"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.
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.
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.
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.
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.
"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.
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.
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 [email protected]. 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."
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.
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.
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).
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.
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?
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. :)
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.
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.
"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.
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.
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?
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].
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?
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.
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.
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)
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.
[+] [-] nsphere|12 years ago|reply
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?
[+] [-] sampo|12 years ago|reply
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.
[+] [-] john_b|12 years ago|reply
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
[+] [-] sp332|12 years ago|reply
[+] [-] jackmaney|12 years ago|reply
Yes, if only because many (if not most) don't want access. Been there, done that, got the medications to prove it.
[+] [-] dagw|12 years ago|reply
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.
[+] [-] agumonkey|12 years ago|reply
[+] [-] realitygrill|12 years ago|reply
So there is this idea of different arts being more or less accessible to the general population.
[+] [-] softbuilder|12 years ago|reply
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.
[+] [-] gxs|12 years ago|reply
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.
[+] [-] arkj|12 years ago|reply
[+] [-] bridgeyman|12 years ago|reply
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 [email protected]. 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.
[+] [-] codyb|12 years ago|reply
"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
http://www.amazon.com/Number-Language-Science-Tobias-Dantzig...
[+] [-] westoncb|12 years ago|reply
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.
[+] [-] nsphere|12 years ago|reply
http://amzn.com/039306204X
[+] [-] Someone|12 years ago|reply
Lots of interesting stories, many of applied mathematics. And, to my surprise, available for free from archive.org.
[+] [-] Nate75Sanders|12 years ago|reply
[+] [-] prestonbriggs|12 years ago|reply
[+] [-] thetwiceler|12 years ago|reply
[+] [-] aet|12 years ago|reply
[+] [-] dsego|12 years ago|reply
[+] [-] dsego|12 years ago|reply
[+] [-] jobenjo|12 years ago|reply
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.
[+] [-] ctl|12 years ago|reply
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?
[+] [-] jfarmer|12 years ago|reply
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. :)
[+] [-] gizmo686|12 years ago|reply
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.
[+] [-] nullc|12 years ago|reply
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.
[+] [-] jfarmer|12 years ago|reply
One of the major themes is the relationship children have with mathematics and ways teachers can change it.
[+] [-] BgSpnnrs|12 years ago|reply
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.
[+] [-] rtfeldman|12 years ago|reply
It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.
[+] [-] gizmo686|12 years ago|reply
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.
[+] [-] adultSwim|12 years ago|reply
[+] [-] jackmaney|12 years ago|reply
"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."
[+] [-] jfarmer|12 years ago|reply
[+] [-] RogerL|12 years ago|reply
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.
[+] [-] mathattack|12 years ago|reply
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?
[+] [-] ef4|12 years ago|reply
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?
[+] [-] saraid216|12 years ago|reply
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.
[+] [-] dominotw|12 years ago|reply
[+] [-] dwaltrip|12 years ago|reply
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.
[+] [-] dnautics|12 years ago|reply
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.
[+] [-] saraid216|12 years ago|reply
That's not actually true.
[+] [-] lmartel|12 years ago|reply
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?
[+] [-] abecedarius|12 years ago|reply
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.
[+] [-] westoncb|12 years ago|reply
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.
[+] [-] gizmo686|12 years ago|reply
[+] [-] ArbitraryLimits|12 years ago|reply
“Physics is like sex: sure, it may give some practical results, but that's not why we do it.”
[+] [-] _bfhp|12 years ago|reply
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)
[+] [-] kazagistar|12 years ago|reply
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.
[+] [-] calebm|12 years ago|reply