At the second attempt I returned to Jo Boaler’s book Mathematical Mindsets. I really wanted to get an insight into what she is proposing because I have no doubt that the curriculum designers are planning to incorporate Boaler’s ideas into what will be Ireland’s revised primary school curriculum. And what happens at primary school level eventually affects us in higher education.
Before reviewing the book in detail I think it’s important to state what I believe about maths teaching.
I believe that mathematics is best seen as a language in which the ‘grammar’ is not arbitrary but follows logically from a small number of axioms.
I believe that all rules of the mathematical language should be explained and justified when first encountered, and not just presented as arbitrary rules to be ‘learned off’. However, I believe that mathematical fluency can be (and usually is) achieved by remembering and applying those rules even when the proof of those rules has been forgotten. If you are bothered by a lack of understanding at any time, then you can always go back and refresh your memory.
I believe that mathematical fluency is an essential component of mathematical problem-solving. A lack of fluency leads, in my opinion, to cognitive overload and frustration.
I believe anyone can become competent at mathematics through effective, enthusiastic teaching and a lot of practice but I’m not sure how far you can push this. Is everyone capable of getting a degree or a PhD in engineering or physics, not to mention maths? Perhaps, but maybe for most people the process would be so slow that the opportunity cost involved in pursuing maths-intensive study might be too great. But I also believe that only some people – perhaps a small fraction of the population – have the innate talent for the abstract thinking required to become high-achievers in math. I think maths is like singing or painting or writing or any other academic, athletic or artistic endeavours. Most of us can reach some degree of competence but true excellence is likely to be beyond us no matter how hard we practice.
I think it’s understandable and no great mystery that many people just don’t like mathematics. In my view it is hard to construct an evolutionary argument as to why human beings would have evolved to the extent that most of us were able to think in the abstract way required to progress in maths (see this paper by Steven Pinker).
I think people are turned off by maths for two reasons: (i) they didn’t really like it in the first place (and shouldn’t have to explain themselves for not doing so) or (ii) they fell behind at some stage of their learning, failed to catch up, and lost all motivation. Maths is a very hierarchical and unforgiving subject.
People who really like mathematics love what many modern maths gurus describe disparagingly as ‘drills’. Every problem is seen as a challenge. Every problem is there to be solved whether it is ‘authentic’ or not. ‘Maths people’ need to be nurtured and challenged just as much as the people to whom maths has to be ‘sold’.
Anyway, given that background, I set off on the Boaler book which despite the name is really about the teaching of arithmetic and basic geometry. It’s not about algebra or calculus or differential equations. My impression from leafing through the book was that Boaler was over-complicating things in the name of enhancing ‘understanding’ but I was willing to be convinced otherwise. I should say that Boaler seems to advocate a very visual approach to mathematics and visualisation wouldn’t be my default way of thinking about anything. So I do have to acknowledge my own built-in biases
I also have to say that when I first encountered Jo Boaler it was through this article in Scientific American in which she seems to seriously misinterpret the PISA 2012 findings on the teaching of mathematics (See Greg Ashman’s analyses which begin here.) So I have some trust issues!
Anyway here is what I thought of the first three chapters.
Chapter One: The brain and mathematics learning
The purpose of this chapter seems to be to present the proposition that, with the right mindset, everyone can become expert at mathematics. “Mindset” here is the growth mindset of Carole Dweck fame and to me it seems to be little more than a ‘sciencey’ way of talking about what you or I might call a positive attitude. It seems to be a very American concept, an idea that you can accomplish anything if you just try hard enough and are not fazed by failure.
Chapter One is fundamentally about making the case for the primacy of nurture over nature. It’s inspired by ideas like brain plasticity which most of us have heard about, but also a curious thing called “brain growth”. My impression of the chapter was that the discussion was superficial and ignored vast swathes of literature in which nature versus nurture has been debated, often quite ferociously. A lot of the argument in Boaler’s book seems to be based on anecdotes, opinion and a few fMRI experiments, many of which seem to be a bit on the old side.
The key point in the chapter for me was this anecdote:
The PISA team not only administer maths tests; they also survey students to collect their ideas and belief about mathematics and their mindsets. I was invited to work with the PISA team after some of the group took the online class I taught last summer. One of them was Pablo Zoldo, a soft-spoken Spaniard who thinks deeply about math learning and has considerable expertise in working with giant data sets. Pablo is an analyst for PISA, and as he and I explored the data, we saw something amazing – that the highest-achieving students in the world are those with growth mindset, and they outrank the other students by the equivalent of more than a year of mathematics.
