A Mathematician’s Lament (2002) [pdf]

This is how people think of Math class:

"Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."

That's very sad and very true.

For some reason, I stumbled upon Hubbard & Hubbard Vector Calculus a few days after first reading this essay and it stroke me as the opposite to this wrong math teaching.

And this is a pretty good description of how it's taught:

"Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school"

I actually disagree pretty strongly with this essay. I think Lockhart is advocating for an educational approach that would cater strongly (and exclusively) to a particular learning style, specifically, those learners who thrive when going from abstract to specific (deductive learning).

For those of us who do best with inductive learning, the type of education he proposes would bring even greater misery to our grade school education. It was not until I took statistics and probability (for science and engineer majors) in college that I truly began to enjoy math once again. I was able to start with concrete ideas and applications, and then work my way back to the theory behind them.

I'm now reading "Concrete Mathematics" and really enjoying it. Knuth's ideas on math education are pretty diametrically opposed to those of Lockhart, as far as I can tell, and they give rise to something very close to my ideal learning environment for math.

Why not allow children to follow which ever of the two math paths that is best suited to them, instead of forcing concrete thinkers into an abstract world, and abstract thinkers into a concrete world?

Edit: I should probably add that I agree with Lockhart that there's a problem in the way math is taught, but I disagree with him on the solution.

Endless drilling, memorization, and focus on pushing numbers through “formulas” is not the same as “inductive learning”. It’s entirely possible to do lots of pattern matching and bottom-up “inductive” problem solving in a way consistent with Lockhart’s recommendations.

Here’s a great book chapter wherein a mathematician teaches some 6-year-olds in what might be called an “inductive” way: http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf (so much better than a standard first grade mathematics curriculum)

Edit: to clarify, I basically think you’re reading something into Lockhart’s essay that isn’t there.

Agreed, which is why I included the note (preceding your comment) adding that I agree with Lockhart that there's a problem in the way math is taught.

Edit: Can you expand on your idea? I did just re-read the essay (for the 3rd time) and Lockhart seems pretty absolute in his belief that math should be taught as a playful, creative abstraction. And he seems to disdain practical applications of math. So how would you teach math in an inductive manner that maintains the level of abstraction and "playfulness" he's advocating for?

By the way, love the book chapter! That's exactly how I'm trying to teach my kids. Not always easy, but they respond well to it.

And it's entirely possible I'm reading something into the essay that isn't there. But I'm not yet convinced.

I’ll let Lockhart speak for himself:

> Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up half their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you.

> By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it’s actually much worse than this— at least it’s understood that there is an art of sculpture that I am being prevented from appreciating).

> By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

I think you should check out the math book he wrote, "Measurement," and you'll find in it that he does not teach in the style you describe (abstract to specific).

Thanks, I'll check that out. It's very possible that I got the wrong impression here, but he does explicitly talk about teaching deductively in this essay:

SIMPLICIO: Then what should we do with young children in math class? SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.

I think maybe he means something different from you here. I interpret deductive reasoning here as, "situations where you need to use logical arguments instead of being told what to do and repeating it with slightly different numbers."

I have no idea where you get the idea that Don Knuth opposes Lockhart's approach.

I guess we'd not know without asking him, but having waded through much of TAOCP, Concrete Mathematics (a book filled to the brim with the kind of delightful discovery of pattern that Lockhart describes as optimal in learning math), Knuth's marvelously playful book "Selected Papers on Fun and Games"[2], and the novel he wrote about Conway's astonishing "Surreal Numbers"[3]...listening to him lecture on "importunate permutation" at the last local (SF) Joint Meeting of the American Mathematical Association, hearing about Knuth's thoughts on the mathematics of pipe organs, and even seeing the play/pattern-making that went into the entrance mosaic in his home [1]....I think you're way off base about what you think Knuth thinks about math education.

Lockhart: "...if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education."

Knuth [Preface to Concrete Mathematics]: "Some people think that mathematics is serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive."

I make an annual pilgrimage to Palo Alto for Knuth's Christmas Tree lecture[4], which content continuously emphasizes exactly the kind of joy in experimenting, discovering, and learning real math that Lockhart is talking about in his paper.

Everything I know about Don Knuth speaks to his amazing playfulness and joy in pattern finding and making - a delight in the music of math...and a denial of the value of making sure everyone's labeled their axes and memorized their circle of fifths.

