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A Mathematician’s Lament (2002) [pdf] (maa.org)
152 points by kjhughes on Aug 9, 2013 | hide | past | web | favorite | 119 comments

"There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun."

As a Mathematician I've always identified as a creative person, yet struggle to convince "artsy" persons that what I do is creative. As soon as I utter the word math they convulse and shutter as if they were afflicted with a PTSD flashback. I try explaining what the idea involves and draw a diagram or three, but all they do is just nod and say "yeah, yup, okay.." Are the ideas of math that inaccessible to the general population when compared to a work of Art?

"Are the ideas of math that inaccessible to the general population when compared to a work of Art?"

I'd guess at least some fine arts (composing classical or modernist orchestral music, modern abstract painting and sculpting) would also be pretty inaccessible to general population.

Yes, but when a classical composer converses with you and says something along the lines of "There actually is, with all the creativity, a sort of formula I follow to produce my best pieces. You see I do this first, revise it three times, etc, etc." and no one will sit and shutter and have the PTSD style flashback the original comment talked about because they do not know and they do not have preconceived, strongly held, notions in mind about the subject.

Or at least I think that'd be true.

Exactly! With the exception that most of the artist's I've conversed with don't like the word "formula", but liberally us the word "process". Also anyone is Science seems to favor "methods". They all are talking about the same activity of concentrating on an action their passionate about. Language can be a funny thing.

But if the composer would use a more geeky language, and talk about e.g. E7#9 chords, mixolydian modes and chromatic passages? How long would a layperson bother to listen.

(Well I used rock/jazz terms, since I am not familiar with classical music.)

I think that language (at least English) fails us here. The word "creative" can be used to mean both "relating to or involving the imagination or original ideas, especially in the production of an artistic work" [1] or "resulting from originality of thought, expression, etc.; imaginative" [2]. It seems that most people identify the word more strongly with the artistic sense of the word, in which creativity is a proxy for a kind of self-expression that is not bound by any rules, logic, or structure. In doing so, they seem to mistake one of the more visible manifestations of creativity with its essence, which really lies in the "originality of thought" and "imagination" of the creative person.

If we had a specific, unique, and widely used word for the "artsy" free-expression type of creativity, I think the confusion many people express when you try to convince them that mathematics is a creative endeavour would be greatly diminished.

[1] http://oxforddictionaries.com/us/definition/american_english...

[2] http://dictionary.reference.com/browse/creative

> creativity is a proxy for a kind of self-expression that is not bound by any rules, logic, or structure

I do not find this to be the case. Art is full of structure, rules and logic. When artists "break" the rules, it usually means they been able to operate using underlying rules, and understand well the rules they break.

Or they're creating new rules.

Yes, indeed. For instance, the creation of bebop in jazz. Of course Bird could only do this by a comprehensive understanding of the existing rules and seeing deeper.

The way I read the essay, it seems to me that math is compatible with [1]. You don't have to follow any given rules. Make up your own ideas and see what they lead to. What if square roots of negative numbers exist? What if the sum of angles of a triangle is not 180 degrees? What if a proposition can be both true and false at the same time?

There have been campaigns to put logic above all other human pursuits. As if only logical things are pure, and anything less rational is beast-like or uncivilized. Artists and poets naturally rejected this idea, but many also rejected the whole subject as hostile.

"Are the ideas of math that inaccessible to the general population when compared to a work of Art?"

Yes, if only because many (if not most) don't want access. Been there, done that, got the medications to prove it.

"got the medications to prove it."

Please, elaborate?

yet struggle to convince "artsy" persons that what I do is creative.

As another mathematician I've almost found the exact opposite. As soon as I mention math to an arts person they instantly start babbling about fractals and chaos and Fibonacci and all kinds of other vague pop-culture terms they've heard of but don't really understand. Artsy types almost seem to find math much more artistic than I do.

They look at fractals and spirals and enjoy seeing the pretty pictures with all the symmetry and colors. However when an explanation of how the series is generated and how it can vary is presented, interest is feigned and the core concepts still elude them. It's almost as if they don't like getting their hands dirty in a different medium.

Come on, that's a gross generalization.

There is a different between artsy and creative. Artsy (roughly) refers to things that appeal to the senses. Creative is a much more general notion of creating. In math, you defiantly create things. And you arguably create beautiful thing. But the beauty is not in the senses, it is in the mind. The senses are involved only as a form of communication.

It's the same kind of beauty you'd find in a well-stated argument.

