"Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely.
One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."
For some reason, I stumbled upon Hubbard & Hubbard Vector Calculus a few days after first reading this essay and it stroke me as the opposite to this wrong math teaching.
I actually disagree pretty strongly with this essay. I think Lockhart is advocating for an educational approach that would cater strongly (and exclusively) to a particular learning style, specifically, those learners who thrive when going from abstract to specific (deductive learning).
For those of us who do best with inductive learning, the type of education he proposes would bring even greater misery to our grade school education. It was not until I took statistics and probability (for science and engineer majors) in college that I truly began to enjoy math once again. I was able to start with concrete ideas and applications, and then work my way back to the theory behind them.
I'm now reading "Concrete Mathematics" and really enjoying it. Knuth's ideas on math education are pretty diametrically opposed to those of Lockhart, as far as I can tell, and they give rise to something very close to my ideal learning environment for math.
Why not allow children to follow which ever of the two math paths that is best suited to them, instead of forcing concrete thinkers into an abstract world, and abstract thinkers into a concrete world?
Edit: I should probably add that I agree with Lockhart that there's a problem in the way math is taught, but I disagree with him on the solution.
Endless drilling, memorization, and focus on pushing numbers through “formulas” is not the same as “inductive learning”. It’s entirely possible to do lots of pattern matching and bottom-up “inductive” problem solving in a way consistent with Lockhart’s recommendations.
Here’s a great book chapter wherein a mathematician teaches some 6-year-olds in what might be called an “inductive” way: http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf (so much better than a standard first grade mathematics curriculum)
Edit: to clarify, I basically think you’re reading something into Lockhart’s essay that isn’t there.
I think you should check out the math book he wrote, "Measurement," and you'll find in it that he does not teach in the style you describe (abstract to specific).
I have no idea where you get the idea that Don Knuth opposes Lockhart's approach.
I guess we'd not know without asking him, but having waded through much of TAOCP, Concrete Mathematics (a book filled to the brim with the kind of delightful discovery of pattern that Lockhart describes as optimal in learning math), Knuth's marvelously playful book "Selected Papers on Fun and Games"[2], and the novel he wrote about Conway's astonishing "Surreal Numbers"[3]...listening to him lecture on "importunate permutation" at the last local (SF) Joint Meeting of the American Mathematical Association, hearing about Knuth's thoughts on the mathematics of pipe organs, and even seeing the play/pattern-making that went into the entrance mosaic in his home [1]....I think you're way off base about what you think Knuth thinks about math education.
Lockhart: "...if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education."
Knuth [Preface to Concrete Mathematics]: "Some people think that mathematics is serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive."
I make an annual pilgrimage to Palo Alto for Knuth's Christmas Tree lecture[4], which content continuously emphasizes exactly the kind of joy in experimenting, discovering, and learning real math that Lockhart is talking about in his paper.
Everything I know about Don Knuth speaks to his amazing playfulness and joy in pattern finding and making - a delight in the music of math...and a denial of the value of making sure everyone's labeled their axes and memorized their circle of fifths.
The preface states "I've never been able to see any boundary between scientific research and game-playing. ... The topics treated here were often inspired by patterns that are visually compelling, or by paradoxical truths that are logically compelling, or by combinations of numbers and/or symbols that fit together just right. These were papers that I couldn't not write.
I believe that the creation of a great puzzle or a great pattern is a scholarly achievement of great merit, an important contribution to world culture, even though the author of such a breakthrough is often an amateur who has no academic credentials. Therefore I'm proud to follow in the footsteps of the pioneers who have come up with significant new “mind-benders” as civilization developed.
Many years ago I wrote an essay that asked “Are toy problems useful?” [reprinted as Chapter 10 in Selected Papers on Computer Science] in which I discussed at some length my view that students are best served by teachers who present them with well-chosen recreational problems. And I've carried on in the same vein ever since, most recently on pages 7--9 of The Art of Computer Programming, Volume 4A, in a section entitled “Puzzles versus the real world.”