[1]: https://www.youtube.com/watch?v=v678Em6qyzk

[2]: http://www-cs-faculty.stanford.edu/~uno/fg.html

The preface states "I've never been able to see any boundary between scientific research and game-playing. ... The topics treated here were often inspired by patterns that are visually compelling, or by paradoxical truths that are logically compelling, or by combinations of numbers and/or symbols that fit together just right. These were papers that I couldn't not write.

I believe that the creation of a great puzzle or a great pattern is a scholarly achievement of great merit, an important contribution to world culture, even though the author of such a breakthrough is often an amateur who has no academic credentials. Therefore I'm proud to follow in the footsteps of the pioneers who have come up with significant new “mind-benders” as civilization developed.

Many years ago I wrote an essay that asked “Are toy problems useful?” [reprinted as Chapter 10 in Selected Papers on Computer Science] in which I discussed at some length my view that students are best served by teachers who present them with well-chosen recreational problems. And I've carried on in the same vein ever since, most recently on pages 7--9 of The Art of Computer Programming, Volume 4A, in a section entitled “Puzzles versus the real world.”

[3]: http://www-cs-faculty.stanford.edu/~uno/sn.html

Surreal Numbers: "How two ex-students turned on to pure mathematics and found total happiness" - In 1973 during a week of relaxation in Oslo, Knuth wrote an introduction to Conway's method in the form of a novelette. ... I believe it is the only time a major mathematical discovery has been published first in a work of fiction. ... The book's primary aim, Knuth explains in a postscript, is not so much to teach Conway's theory as ``to teach how one might go about developing such a theory.'' He continues: ``Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded their false starts and frustrations as well as their good ideas. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story as I was actually doing the research myself.'' ... It is an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other ``real'' value does. The system is truly ``surreal.''

[4]: http://www-cs-faculty.stanford.edu/~uno/musings.html

OK, my argument against Lockhart shouldn't have included the words "playfulness", because now everyone has latched on to that and portrayed me into the Grinch who Stole Math. As a personal anecdote, I hated the way math was taught in elementary and high school, as a dry series of formulas to be memorized (particularly geometry, which I took in the 7th grade). Math should be fun, but for many students, it shouldn't be a series of abstract problems unconnected to the real world.

My main complaint here is about learning in the abstract vs learning in specifics. That's my sole argument against Lockhart's piece. I regret ever using the word "playfulness", because that has nothing to do with my main argument (and if you re-read my top comment, I think you'll understand this). My concern is about abstract-to-specific versus specific-to-abstract learning.

But let's quote a little more from the preface of "Concrete Mathematics":

"Abstract mathematics is a wonderful subject, and there's nothing wrong with it: It's beautiful, general, and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worth of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance".

So, that basically sums of my feelings about how I was taught math in college. I'm not going to pretend to know what Knuth or Lockhart believe, but that paragraph describes my personal concerns about math education.

I'm not a professional mathematician, nor even a math major, but I am an educated parent and informed citizen, and so fall directly into Lockhart's intended audience for this piece. I've now read it 5 times, and each time I get more and more the feeling that he wants mathematics to be taught deductively, as an abstraction. I base this interpretation on how much "pure" and "abstract" learning is emphasized in the essay, and how much disdain he has for practical math applications. I'm not going to pull out all the quotes, but if you disagree, then I don't know what to say.

Unfortunately, I honestly can't afford to spend any more time on this argument, so I'll have to read your rebuttal (if any) and stop there.

I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.

That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.

With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.

A beautifully written and tragic essay.

(Note: What follows is US-centric.)

After nine years of teaching mathematics courses (one semester as an undergraduate, 4.5 years as a graduate student, and 4 years as an assistant professor) and navigating university politics, I'm convinced that this is, at its heart, a cultural issue.

There's a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).

>There's a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).

Could you explain further? It seems like you believe that something should change, but it is not clear if you are saying that the "mathematics" or the mainstream American culture should change.

Lockhart's gist is that of course "everyone" hates "mathematics", because they are being presented with a dried up empty shell that only superficially resembles real "mathematics". Are you disagreeing with Lockhart, and think that the culture should embrace the "mathematics" as-it-stands? Or are you saying that it is unlikely that reforms will be possible to change the system to present "mathematics" in the way Lockhart would approve of? Or are you saying that even Lockhart's "mathematics" is and will always be hated by mainstream American culture?

