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Jo Boaler and a common fallacy about maths teaching

Yesterday, the British newspaper, The Telegraph, published a piece by Jo Boaler, Professor of Mathematics Education at Stanford University. In the article, Boaler celebrates the achievement of Maryam Mirzakhani, the first female winner of the Fields Medal – often described as the maths equivalent of the Nobel Prize – and she laments the gender gap in PISA maths achievement in England. Those who have followed Boaler’s arguments since the 1990s will not be surprised by her solution. It is a proposal that has echos of Herbert Spencer, John Dewey and the rest, although the focus on gender is more recent. And it is a proposal that is wrong.

Talking about a PhD viva that she recently attended, Boaler notes, “The young woman defending her PhD that day paced the room, sharing conjectures and connecting different theories. The mathematics we saw was visual, creative and alive. Sadly, however, the approach of Maryam and her students is far removed from that taught in British classrooms, and it is this chasm that stands in the way of gender equity in maths in Britain, and of better maths education in general.”

Boaler continues, “I am British and was educated in Britain. In our maths classrooms today, students do not make conjectures, or learn creatively. Instead they sit watching teachers demonstrate standard methods, which they are forced to reproduce. Of course, the mathematics at PhD level and in a school classroom are not the same. But even so, the classroom has something vital to learn from the way Maryam practises mathematics.”

In order to better emulate the mathematics practised by professional mathematicians, Boaler prescribes that, “We need a maths revolution in schools. We need to stop implying that female students are incapable; we need to stop offering a mathematics that is procedural; and we need to start teaching problem-solving mathematics better for students in general and for girls in particular.”

Setting aside for the moment the notion that we are somehow implying that female students are incapable, this argument follows a classic structure: Experts behave in a certain way and so, in order to become experts, we need novices to emulate these behaviours – the route to expertise is the emulation of experts.

This is wrong. The differences between novices and experts are profound. Experts have vast stores of knowledge in their long term memories – both knowledge of principles and concepts and episodic knowledge, the knowledge gained from the experience of solving many problems in the past. This enables experts to pace the room, sharing conjectures and connecting different theories. Novices don’t have theories to connect together and so the many and various attempts to make children learn maths this way over the past century have failed.  And yet professors of maths education like Jo Boaler keep popping up to suggest them.

This fallacy is the subject of Chapter 6 of Dan Willingham’s book “Why don’t students like school?”. He asks the question, “What can be done to get students to think like scientists, mathematicians and historians?” Before answering that, “a flawed assumption underlies the logic, namely that students are cognitively capable of doing what scientists or historians do… Cognition early in training is fundamentally different from cognition late in training.”

Willingham discusses the seductive research that implies that skilled readers make fewer eye movements than novices and how this led to some  to conclude that we need to teach children to recognise whole words rather than sounding them out – a flawed approach that has led to a downgrading of the teaching of phonics with a cost in more children developing reading difficulties.

The novice versus expert difference is the reason for the traditional form of maths teaching that Boaler criticises. Teachers show students how to solve problems and then ask them to practise doing this because, at this stage, students are not able to work out fundamental principles of arithmetic or algebra or whatever for themselves. Indeed, when I teach my advanced students about the long division of polynomials, I don’t teach the subtraction of fractions – a necessary part of the procedure – because they can already do this. Here we have an example of a transitional stage; students who are expert in some areas but not others. And students will pass through many such transitional stages on the path to expertise.

What conjectures, exactly, would our students be developing in the absence of sound procedural knowledge? What connections would they be making? Such an approach either doesn’t work at all or, more commonly, large amounts of time will be spent developing false conjectures about trivial problems.

The gender part of Boaler’s argument is also interesting. Apparently, “Research shows that girls need to explore subjects in depth, while boys are more prepared to accept rote learning. Girls want to understand, not just follow rules.” I am not sure what research Boaler is referring to and it would be good to see it. I have to say that this statement seems rather dismissive of boys and provokes something of an emotional response in me.

The other key issue around gender is, “the routine advice, intended to be helpful, of primary school teachers to young girls that ‘maths might not be for them’.” I didn’t realise that such advice was so routine in primary schools. There is an overwhelming majority of female primary school teachers and so it seems odd that they would set out to crush girls’ dreams in this way. Do we have the evidence?

I am not downgrading the issue of a gender disparity in PISA. There is a similar, worrying difference opening up in Australia. I am deeply sceptical of Boaler’s diagnosis and suggested solution.