Surreal Numbers: "How two ex-students turned on to pure mathematics and found total happiness" - In 1973 during a week of relaxation in Oslo, Knuth wrote an introduction to Conway's method in the form of a novelette. ... I believe it is the only time a major mathematical discovery has been published first in a work of fiction. ... The book's primary aim, Knuth explains in a postscript, is not so much to teach Conway's theory as ``to teach how one might go about developing such a theory.'' He continues: ``Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded their false starts and frustrations as well as their good ideas. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story as I was actually doing the research myself.'' ... It is an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other ``real'' value does. The system is truly ``surreal.''
I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.
That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.
With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.
After nine years of teaching mathematics courses (one semester as an undergraduate, 4.5 years as a graduate student, and 4 years as an assistant professor) and navigating university politics, I'm convinced that this is, at its heart, a cultural issue.
There's a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).
Such a wonderful essay. It's very applicable to the teaching of computer science/software engineering as well. So much of the problem is the misunderstanding people have about the field. It's a creative, constructive discipline, and so much of the instruction is consumption, mimicry, and repetition.
Solving well defined problems is relatively easy. Our real problem is that real problems are not well defined.
I think I agree but isn't your last line a bit like saying "hammering in a nail is easy. The problem is that these screws aren't nails."
Chuck Close:
"I think while appropriation has produced some interesting work … for me, the most interesting thing is to back yourself into your own corner where no one else’s answers will fit. You will somehow have to come up with your own personal solutions to this problem that you have set for yourself because no one else’s answers are applicable." ...
"See, I think our whole society is much too problem-solving oriented. It is far more interesting to [participate in] ‘problem creation’ … You know, ask yourself an interesting enough question and your attempt to find a tailor-made solution to that question will push you to a place where, pretty soon, you’ll find yourself all by your lonesome — which I think is a more interesting place to be."
http://www.brainpickings.org/2012/12/27/chuck-close-on-creat...
FWIW one of our professors here at uwaterloo taught a first year abstract algebra / number theory class in a very Lockhart-esque way (Math 145; he even quoted Lockhart on one of the assignments). I learned a lot of math and enjoyed myself, but the main problem I observed was figuring out how to fairly grade students, and the fact that the homework took a lot more time than a class taught normally.
I have a lot of sympathy for his point of view. I loved Math growing up. High school drove the interest out of me, and I didn't get it back until senior Calculus, when I started doing well again. Then I learned to appreciate CS theory, economic theory, etc. Trying to figure out how to break the cycle for my kids: Stats for practical work, and math for curiosity.
As a Physicist, I feel obliged to mention that black holes were first hypothesised by Physicists (contrary to the essay), albeit through Mathematical enquiry.
I would assume that he means that mathematicians worked with singularities, before we knew, or hypothesised, that singularities arises in the real world.
Pierre Simon de Laplace was the first person to hypothesize black holes. I think he was a mathematician, although such distinctions were not so clear in the eighteenth century.
I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.
However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.
Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.
Geometry, if I recall correctly, was one of the exceptions in early math where you are allowed to veer off the path a bit. Is everything else algorithmic until college?
With that being said, I've always loved this essay. As of recently, I've viewed it as relevant to the recent argument that programming should be a requirement in American public schools, either as a tool in math and science classes or a free-standing course. This kind of mathematical reform might actually be a prerequisite for programming and computer science, given that it would develop mathematical maturity much more effectively than the current system does.
Yes, but Lockhart is arguing against any practical applications of math in the early years. I'm almost certain he would oppose any attempt to connect math to programming, for the reasons outlined in his essay (roughly, "math should be about imagination and playing, not applications"). In fact, I disagree with this essay, for reasons outlined in [1].
I think for most people, math is best learned in the context of some application that they care about (the last four words are very important). Few people appreciate the beauty of the abstract game itself.
For example, most people who play poker online quickly learn about expected value, probability, and variance.
Precisely. Poker is one of the most advanced games that legitimately teaches you what you need to know! The language used by poker players should be adapted wholesale.
As a former high-school math teacher and big proponent of Lockhart's philosophy, I highly recommend Phillips Exeter's math curriculum (http://www.exeter.edu/academics/72_6539.aspx).
