……结合在一起。在这些情况下，我似乎能迅速完全掌握该工具的运行机制和本质。我真希望教育也能建立在这一原则之上。有点像 Paul Lockhart 在《一个数学家的叹息》（A Mathematician's Lament）中所倡导的那样（https://www.maa.org/external_archive/devlin/LockhartsLament....）
…together. In these cases, I seem to quickly get a full grasp of the operation and essence of the tool. I wish education was based around this principle. A bit like what Paul Lockhart advocated for in "A Mathematician's Lament" (https://www.maa.org/external_archive/devlin/LockhartsLament....)
……无一例外全是非数学专业的人在教，而且是以一种极其非数学的方式在教——说实话，这种方式甚至与这门学科的核心本质背道而驰。请去 Google 搜来读一读 Paul Lockhart 写的名为《一个数学家的叹息》（A Mathematician's Lament）的文章，看看你是否真的理解什么是数学（还是说你学生时代的有害观念仍在继续蒙蔽你，让你看不到纯数学那种质朴的美）。我还要补充的是……
…invariably taught by non-mathematicians, in a highly non-mathematical way--in fact, in a way that is antithetical to the very core of the subject. Please google and read Paul Lockhart's essay titled "A Mathematician's Lament" to see if you really understand just what mathematics is (or if deleterious notions from your schooldays are continuing to blind you from the simple beauty of pure mathematics). I will add as well the…
……倒不是因为这些改革有多好，而是因为它们太微不足道、无关紧要了。“学校数学”早就成了对真正数学的荒唐拙劣模仿（免得我再重复论点，去看看 Paul Lockhart 的《一个数学家的叹息》吧）。而现在居然要我为换了一种稍微不同的长除法算法就大发雷霆？认真的吗？这些家长和老师对数学无知到连话都说不利索……
…Not because the reforms are any good, but because they're so tiny and insignificant. "School math" is already an absurd parody of the real thing (rather than repeat the argument, see Paul Lockhart "A Mathematician's Lament"). And I'm supposed to get up in arms about switching to a slightly different long-division algorithm? Seriously? These parents and teachers are so ignorant of mathematics that they can't even form coherent…
Cool, i also had an "aha" moment while reading: “It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two…
This is why math teaching pedagogy is important. I'm a fan of first principals and pattern finding for learning math (see Mathematician's Lament by Lockhart [0]). Most kids in the US are historically taught memorization tricks. You have a kid who can't recall if x^1 = 0 or 1 or if it was x^0 = 1 or 0. They can't _remember_ some fact like a needle…
我同意。我推荐 Paul Lockhart 写的《一个数学家的叹息》（A Mathematician’s Lament）。那是一篇关于数学教育的绝佳文章。整篇论文虽然很长，但光看前 5 页就非常棒。它涵盖了方方面面，从数学中不为人知的创造性……
I agree. I recommend "A Mathematician’s Lament" by Paul Lockhart. It's an excellent essay on math education. The entire paper is long but even the first 5 pages are great. It covers everything from the unacknowledged creativity that is involved in…
……结合在一起。在这些情况下，我似乎能迅速完全掌握该工具的运行机制和本质。我真希望教育也能建立在这一原则之上。有点像 Paul Lockhart 在《一个数学家的叹息》（A Mathematician's Lament）中所倡导的那样（https://www.maa.org/external_archive/devlin/LockhartsLament....）
…together. In these cases, I seem to quickly get a full grasp of the operation and essence of the tool. I wish education was based around this principle. A bit like what Paul Lockhart advocated for in "A Mathematician's Lament" (https://www.maa.org/external_archive/devlin/LockhartsLament....)
……无一例外全是非数学专业的人在教，而且是以一种极其非数学的方式在教——说实话，这种方式甚至与这门学科的核心本质背道而驰。请去 Google 搜来读一读 Paul Lockhart 写的名为《一个数学家的叹息》（A Mathematician's Lament）的文章，看看你是否真的理解什么是数学（还是说你学生时代的有害观念仍在继续蒙蔽你，让你看不到纯数学那种质朴的美）。我还要补充的是……
…invariably taught by non-mathematicians, in a highly non-mathematical way--in fact, in a way that is antithetical to the very core of the subject. Please google and read Paul Lockhart's essay titled "A Mathematician's Lament" to see if you really understand just what mathematics is (or if deleterious notions from your schooldays are continuing to blind you from the simple beauty of pure mathematics). I will add as well the…
……倒不是因为这些改革有多好，而是因为它们太微不足道、无关紧要了。“学校数学”早就成了对真正数学的荒唐拙劣模仿（免得我再重复论点，去看看 Paul Lockhart 的《一个数学家的叹息》吧）。而现在居然要我为换了一种稍微不同的长除法算法就大发雷霆？认真的吗？这些家长和老师对数学无知到连话都说不利索……
…Not because the reforms are any good, but because they're so tiny and insignificant. "School math" is already an absurd parody of the real thing (rather than repeat the argument, see Paul Lockhart "A Mathematician's Lament"). And I'm supposed to get up in arms about switching to a slightly different long-division algorithm? Seriously? These parents and teachers are so ignorant of mathematics that they can't even form coherent…
太酷了，我在读到这段时也有同感的大悟时刻：“直到研究生毕业后，我才（从 Lockhart 的书《一个数学家的叹息》中）学会把自然数看作可以排列成各种图案的石子，用这些图案来直观展现一个数的不同性质。例如，偶数就是可以排成两……的石子堆”
Cool, i also had an "aha" moment while reading: “It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two…
这就是为什么数学教学法至关重要。我非常推崇通过第一性原理和寻找规律来学习数学（参见 Lockhart 的《一个数学家的叹息》[0]）。从历史上看，美国的大多数孩子接受的都是死记硬背的套路教学。你常会遇到一个孩子想不起 x^1 是等于 0 还是 1，或者到底 x^0 是等于 1 还是 0。他们根本无法像大海捞针一样去_硬记_某个死知识点……
This is why math teaching pedagogy is important. I'm a fan of first principals and pattern finding for learning math (see Mathematician's Lament by Lockhart [0]). Most kids in the US are historically taught memorization tricks. You have a kid who can't recall if x^1 = 0 or 1 or if it was x^0 = 1 or 0. They can't _remember_ some fact like a needle…
我同意。我推荐 Paul Lockhart 写的《一个数学家的叹息》（A Mathematician’s Lament）。那是一篇关于数学教育的绝佳文章。整篇论文虽然很长，但光看前 5 页就非常棒。它涵盖了方方面面，从数学中不为人知的创造性……
I agree. I recommend "A Mathematician’s Lament" by Paul Lockhart. It's an excellent essay on math education. The entire paper is long but even the first 5 pages are great. It covers everything from the unacknowledged creativity that is involved in…