> Could you explain further? It seems like you believe that something should change, but it is not clear if you are saying that the "mathematics" or the mainstream American culture should change.

The culture needs to change. Full stop.

I just have no idea how to do it.

> Are you disagreeing with Lockhart

Nope.

> Or are you saying that it is unlikely that reforms will be possible to change the system to present "mathematics" in the way Lockhart would approve of?

Not exactly what I was saying, no. Changing "the system", however, will be extremely difficult at best, though.

> Or are you saying that even Lockhart's "mathematics" is and will always be hated by mainstream American culture?

That remains to be seen.

Interesting. I would have thought that "the culture needs to change" is in direct opposition to Lockhart's thesis. My reading of his essay is that people are rightly justified in hating the "mathematics" that was forced upon them.

I wonder if this difference in reading is some sort of confirmation bias, reading into an essay what we each want to hear? On a more meta-level, I wonder if that quality improves or detracts from an essay. Or maybe that quality isn't a part of the essay as much is it is attached to the subject or the reader? Something I'll have to ponder some more.

I studied mathematics as an undergraduate and it does seem that the very mention of math tends to scare and alienate some people.

If you don't mind, I'd be very curious to hear you elaborate further on this subject.

>There's a hatred of mathematics in mainstream American culture that runs very, very deep

Completely agreed. As an immigrant, I can say it is very much a US thing. Haven't seen this much math-hate, but more importantly, math-utilitarianism, as in the US. In Asian countries & in Europe ( UK, France especially), people don't constantly fixate on stupid questions like "what is it good for ? ", which is ultimately a proxy for "how do I make money with this thing ?". But when I taught math here in the US as a graduate student TA, the majority of questions focussed on this single metric - usefulness.

So math texts here are forced to invent bogus problems like "You want to house pigs with 500 feet of fencing. What dimensions of your rectangular pen will house the most hogs ?". Then the American kid says, Ah! Now I see the point of all this! Let l be the length of my pen and b its breadth. You want me to maximize the area of my pen lb subject to 2l+2b=500 so I can house the most pigs! Ok so I see that b = 250-l, so lb is 250l - l^2, so I take its derivative & equate to zero & l=125, b=125, and that's the biggest pen that can house the most pigs. Very nice!

In other countries, you simple wouldn't come up with all these sort of bogus utilitarian problems in animal husbandry. Students here learn exponentials & Taylor expansion as part of "how do I compute compound interest on my bank account", because that's supposedly the only legitimate use of e^x !

I honestly found teaching pre-calc, calc-1 & calc-2 a complete travesty, because the theorems & the entire courseware was essentially perverted - it was all in service of how to make use of the math for some bogus application, rather than learn it for its own good. The worst was when I had to teach how the horizontal range of parabolic trajectories varied - the textbook had examples of the US bombing Japan, & the students went to work computing the best possible angle for firing the missile, so that it would fly across in a parabolic trajectory and land the farthest thus maximizing its horizontal range & kill the most number of Japanese! There was no thought given to how violent & nasty this was.

I actually have very radical ideas about how things should be taught - like you must learn Rolle's theorem before learning shit like pre-calc. Learn as much of undergrad real analysis before you get into application oriented shit like calc-1, calc-2 etc. Don't bring in garbage like LCR circuitry into pde's, even though yes, you can use a third order differential equation to compute current through an LCR circuit.

Applications have their place, but such an overemphasis on application is simply not healthy. It actively distorts the culture & the body politic. Note that American students don'r ask "what use is rock and roll ? otr what use is hbo ? or what use is literature ?" all those things are given a free pass. But when it comes to math, suddenly use becomes the primary criterion. Read pages 6-7, & especially 12 of Lockhart's lament, where he chooses to parody this point of view via Simplicio.

I'm also a math teacher, and agree that there's something about American culture that's really messing with mathematics education. As a student, I was always annoyed by applications questions. Not because I have anything against applications, but because they were usually contrived, and often outside of my field of study. There were all sorts of physics questions that would pop up in the lower math classes (Never took physics) that I could do the math for, but had no reasonable physical intuition about how the system in question should actually behave. I was mostly just crunching numbers.

Would you elaborate on how you think things should be taught? I fell off around the time of the US pre-calc curriculum, but later enjoyed learning basic calculus and basic abstract algebra, so I'm pretty curious.