It's entirely problem-based; each night the students are assigned a set of about ten problems, and they do the best they can with them. The next day, they meet in groups of twelve with a teacher (more like a facilitator) who leads them in a discussion of how they approached the problems and their solutions. Although the problems themselves are often quite practical, they are designed to illustrate and reveal over time the abstract concepts behind them, and they lead the students towards finding abstractions and generalities as they go.
It's free and publicly available; I recommend downloading them and reading through a few problem sets to see what I mean.
I made a little curated list for my friends who are starting to raise kids of their own and wanted to know about resources. I think most of what's out there is math education phrased as a game (to develop mechanical skills without being boring), and later translates into Lockhart-style exploratory "joy of math" stuff. The latter doesn't have much besides, Lockhart's own work, though, from what I can find.
I would look at actual math education research and research-based materials and lessons and tools, such as at sites like NCTM and MAA. Most educational research is not consistent with Lockhart's views.
Polya's "How to solve it" is pretty good in that respect. I asked this question on math.stackexchange.com a while ago but haven't received many answers yet
I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.
That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.
With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing buggy programs is easy.
I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.
That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.
With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.
Love the bit about the misconception that Mathematics is mainly about utility. I remember reading something about G.H. Hardy (which I can no longer find) in which he said he would get a little bit disappointed if he found that one of his results ended up finding a practical use.
He actually argues quite the opposite in [1] (see footnote, pg 33) - a good read in itself. The quote you're probably referring to is this:
"a science is said to be
_useful_ if its development tends to accentuate the existing inequalities in the distribution of
wealth, or more directly promotes the destruction of human life" (emphasis mine).
And Hardy's response (excerpt):
> It is sometimes
suggested that pure mathematicians glory in the uselessness of
their work, and make it a boast that it has no practical applications.
> I am sure that Gauss’s
saying (if indeed it be his) has been rather crudely misinterpreted.
If the theory of numbers could be employed for any practical and
obviously honourable purpose, if it could be turned directly to the
furtherance of human happiness or the relief of human suffering,
as physiology and even chemistry can, then surely neither Gauss
nor any other mathematician would have been so foolish as to
decry or regret such applications.
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.
However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.
Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.
[+] [-] mrcactu5|11 years ago|reply
"Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."
[+] [-] nextos|11 years ago|reply
For some reason, I stumbled upon Hubbard & Hubbard Vector Calculus a few days after first reading this essay and it stroke me as the opposite to this wrong math teaching.
[+] [-] Thrymr|11 years ago|reply
"Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school"
[+] [-] jnbiche|11 years ago|reply
For those of us who do best with inductive learning, the type of education he proposes would bring even greater misery to our grade school education. It was not until I took statistics and probability (for science and engineer majors) in college that I truly began to enjoy math once again. I was able to start with concrete ideas and applications, and then work my way back to the theory behind them.
I'm now reading "Concrete Mathematics" and really enjoying it. Knuth's ideas on math education are pretty diametrically opposed to those of Lockhart, as far as I can tell, and they give rise to something very close to my ideal learning environment for math.
Why not allow children to follow which ever of the two math paths that is best suited to them, instead of forcing concrete thinkers into an abstract world, and abstract thinkers into a concrete world?
Edit: I should probably add that I agree with Lockhart that there's a problem in the way math is taught, but I disagree with him on the solution.
[+] [-] jacobolus|11 years ago|reply
Here’s a great book chapter wherein a mathematician teaches some 6-year-olds in what might be called an “inductive” way: http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf (so much better than a standard first grade mathematics curriculum)
Edit: to clarify, I basically think you’re reading something into Lockhart’s essay that isn’t there.