However, there clearly are problems in maths teaching. The main issue is that maths graduates and those graduates of degree courses that lend themselves to the teaching of maths – such as the physical sciences and engineering – are in high demand in industry and can earn much more money there than in teaching. Many students therefore do not have a specialist maths teacher in secondary school and hardly any have one in primary school. Indeed, some professors of maths education who are teaching the next wave of maths teachers do not hold maths degrees themselves. So, the situation is endemic and self-perpetuating.

I would assert that all students want and need to understand maths, not just the girls. The reason why confusion often reigns in the maths classroom is not due to the teaching methods that are used but due to the thin grasp that many teachers have on the maths that they are teaching. This means that they struggle to explain it and resort to just listing the steps. Confusion, difficulties and anxiety ensue and this is a real problem.

I’m with Hung-Hsi Wu on this. Rather than abandon traditional approaches, we need a better educated group of maths teachers and we should start with training primary teachers on how best to explain the standard algorithms.

Perhaps we should be providing our maths teachers with professional development on how to explain maths better instead of constantly calling for a revolution in teaching methods based upon a misconceived philosophy.

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29 responses »

  1. Great article Harry.

    Your point about the different needs of novices vs experts is true at many levels (e.g. also at a micro-level – kids who have been learning the 2-digit x 2-digit for a few weeks benefit from different strategies than those who are learning about it for the first time today.) I find the SOLO taxonomy useful – it views knowledge acquisition as an essential foundation – yes you build on it, but not until its ready.

    The research I have read also concurs with the idea that good teaching is good teaching – be it for boys or girls. Novice vs intermediate vs expert stages are worth differentiating for, but gender is not.

    Re teachers knowledge of subject-matter, my common sense says yes its important. Yet, the research I have read is more subtle. It shows a) basic knowledge is essential, but still doesn’t average fro great teachers b) depth of knowledge does – which supports your statement about teachers no more than just the steps involved.


    • Trying again because my phone died.

      Gender is not worth differentiating for in terms of content but maybe in terms of how you interact with the class and how they interact with each other and you. This might be an interesting RCT using video I suspect.

      Group dynamics in the primary school are very different to secondary, puberty notwithstanding.

  2. As one who has attended many PhD defences, who has supervised graduate students, who works every day with graduate students, and who was one himself, I can say that the student Boaler observed would not even be in that room had she not already mastered the basics. And by “the basics” I mean not only elementary and high school math, but the entire undergraduate curriculum including several levels of calculus and linear algebra, all very algorithmic in nature.

    At most respectable institutions, a student does not proceed to thesis work until he/she has passed comprehensive exams. These are exams testing their mastery of the undergraduate curriculum: Calculus, analysis, algebra, topology, Differential Equations, complex analysis, etc. For many graduate students simply studying for Comps consumes a year or so of their lives.

    As for creativity, critical thinking, learning to “conjecture”, and so on, these are also absolutely essential. The skills grow, and become expectations in a student’s work, as they gain mastery of basics — not the other way around. In the end original thinking is what defines a PhD student. It is regarded as the gold standard for PhD-level mastery: Can you develop original, new, publishable mathematics and present it convincingly to experts in the field?

    So it is no surprise that Boaler observed a PhD student exhibiting original, creative thinking. That, indeed, is what a PhD thesis is all about. Anything less, and the student should not be there.

    But K-6 students are — surprisingly — NOT PhD students.

  3. I was always useless at maths and I used to say that I was OK as long as I understood it, just as Boaler suggests. When I started teaching my own children maths I realised that wasn’t quite my problem. My fundamental lack of fluency in even basic areas meant nothing was automatic. I had to remember all the different stages of a problem but would forget how to do some bits along the way. What I had was a working memory failure and what I was really saying was that I could not do maths when I forgot how to do some parts of a procedure. My children don’t tend to forget because all except the one small new thing being taught is pretty automatic.

  4. I think there’s a confusion, here, between comparing expertise at both levels and the teaching of people to be curious about problems and then them following through by finding the motivation to access that knowledge independently and work on it. The conditions for that vary for each individual.

    Of course this is based on both exceptional talent and access to knowledge in the cases that stand out. That, I found, when teaching in primary school distinguished those children who needed multiple strategies to get the basics and those who seemed to be fluent from day one with what seemed minimal input and who responded well to more informal strategies – they already had all the basics!