IMO, Mathematics are abstract aesthetics, invisible unless you to forget your senses for a minute and see in relationships, recurrences, evolution .. patterns. Add the fact that the mathematical culture is cryptic .. (centuries of abstraction stacked and compressed in a symbol, unspoken principles, ...) and that it's very badly taught in the first years of school, you get inaccessibility.

"Music is a stupid way of art, usually for stupid people. If you are writing literature or poetry, then you should be an intellectual; as a really good musician, that's not a must." -Holger Czukay

So there is this idea of different arts being more or less accessible to the general population.

There are lots of stupid musicians, just not any I want to hear. But be careful comparing writing to performing. Yes, you can be a mindless automaton and still sound pretty good. But composers are generally intellectual, and in my experience the set of good composers who are not blindingly intelligent is very small.

"but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.”

So painfully spot-on. My mathematical education was horrible. Meanwhile I had been writing code since I was a little kid. It wasn't until I was an adult that I realized how much math I had been learning while programming. And worse, that I had been completely miseducated about what math actually is.

Interestingly, the converse is also true.

I was always into computers and tech but never really programming.

I was a math major in college, however, and after graduating I started programming. The transition was almost seamless, I picked up programming really quickly, it was surprising to me how much the "ways of thinking" are alike.

One of the key mistakes of educational strategy is dividing up knowledge into a bunch of subjects like "Language", "Math", and so on. You can find linkages in just about everything; this siloing really only serves to make people think they're particularly good at one thing and not another, when really it's a preference for one perspective over another.

So the pragmatists question is then naturally, how do you teach it? How do you shard the curriculum if not by subject? It seems a bit overbearing to ask each teacher to have the proper renaissance man's education of knowing a bit of everything.

To me, a teacher's proper role isn't having knowledge and dumping it into someone's brain. It's knowing where to find that knowledge. My three Rs are "reading, research, and reflection": a teacher's job is to (1) provide useful material to consume, via lecture or homework or whatnot, (2) point towards larger resources for further exploration, and (3) guide thought processes to make useful conclusions.

A teacher's job is not to teach. It's to provide a space in which a student can learn. A focus is useful for this, but the focus doesn't need to be an abstract subject. It's a MacGuffin; it can be anything.

Classes are already discrete, and can thus be taught by different teachers.

As I understand, classifying classes into "disciplines" is for the convenience of administrative systems, like university deans. This classification is not a law of the universe, nor of the human mind. One can imagine the negatives of fitting learning into hierarchical administrative models.

I completely agree! By other's perspectives my hardest semester was when I took PDEs, Geometry Space-time (Minkowski), Algorithms, P-chem (Quantum Mechanics), and Analysis. However there was so much over lap with all the materials and topics. A few weeks after midterms all the course work just fit together like a 100 piece puzzle.

Taking as many physics courses as I have, I was getting worried for a while that there was nothing more to life than solving a boundary value problem in some domain or another.

Can't vouch for this book, but coincidentally just read about it today-- "The Essential Knuth" by Knuth, Daylight, and DeGrave.

Donald E. Knuth lived two separate lives in the late 1950s. During daylight he ran down the visible and respectable lane of mathematics. During nighttime, he trod the unpaved road of computer programming and compiler writing. Both roads intersected! -- as Knuth discovered while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961. "Chomsky theories fascinated me, because they were mathematical yet they could also be understood with my programmer's intuition. It was very curious because otherwise, as a mathematician, I was doing integrals or maybe was learning about Fermat's number theory, but I wasn't manipulating symbols the way I did when I was writing a compiler. With Chomsky, wow, I was actually doing mathematics and computer science simultaneously."

> (...) while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961

Which makes you wonder, what kind of person reads this stuff on the honeymoon.

Someone who chose the right person to marry.

In India this too never happens. The article is a 1000 times valid here.

I read this a couple weeks ago and decided to try and find a math book that incorporates some history in it. I found Journey through Genius: The Great Theorems of Mathematics with some great reviews.

I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading the book with someone, email me at bridger@understudyapp.com. I could really use some beta testers for the app!

If you have already read it, you could still mentor someone in it to review the material again.

To add to the suggestions here, "Number" by Tobias Dantzig is absolutely a wonderful history of mathematics. I mean, shoot, it's actually got a quote from Einstein:

"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."

—Albert Einstein


I've spent a lot of time looking for books that'd teach some mathematical ideas while keeping the original progression of motivations/concepts in tact.