[+] [-] j2kun|11 years ago|reply
[+] [-] keithflower|11 years ago|reply
I guess we'd not know without asking him, but having waded through much of TAOCP, Concrete Mathematics (a book filled to the brim with the kind of delightful discovery of pattern that Lockhart describes as optimal in learning math), Knuth's marvelously playful book "Selected Papers on Fun and Games"[2], and the novel he wrote about Conway's astonishing "Surreal Numbers"[3]...listening to him lecture on "importunate permutation" at the last local (SF) Joint Meeting of the American Mathematical Association, hearing about Knuth's thoughts on the mathematics of pipe organs, and even seeing the play/pattern-making that went into the entrance mosaic in his home [1]....I think you're way off base about what you think Knuth thinks about math education.
Lockhart: "...if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education."
Knuth [Preface to Concrete Mathematics]: "Some people think that mathematics is serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive."
I make an annual pilgrimage to Palo Alto for Knuth's Christmas Tree lecture[4], which content continuously emphasizes exactly the kind of joy in experimenting, discovering, and learning real math that Lockhart is talking about in his paper.
Everything I know about Don Knuth speaks to his amazing playfulness and joy in pattern finding and making - a delight in the music of math...and a denial of the value of making sure everyone's labeled their axes and memorized their circle of fifths.
[1]: https://www.youtube.com/watch?v=v678Em6qyzk
[2]: http://www-cs-faculty.stanford.edu/~uno/fg.html
The preface states "I've never been able to see any boundary between scientific research and game-playing. ... The topics treated here were often inspired by patterns that are visually compelling, or by paradoxical truths that are logically compelling, or by combinations of numbers and/or symbols that fit together just right. These were papers that I couldn't not write.
I believe that the creation of a great puzzle or a great pattern is a scholarly achievement of great merit, an important contribution to world culture, even though the author of such a breakthrough is often an amateur who has no academic credentials. Therefore I'm proud to follow in the footsteps of the pioneers who have come up with significant new “mind-benders” as civilization developed.
Many years ago I wrote an essay that asked “Are toy problems useful?” [reprinted as Chapter 10 in Selected Papers on Computer Science] in which I discussed at some length my view that students are best served by teachers who present them with well-chosen recreational problems. And I've carried on in the same vein ever since, most recently on pages 7--9 of The Art of Computer Programming, Volume 4A, in a section entitled “Puzzles versus the real world.”
[3]: http://www-cs-faculty.stanford.edu/~uno/sn.html
Surreal Numbers: "How two ex-students turned on to pure mathematics and found total happiness" - In 1973 during a week of relaxation in Oslo, Knuth wrote an introduction to Conway's method in the form of a novelette. ... I believe it is the only time a major mathematical discovery has been published first in a work of fiction. ... The book's primary aim, Knuth explains in a postscript, is not so much to teach Conway's theory as ``to teach how one might go about developing such a theory.'' He continues: ``Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded their false starts and frustrations as well as their good ideas. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story as I was actually doing the research myself.'' ... It is an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other ``real'' value does. The system is truly ``surreal.''
[4]: http://www-cs-faculty.stanford.edu/~uno/musings.html
[+] [-] drcomputer|11 years ago|reply
That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.
With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.
[+] [-] jackmaney|11 years ago|reply
(Note: What follows is US-centric.)
After nine years of teaching mathematics courses (one semester as an undergraduate, 4.5 years as a graduate student, and 4 years as an assistant professor) and navigating university politics, I'm convinced that this is, at its heart, a cultural issue.
There's a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).
[+] [-] ColinDabritz|11 years ago|reply
Solving well defined problems is relatively easy. Our real problem is that real problems are not well defined.
[+] [-] theoh|11 years ago|reply
Chuck Close: "I think while appropriation has produced some interesting work … for me, the most interesting thing is to back yourself into your own corner where no one else’s answers will fit. You will somehow have to come up with your own personal solutions to this problem that you have set for yourself because no one else’s answers are applicable." ... "See, I think our whole society is much too problem-solving oriented. It is far more interesting to [participate in] ‘problem creation’ … You know, ask yourself an interesting enough question and your attempt to find a tailor-made solution to that question will push you to a place where, pretty soon, you’ll find yourself all by your lonesome — which I think is a more interesting place to be." http://www.brainpickings.org/2012/12/27/chuck-close-on-creat...
