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Paul Lockhart said it better in his Lament - http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf This article and resea

Paul Lockhart said it better in his Lament - http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf This article and research is a non-starter. It doesn't address the core problem. Mathematics is a beautiful way of thinking about the abstract, yet it's taught as nothing more than a procedural tool. Teach it like music. Instill a love of mathematics as the art that it is. Reconstruct how we present mathematics to children and they will have a chance to love it, like they might love music or art or science. These micro-problems are just hairline cracks in the broken pieces of a shattered cup. Why patch the cracks if the cup will never hold water? We need to forge a new one. "I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers—the kind of thing a real mathematical education might provide." -- Paul Lockhart

impendia 14y ago
> Teach it like music. I remember my elementary school music classes with deep chagrin. The teacher had a deck of cards with photographs of musical instruments. "This is an oboe. It has a double reed." And so on, for fifteen minutes or so.
tokenadult 14y ago
Teach it like music. Any competent music teacher (I write as the husband of a piano teacher) can perform music with musical expression, and thus show students examples of the beauty of music. But the very best music teachers are also intimately familiar with all the isolated subskills that build into understanding a piece of music, and controlling the performer's muscle movements, and responding to the audience in a live setting to build a coherent, musically expressive performance. My wife teaches skills such as "music mapping," http://www.amazon.com/Mapping-Music-Learning-Teachers-Students/dp/0895793970/ http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen... proper hand position, http://www.amazon.com/Playing-Less-Hurt-Prevention-Musicians/dp/1423488466/ http://www.amazon.com/Playing-Less-Hurt-Prevention-Musicians... and how to tie those and many other skills together http://www.amazon.com/Art-Piano-Playing-Heinrich-Neuhaus/dp/1871082455/ http://www.amazon.com/Art-Piano-Playing-Heinrich-Neuhaus/dp/... as part of a comprehensive process of teaching making music. The basic problem with mathematics education at the elementary school level in the United States (see my previous reply to this thread http://news.ycombinator.com/item?id=4034248 http://news.ycombinator.com/item?id=4034248 generally commenting on the submitted article) is that elementary school mathematics teachers can do NONE of the comparable things with mathematics that a good music teacher can do with music. They cannot isolate and focus on useful techniques, they cannot put on an example performance of solving an interesting, challenging problem, and they cannot make connections between their (poor) understanding of the problems found in elementary mathematics and their students' (often better, but different) understanding of the same problems. In the United States, people mostly seek music instruction for young learners from the private enterprise system. My wife gains most of her new clients, who are crammed into her busy teaching schedule as previous clients graduate from secondary school and go off to university, from friends of current clients who are happy with her work. By contrast, an elementary school teacher in a typical government-run school in the United States teaches on a take-it-or-leave-it basis, with attendance being compulsory in default of a government-approved alternative, and funding guaranteed to the school, and thus employment guaranteed to the teacher, whether the learners learn mathematics or not. Systemic change is necessary to get mathematics taught as music is taught to elementary-age pupils in the United States. For an eye-opening look at how elementary mathematics teachers could be prepared, and how that would help learners, see Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma. http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... That book is a very enjoyable--but rather shocking--read, full of information about how to teach mathematics for "Profound Understanding of Fundamental Mathematics."
Daniel_Newby 14y ago
> Mathematics is a beautiful way of thinking about the abstract, ... Most people do not have the intelligence for much abstract thought. Look at how much trouble CS students have with recursion, and they are in the top few percent of intelligence. > Teach it like music. The majority of people do have enough tone perception, coordination, and muscle memory to make music. It does not compare to math.
- simonbrown 14y ago
  they are in the top few percent of intelligence Citation needed
alexqgb 14y ago
From Lockhert's Lament, in which the author debates a fictional idiot (this should be required reading for anyone who has anything to do with curriculum design). "What about geometry? Don’t students prove things there? Isn’t High School Geometry a perfect example of what you want math classes to be?" His reply "There is nothing quite so vexing to the author of a scathing indictment as having the primary target of his venom offered up in his support. And never was a wolf in sheep’s clothing as insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it is school’s attempt to introduce students to the art of argument that makes it so very dangerous. Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way. The mechanism behind this is subtle and devious. The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.” As Lockhart notes "Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions."