I'd first recommend "Men of Mathematics" by E.T. Bell. It's a collection of short biographies on 20 or so Mathematicians, also discussing a few of the most salient points of each's work. It's an enjoyable introduction, useful for getting a broad view of what math is made of and how mathematicians think. Bell was a serious mathematician himself (not of the rank of anyone he's writing about, of course), as well as a sci-fi author apparently :)

edit: could also try "Mathematics and the Imagination" as an alternate introduction.

After that would be "What is Mathematics?," by Richard Courant and Herbert Robbins. This one's a bit tougher, and I have to admit I had the experience of being perplexed at the selection of topics, and that it didn't tell me immediately what mathematics is -- but! Without too much time passing, I now appreciate the selection and think it could be read profitably by trusting that the selection is good and trying to answer the question why that's the case while reading.

At the moment I'm trying my second book from E.T. Bell, The Development of Mathematics, and like it quite a lot so far, though it assumes a little more math knowledge. This one's probably great if you did a mathematics undergrad, or similar, but would like to see the various topics related and given context.

Another I believe worth checking out, if none of the others fits exactly, is William Kingdon Clifford's "Common Sense of the Exact Sciences." I've only skimmed sections in this one, but it looks extremely promising; and from what I've read about it and about Clifford, I think it could be an important piece of pedagogy along the lines of what Lockhart's into. Not too long and pretty accessible I think.

Another suggestion: http://archive.org/details/TheWorldOfMathematicsVolume1 (and volumes 2, 3, and 4).

Lots of interesting stories, many of applied mathematics. And, to my surprise, available for free from archive.org.

Check out this book as well.


Thanks! It is next on my list now.

Journey through Genius is an absolutely fantastic book. I took History of Math from a professor in undergrad who had a conjecture named after him and this was his choice of textbook (along with a few other materials) for the course.

You might enjoy Lockhart's book, Measurement. http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755...

Yes, a million times! Journey through Genius is an excellent book (I, too, had this book for a History of Math class). It's really appropriate for people of all different levels of mathematical maturity. It's aimed to be read by a pretty much lay audience. But it covers some interesting t material that you're likely not to have seen in an undergraduate curriculum (Heron's formula, cubic/quartic equations, Euler's windmill proof).

Sounds like an interesting idea. Learning is much easier if you can find someone equally interested and at your same level.

Still haven't finished reading this and all the painful memories came back from middle school. Not math (although it too wasn't a fun experience) but art. For four years we had a teacher that thought music and visual arts by dictating. For 45 minutes we would write down everything word for word. Then after a few weeks she would ask us broad questions and we'd have to recite everything back to her. I still remember (I'm almost 30 now) rote learning from my notebook, walking around the house, repeating those stupid sentences in my head, memorising descriptions of works of art I've never seen, painting techniques I've never witnessed, music I've never heard. Wow, just talking about it makes me angry, makes me want to to choke the bitch's neck. Sadly, that's the state of schools in Croatia, people equate schools with rote learning and memorising. You have to do it to get into a good university and get a good job. Who will create those jobs? No wonder the country is in the shitter.

oops, meant taught instead of thought.

Love this essay. I read it years ago when my brother was working with Paul Lockhart, who deeply influenced him as a math teacher.

My brother and his wife have since started an organization called Math For Love (http://www.mathforlove.com) focused on changing the way math is taught. They run workshops for teachers and provide great material for students.

If you're in Seattle and interesting in pedagogy and math, you should check them out.

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum?

I'm really asking -- my friend is about to start as a high-school math teacher.

I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Lockhart sense -- real puzzles, not exercises.)

But how often are "techniques" actually taught in high school math, especially algebra and precalculus? A lot of high school math consists of digesting new definitions, or the generalization of old definitions. A fair amount of it consists of learning theorems that go unproven, or that are proven (by the teacher) too quickly for students to understand where they come from -- and in general it isn't satisfying to solve a puzzle with a theorem that one doesn't actually understand.

On top of that... students have to spend time with problems before they become genuinely interested in their solutions, so progress would be slower with this method. It's not clear that you could teach a whole year's curriculum in one year like this. (And if you fail to do that you'll eventually get fired.)

Any insight? I believe that it's possible to teach math, even standard high school curriculum, in such a way that students are at all times intrinsically interested in what's presented. But it would be awfully hard to do at scale, at the standard pace, as a high school teacher would have to. How might a teacher start in that direction?

One thing you could do is take an interesting piece of subject matter from later curriculum (like next years or later in the current year) and present it as a puzzle for the students to explore with the inquisitive techniques presented in the paper.

Award participation credit, etc as relevant to help keep people engaged who need it. The fact that it's "future" material can also help students who need the extra goals pay attention.