America, Britain, France, Italy, etc. education systems built upon "getting results" seem to end up ruining the subject they're trying to teach.

How do you get results then? Ignore the results?

"Get results" is code for "reduce people to statistics, then optimize for a convenient statistic".

It's like dealing with a homeless problem by shipping them all to some other town.

Encourage freedom of thought instead of preparing students for some standardized test.

It's crazy how much the AP/IB and SAT tests dictate the direction and content of high school curricula.

It gets much better at the college level, because teachers don't have to prepare students for anything in particular.

By developing the argument. The correct answer isn't nearly as important as forming a strong argument, at least in the process of learning, as a well written argument will naturally lead you towards the correct answer. Its the reason why "showing your work" is taken so strongly. The flaw being that you're not showing your work so much as you're showing your ability to follow a defined series of actions. The grade school test doesn't ask you to think of a procedure to resolve some stated question. It asks if you already have it memorized, as the formula was handed to you in a prior lecture. Ideally, the test should ask the student to form an argument, thus showing mastery over the knowledge that precedes its creation. This would be judged, and of the students ability to consistently dot the i's and cross his t's, expected but given a minor role (as it is in "real life", mistakes are inevitable and its why we check our work twice, and then give it to others to check again.)

Invest in the infrastructure.

Such a wonderful essay. It's very applicable to the teaching of computer science/software engineering as well. So much of the problem is the misunderstanding people have about the field. It's a creative, constructive discipline, and so much of the instruction is consumption, mimicry, and repetition.

Solving well defined problems is relatively easy. Our real problem is that real problems are not well defined.

I think I agree but isn't your last line a bit like saying "hammering in a nail is easy. The problem is that these screws aren't nails."

Chuck Close: "I think while appropriation has produced some interesting work … for me, the most interesting thing is to back yourself into your own corner where no one else’s answers will fit. You will somehow have to come up with your own personal solutions to this problem that you have set for yourself because no one else’s answers are applicable." ... "See, I think our whole society is much too problem-solving oriented. It is far more interesting to [participate in] ‘problem creation’ … You know, ask yourself an interesting enough question and your attempt to find a tailor-made solution to that question will push you to a place where, pretty soon, you’ll find yourself all by your lonesome — which I think is a more interesting place to be." http://www.brainpickings.org/2012/12/27/chuck-close-on-creat...

FWIW one of our professors here at uwaterloo taught a first year abstract algebra / number theory class in a very Lockhart-esque way (Math 145; he even quoted Lockhart on one of the assignments). I learned a lot of math and enjoyed myself, but the main problem I observed was figuring out how to fairly grade students, and the fact that the homework took a lot more time than a class taught normally.

I have a lot of sympathy for his point of view. I loved Math growing up. High school drove the interest out of me, and I didn't get it back until senior Calculus, when I started doing well again. Then I learned to appreciate CS theory, economic theory, etc. Trying to figure out how to break the cycle for my kids: Stats for practical work, and math for curiosity.

A very similar piece by V.I. Arnold[1]: http://pauli.uni-muenster.de/~munsteg/arnold.html

[1]: http://en.wikipedia.org/wiki/Vladimir_Arnold

As a Physicist, I feel obliged to mention that black holes were first hypothesised by Physicists (contrary to the essay), albeit through Mathematical enquiry.

I would assume that he means that mathematicians worked with singularities, before we knew, or hypothesised, that singularities arises in the real world.

Pierre Simon de Laplace was the first person to hypothesize black holes. I think he was a mathematician, although such distinctions were not so clear in the eighteenth century.

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.

However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.

Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.

Geometry, if I recall correctly, was one of the exceptions in early math where you are allowed to veer off the path a bit. Is everything else algorithmic until college?

https://news.ycombinator.com/item?id=6187014

With that being said, I've always loved this essay. As of recently, I've viewed it as relevant to the recent argument that programming should be a requirement in American public schools, either as a tool in math and science classes or a free-standing course. This kind of mathematical reform might actually be a prerequisite for programming and computer science, given that it would develop mathematical maturity much more effectively than the current system does.

Yes, but Lockhart is arguing against any practical applications of math in the early years. I'm almost certain he would oppose any attempt to connect math to programming, for the reasons outlined in his essay (roughly, "math should be about imagination and playing, not applications"). In fact, I disagree with this essay, for reasons outlined in [1].

