27 into 20 + 7. The traditional algorithm will set up the required computation in columns. By contrast, the area model will picture each component product as the area of a rectangle, and then the “areas” are summed, as helpfully and elegantly illustrated in the linked AMSI PDF:

There are a number of things to be said about this box method. First of all, the method is quite commonly (pseudo)taught now and will be unfamiliar to many parents. As such, the examples fits well with the Incredibles joke. Secondly, there are non-trivial arguments for teaching multiplication with boxes: the boxes make visual and intuitive the commutativity and distributivity of arithmetic, preparing the path to algebra. Thirdly, there is nothing remotely “inventive” about kids learning the box method, either instead of or as well as the traditional method. Fourthly, notwithstanding the pre-algebraic arguments for teaching the box method, the arguments against teaching it are much stronger: it is a hideous method (something AMSI forgot to tell us).

The fundamental thing one desires from a mathematical technique is that it works clearly and efficiently for the purposes at the time the technique is being taught, not for purposes that will emerge three years later. And yes, the box method is vaguely efficient on examples such as 8 x 27, but these are examples that are simple enough that they should be learned to be done mentally. Going on to examples such as 38 x 47 or 315 x 526 and the box method quickly becomes a nightmare. What this means is that the box method should not be taught as the fundamental algorithm; if it is to be taught at all, it should come after the traditional column method. But, fifthly, showing students the box method after the column method may have some merit, but even then it will likely also create serious problems: since most kids will be insufficiently practised in the traditional algorithm, introducing a second method is more likely to confuse than anything else.

Finally, it must be noted that the box method as an Incredibles example, of the teaching of a new and unfamiliar method, is being way oversold. Such new methods are rare. Amusingly, the example in the Incredibles is of solving linear equations, which is done exactly the same as it ever was. What has changed, dramatically, is the manner in which these methods are presented and the manner in which students are instructed that these methods be (under)practised.

The penultimate section is,

The world is changing

This section is more of the 21st century skills nonsense, together with the selling of dynamic software and the like as technological saviour. Nothing new, and nothing of sense. Then, the article closes with its final, “don’t worry” section:

How can parents help?

Parents can help by not buying the nonsense offered by Eddie Professor of Practice Woo and his fellow maths ed nitwits. Parents can help by ensuring that their kids learn traditional techniques and practise these techniques in a traditional, solid manner. That may not be perfect and it may not be sufficient, but it is where to start. Without the solid knowledge of clear facts and a solid facility with efficient techniques the kids are doomed, the same as it ever was.

As for taking lessons from the movies, that is fine. Eddie and his mates have simply chosen the wrong movie:

50 Replies to “Eddie Woo’s Mental Connections”

  1. Dear Parents, Eddie Woo and anyone else who thinks they know better,

    1. If you can teach your kids everything they need to know, go for it. You have my blessing.
    2. Point (1) does not give you permission to EVER say that what the teacher is doing is wrong unless you have strong evidence, in which case, I would prefer if you presented it in a calm and polite manner rather than on the group-chat-thing.
    3. If you EVER say the words “I wish Eddie was my teacher” I will immediately stop listening.


    The person who is doing their damned best to interpret the weird-as-bananas-curriculum and get your kids through the rather unusual world that schools have become.

    1. RF, many teachers are screwing up and you know it. And even if the teachers are not screwing up, the system is. Parents may not have a right to hector, but they have every right to be pissed off.

      1. Totally, but there is a right way to do it.

        Just as there is a right way for a school to get teachers to lift their game.

        1. Funnily enough, I’m in no mood right now to be considering teachers’ fragile egos.

          I loathe whiny parents, and I appreciate that teachers are screwed over by such parents, making good teachers unwilling to risk trying anything outside ACARA’s/VCAA’s idiotic plans. But the majority of maths teachers have no proper sense of how screwed things are. The majority of maths teachers are not teaching maths. With that reality, everything else is secondary, including teachers’ feelings.

          1. Sure, and I do agree to an extent.

            Except, I’m not sure these teachers know they are teaching crap and schools are not exactly good at helping/allowing them to see this.

            So while I cannot disagree that teachers are a large part of the problem (myself included), I do not think they are the source.

    2. RF
      I think teachers need to explain why they are not complaining on mass that they are being asked to breach their code of ethics.
      These generally include something about adherence to high standards of practice.
      So anyone asking them to do something that doesn’t have evidence to support it being an improvement or worse where a Greg or Marty or others has pointed out problems should immediately be challenged for either wasting their time or breaching the code of ethics.
      It’s not like teachers are unable to speak up about things they care about with an effective union to represent them if needed.

      Until teachers make a bigger fuss then what should limit parents dim view of the profession? And why not chat about it here?