    Of course the ones who appeared to be getting minimal input but making amazing progress were those who had extremely rich (as in cultural and other capital) backgrounds in nearly every case although there were some notable exceptions and in nearly all the children I taught it was to do with remembering fluently.

    People like Jack Andraka do come along – but they have had the basics years ago and so that is why they are early achievers – also there is an element of being driven by family tragedy in a surprising number of cases. They develop and expertise and make connections purely because they have put in hours and hours of learning.

    I saw it with my own son over the years – he was able to work out the basics in maths – including tables – without ever having to learn them by rote 1) Because he had an amazing memory 2) He played all sorts of maths games and had a intrinsic interest in maths that was fostered at every opportunity in his home life. The enrichment he got was never forced but the modelling and input of knowledge was there in an extremely informal way. It’s no surprise he excelled in the subject.

    Of course K-6 students aren’t PhD students but some of them have the conditions in their home backgrounds to enable them to be absolutely versed in the basics before they get anywhere near school and I’d think that they might need a greater challenge than other children for whom all the knowledge is completely new and (in many cases daunting).

    • Good distinction! Students can be exploratory in elementary maths without either exceptional talent or great knowledge (or strong mathematical reasoning strategies).

      This does not mean that learning should therefore be by ‘discovery’ or ‘investigation’. However non-routine problem solving (and investigatory work) are a useful part of standard mathematics teaching and learning – and contribute to the development of mathematical reasoning strategies and positive attitudes, alongside the routine learning of facts and skills, and developing conceptual structures. The Cockcroft Reportr of 1982 versy sensibly proposed a balanced experience.

      Mathematics teaching at all levels should include opportunities for
      * exposition by the teacher;
      * discussion between teacher and pupils and between pupils themselves;
      * appropriate practical work;
      * consolidation and practice of fundamental skills and routines;
      * problem solving, including the application of mathematics to everyday situations;
      * investigational work. (Cockcroft 1982, paragraph 243)

  5. Whilst I doubt that many (any?) primary teachers would suggest to girls that they couldn’t do maths, I do wonder about unconscious bias amongst teachers. Whether the fact that many primary teachers are not maths or science specialists leads to them unconsciously treating kids differently, or playing up the difficulty of the subjects. I’m not sure if anyone has looked into this, but it would be interesting.

    • There is definitely an issue. I work with a very glamorous female science teacher. She was asked to teach a physics unit (physics is not her specialism). Her opening gambit was to joke with the students about how bad she was at physics.

      • Your science teacher friend may be simply playing for the drum roll, knowing the stereotype. Or she may not have had her gender in mind in the first place when she made that joke, and might be put off to know that in your mind it came across as a “girls don’t do physics” quip. I, for example, often confess to having done poor in Chemistry. It’s simply a fact, nothing to do with my chromosomal make-up as far as I know — but how would I know, chromosomes are chemicals and I did poorly in Chemistry.

        In my experience top-level performing math students of the female persuasion are almost as common as those of the male persuasion. Many of our best — maybe the majority here, in recent years — are female. These are very impressive kids, and as a teacher you generally stop bothering to put them into gender classes mentally.

        I direct our provincial Grade 12 math contest, and while the guys outnumber the girls in the winner’s circle, not by a whole lot. And it is quite common for girls to take the top spot. There does seem to be a perceptible predisposition among boys toward competitive mathematics that girls don’t have, and in my view it is not a question of ability as much as enjoyment. Once you reach the top undergraduate competitions at extreme levels of competition you see the girls dropping out of the activity and those who stay don’t reach the highest levels (though a few are very impressive!). Again, I think this has to do with predisposition, not raw talent.

        Each year, by the way, when applying for our contest grant, we do NOT check off the box stating that we have special measures to encourage the participation of girls. We choose to be gender-blind. We are also race-blind (etc.). We probably miss out on extra funding for that reason. (We do have an incentive for francophone students, but for different reasons than you might have for race, gender etc.)

        I think girls find it hard to motivate themselves to do, for example, the Putnam Competition, which is written by 4000 of the top North American students every December. This contest is 6 hours long and consists of 12 questions written in two 3-hour seatings of 6 questions each, with a 2-hour lunch break. Each question is worth 10 marks. The questions are so involved and such complexity might appear in a correct answer that it takes until April before scores are ready — a test of patience! For their reward, what kind of scores do students receive? Well, a very small minority will get scores above 50%. VERY small minority. In most years, the MEDIAN score is a 0 or a 1. Let me put that in simpler terms: generally speaking 2000 or more students (4000 of the best in our continent, remember!) end up with a score of 0. Yes, out of 120. A score of 10 is far above average. It is not recommended that you even attempt the putnam without 3 months of preparation and practice. And most people get 0 in their first year anyway. To do well requires years of persistence.

