I wish I could remember who linked to David Bessis’s Twitter thread explaining that we teach math wrong, as that thread and this one on why math talent isn’t primarily driven by genetics convinced me to buy his book, which was a delight to read.
Bessis believes that the way we talk about math is reversed. Math is expressed in abstruse logical terminology about ideas and concepts that seem to exist outside of reality (aka Platonic ideals), so people think math is about those logical manipulations. If somebody struggles with visualizing those concepts or with the formal language of math, they believe they are just bad at math and give up (what Carol Dweck would call a fixed mindset).
Bessis disarmingly reveals that professional mathematicians feel similarly frustrated with new math concepts all the time. Bessis was feeling shame that he was struggling to read a book about math, and asked for help from his officemate, who responded: “Didn’t anyone ever tell you that you should never read math books? Didn’t they tell you they’re impossible to read?” He later compares such books to the user manual of a toaster, a reference guide that you might consult if you have a problem, but not a book designed to be read the way we read a novel or even a business book because “Math books aren’t written in the language of humans”. The way to really learn math is to practice seeing the world differently, perhaps by getting a pointer from another mathematician who’s already shifted their perspective in that way.
A turning point for Bessis’s career was giving a lecture where a famous mathematician, Jean-Pierre Serre, was in attendance. At the end of the seminar, Serre said “You’ll have to explain that to me again, because I didn’t understand anything.” Bessis was floored. He knew that Serre was more skilled and intelligent than him, so the concepts and reasoning couldn’t have been a problem. Instead, Bessis realized that he had presented conclusions without the intuition and understanding behind the concepts. Serre was asking him to explain “why what I [Bessis] had explained was true”.
Bessis was inspired to try the technique himself, to embrace this “radical form of curiosity and indifference to judgment”. Instead of nodding along to presentations or explanations, acting as if he already understood to “prove” his intelligence, he started revealing his confusion and asking questions. And he found that when he talked to the presenter directly, they told him how they had come to their understanding through intuition and analogy; the complicated symbols presented were the formal proof of their logic, but that was not where they started.
Bessis realized that what we think of as “math” is all wrong – the formal proofs that we see are not the point; they are just a byproduct output, much like a factory produces valuable goods and often creates industrial waste in the process. The central product of math is developing our intuition to wrestle with these abstract concepts, which has absolutely nothing to do with the logical formalism and theoretical language of math. He ends the book with the story of Ramanujan, an Indian math prodigy who clearly saw results and wrote them down, but never developed the logical step-by-step proofs to formally confirm them. He is widely considered one of the greatest mathematicians even though he didn’t prove his conjectures, because they were all proven correct eventually, even though it took others almost a hundred years to develop those formal proofs. The insight was what made him a great mathematician, not the formal proofs.
Developing that intuition is not a matter of learning information from others; it’s about training one’s brain to expand what one can conceive of. He gives a non-math example of Ben Underwood, a blind boy who learned to “see” by clicking his tongue and listening to the echo. He didn’t know he “couldn’t” navigate effectively through echolocation, so he just did it to a degree that nobody else thought was possible. Bessis points to this as an example of mental plasticity, the ability of our brain to adapt if we give it the opportunity; we might think of Underwood’s ability as extraordinary, but Bessis believes (as do I) that all of us have this capability if we are willing to push through the discomfort of feeling stupid as we try something new. He gives the example of children learning to walk, to talk (even other languages!), to swim – every child learns to do these activities by trying, failing and trying again, not by sitting in a classroom receiving information. He even suggests that any of us could learn basic echolocation if we committed 10-20 hours to actually trying it.
The shift to this growth mindset requires believing in two things: the starting point is insignificant (natural talent matters far less than deliberate practice in skill building) and progress is slow and almost imperceptible (small steps add up!). In this framework, mental plasticity “transforms audacity into competence”, but “you need a lot of self-control and self-confidence to commit to a process that’s confusing, slow, and uncertain”. He points out this unwillingness to experiment leads most people to limit themselves to what they are taught or what they see others do, dismissing anything else as “impossible” until they see somebody like themselves do it. But those that embrace this mindset develop “the freedom to ceaselessly refashion our way of seeing and thinking, and to construct our own intelligence day after day”. In other words, You Have A Choice.
I also like Bessis’s description of _how_ to develop your mathematical intuition. Daniel Kahneman set up a binary choice between System 1 (intuition) and System 2 (logic and rationality) in his book Thinking Fast and Slow. Rather than trust our intuition unreservedly (although it’s faster and less effortful), Kahneman recommends slowing down and using System 2 when one’s intuition might be miscalibrated (especially in new situations). Bessis takes it a step further and suggests a “System 3” to “establish a dialogue between intuition and rationality”, where he mediates between the two systems in a way that allows his intuition to grow and learn: “When I force it [my intuition] to listen to what logic is saying, it takes that into account and adjusts its position”. By constantly testing his intuition against rationality and the real world, he systematically expands its capabilities, and that is what led to his success as a mathematician. He even summarizes the beliefs he uses to confront an idea he doesn’t understand:
- You can reprogram your intuition.
- Any misalignment between your intuition and reason is an opportunity to create within yourself a new way of seeing things.
- Don’t expect it all to come at once, in real time. Developing mental images means reorganizing the connections between your neurons. This process is organic and has its own pace.
- Don’t force it. Simply start from what you already understand, what you can already see, what you find easy, and just play with it. Try to intuitively interpret the calculations you would have written down. If it helps, scribble on a piece of paper.
- With time and practice, this activity will strengthen your intuitive capacities. It may not seem like you’re making progress, until the day the right answer suddenly seems obvious.
I love this perspective, mostly because I agree with Bessis about the limits of rationality – as he puts it, “rationality should be used as a guide rather than an ultimate judge. The reality that’s before our eyes always merits more attention than the certitudes in our heads.” This blog exists because it is my instrument for testing my intuitions against logic – I may have what seems like a great insight, but I don’t trust it unless I can write it out logically in a way that others understand it, and then test it against the real world. I have a tremendously honed intuition partially because I was rigorously trained as a scientist; I won’t let sloppy intuition go uncorrected, and as a result, my intuition is highly trained at pattern matching and finding commonalities in different situations while also staying alert for exceptions.
Bessis’s call to action is that “Learning math should be like learning any other motor skill, like learning to swim or ride a bike, and it should be accessible to everyone.” But this is hard, because “we learn precisely when we force ourselves to imagine things that we don’t yet understand, which unfortunately is the same exact thing that most people run away from … the fastest way to learn is to follow the path of maximum perplexity.” “Progress is slow because the body needs time to transform itself. It doesn’t help to force it, which may end up hurting you. You just need to commit to a regular training schedule, keep your cool, keep going even when it seems you’re not making any progress.”
I’ll extend that idea to every skill in life, whether it’s math, business, leadership, skiing, or empathy; I have not yet found something where I didn’t improve once I started investing in regular training of that skill. Bessis came to his understanding of this belief through mathematics, but Daniel Coyle says much the same thing in his book The Talent Code that deep practice, commitment and coaching are the ingredients of world-class talent. Like Bessis, I believe that “the power of our mental plasticity is profoundly shocking and almost supernatural”, and love this book for providing new perspectives on how to apply that insight to our own development.
