A Mathematician’s Lament: 25 pages of ruthless, brilliant, deeply meaningful lamentation about what has become of arguably the most beautiful of all the subjects – maths. Though Lockhart focuses almost exclusively on how modern education has bleached the insides of the subject that Gave Us The Information Age, the light is also unavoidably shone on the tragic state of all education: the teaching methods and materials used in science, philosophy, english, food, and just about every other subject found on your typical K-12 curriculum. And good. To say it’s a shambles is a shambles. More accurate but still shallow is to say that 21st-century schooling is criminal. Perhaps back-a-day there were valid excuses — life was hard, opportunity scarce, information was hard to get hold of, there was less knowledge— but today there is no justifiable reason. How can such slow, prolonged suffocation of the Golden Human Trait, namely curiosity, not be criminal. There is no question.
What exactly is the problem with today’s education system? There’s little point repeating the far more beautiful expressions of those whom you’ve learned from unless you’re convinced you’ve got something original to add. Hence my recommendations that you read:
- Mason Hartman, who’s particularly attuned and incisively expressive on just what the typical K-12 schooling is doing to young minds
- David Deutsch, theoretical physicist, founding father of Quantum Computation, and former co-head of Taking Children Seriously
- Eric Weinstein, who on this podcast draws a sorry picture of how we’re digging ourselves into a hole by systematically categorising our super-learners as ‘persons with learning difficulties’, thereby depriving the world of God-knows how much progress.
- About the life of Richard Feynman and a guy called Albert Einstein — two people who actively rebelled against the education norms whilst being deeply entrenched in it.
At the risk of repeating myself, I won’t repeat any of Paul Lockhart’s words here — I’ll add it the the recommendations list — but instead share some passages that spoke to me, with a few words of my own.
Lockhart begins the essay by drawing an analogy first to music, then to painting. Imagine trying to learn the piano by studying only note symbols, key signatures, and the infrastructure of an organ. Insane, right? Just as insane as learning to paint by naming colours and monotonous exercises like ‘Paint-By-Numbers’1. Unfortunately, this pretty much describes modern education: the conversion of art forms to military-style programmes based; the nullification of creativity; curiosity’s castration.
The analogy to painting ends in a dialogue:
“Why do colleges care if you can fill in numbered regions with the corresponding color?”
“Oh, well, you know, it shows clear-headed logical thinking. And of course if a student is planning to major in one of the visual sciences, like fashion or interior decorating, then it’s really a good idea to get your painting requirements out of the way in high school.”
“I see. And when do students get to paint freely, on a blank canvas?”
“You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there abstract stuff. I’ve got a degree in Painting myself, but I’ve never really worked much with blank canvases. I just use the Paint-by-Numbers kits supplied by the school board.”
I’ve made bold the most critical part: I can’t fathom the amount of times I’ve been accused of going ’too deep’ or getting ’too abstract’ after raising a basic point such as the fact that prescriptions (like that of anything resembling a textbook) are destroying young minds; that the mind can be better understood as an organism which learns by trial and error, which requires stressors, and most importantly which requires a reason. This stuff is not ‘really way-out-there’ abstract stuff. It isn’t detached from reality; what’s detached is our ideas of and practices with regards to teaching, learning, creativity and the growth of knowledge.
The fact that one might get accused of being ’too abstract’ is itself a symptom of the problem: because we’re taught at school to follow steps, that This Way is The Way to learn, protocols based on strict obedience, and all that, we develop a fear of doing anything outside of what is known, or which generates slight discomfort, or which is against the norm. We develop a fear of exactly the kind of thinking that creativity requires. And without creativity, we can’t learn anything; progress is made by creative engagement with problems: we have a problem, propose an original solution, test it, and regardless of whether we fail or succeed, we have learned: progress is being made. All stages of problem-solving are critical, but the generation of an original solution is the most important; here also lay the origins of the term ’think outside the box’. But we have to ask ourselves, why get inside the box? Modern schooling makes us scared to Think.
Lockhart cleverly punches his lamentation with soundbites and short rambles that at minimum shake up your definition of what mathematics actually is, and at most, turn it completely on its head
Some surgically-removed favourites:
- ‘[In mathematics], things are what you want them to be. You have endless choice; there is no reality to get in your way.’
- ‘…math, like painting or poetry, is hard creativity work. … Mathematics is a slow, contemplative process. … Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognise it when they see it.’
- ‘… there is nothing more there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. … Mathematics is the purest of the arts, as well as the most misunderstood.’