        For some reason, guys have a greater tendency to subject themselves to that kind of regime than girls. But I don’t believe, from my experience, that they are intrinsically better at math than girls in any way that could be discerned without straining gnats using large-scale statistics.

      • Dubious stereotypes apart (I thought we were interested in evidence based education?), all this would really prove is that motivation is a significant factor in learning, it is not just about cognition but attitudes, preferred learning contexts, perceived relative difficulty, the importance of understanding the alternative knowledge schema children bring with them to lessons and the prevailing culture. Most people in the media give the impression its OK to be bad at maths and science but a heinous crime to make a grammatical or spelling error. So this teacher is only anecdote is a whole prevalent culture.

    • Good point! There is research showing that teachers treat students according to their own stereotypes – often despite trying to be gender-fair, or trying not to have low expectations of ‘slow learners’. Such research has been around since the 1980s including work by V Walkerdine and K Ruthven

  6. I think Hattie asks an interesting question “Is knowledge an obstacle to teaching?” as he points out “teachers’ actual depth of knowledge of the content of what is being taught bears little relationship to the attainment level of their students”.

    • I would guess it becomes more important the more conceptually demanding a subject is.

      • Agreed. Imagine a mathematical ignoramus trying to teach Complex Analysis. But I think the same applies in elementary school math. I believe what Hattie is observing is that the ability level of the teacher is occluded in outcome data by other, more significant factors. If we cleaned up other problems in the way teachers are pressured to teach and let teachers be the experts, then their level of expertise would surely bring more to bear on the student outcomes.

  7. However DrDav was talking about primary school pupils

  8. Children certainly do not have the same capacities as professionals in any field otherwise why bother teaching them anything. However, children are motivated by emulating grown ups. I don’t think Willingham is saying don’t ever teach anything in the mode of a professional, he’s saying be careful about why you are doing anything and what you expect to achieve. Motivation is a big part of learning. It’s not just about cognition. While it is probably true that there is quite a lot of bad practice in terms of lack of focus on essential knowledge, ignoring motivating contexts for learning, especially in maths is just as dangerous.

  9. Again I agree with your arguments, Harry.

    In response to your question as to whether girls really are discouraged by their primary school (predominately female) primary school teachers, in my experience, as a mother to 2 girls, I would answer, yes, at least in our case. Thus far my girls have had female teachers who start the year telling the students how they don’t like maths and are not good at it. Not very good role modeling for either gender but perhaps particularly for girls. In addition they have told my girls on numerous occasions “maths just isn’t your thing, don’t worry you’re great at English”. This was very distressing to me, as I knew there was nothing wrong with my girls ability, just their teaching of mathematics. Once my eldest was equipped with the basics, thanks to Kumon tutoring, she excelled at maths. This turnaround, could also be explained in part, because she had 2 male teachers, back to back, both of whom liked maths and perhaps were better equipped to teach it. Of course my story is purely anecdotal.

    Something that drives me batty at the girls school is the way they insist on teaching “strategies” such as spilt, amongst various others, rather than the standard algorithm. I don’t see the point? These strategies are often very convoluted and, as a result are more likely to lead to an error during calculation.

  10. Training for Primary Teachers to become Maths specialists has been available for the past five years under the MaST programme. It was originally fully funded but now has to be paid for by schools. However the programme is still thriving and helping Primary teachers who are traditionally generalists to gain a deeper understanding of Maths including the need and use of algorithms. I have certainly never told any of my girls that Maths is not for them!!

  11. I would like to make a case for Jo Boaler’s idea about fostering creativity and problem solving in the Elementary school math classroom. In a more problem based educational model, students are given a big problem, then the students ask the questions about how to arrive at the answer. In this sense, as previous commenters have remarked, students have a great motivation for learning the methods and algorithms necessary to answer a problem. Not only does this increase motivation in the math classroom, but it allows elementary students to practice asking questions and critically thinking, which are “professional” skills.