Most likely you can't actually cover the required curriculum like this— as you note, a lot of things do not lend themselves to compact discovery. (It's all ashame, it's not like the students actually retain into adulthood all those procedures that they don't really understand in any case :( ) But maybe you can still inspire people with a few things which do lend themselves to compact discovery, and that inspiration may also make the rest of the subject more accessible to them. "This things have a reason and a pattern to them, even if I don't know what it is right now."

I had some challenges in math in school because I studied calculus, analysis, linear algebra, discrete math, etc. on my own and would derive solutions— sometimes the same as they wanted me to memorize, sometimes not— on my own instead of memorizing the fixed routines, and this was unwelcome. It would be nice if more teachers made an effort to at least not penalize students that were independently interested.

I was a student from a small private school that literally wrote their own math book, so I have no idea how generally applicable this is. As you suggest, a common technique my teachers employed is setting us loose on problems we did not yet have the tools to easily solve (but which were within reach). We would typically work in small groups, and if necessary the teacher could speed up progress by dropping us hints. We inevitably (in the beginning) would come up with week/non-rigourous solutions, which would often lead to debate as a class, pushing us to formalisation. As far as learning new techniques/generalizations/ETC, we would almost always 'learn' them after we have already been using them.

One thing I noticed during in math classes is that I don't really need to know anything. For me, and most of my classmates, most of formulas could be easily derived from simple and intuitive principles. For example, almost no one in my class actually 'knew' the quadratic equation, or the common trig values (ie. sin(30)). What we did know was how to quickly find those if we needed them.

As to your question of a standard pace, groups tend synchronize themselves. If every comes in at a similar place, and you have alot of group work and full class collaboration, then the slower students will gennerally still be able to follow the groups discovery trail, even if they do not contribute as much. The important thing here is you make sure that students are comfortable to ask questions, and that you do not have a few students dominate the discussion such that they loose the rest of the class.

Again, that comes from the perspective of a student at a school where the teachers had a lot of leeway in how and what to teach.

Would it be possible to get that math book they wrote and perhaps the teachers' notes to go with it?

Also, your friend should read Mindstorms: http://www.amazon.com/Mindstorms-Children-Computers-Powerful...

One of the major themes is the relationship children have with mathematics and ways teachers can change it.

I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult, even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy, have to teach to a common set of state-wide standards, and have students of varying levels of interest.

Here are my scattered thoughts, though. I'm going to try to not suggest a pie-in-the-sky solution like "new curriculum!"

First, I majored in mathematics at the University of Chicago, but I hate, hate, hated mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication?

And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work.

Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it.

So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on.

If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them.

Second, I think the way math is taught is overly symbolic. What most non-mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed.

I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa.

Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason

Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations.

Hey class! Look at this expression: 45+6. What does it equal?

A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer."

A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like...

Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong per se. We could have chosen to live in either world, but we have to choose one consistent set of rules.

These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and . So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it.

It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't.

I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)

> when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent.

Might we benefit from a different set of symbols that actually convey the geometrical meaning behind them? If instead of π, we used a glyph that shows a circle over a diameter, instead of x for a variable, we show an empty rectangle that shows that it's a placeholder for a value?

> Students who said the answer was 44, how did the students who said 26 arrive at their answer?

I came across a great example of this approach in Chess: The Complete Self-Tutor by Edward Lasker. Instead of just showing the right answer for a chess puzzle, he tells you what you did right in your answer and what you missed in getting an even better answer. This was a printed book that was completely interactive.

People should note that * are turned into italics, when text is between them. So, the original equation is 4 * 5 + 6.

(Here, I use spaces to "escape" the * . '\' doesn't work as an escape character.)

"Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind."

That's almost Douglas Adams levels of dry wit.

This is a fascinating article, as someone who's never really contemplated the playfulness of maths....I mean, for sure the wonder of maths or the power of maths - I've just never really put the pieces together to link that back to my grounding in maths, beyond the practical, functional stuff I incorporate into my daily life, without considering it's mathiness. Very interesting.

This is intensely thought-provoking and beautifully well-written. It's not directly about hacking, but it's the type of treasure that hackers love to stumble upon.

It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.

>It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.

At risk of getting to meta, this piece seems to pop up everywhere. I don't mean that in a bad way, but this is at least the fifth time over the course of many years that I have seen this piece pop up on completely unrelated sites.

Not to mention it's been here on HN at least 8 times, probably 10 or more. I've stopped bothering to direct people to previous conversations - they just say the same things over and over again anyway.

I thought I had been taking math my whole life until I finally took a math class.