1. https://news.ycombinator.com/item?id=8847132

Who do you think mathematical programming has to be practical?

You're taking away the wrong thing from the grandparent post.

If you teach math as a prerequisite for something else, you're naturally going to be undervaluing it. Math should be taught for the sake of learning math, not because it's the gateway to something else. It is, yes, but that's not the point.

You know the problem with that sentiment? Most people do not find abstract patterns and rote symbolic manipulation to be interesting. They simply don't, and if you teach math class mistakenly assuming they do, 90% tunes out.

If you can show them how those abstract patterns reflect things in nature, and how those symbolic manipulations represent ideas and algorithms and systems, and can explain complicated things, that's something else.

e.g. You don't need computer graphics to teach linear algebra, but how many people who 'know' matrix algebra know that the columns of a matrix are the basis vectors for the principal axes, and that matrix multiplication is the (affine) transform tool from photoshop?

I would also suggest that if you want to get kids interested in trigonometry and you fail to mention that every single thing in a 3D video game is made out of triangles, you are a bad teacher and you should feel bad.

Rote symbolic manipulation is not inherent to solving the mathematics described in the paper. That's part of the point.

I'd argue that the endless swarms of people playing bejeweled variants suggests pattern analysis is extremely popular.

Mathematics is currently taught with the excuse "you'll need this later, it's useful - honest". If you're advocating an additional helping of that, why is THIS future possible application relevant in a way that the others aren't?

I'm not interested in teaching math to people who don't want to learn math, honestly. It baffles me why so many people think that if someone is uninterested in math, they must be tricked into being good at it anyways.

Teaching shouldn't be so dishonest.

Why do you think creating interest from real life is going to make them not "want to learn math"? That makes no sense.

Teaching is all about benevolent dishonesty. The first thing a teacher does is dumb themselves down to their students' level.

> Why do you think creating interest from real life is going to make them not "want to learn math"? That makes no sense.

I don't. I think it's entirely possible to cram knowledge into people's minds unwillingly.

But you're making an unnecessary assumption: that you have to teach everyone math. Why is that?

Why do we teach everyone history? Because it's valuable to know where we came from, and to realize that whatever's going on today, it's probably happened before.

Why do we teach everyone math? Because it creates quantitative literacy, because it helps people deal with systems and complexity, to break down problems logically, and to not let biases get in the way of the truth. Or at least that's what I wish it would be focused on, instead of producing droids who know how to execute symbolic algorithms.

> Because it creates quantitative literacy, because it helps people deal with systems and complexity, to break down problems logically, and to not let biases get in the way of the truth.

None of these things are math. These things can take advantage of math, yes, but they have as much to do with math as figuring out how to use Microsoft Word does.

> Or at least that's what I wish it would be focused on, instead of producing droids who know how to execute symbolic algorithms.

You want to know how to get them to focus on it? Ask them to.

Stop asking people to teach math. Ask them to teach quantitative literacy. Recognize that this isn't necessarily math. Ask them to teach systems theory and complexity theory. Recognize that math is not the best vehicle for understanding those things, especially for grade schoolers. Ask them to teach logic. Recognize that set theory isn't covered in grade school at the moment, and that learning logic isn't going to happen through math. Ask them to teach ways of discerning truth despite bias. That means covering the scientific method, covering statistics, covering research strategies, covering fact-checking.

Be. Honest. With. Your. Goals.

Your goal isn't "Students should know how to derive polynomial expressions." You've stated your goals. Recognize them for what they are. Stop asking math teachers to carry all that weight for you. Stop hoping that students will magically gain "quantitative literacy" from geometry proofs about angles.

You get "droids" because you've asked for "droids".

Something like MIT Scratch can be both 'imagination and playing' and 'applications' (programming).

I think for most people, math is best learned in the context of some application that they care about (the last four words are very important). Few people appreciate the beauty of the abstract game itself.

For example, most people who play poker online quickly learn about expected value, probability, and variance.

Precisely. Poker is one of the most advanced games that legitimately teaches you what you need to know! The language used by poker players should be adapted wholesale.

For such people, is there a good reason to learn math?

HN is big on curated lists lately. Is there one for resources to aid in teaching mathematics the way Lockhart would prefer?

I've seen this posted so many places so many times that surely there's a market for materials and support for it. Where are they?

With a quick search, here's the closest thing I could find:

http://mathoverflow.net/questions/5074/are-there-elementary-...