- ‘So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin
with G.H. Hardy’s excellent description:So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language and culture? No, not usually. These things are all far too complicated for most mathematicians’ taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”
If mathematics is about seeing the beauty in simplicity, Confucius isn’t surprised by the today’s state of affairs:
“Life is really simple, but we insist on making it complicated.”
“Make everything as simple as possible, but not simpler.”
The Postponement of Interest
Lockhart sings a similar song to David Deutsch on a few points, one being the question of how subjects should be learned. And to everyone’s surprise, it’s not via ‘first principles’. Richard Feynman is these days deservedly spoken much about by scientists, engineers and entrepreneurs alike. As far as conservative educationists from recent centuries are concerned, Feynman is the prime anti-example.You may have heard of the Feynman Technique: a sort of non-typical systematic approach to learning the important bulk of pretty much any topic in ridiculously quick fashion by starting with the most fundamental truths or axioms and building your way up. These fundamental truths or axioms are what people mean when they speak of ‘first principles’. Perhaps you’ve tried the first principles approach to learning? If so, you’ll be familiar with the deep dissatisfaction upon the realisation that it’s not quite as easy as it’s made out to be. Why was it so easy for Feynman? How come Musk can build rockets by taking a first principles approach and I can’t even grasp Macroeconomics 101?
The first principles exponents violate their own rules when they leave out what must precede a first principles approach if what is being learned is being learned with the intention of being remembered and soon thereafter utilised, which of course it should be2. What they fail to address is the critical role of interest. How interested you are in topic X has profound consequences for how well you’ll be able to learn topic X, how well you’ll remember it, and what you’ll be able to do with your knowledge.
Richard Feynman took a first principles approach to learning physics, sketching, dance, safecracking, the frigideria, and seduction. But his approach to all of these domains was preceded by interest. For evidence that his interest was not externally imposed — i.e., forced upon him by teachers, fellow academics, or his father — pe into one of the many books written about his life. The best introduction as always is the written by the person themselves3.
Humans are problem-solving beings. We’re at our best when we have a problem and are actively trying to solve it. Devoid of obvious, definable problems, we’re unsurprisingly very adept at finding them, and failing that, manufacturing them.4 David Deutsch argues that philosophy shouldn’t be studied in isolation. Lockhart says the same is true for maths:
‘A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?
I can understand the idea of training students to master certain techniques— I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process— that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.’
The phrase ‘learn to code’ is perhaps the most pertinent modern example. Baggage aside, it stems directly from the misconception that we create sharp, powerful, able minds by filling them with ideas, questions and knowledge that will one day be relevant. The idea of ‘filling minds’ is also another misconception5. How much do you remember of your schooling? Likely, what you do remember was taught by a teacher for whom you have at least some fond memories of; in other words, it probably wasn’t the material that you remember. Believing that people should learn to code today is as fallacious as the pre- or post-calculator-era belief that kids must learn calculus or that people should study philosophy to properly learn how to think. The approach is backwards. The correct approach is pious emphasis on problems and problem-solving:
- You have a problem.
- You identify some possible solutions.
- Executing on those solutions brings you to topics X, Y and Z.
- X and Y and Z might help you solve your problem, it might encourage you to drop your problem in favour of learning more about X and Y and Z, or it might lead to a better definition of your problem, which leads to topics A, B and C.
- And so on.
Take two seemingly world-apart coding examples:
a) You have a particular business idea that requires a ton of original software.
b) You are a writer and need some kind of tool to help you identify repeated phrases and words.
In both cases, you don’t have a world-class dev team nor the funds to access such expertise. Besides, you’re not even sure exactly what it is that you want. What to do… what to do? Granted, you might try and talk your way into some venture capital, which has its own merits, but you also might take the programming task upon yourself. Patent or not, your have a problem that you are trying to solve. If there was ever a try method for hacking learning, this is it.
The meaning crisis is real. We have millions of adults wandering around aimlessly, correctly wondering why their qualifications are failing them — why all those promises made by their primary school teachers and university professors about X being important for ‘your future’ are not being resolved. They can never be resolved, because they were based on false premises. In a fictional dialogue with an opponent to his ideas, Lockhart responds to the same tired sentiment that ’surely there is some body of [mathematical/cultural/historical — fill in your favourite topic] facts of which an educated person must be cognisant’:
‘Yes, the most important of which is that mathematics is an art form done by human beings for pleasure! Alright, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them and you remember what matters to you. We have millions of adults wandering around with “negative b plus or minus the square root of b squared minus 4ac all over 2a” in their heads, and absolutely no idea whatsoever what it means. And the reason is that they were never given the chance to discover or invent such things for themselves. They never had an engaging problem to think about, to be frustrated by, and to create in them the desire for technique or method.’
