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flexibeasti just finished reading Lockhart's Lament [PDF], more formally titled "A Mathematician's Lament". It's an excellent essay critiquing the way mathematics is often taught - in the US in particular, although i feel it applies to maths education here in Australia, and probably in other Anglophone countries as well.
Lockhart invites us to imagine worlds in which music or painting is taught to children the same way we teach them maths, demonstrating how these subjects would probably seem as sterile and pointless as mathematics is to many (most?) people in our world. He then goes on to argue that we should not be teaching mathematics as simply an arcane tool requiring a variety of mysterious incantations in order to solve particular problems, but as a form of art that should be pursued for its own sake, regardless of its practical applications (thus echoing the sentiments of the famous essay by G.H. Hardy, "A Mathematician's Apology"). That is, just as we don't teach children about music primarily in terms of music notation, or in terms of being able to write music for ads and movies, we shouldn't talk to children about mathematics unnecessarily formally, or in terms of "real world" examples (e.g. being able to work out how much money we actually end up spending when we buy something on credit).
As someone who takes pleasure in certain forms of mathematics; as someone who feels that my society is poorer for its lack of popular interest in mathematics1; and as someone who has in recent times had the privilege of tutoring a friend's teenage daughter in maths, i found Lockhart's essay very thought-provoking. The teen in question often complains about how she feels that none of what she's having to learn is relevant, and i've usually hastened to provide real-world examples of what the maths she's learning is good for. However, irrelevance per se is not actually the issue, as she has no problem (for example) learning to sing, which might or might not end up being relevant to her career. The problem is that it's irrelevant and it's boring. It's boring because the primary emphasis is on learning what seems to be set after set of rules, rather than trying to get student engaged in the overall idea of mathematics; and attempts to make learning set after set of rules 'fun' is as lame to kids as is parents trying to relate to their teenage children by using the teen lingo du jour.
So Lockhart effectively wants the teaching of mathematics to be 'humanised', so to speak: to involve history, personalities, opinions, debates, imagination and intuition. Young people often have active imaginations; but as they interact with the world, and develop a sense of the internal lives of people other than themselves, they also tend to develop rules and intuitions about how the world, and various human interactions, 'work' - rules and intuitions which they then go on to apply in imaginary scenarios. Isn't it possible that maybe, by encouraging kids to play with various mathematical objects, they can, of their own accord, develop at least some rules and intuitions about mathematics, and thus develop a good foundation on which to build further mathematical knowledge at the time they actually need it?
1. And indeed, the active pride that some people take in not having a clue about mathematics, the history of mathematics or significant mathematicians. To paraphrase something i once read, if a person boasted about not knowing who these "Beethoven" or "Picasso" people were, they'd often be labelled a philistine; but not knowing about Euler, Gauss or Gödel is fine.
Lockhart invites us to imagine worlds in which music or painting is taught to children the same way we teach them maths, demonstrating how these subjects would probably seem as sterile and pointless as mathematics is to many (most?) people in our world. He then goes on to argue that we should not be teaching mathematics as simply an arcane tool requiring a variety of mysterious incantations in order to solve particular problems, but as a form of art that should be pursued for its own sake, regardless of its practical applications (thus echoing the sentiments of the famous essay by G.H. Hardy, "A Mathematician's Apology"). That is, just as we don't teach children about music primarily in terms of music notation, or in terms of being able to write music for ads and movies, we shouldn't talk to children about mathematics unnecessarily formally, or in terms of "real world" examples (e.g. being able to work out how much money we actually end up spending when we buy something on credit).
As someone who takes pleasure in certain forms of mathematics; as someone who feels that my society is poorer for its lack of popular interest in mathematics1; and as someone who has in recent times had the privilege of tutoring a friend's teenage daughter in maths, i found Lockhart's essay very thought-provoking. The teen in question often complains about how she feels that none of what she's having to learn is relevant, and i've usually hastened to provide real-world examples of what the maths she's learning is good for. However, irrelevance per se is not actually the issue, as she has no problem (for example) learning to sing, which might or might not end up being relevant to her career. The problem is that it's irrelevant and it's boring. It's boring because the primary emphasis is on learning what seems to be set after set of rules, rather than trying to get student engaged in the overall idea of mathematics; and attempts to make learning set after set of rules 'fun' is as lame to kids as is parents trying to relate to their teenage children by using the teen lingo du jour.
So Lockhart effectively wants the teaching of mathematics to be 'humanised', so to speak: to involve history, personalities, opinions, debates, imagination and intuition. Young people often have active imaginations; but as they interact with the world, and develop a sense of the internal lives of people other than themselves, they also tend to develop rules and intuitions about how the world, and various human interactions, 'work' - rules and intuitions which they then go on to apply in imaginary scenarios. Isn't it possible that maybe, by encouraging kids to play with various mathematical objects, they can, of their own accord, develop at least some rules and intuitions about mathematics, and thus develop a good foundation on which to build further mathematical knowledge at the time they actually need it?
1. And indeed, the active pride that some people take in not having a clue about mathematics, the history of mathematics or significant mathematicians. To paraphrase something i once read, if a person boasted about not knowing who these "Beethoven" or "Picasso" people were, they'd often be labelled a philistine; but not knowing about Euler, Gauss or Gödel is fine.