Back when I was in academia, one of my favorite student evaluations from an abstract algebra class had the following comment (paraphrased from memory):

"This course was like climbing Mt. Everest. It's difficult, and sometimes slow going, but at the end, it's breathtaking how much you've learned and accomplished."

Hah, yes! I hated math in high school but wound up graduating with a BS in mathematics from the University of Chicago. It wasn't until I took my first math course at Chicago using Michael Spivak's Calculus that I thought, "Wait, if this is math, what was I studying all through high school?"

The article resonated with me on some level, because it does take a long time to learn how to actually do math. If you are at the point of just doing algebraic manipulations on equations to try to figure something out, you've lost the battle (as opposed to using algebraic manipulations to encode your thoughts, and work out the details).

On the other hand, I think everybody really did see the beauty in geometry. Yes, the initial manipulation to prove something about symmetric angles is a bit silly, but you aren't being taught symmetric angles here, you are being taught how to do a geometric proof. Which is not easy to learn, so you start with something super simple and obvious. I didn't observe anyone in my classes (well this was back in 1982, so memory is a challenge here) confused about that point. And, as soon as we learned it, the requirement for formalism was dropped. It was the same in algebra. In the first weeks you weren't allowed to go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 = 1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using written out. Annoying yes, once you grasp it, but once you proved you grasped it that was the end of that, and we never had to do it again.

But, I had good teachers that always tried to explain the 'why' of what we were doing, and did not make us engage in pointless formalisms. But you need to understand terminology like ABC in geometry; when you get to tough problems you'll be using it. So, learn it with the easy problems.

My best math teachers never made us engage formalisations. What they would do is manipulate us into having an arguement, or construct an apparent contradiction. We then discussed the problem until we all agree (often without the Teacher talking). The inevitable result of this is that everyone learned formalisation: because it makes for really convinving arguements.

I hated math until I got to calculus. I never knew why, but I might as well quite this mathematician... They took an exciting topic that interested me as a youth and killed it with rote repetition.

The question I struggle with is... How do you really get by without math as a mandatory topic? How can you teach science? How can you teach someone to balance a checkbook? The current system is awful, but do we push the subject to electives as a result?

> The question I struggle with is... How do you really get by without math as a mandatory topic?

That's the wrong question, because "mandatory learning" is a contradiction in terms. It very clearly doesn't work. Beyond the most primitive pavlovian conditioning, you can't force people to learn if they don't want to. Intrinsic motivation is the dominating factor in how well a person can master an intellectually demanding task.

The right question is, how do you inspire people to want to learn?

True enough. And that's a tough question too. I waited from 4th grade to calculus to get inspired again. And then I think it was the material more than the teaching method.

This is one of my current problems in curriculum design. I agree fully with Lockhart, and I strongly feel that requiring math is a mistake. But at the same time, you do need significant math skills in order to appreciate science with any meaningful depth, and I feel that science is a requirement.

The proper next step, it seems to me, is to figure out what bits of science we really want to teach and then figure out how to explain math through that.

> How can you teach someone to balance a checkbook?

Outside of America, no one uses checkbooks anymore. There's no reason to use exact change for tipping, either: just estimate a percentage and then round up. Or splitting the bill: the servers virtually always have a calculator at their disposal.

The author of this article teaches a course on coursera that I can't recommend enough, he put so much effort into that course. https://www.coursera.org/course/maththink As you might have guessed its not really about 'math' in traditional sense.

This article just blew my mind. I feel little silly, especially since I majored in my math. I have never seen these types of ideas expressed so clearly and powerfully.

He is 100% right. We should not be teaching the formulaic rules of basic arithmetic until high school, and only then as a part of life skills class. It should be taught as the art form that it is.

As an erstwhile math major (I couldn't hack the honors basic algebra class - the difference between a euclidean domain and a principal ideal domain got too confusing; but I rocked proofs in analysis) I have to say that the author is confusing Mathematics (which is an art) and Arithmetic (which is a skill). Part of what make the opening farce absurd is that musical skills and painting are not terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death; whereas understanding sums and compounding processes ARE.

While SALVIATI is completely correct in his analysis of the situation, he offers no solution, and I identify with SIMPLICIO more.

Perhaps the problem is that in our schools we conflate arithmetic with mathematics. Surely, they are related, but perhaps they need to be delineated and the difference understood.

> whereas understanding sums and compounding processes ARE [terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death].

That's not actually true.

in an ideal world, I would agree with you. We do however live in the real world.

Please give one example applicable to a reasonable majority of human beings on the planet at this current time where "understanding sums and compounding processes" are a matter of life and death.