      1. Interesting – I’ve never thought of it that way.

        Maybe because I’m not sure many teachers think of themselves as adhering to a code of ethics. The whole VIT code of practice is not much more than a box-ticking exercise and then back to surviving as best we can until the next break.

        If some good teaching happens to occur, excellent.

        1. IF the VIT followed it’s own VIT code of practice, the teaching shortage in Victoria would not be as diabolical as it is. The conduct of the VIT towards teachers is something the Parliamentary Inquiry into the State Education System in Victoria

          An Education Review: The State of the State


          must probe. Three general issues tell the story:
          1) the VIT has made it unreasonably difficult (to the point of nearly impossible) for retired teachers to return and help out during the severe teacher shortage. (This is despite the ‘teaching pool’ spin that Education Ministers trot out every so often).
          2) The VIT has significantly increased the work load of new teachers entering the work force by creating numerous obstacles and tasks.
          3) The re-registration process is farcical.
          The VIT makes Utopia look like a drama series.

          (Wow, the above could make a decent submission to the Inquiry.
          Update: But use different words because these words are taken).

  2. The times tables point is not made loudly enough. I feel like sitting down with Woo opposite and just removing that ridiculous notion from his brain. Hard boiled. Just sitting in a room with a plastic desk and a swinging exposed bulb. Expose the claim for what it is and force him to see it for what it is.

    It is completely illogical. Times tables are so incredibly fundamental, so indispensable, and so important for confidence building. Such a comment in an article he has authored should not be forgotten. And until he publicly shouts from the rooftops that times tables must be taught, that they must be learnt by heart, he should not be forgetting it either.


    1. “I feel like sitting down with Woo opposite and just removing that ridiculous notion from his brain. Hard boiled. Just sitting in a room with a plastic desk and a swinging exposed bulb.”

      Watch it, Glen. Not everyone understands metaphoric language.

      More seriously, the problem we have here is not unlike trying to argue with a climate change skeptic. A quick google search finds a number of possibly useful articles. For example: https://www.nytimes.com/2020/01/02/opinion/climate-change-deniers.html

      I believe some of these techniques could be transferable and used to try and convince ‘times-tables’ skeptics. Then again, some people just want to be loved and adored and appear wooke and will deny to their dying breath that rote learning things like the times-tables is essential.

      1. The difference is, with climate change the authorities are shouting the truth. With maths ed, the authorities are shouting lunacy.

        1. Fair point. How to get/force the authorities to stop listening to the Rasputin’s of mathematics education? That’s the 3×23 dollar question.

  3. Although I agree with much of this article I do take exception to your dislike of the area model. It is a really good introduction to algebra. Not saying that it should be the only method but showing it and making the connection to the vertical algorithm can be really useful.

    1. The area model is great in Grade 7 or 8 when explaining distributivity. It is idiotic to use it when teaching multiplication; the concept of area is much deeper than that of a product of two natural numbers. Kids who don’t understand the connection will not learn how to multiply, and they will dislike mathematics for the rest of their lives.

      1. To suggest Mathematics is being taught at a superior level to 30 years ago is rubbish.

        My nephew said he’s struggling with Maths and really wants to improve. I asked if he’s memorised his times tables… no… he struggles with division… i wonder why? his primary teachers said they don’t practice times tables.

        These Maths Gurus like Eddie Woo need to crawl back into their hole and fit a muzzle whilst they’re at it…

    2. Hi, Vicky. What Franz says.

      Of course I’m not bothered with you disagreeing with part of (or all of) the post, but I don’t think you do.

      As a method of picturing what multiplication is doing, the area thing is great and pretty much mandatory. But as a method of multiplying, of actually computing, it sucks. I don’t know that you appreciate the madness of the manner in which this stuff is used in primary schools.