That text was followed by a figure like this:
And that was it. That was the PISA evidence presented. There was no mention of methodology or controls or uncertainties – no scatter plots, no correlation coefficients. The idea that mindset could be worth a year of mathematics teaching is an extraordinary claim and as Carl Sagan said, extraordinary claims require extraordinary evidence. And I saw none.
Chapter 2: The Power of Mistakes and Struggle
The key idea being presented in this chapter is that making mistakes is good because they make your brain “grow”. I’m not quite sure what this means but, to be honest, I’m always wary of claims made about learning on the basis of experiments in which parts of the brain are seen ‘lighting up’ after some event or other.
The idea that making a mistake can be a useful learning experience is incontestable but a core assumption in this chapter seems to be that traditional maths teachers abhor mistakes and create a climate of fear and anxiety which is exacerbated by relentless drilling, repetition and testing designed to eliminate errors. If that’s what maths teacher sdo, then Boaler is quite right to be concerned. But whatever about primary school, the teaching of mathematics at higher levels often involves not so much drilling as simply solving lots of problems; not problems of the same type or of uniform difficulty but a variety of problems of escalating difficulty. This approach to the teaching of advanced mathematics (in Russia) is well described in Mashsa Gessen’s fascinating book on Grigori Perelman , an extremely introverted character who solved the Poincare Conjecture, after which he withdrew from the world without claiming his million dollar Millennium Prize.
So while the basic point being made in this chapter is that maths classes should be ‘safe spaces’ where students are not afraid to make mistakes and where mistakes are seen as an opportunity to learn, I think the impact of the chapter depends to a large extent on presenting a caricature of maths teaching, one that I don’t recognise either in the “Mental Maths” books used currently in Irish primary schools or in the way I was taught maths at all levels of the system.
Chapter 3: The Creativity and Beauty in Mathematics
I completely agree with Boaler’s claims that mathematics can be a wonderful discipline and if I were to recommend one book that shows the deep connections that exist between the different sub-disciplines of maths, it would be this one.
Anyway, Chapter Three moves away from primary school arithmetic to advanced mathematics. The late (and tragically so) Maryam Mirazakhani, Fields Medal winner for her work on that most abstract of maths disciplines – topology – is mentioned, and the general point is made that mathematics often involves exploration, journeys down blind allies and admissions of ignorance. But later in the chapter the signals become a little mixed.
Boaler says that “over the years, school mathematics has become more and more disconnected from the mathematics that mathematicians use and the mathematics of life”. I would have thought that the reason for this is that mathematicians are forging ahead, creating whole new fields of mathematics, and new computational tools, while primary school pupils still have to learn the basics. But Boaler’s solution to this ‘problem’ is to seek advice from someone who is the polar opposite of Maryam Mirazakhani, namely Conrad Wolfram. Wolfram’s main achievement has been to create the Mathematica computation package (and it is fantastic to be fair) and, like Lego executives who advocate for more learning through play, he advocates the following recipe for maths education::
- Posing a question
- Going from the real world to a mathematical model
- Performing a calculation (using computational software)
- Going from the model back to the real world to see if the original question is answered.
This is utilitarian mathematics, it’s not the doing of maths because maths is a beautiful and worthwhile thing in itself. Indeed, if truth be told, a lot of mathematical models are pretty ugly; they often contain ‘fudge factors’ that destroy any beauty there might have been in the original, simplest version of the model.
The chapter spends a little bit of time on those lists of 21st century skills that the World Economic Forum regularly churns out, before ending with what seems like a call for maths teaching to mimic how maths is used in the real world. This idea is commonplace in education these days especially in the sciences where it is widely believed that to learn science, students should spend their time behaving like scientists, i.e. by conducting research and enquiry-based activities rather than acquiring scientific knowledge.
It is hard to understand why this belief has become so popular in education because it has not gained traction in most other areas of life including sport, music, painting, sculpture, acting, writing and singing. In all of these endeavours, it is accepted that practice and performance can take very different forms. Rugby players practice individual and small-group skills far more than they play 15-a-side full contact rugby; the great renaissance painters spent many years drawing in pencil before being allowed to come within a mile of paint, and most actors have to undergo all sorts of embarrassing training exercises in acting school even before getting a bit part in a dodgy TV series.
For me, the take home point from the first three chapters is this: teachers should be nice to kids, create an encouraging and supportive class environment, and make sure that the kids are not afraid to get answers wrong. And that’s about it.
Next time: Chapters 4 and 5