It seems probable that the subprime mortgage crisis was a contributing factor in a number of deaths, via suicide, stress-induced illness, or, with the help of alcohol, violent or vehicular incidents.


(1) The subprime mortgage crisis could not have been averted by a larger percentage of the population understanding sums and compounding processes.

(2) The advent of civilization was a contributing factor to everything that's happened in the last several thousand years, including the paper cut I just got.

you understand that interest-bearing loans are a compound process, right - and that an understanding thereof might be of relevance to someone entering into a loan agreement which they probably, if they really understood it, were not going to be able to repay...

But let's not only blame the borrowers of subprime loans, let's also ask whether the banks didn't entirely understand the risk models of the derivatives they were compiling out of subprime mortgages because many of their senior managers also didn't understand compounding processes, or, possibly, sums...

> an understanding thereof might be of relevance to someone entering into a loan agreement which they probably, if they really understood it, were not going to be able to repay...

Oh, it's relevant. It's sort of like how being able to load and fire a gun is relevant to the decision whether or not to commit suicide. Sure, it can modify one of the many, many details, but some people just make a noose and hang themselves.

Keep in mind, this is your argument: if more people in the world understood sums and compounding processes, this would have prevented the subprime mortgage crisis and thus the many deaths that were inspired by the resultant fallout. There is no possibility that anything else caused the crisis, and no possibility that anyone is at fault for these deaths other than parents and teachers.

This claim, if true, actually absolves the lenders that you talk about below, because they can be held responsible only for their own understanding of sums and compounding processes.

Amusingly, if you accept your argument, you can also make an interesting inverse version. The fact that lenders understood sums and compounding processes led to their employment at unscrupulous institutions which then mandated their sign-off on high-risk mortgages, which then caused the deaths of all those people.

Math, apparently, kills.

did you not read what I wrote? "or quality of life [or quality of death]". Nor did I proclaim that the life-or-death situation was applicable to a majority of people.

Your point is lost if it's not applicable to a significant majority. If it's only applicable to a minority, then that minority ought to be identified and specially trained, like we do with peanut allergies.

> The difference between a euclidean domain and a principal ideal domain

A Euclidean domain is an integral domain where the Euclidean algorithm works. For the Euclidean algorithm to work, you need to be able to divide two elements and produce a remainder that's smaller than the divisor.

So an ED requires a notion of "smallness" (the Euclidean norm) which interacts with division in a way that makes the Euclidean algorithm work (remainder is always smaller than divisor).

A principal ideal domain is an integral domain where every ideal is principal (can be generated by one element). It can be useful for you to know that certain situations cannot happen, e.g. in a PID you can say, "Let I be an ideal of D, then I = <g>..." and do something with the generating element g. It lets you pass from an ideal to a single generating element in a proof, which may be a useful capability. The PID concept is also part of a taxonomy, since Euclidean domains ⊆ principal ideal domains.

yes, I know what the technical definition is, but I never felt like I understood in a deep, mechanistic way, why all EDs are PIDs, beyond the constructive proof, and while I could prove that some wierd Q[some element] was a PID, but !ED, I never felt like I truly understood why. Mathematical taxonomy always wierded me out, maybe I took the Rutherford quote "all science is either physics or stamp collecting" too seriously.

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it.

I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?

http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755... (by guess who) looks like the best starting point since this resonated with you. (I've dipped into it but not seriously tackled it yet.)

It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.)

More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.

Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not.

I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.

Courant and Robbins is really great. I'd recommend checking it out after the Lockhart book (or together with it), as it's more textbookish; there's more danger of bouncing off.

During high school, at the end of the year everyone in my Calculus class had to make a math presentation (to go along with a paper). During lunch one day we stood with our posters to explain our project the the students that choice/were encouraged to attend. I explained my topic (re-ordering conditionally convergent series) to a group of students. They seemed to understand what I was saying, showed awe at some of the key insights/tricks, and even interupted me to excitadly finish the arguement. Then, when I was done, they pointed to the equations and asked me to explain them. My response was that those equations are exactly what I had just said, and I repeated the arguement while showing them where it is in the equations.

Richard P. Feynman

“Physics is like sex: sure, it may give some practical results, but that's not why we do it.”

We really need to teach people _how_ to teach induction, which is only done right when you put quotes around your Boolean statements; the "implies" symbol gets jumbled up with everything else otherwise, and not using it at all is passing up on a great tool. One can do simple proofs-by-induction without a single English word, completely symbolically, and have it be understood easily, if one uses quotes and correct LaTeX formatting (or good handwriting)

Induction doesn't just involve numbers and equality signs, it involves _statements_ with variables inside of them, and non-programmers need to be made well-aware of this (and taught Boolean logic early, PLEASE)

There was a time people were taught logic and geometry and math and history all at the same time. Also, philosophy.