  4. I think that the problem with the Woo essay is its triteness. At some level, you can probably agree with everything he says, but he gets into trouble when he prescribes how understanding and all the other good things are supposed to happen. Basically, the teacher sets the scene but has to be opportunistic to take advantage of openings that present themselves through questions that children have and so forth. i see learning as a three stage operation: the introduction stage where the students become familiar with the ideas and begin to come to grips with them, the formalization stage where students learn the protocols necessary for efficient learning and utilization, and the consolidation stage where students apply whatever they have learned to applications and problems. if the teacher handles these stages properly, then the understanding, appreciation of structure, resilience and divergence will infuse over the whole process. In arithmetic for example, pupils begin with familiar situations that involve numbers in their guises in counting and measuring for example, and understand what the basic operations of addition, &c connote, even developing their own idea as to how to operate on them. Having laid the foundation, you then systematize arithmetic, selling the standard algorithms on the basis of efficiency, automaticity and the satisfaction of being in control. Finally, there are innumerable ways in which this intervenes in applications that are meaningful to the kids — after all they play games, buy things at the store, maybe even earn money and keep a bank account. Rote and memorization are parts of traditional elementary education, but they still have their place. The learning of the tables and standard p-and-p algorithms also underscore the underlying structure of mathematics, so that the memorization and rote part of it is buttressed by being able to play off the structure. By the way, I am in great favour of having students get in the habit of checking their work, if possible, in an independent way. This also promotes the divergent thinking we would all like to see. Another way to encourage judgment and inventiveness is to make mental arithmetic an important part of the syllabus; unlike the use of an algorithm, how you perform an operation mentally depends on the characteristics of the numbers involved and is open to useful “short cuts”.
    However, we all have to remember that every teaching method comes with a probability of success attached that is highly dependent on the context. You can’t teach out of a can, which Woo seems to think. Good teachers make that probability as high as possible, but they need to be sensitive to the situation, and this is what our teaching formation programmes should seek for. For an analogy, imagine you are learning to play a musical instrument or a sport such as tennis; this may help to understand the balance of fun, diligence, hard work and gratification that is need in a mathematics syllabus.

    1. Thanks very much, Ed. You’re much more knowledgable than I, and I’ll give your comment a careful read later today. But I think you’re way underselling the badness of Woo’s article. It is not just trite, and smug, it is bad. And, it is bad from the most respected maths ed commenter in Australia. which makes it ruinous.

      Your point that we cannot teach out of a can is a very good one. But we still have to start with cans, to base the teaching on templates or models or whatever. Woo and his mates are promoting a very standard and very bad can.

      1. Your point about cans is a good one, but it depends on what you want to teach. i suspect that when I went to school, most public school teachers probably taught the basics of arithmetic the same way. But this was in the days when a lot of emphasis was put on kids learning a set syllabus. The old joke was that on a given day, every kid in Ontario was on the same page of the same text. But I think that this will work only when the task to be taught is sufficiently well-defined. However, not every kid was in the same place with respect to understanding and resilience, even if they could go through the motions.
        The difficulty comes when the mandate is to teach more affective things and foster creativity, judgment, originality. Some of the talks i have attended at education meetings and article in journals have been by people who put forward in effect a script for how kids may discover things. Here the “can” might contain not a recipe but ideas that teachers might start with to get the ball rolling.
        Basically, a good teacher is a witness to her subject (to use an evangelical term), and you cannot be a proper witness unless you have some base of knowledge and experience, and this is what you hope to endow them with in their formation. If teachers in their training have to tackle open-ended problems and become familiar with the intuition and sort of reasoning involved, then they are in a better position to understand what is going on with their students and adapt to the situation.
        One thing I found useful in a class of teachers is what I call “convening”. You assign a set of problems, and when the solutions come in, you give all the solutions to a given problem to one of the class. This person has to go home, read all the solutions, and give a report of what the class did with the problem. They can also make corrections and comments on the script. The idea is for them to get into others people’s thinking and understand it well enough that it can be included in their report to the whole class.
        Another useful thing is to ask them to take a standard problem from a text, work out the “expected” solution and then find another different solution (they pick the problem). For example, the following problem appeared in an algebra text: A man in a theatre line finds that 1/7 of the line is behind him and 5/6 is in front. How many people are in the line? The context of the problem dictates you set up an equation in x and solve it. However, an alternative way of looking at it is to realize that the number must be divisible by both 6 and 7, and therefore a multiple of 42. Moreover, *if the answer exists*, it has to be the smallest multiple of 42. So you check it, and bingo, you are in business. Now this is a very interesting situation and logically takes a little bit of unpacking. One route of further investigation might be to ask, suppose you replace 1/7 and 5/6 by other fractions; what can you say about when the problem has a solution?
        What you are after is to foster a kind of mindset, so that when the teacher gets into the field, she can interact with and learn from her colleagues, in somewhat the same way that Liping Ma describes of Chinese elementary teachers in her book comparing American and Chinese teachers. However, much depends on the mathematical characteristics of those in education faculties teaching the cadet teachers, and it seems to me that the basic criticism of Woo might be that he maybe never sat down and wrestled with mathematics, either in just messing around and see something interesting or solving puzzles and problems and reflecting on them.

        1. Thanks again, Ed. I still don’t agree. You wrote,

          “The difficulty comes when the mandate is to teach more affective things and foster creativity, judgment, originality.”

          Yes, that’s the mandate. But that mandate, or at least the extent of that mandate in primary school, is stupid. It’s a bad can.