Revelling in Pointless Nomenclature
Those familiar with Feynman will know of the story of how he came to look at the world through a lens of meaning as opposed to one of labels. Labels, as useful as they can be, tell us nothing about The Thing, what it does, why it exists, why we should even give a damn, and so on. Explanations are required — and simple ones, at that. Even Confucius, for all his Zen and wisdom would contort in his grave on being made aware of how highfalutin, inaccessible, and horribly complicated beauty as a topic of serious study has become.
‘How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use words like “conjecture,” and “counterexample.” High school students must learn to use the secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’ (a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” ) That this particular shorthand, a holdover from fifteenth century nautical tables, is still with us (whereas others, such as the “versine” have died out) is mere historical accident, and is of utterly no value in an era when rapid and precise shipboard computation is no longer an issue. Thus we clutter our math classes with pointless nomenclature for its own sake.
In practice, the curriculum is not even so much a sequence of topics, or ideas, as it is a sequence of notations. Apparently mathematics consists of a secret list of mystical symbols and rules for their manipulation. Young children are given ‘+’ and ‘÷.’ Only later can they be entrusted with ‘√ ̄,’ and then ‘x’ and ‘y’ and the alchemy of parentheses. Finally, they are indoctrinated in the use of ‘sin,’ ‘log,’ ‘f(x),’ and if they are deemed worthy, ‘d’ and ‘∫.’ All without having had a single meaningful mathematical experience.’
Transforming Areas of Interest into Cryptography
‘Mathematics is about removing obstacles to our intuition, and keeping things simple.” Without mention of the largely fallacious topic of study known as cognitive biases, give or take behavioural economics, contrast this sentence with how we go about teaching most subjects in school. I remember history class being exclusively about memorisation of facts that had zero relevance to anything I knew and that I couldn’t care less about. In science, despite a fascination with the way the world works and the occasional lesson I deeply enjoyed, it was always far removed from my interest, and in fact temporarily destroyed my passion for the subject. In food studies we spent an insane amount of time studying safety, hygiene, rules and regulations, and micronutrients and macronutrients; consequently, we spent little time actually cooking, and developing our palates, learning about the history of food, and so on. And worse, when we did cook, it was either cake, sponge, cheese and potato pie, or cookies. Not only did this not square with all we were learning about the importance of calories and the nutritional makeup of food, it was also extremely boring. English was pretty much as torturous; the only difference here being that I was fortunate enough to encounter a good teacher or two6 French and German? Don’t get me started on that uno.
Perhaps my favourite part of the essay is the section titled ‘High School Geometry: Instrument of the Devil’, in which Lockhart completely dismantles the shambolic manner in which geometry is taught to young, ignorant, once-budding-mathematicians-but-by-now-most-probably-hopelessly-at-a-loss minds. The weight given to fancy jargon, the pre-agreed definitions, the depressing length — it’s all horribly, shamefully, tragically backwards. But I bring this part up only to give context to my favourite line from the entire lamentation:
‘A proof should be an epiphany from the Gods, not a coded message from the Pentagon.’
As a sentence, this encapsulates the entire problem with modern education:
- it’s almost completely detached from the real world
- like a prison, it ill-prepares it’s customers for their release into the real world
- it values loquaciousness over succinctness, jargon over discourse, labels over meaning
- it destroys the joy in finding things out by ensuring students know in their bones that:
and most of all, that you will one day be rewarded if you live out these rules.
- Description cites the description: ‘This prepares students for Advanced Placement Classes by teaching them such critical things as how to dip a brush in paint, wipe it clean, and so on.’
- If memory isn’t a concern, what is the point? If immediate utilisation isn’t a concern, why learn X now and not when X is needed?
- Technically this is false because Feynman used his neighbour as a scribe.
- See Tony Robbins
- The Bucket Brain Fallacy: the false idea that knowledge can be poured into a mind, by the culture, teachers, parents, books, or whatever other means.
- Teachers do and will always have a critical role; reforming the education system in the manner suggested by the likes of Hartman, Deutsch and Lockhart will not make them obsolete, it will just change their role.