    • That makes an intuitive kind of sense but it isn’t really supported by the evidence. For instance, in Project Follow Through, a number of teaching approaches were trialled including similar ones to the approach you describe. Direct Instruction – a form of fully guided instruction – led to the greatest gains not only in basic skills but in problem solving and even self esteem. Motivation comes from getting better at maths and the best way to do this is via explicit instruction.

      Dan Willingham is good on Critical Thinking skills.

  12. The expert fallacy is a really pernicious one that seems to crop up time and time again in various disciplines. I really wish more people were familiar with research examining differences between experts and novices. The key difference is the amount and way in which their knowledge in long term memory is organised. Almost all of this knowledge is automatic and the expert in question isn’t consciously aware of it which is also why they tend to suggest expert orientated approaches that tend to fail with pupils lacking the necessary background.

    Attempting to engage novices in complex processes in particular areas without the appropriate background knowledge is doomed to failure. Their working memory becomes massively overloaded and they retain very little of the experience, a necessary condition for learning. One way to see this for yourself is to ask a pupil what they learned after engaging them in a complex problem solving exercise later on in the week. Most of the other teachers I’ve spoken to have who’ve tried this have said that their pupils could remember very little of the experience other than the difficulty and how much they had struggled.

    One of the very first exercises we do as part of the Professional Learning And Networking for Computing CPD programme is to unpick how much background knowledge is needed to understand natural language instructions before moving over to doing the same for a small piece of code. Once teachers realise how much background knowledge novices actually need they soon see the importance of adopting a more comprehension orientated approach to programming in order to develop that knowledge more effectively.

    The learning process in general can be very counter-intuitive and I’d always be very skeptical of anyone who hadn’t had a serious look at the relevant literature first before advocating a particular approach to learning and teaching- sellers of snake oil would be the most polite way of describing them.

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  14. Entering this thread very late in the day, I completely agree with your analysis of “the expert fallacy”. But I am sceptical about your conclusion that the only solution is to recruit or train more Maths teachers who are themselves really expert in Maths. And Computing teachers who are expert in Computing and Physics teachers who are expert in Physics and Mandarin teachers who are experts in Mandarin? You never will.

    That, in my view, is why we need to understand better how people learn and produce better teaching resources that encapsulate that understanding, supporting teachers who will not necessarily be at the top of their subject. It is not that I think the role of the teacher can be replaced (agreeing with Ian Lynch that emulation and motivation are important) – but activity, process and feedback are also important as well as being labour-intensive – and in these areas we can do much more to support teachers with the “tools of the trade” that are generally missing at the moment.

    • …and having now listened to Jo Boaler’s BBC interview, I think you may be being a little hard on her in some respects. However valid your criticisms of “the expert fallacy”, I think she also says important things about not accepting the theory of innate inability (an important conclusion from PISA) and not being content with learning procedures alone. I have always thought that “procedural knowledge” was not a sufficient description of what it means to know *how*. For example, knowing how to be a leader of men is not achieved by memorizing procedures. “Procedural learning” in fact confuses knowing *how* with knowing *that*.

      As so often, I do not think this is an either-or. Just as we might start by learning some facts, so we might progress to learning some mechanical procedures, before ultimately developing the sort of sophisticated framework of understanding that allows us to deploy both knowledge and procedures flexibly (or to improvise) as circumstances require. So I think that the sort of playful testing of hypotheses is useful part of the repertoire, once some basic mastery has been achieved.

      At the same time (and although I do not teach Maths) I am pretty sceptical of her belief that you can get by without the basics (such as learning your multiplication tables) or that it helps in the early stages to learn multiple, parallel methods of doing the same thing. This strikes me as having more to do with bogus theories of learning styles than giving young children a step up to critical thinking. Her argument against setting should be compared with Dylan Wiliam’s account of the same evidence at ResearchEd, which he says shows problems not with setting in principle but with the way that it is often implemented ( at 13:28). In this section of the interview, Jo Boaler deliberately misrepresents the evidence, ducking one question and conflating her anecdotal experience with the peer-reviewed research, which according to Wiliam shows the exact opposite of what she is saying. at 22:30.

      So I still support your analysis Harry. I am just saying there are a couple of babies in the bath that need to be fished out before the water is thrown away.

      • Seem to be agreeing a lot these days Crispin ;-) The thing about being successful in life is that it is multi-faced. Certainly knowledge plays a big part but it certainly isn’t the only consideration and even success itself is judged differently by different people.

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