The reductionist approach of our education is it's major failure.

A classic issue that mathematicians, philosophers, and even computer "scientists" have is the idea that they can somehow reason their way to the truth. Oh, he makes a good argument. But the greeks made great arguments about how the sun goes around the earth.

I could say that this or that argument is flawed. But really, the only valid arguments are data. His data is severely lacking, and many modern methods of teaching are much more supported by experimentation. Logic is a tool for finding logically consistent imaginary realities. Science is a tool for finding out about this reality.

I had to overcome my school-age brainwashing in order to enjoy math. One book that helped me was "Who is Fourier? A Mathematical Adventure" (http://www.amazon.com/ho-Fourier-Mathematical-Adventure-Edit...). I highly recommend it for those looking to find enjoyment in math.

Attempts to present mathematics as relevant to daily life inevitably appear forced and contrived: “You see kids, if you know algebra then you can figure out how old Maria is if we know thatshe istwo years older than twice her age seven years ago!” (Asif anyone would ever have access to that ridiculous kind of information, and not her age.)

I wince every time I see these examples in my niece's textbook. So ridiculous.

For music, the formal curriculum does involve memorizing songs and scales and whatnot. The problem is that the best way to start a subject is to fuck around with it (playing random songs you like on your guitar, drawing cool shapes and noticing patterns in them, etc), but our schools are ill equipped for such things, and enabling this style is difficult and incredibly expensive.

Reflects my experience. During school I had interest and good grades in all classes, including physics and geometry (those are separate from math in Brazil), the only exception being math (which covers algebra, polynomials, etc.). How a kid can get an A in physics and a D in math is beyond my comprehension, but I did. I only started appreciating math much later in life.

"The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.”"


Also s/math/science/ +1

In contrast, my 5- and 3-year-olds are enjoying DragonBox on the iPad, not knowing they're learning Algebra.

This is how I feel about doing most coding tutorials versus the MIT intro to CS through python, or project euler.

When coding is presented as: - here is a problem - how would you solve this problem? - here are some hints to get you started

it is incredibly fun for me. when it is presented as:

- follow along - look what you did!

it can be a bit dry.

Hi whiddershins, here's a little present for you then: https://github.com/darius/regexercise

(It's not quite finished and I'd love to get feedback.)

Please read the article with a critical eye, some of it is complete non-sense, for example:

CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned.

"Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox:


The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it:


This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox.

If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first!

One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.

The point he's making is against unnecessary rigorization of introductory calculus and I think you are getting a bit too hung up on the "function-based approach". I repeat - introductory calculus. There's lot of time and space to make things more rigorous in a class like Analysis.

When I help students with calculus most of them have no trouble with the ideas but the implementation that they are required to perform. I spend a lot of time getting them to understand the simple particulars. There are ways of teaching calculus that would dispense with some of the more rigorous aspects and make it a much more bearable experience for most.

Introductory calculus classes are hardly ever rigorous. The only formalism you see is functions and limits, and that's a very useful one, far from unnecessary, it might just not always be motivated appropriately by poor teachers who themselves have little understanding of its usefulness. Your interpretation is also very far from what he has actually written. I edited my parent comment to make what I mean more clear.

it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)

you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.

it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.

don't led your initial fascination with rigour (it can be addicting) get in the way of your intuition. rigour is necessary, but it comes after - sometimes to the chagrin of some. look at the teaching of modern algebra. pure abstraction and rigour, with complete detachment of all the wonderful ideas, and experiences that gave it rise.

up with triangles i say :)

it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)

Now any hard-working student of university calculus can without problems understand and reproduce the results of those treatises. I think this would not be possible without the modern systematic methods, including the epsilon-delta approach.

you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.

I mentioned fourier in another comments. Those weird functions were postulated in the discussion that arised somewhere in the same time period. I don't know what the author is referring to, because he is irritatingly vague, especially for a mathematician.

it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.

If you are familiar with the concept of a limit, you can notice that Zeno considers a limiting process of two related quantities, time and the difference in the position of Achilles and the Tortoise. Since in this limiting process time get arbitrarly close to some definite value (the meeting time of Achilles and Tortoise), but never gets equal to it, it stops being so surprising that the distance between them never reaches zero, though it gets arbitrarly close. The formalism clarifies what seems parodoxical when described in natural language. I find this quite convincing, and I never found a better explanation, altough I know philosophers still dispute this.