[+] [-] zodiac|11 years ago|reply
[+] [-] mathattack|11 years ago|reply
[+] [-] 4ad|11 years ago|reply
[1]: http://en.wikipedia.org/wiki/Vladimir_Arnold
[+] [-] samwilkinson|11 years ago|reply
[+] [-] tokai|11 years ago|reply
[+] [-] ninguem2|11 years ago|reply
[+] [-] WhitneyLand|11 years ago|reply
However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.
Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.
Geometry, if I recall correctly, was one of the exceptions in early math where you are allowed to veer off the path a bit. Is everything else algorithmic until college?
[+] [-] MaxScheiber|11 years ago|reply
With that being said, I've always loved this essay. As of recently, I've viewed it as relevant to the recent argument that programming should be a requirement in American public schools, either as a tool in math and science classes or a free-standing course. This kind of mathematical reform might actually be a prerequisite for programming and computer science, given that it would develop mathematical maturity much more effectively than the current system does.
[+] [-] jnbiche|11 years ago|reply
1. https://news.ycombinator.com/item?id=8847132
[+] [-] blahblahblah3|11 years ago|reply
For example, most people who play poker online quickly learn about expected value, probability, and variance.
[+] [-] eruditely|11 years ago|reply
[+] [-] saraid216|11 years ago|reply
[+] [-] ashark|11 years ago|reply
I've seen this posted so many places so many times that surely there's a market for materials and support for it. Where are they?
[+] [-] GregBuchholz|11 years ago|reply
http://mathoverflow.net/questions/5074/are-there-elementary-...
...but I too would like to see what you are looking for, at an elementary school age level.
[+] [-] alexbasson|11 years ago|reply
It's entirely problem-based; each night the students are assigned a set of about ten problems, and they do the best they can with them. The next day, they meet in groups of twelve with a teacher (more like a facilitator) who leads them in a discussion of how they approached the problems and their solutions. Although the problems themselves are often quite practical, they are designed to illustrate and reveal over time the abstract concepts behind them, and they lead the students towards finding abstractions and generalities as they go.
It's free and publicly available; I recommend downloading them and reading through a few problem sets to see what I mean.
[+] [-] nextos|11 years ago|reply
At undergrad level, I love Axler but I haven't found a similar calculus book.
[+] [-] j2kun|11 years ago|reply
http://mcsforkids.herokuapp.com/
[+] [-] edtechdev|11 years ago|reply
As an example, here's a summary of research-based best practices for teaching Calculus: https://edtechdev.wordpress.com/2014/06/03/calculus/
[+] [-] murbard2|11 years ago|reply
http://math.stackexchange.com/questions/1040232/can-you-reco...
[+] [-] andystanton|11 years ago|reply
[+] [-] murbard2|11 years ago|reply
[+] [-] murbard2|11 years ago|reply
[+] [-] murbard2|11 years ago|reply
[+] [-] prestonbriggs|11 years ago|reply
[+] [-] mushishi|11 years ago|reply
[+] [-] unknown|11 years ago|reply
[deleted]
[+] [-] drcomputer|11 years ago|reply
That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.
With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing buggy programs is easy.
[+] [-] drcomputer|11 years ago|reply
That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me.
With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.
[+] [-] krazydad|11 years ago|reply
[+] [-] gshrikant|11 years ago|reply
"a science is said to be _useful_ if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life" (emphasis mine).
And Hardy's response (excerpt):
> It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications.
> I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications.
[1] http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%2...
[+] [-] Retra|11 years ago|reply
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
[+] [-] tujv|11 years ago|reply
Not sure how he'd feel that now that his own specialty of Number Theory has turned out to be very practical.
[+] [-] Retra|11 years ago|reply
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
[+] [-] Retra|11 years ago|reply
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
[+] [-] Retra|11 years ago|reply
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
[+] [-] Retra|11 years ago|reply
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
[+] [-] Retra|11 years ago|reply
It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking.
Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
[+] [-] aaronem|11 years ago|reply
[+] [-] WhitneyLand|11 years ago|reply
However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.
Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.