          1. Marty: Not at all. I think the ultimate goal is for the children to understand that mathematics is a coherent system with many parts that fit together, connected by structure and reasoning. Also, you want to convey the utility of mathematics. As for the creativity side, this also has to be in the mix in some way. Sometimes these goals are better reached in an extracurricular environment, where the stress of grading and implementing a syllabus is absent. Sometimes the points may be made in the teaching of another subject, such as geography, science, civics and financial literacy, where the mathematics occurs naturally among other relevant factors.

            However, all this has to be done with a mind to the context the pupils find themselves in and their capabilities and level of maturity. This is why in the formation of teachers, you need courses where they really have to engage in the mathematics and have the opportunity to talk about it in their classes and among their peers. University courses are often too advanced or specialized to allow this, and education courses often have other goals (e.g.education theory, measurement, psychology, education law), are sometimes riddle with ideology and the instructors often do not have a deep enough background in mathematics to handle it. The New Maths of the 1950s and 1960s is a way not to do it.

            What I meant by a different sort of can is that there are ways for teachers to set up a situation where some of these affective things emerge. For example, if a student makes a mistake, to encourage discussion of it and how it might have arisen, which can be a dandy way to find a misconception or uncover something that needs more explanation.

            If you want your rocket to reach outer space, you align its trajectory so that it comes near and gains momentum from planets in its path.

            1. Thanks agin, Ed. I doubt we really disagree.

              Of course the ultimate goals are the goals. But of course it is not enough to have a goal without some reasonable means to achieving that goal. In Australia at least, the Creativity Can is 99% goal and 1% means.

  5. Yesterday I came across a multiple choice problem in the 2015 paper for students in Years 7-8 in the UK mathematics competition. The answers and explanations from my students were interesting.

    In the expression 1 & 2 & 3 & 4 & 5, each & is replaced by either + or x.

    What is the largest value of all the expressions that can be obtained in this way?

    A 10 B 14 C 15 D 24 E 25

      1. I get 1 + 5! = 121. Looking at the possible answers, I don’t understand Terry’s question.

        Maybe it should have been phrased “Which of the following numbers is the greatest obtainable in this way?”.

        1. That’s nice, Glen! 121 beats 120 (except in golf).
          (And I agree with your re-wording of the question. That crossed my mind after posting those earlier comments)

    1. I made an error. The question was

      In the expression 1 & 2 & 3 & 4, each & is replaced by either + or x.

      What is the largest value of all the expressions that can be obtained in this way?

      A 10 B 14 C 15 D 24 E 25

      Mea culpa.

        1. I have now tried the above question on two classes. In each class there was a variety of responses. When I asked the students how they arrived at their answers, everyone appealed to BODMAS – irrespective of their answer.

            1. Yes – more politely – but perhaps not firmly enough. BODMAS is ingrained in their memories (in some form of other).

              BTW, I just bought Todhunter, I. (1870). “Algebra for beginners”; first edition; 16 cm x 10 cm – pocketbook size; excellent condition; no colour pictures; in about 300 pages, it contains everything on algebra that the Australian Curriculum would expect.

              I came across an interesting post on BODMAS yesterday:

  6. Did he not say he was bad at maths?
    No need to listen to him if he still is not.
    Surely he has not idea what naths was like a generation or two ago.
    But if he can fire up students’ minds…let him dò it

    1. Would be a fair point except I don’t agree with the contention that he fires up students’ minds,.

      Too many students watch mathsy video clips and then think they understand. The need to read, think and practice cannot be understated.

      As to why the subject is taught differently to 30 years ago… 30 years ago, there were different methods for preparing teachers. End of story.

    2. @Banacek Spaces: When you say “he” are you referring to Eddie Woo?

      Eddie Woo has often made it clear that he is not a mathematician. I infer (from Wikipedia) that he completed a B. Ed. (Hons) in secondary mathematics and IT from the University of Sydney and went on to become a secondary teacher. I suppose that secondary mathematics and IT were his two methods.

      In some universities, some of the mathematics subjects that a student covers in an education degree are also the same as the subjects that he or she might cover in a mathematics degree; same mathematics, same lecturer, same classroom. It’s possible that this was the case in Eddie’s degree.

      1. The problem isn’t that Woo isn’t a mathematician. The problem, and not the only problem, is that Woo is willing to say thoroughly idiotic things and that no one ever calls him on it.

  7. Marty writes: “Um, times tables are not procedures, they’re related and naturally organised facts.”

    Charles Dickens opens “Hard times” with:

    ‘Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!’

      1. I suppose with all the censorship currently taking place on classic novels, the updated PC (*) title is
        Hard Times Tables.

        * I’d use “woke” but I know Marty’s feelings on the word. I’m still looking for a more suitable word …

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