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I actually disagree pretty strongly with this essay. I think Lockhart is advocating for an educational approach that would cater strongly (and exclusively) to

by jnbiche 12y ago

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.

jacobolus 12y ago
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 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.
- deleted 12y ago
  [deleted]
- jnbiche 12y ago
  Agreed, which is why I included the note (preceding your comment) adding that I agree with Lockhart that there's a problem in the way math is taught. Edit: Can you expand on your idea? I did just re-read the essay (for the 3rd time) and Lockhart seems pretty absolute in his belief that math should be taught as a playful, creative abstraction. And he seems to disdain practical applications of math. So how would you teach math in an inductive manner that maintains the level of abstraction and "playfulness" he's advocating for? By the way, love the book chapter! That's exactly how I'm trying to teach my kids. Not always easy, but they respond well to it. And it's entirely possible I'm reading something into the essay that isn't there. But I'm not yet convinced.
  - jacobolus 12y ago
    I’ll let Lockhart speak for himself: > Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up half their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you. > By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it’s actually much worse than this— at least it’s understood that there is an art of sculpture that I am being prevented from appreciating). > By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.
j2kun 12y ago
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).
- jnbiche 12y ago
  Thanks, I'll check that out. It's very possible that I got the wrong impression here, but he does explicitly talk about teaching deductively in this essay: SIMPLICIO: Then what should we do with young children in math class? SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.
  - j2kun 12y ago
    I think maybe he means something different from you here. I interpret deductive reasoning here as, "situations where you need to use logical arguments instead of being told what to do and repeating it with slightly different numbers."
keithflower 12y ago
I have no idea where you get the idea that Don Knuth opposes Lockhart's approach. I guess we'd not know without asking him, but having waded through much of TAOCP, Concrete Mathematics (a book filled to the brim with the kind of delightful discovery of pattern that Lockhart describes as optimal in learning math), Knuth's marvelously playful book "Selected Papers on Fun and Games"[2], and the novel he wrote about Conway's astonishing "Surreal Numbers"[3]...listening to him lecture on "importunate permutation" at the last local (SF) Joint Meeting of the American Mathematical Association, hearing about Knuth's thoughts on the mathematics of pipe organs, and even seeing the play/pattern-making that went into the entrance mosaic in his home [1]....I think you're way off base about what you think Knuth thinks about math education. Lockhart: "...if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education." Knuth [Preface to Concrete Mathematics]: "Some people think that mathematics is serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive." I make an annual pilgrimage to Palo Alto for Knuth's Christmas Tree lecture[4], which content continuously emphasizes exactly the kind of joy in experimenting, discovering, and learning real math that Lockhart is talking about in his paper. Everything I know about Don Knuth speaks to his amazing playfulness and joy in pattern finding and making - a delight in the music of math...and a denial of the value of making sure everyone's labeled their axes and memorized their circle of fifths. [1]: https://www.youtube.com/watch?v=v678Em6qyzk https://www.youtube.com/watch?v=v678Em6qyzk [2]: http://www-cs-faculty.stanford.edu/~uno/fg.html 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 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 http://www-cs-faculty.stanford.edu/~uno/musings.html

by jnbiche 12y ago

jacobolus 12y ago

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 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.

deleted 12y ago
[deleted]
jnbiche 12y ago
Agreed, which is why I included the note (preceding your comment) adding that I agree with Lockhart that there's a problem in the way math is taught. Edit: Can you expand on your idea? I did just re-read the essay (for the 3rd time) and Lockhart seems pretty absolute in his belief that math should be taught as a playful, creative abstraction. And he seems to disdain practical applications of math. So how would you teach math in an inductive manner that maintains the level of abstraction and "playfulness" he's advocating for? By the way, love the book chapter! That's exactly how I'm trying to teach my kids. Not always easy, but they respond well to it. And it's entirely possible I'm reading something into the essay that isn't there. But I'm not yet convinced.
- jacobolus 12y ago
  I’ll let Lockhart speak for himself: > Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up half their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you. > By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it’s actually much worse than this— at least it’s understood that there is an art of sculpture that I am being prevented from appreciating). > By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

j2kun 12y ago

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).

jnbiche 12y ago
Thanks, I'll check that out. It's very possible that I got the wrong impression here, but he does explicitly talk about teaching deductively in this essay: SIMPLICIO: Then what should we do with young children in math class? SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.
- j2kun 12y ago
  I think maybe he means something different from you here. I interpret deductive reasoning here as, "situations where you need to use logical arguments instead of being told what to do and repeating it with slightly different numbers."

keithflower 12y ago

I have no idea where you get the idea that Don Knuth opposes Lockhart's approach. I guess we'd not know without asking him, but having waded through much of TAOCP, Concrete Mathematics (a book filled to the brim with the kind of delightful discovery of pattern that Lockhart describes as optimal in learning math), Knuth's marvelously playful book "Selected Papers on Fun and Games"[2], and the novel he wrote about Conway's astonishing "Surreal Numbers"[3]...listening to him lecture on "importunate permutation" at the last local (SF) Joint Meeting of the American Mathematical Association, hearing about Knuth's thoughts on the mathematics of pipe organs, and even seeing the play/pattern-making that went into the entrance mosaic in his home [1]....I think you're way off base about what you think Knuth thinks about math education. Lockhart: "...if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education." Knuth [Preface to Concrete Mathematics]: "Some people think that mathematics is serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive." I make an annual pilgrimage to Palo Alto for Knuth's Christmas Tree lecture[4], which content continuously emphasizes exactly the kind of joy in experimenting, discovering, and learning real math that Lockhart is talking about in his paper. Everything I know about Don Knuth speaks to his amazing playfulness and joy in pattern finding and making - a delight in the music of math...and a denial of the value of making sure everyone's labeled their axes and memorized their circle of fifths. [1]: https://www.youtube.com/watch?v=v678Em6qyzk https://www.youtube.com/watch?v=v678Em6qyzk [2]: http://www-cs-faculty.stanford.edu/~uno/fg.html 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 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 http://www-cs-faculty.stanford.edu/~uno/musings.html