I never said rigour takes precedence over intuition. I just think it's the inherent difficulty of calculus that stops students from understanding it, and not the epsilon-delta stuff.

Hi, I hope you take my comment well. This need to avoid some theoretical logical contradiction far down the road is similar to explaining the perils of split infinitives before teaching a child to say "I want food."

99.9% of calculus students are there to expand their mind. They will never design a quantum mechanical reactor and think "Wow, I'm getting all these weird results, my calculus education must have held me back."

It's generally preferable to give students a first order approximation at first, the successively refine with additional terms down the road. The fact that most calculus students will have no idea I'm referring to a Taylor series explanatory strategy highlights the problems with an overly rigorous introduction. Math at this high level should primarily expand your thinking, and secondarily your storehouse of previously proven rigorous statements.

The author is criticizing calculus courses. The level of formalism in calculus courses is relatively low, and none of it has anything to do with quantum mechanics or contradictions far down. It is only enough formalism to give a definition of derivative and integral that actually makes sense, that's all.

All of this misses the author's point.

How many students make it through calculus without learning that it is the mathematics of motion at all? I'd wager that it's more than half.

Almost any calculus course goes over accelaration being the derivative of speed and so forth. It doesn't make students suddenly understand calculus.

This kind of approach simply doesn't make sense

It was poorly explained there, but essentially that style of reasoning does work: http://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bel...

And I at least find that approach easier and more useful. (I learned it from the Feynman lectures on physics, where he didn't axiomatize it; the above link does.)

I find the logically sound version of the infinitesimals approach to be much more difficult to understand than the approach using limits. For example, you have to introduce hyperreals to make it work (most common approach):


How would you explain the continued popularity of 103-year-old "Calculus Made Easy" then? The method used in the book might not be rigorous enough to solve all calculus problems but it jives well with a large number of people.

Most of them could get a better intuitive understanding of Calculus. (We should of course conduct a rigorous study comparing the effects of using different approaches to teach Introductory Calculus.) Then those who need to use the limits approach for other courses could use the intuition to learn it faster. In addition, they would understand Calculus from two different angles and could select the more suitable approach for each problem.


The book I linked to uses http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis (a quite different approach). I haven't studied nonstandard analysis.

"... earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" ..."

I don't think so.

99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that. I find the epsilon-delta construction (EDC) to be not only inordinately clunky but to provide no insight whatsoever insofar as motion is concerned. From a pedagogical standpoint EDC is roughly the equivalent of tossing a monkey wrench into the smoothly oiled and finely-working innards of the mathematics of motion. The student's mind grinds to a screeching halt as (s)he attempts to come to terms with a new construct that not even a mathematician's mother could love. I consider any praise of the epsilon-delta construction of calculus to be little more than turd-polishing and while it is unfortunately necessary, there is no need to call attention to it more than once. And once pointed out, best forgotten.

One of the most underrated advancements in mathematics is Descarte's introduction of Cartesian coordinates. The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.

99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that.

There were logical contradictions even in Newtons and Leibniz works, far before anyone considered continuous functions without derivatives etc., they were basically making decisions about when a given operation or transformation can be applied based on intuition alone and not any logical deductions, and it wasn't rare they arrived at incorrect conclusions. Also, Newton himself did epsilon-delta reasonings, he just did not notice their generality:


Again, you are confusing the work of Cauchy and Weierstrass and the epsilon-delta stuff with all the latter even more formal approaches to treat more complicated functions, which by the way were developed in response to Fourier examining heat transfer and trying to describe it mathematically (so it's actually rooted in physics).

People were saying the same things you are just saying about limits about the geometry of Euclid, in fact Newton at one time was of the opinion all the formal development of geometry is useless. He reconsidered after obtaining nonsense geometrical results a few times...

The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.

I wish you luck doing quantum mechanics with Newton-style calculus.

I took courses in graduate level QM and Newton-style calculus sufficed. The only unusual artifact that stands out in my memory was the Dirac delta function.

Perhaps physicists (e.g., your example of Newton above) somehow intuitively step over the "holes" in the underlying analytical frameworks that mathematicians fall into with regularly. Mathematicians are wont to build "manholes" to cover those holes and physicists have no desire to stop them, seeing that it is honest labour and keeps the mathematicians busy.

>"Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox (...) The ideas of Newton and Leibniz were hardly simple

All of which is besides the point (he makes), that they are simpler than the current formalism.

are we upvoting for length of comments now?

Do you have any particular criticism to make?

its copy pasta, use a link.

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