...but I too would like to see what you are looking for, at an elementary school age level.

Now that I've thought about it a bit more, what I'd really love is a directory of sensibly ordered learning resources (free online stuff, videos, links to books on Amazon, whatever), headed by one or two mission-statement type papers (like Lockhart here) for the pedagogical strategy they generally follow, for a whole bunch of topics with several strategies per topic. That way you could pick a topic, read the statements for the various strategies, select one or more that you like and just start working down from the top.

... and I'd also like a pony :-)

As a former high-school math teacher and big proponent of Lockhart's philosophy, I highly recommend Phillips Exeter's math curriculum (http://www.exeter.edu/academics/72_6539.aspx).

It's entirely problem-based; each night the students are assigned a set of about ten problems, and they do the best they can with them. The next day, they meet in groups of twelve with a teacher (more like a facilitator) who leads them in a discussion of how they approached the problems and their solutions. Although the problems themselves are often quite practical, they are designed to illustrate and reveal over time the abstract concepts behind them, and they lead the students towards finding abstractions and generalities as they go.

It's free and publicly available; I recommend downloading them and reading through a few problem sets to see what I mean.

For middle school I'm sure Lockhart would love Gelfand's books: http://gcpm.rutgers.edu/books.html

At undergrad level, I love Axler but I haven't found a similar calculus book.

Spivak? It's an excellent calculus book. Apostol or Courant are good as well.

I made a little curated list for my friends who are starting to raise kids of their own and wanted to know about resources. I think most of what's out there is math education phrased as a game (to develop mechanical skills without being boring), and later translates into Lockhart-style exploratory "joy of math" stuff. The latter doesn't have much besides, Lockhart's own work, though, from what I can find.

http://mcsforkids.herokuapp.com/

I would look at actual math education research and research-based materials and lessons and tools, such as at sites like NCTM and MAA. Most educational research is not consistent with Lockhart's views.

As an example, here's a summary of research-based best practices for teaching Calculus: https://edtechdev.wordpress.com/2014/06/03/calculus/

> Most educational research is not consistent with Lockhart's views.

This sounds suspicious, and I imagine there are differing goals and motivations here. For example, much of math education research also centers around standards and testing.

Polya's "How to solve it" is pretty good in that respect. I asked this question on math.stackexchange.com a while ago but haven't received many answers yet

http://math.stackexchange.com/questions/1040232/can-you-reco...

It's a game rather than a list, but http://euclidthegame.com/ seems to address some of Lockhart's issues with the teaching of geometry.

"How to solve it" is a good book in this direction, by Polya.

"How to solve it", by Polya.

Lockhart has a book, "Measurement", that elaborates in the same vein. Quite nice.

I personally found it inspiring and beautifully written.

I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.

That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.

With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing buggy programs is easy.

I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.

That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.

With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.

Love the bit about the misconception that Mathematics is mainly about utility. I remember reading something about G.H. Hardy (which I can no longer find) in which he said he would get a little bit disappointed if he found that one of his results ended up finding a practical use.

He actually argues quite the opposite in [1] (see footnote, pg 33) - a good read in itself. The quote you're probably referring to is this:

"a science is said to be _useful_ if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life" (emphasis mine).

And Hardy's response (excerpt):

> It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications.

> I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications.

[1] http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%2...

That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.

Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.

I'd be more concerned that such a paper would be a lie. When did utility become more important than truth?

Truth is important because it has utility. It enables consistency and reproducibility in a way that nothing else does.

Really? What consistency and reproducibility does "An individual is a worthwhile human being" have?

He said that the most beautiful mathematics were that which had no practical applications.

Not sure how he'd feel that now that his own specialty of Number Theory has turned out to be very practical.

That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.

Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.

That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.

Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.

That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.

Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.

That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.

Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.

That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.

Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.

Vital context: The Underground History of American Education, free online: http://mhkeehn.tripod.com/ughoae.pdf

Wow. It certainly has an excellent hook in the very first sentence:

"Our problem in understanding forced schooling stems from an inconvenient fact: that the wrong it does from a human perspective is right from a systems perspective."

It gets better from there. Keep reading.

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.

However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.

Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.

However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.

Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.

If the musician woke up in the first sentence, how is he still dreaming in the next one?

My favorite part is "The Standard School Mathematics Curriculum".

Always a good read.