impendia 14y ago

> Teach it like music. I remember my elementary school music classes with deep chagrin. The teacher had a deck of cards with photographs of musical instruments. "This is an oboe. It has a double reed." And so on, for fifteen minutes or so.

tokenadult 14y ago

Teach it like music. Any competent music teacher (I write as the husband of a piano teacher) can perform music with musical expression, and thus show students examples of the beauty of music. But the very best music teachers are also intimately familiar with all the isolated subskills that build into understanding a piece of music, and controlling the performer's muscle movements, and responding to the audience in a live setting to build a coherent, musically expressive performance. My wife teaches skills such as "music mapping," http://www.amazon.com/Mapping-Music-Learning-Teachers-Students/dp/0895793970/ http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen... proper hand position, http://www.amazon.com/Playing-Less-Hurt-Prevention-Musicians/dp/1423488466/ http://www.amazon.com/Playing-Less-Hurt-Prevention-Musicians... and how to tie those and many other skills together http://www.amazon.com/Art-Piano-Playing-Heinrich-Neuhaus/dp/1871082455/ http://www.amazon.com/Art-Piano-Playing-Heinrich-Neuhaus/dp/... as part of a comprehensive process of teaching making music. The basic problem with mathematics education at the elementary school level in the United States (see my previous reply to this thread http://news.ycombinator.com/item?id=4034248 http://news.ycombinator.com/item?id=4034248 generally commenting on the submitted article) is that elementary school mathematics teachers can do NONE of the comparable things with mathematics that a good music teacher can do with music. They cannot isolate and focus on useful techniques, they cannot put on an example performance of solving an interesting, challenging problem, and they cannot make connections between their (poor) understanding of the problems found in elementary mathematics and their students' (often better, but different) understanding of the same problems. In the United States, people mostly seek music instruction for young learners from the private enterprise system. My wife gains most of her new clients, who are crammed into her busy teaching schedule as previous clients graduate from secondary school and go off to university, from friends of current clients who are happy with her work. By contrast, an elementary school teacher in a typical government-run school in the United States teaches on a take-it-or-leave-it basis, with attendance being compulsory in default of a government-approved alternative, and funding guaranteed to the school, and thus employment guaranteed to the teacher, whether the learners learn mathematics or not. Systemic change is necessary to get mathematics taught as music is taught to elementary-age pupils in the United States. For an eye-opening look at how elementary mathematics teachers could be prepared, and how that would help learners, see Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma. http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... That book is a very enjoyable--but rather shocking--read, full of information about how to teach mathematics for "Profound Understanding of Fundamental Mathematics."

Daniel_Newby 14y ago

> Mathematics is a beautiful way of thinking about the abstract, ... Most people do not have the intelligence for much abstract thought. Look at how much trouble CS students have with recursion, and they are in the top few percent of intelligence. > Teach it like music. The majority of people do have enough tone perception, coordination, and muscle memory to make music. It does not compare to math.

simonbrown 14y ago
they are in the top few percent of intelligence Citation needed

alexqgb 14y ago

From Lockhert's Lament, in which the author debates a fictional idiot (this should be required reading for anyone who has anything to do with curriculum design). "What about geometry? Don’t students prove things there? Isn’t High School Geometry a perfect example of what you want math classes to be?" His reply "There is nothing quite so vexing to the author of a scathing indictment as having the primary target of his venom offered up in his support. And never was a wolf in sheep’s clothing as insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it is school’s attempt to introduce students to the art of argument that makes it so very dangerous. Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way. The mechanism behind this is subtle and devious. The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.” As Lockhart notes "Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions."