小学数学的掌握和教学
Knowing and teaching elementary mathematics
Liping Ma · 初版 1999 年
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针对美国教师的研究往往表明他们在数学学科知识方面存在不足。然而，这些研究很少列举教师为了支持教学（尤其是近来数学教育改革所要求的教学）所需的具体知识示例。Knowing and Teaching Elementary Mathematics 阐述了小学教师成长为优秀数学教师所需知识的本质与发展路径，并探讨了为何此类知识在教师正规教育受训较少的中国反而比在美国更为普遍。
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……我家里有很多针对进阶学习者的中文小学数学教材，这些教材都会特意对所有数学概念给出多种表征方式。马立平（Liping Ma）写的那本神作《Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States》（小学数学的知识与教学）http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... 用非常扎实的例子证明了究竟……
…Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book 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-Mathematic... demonstrates with cogent examples just…
……总的来说，任何 Hacker News 读者只要去读读马立平写的《Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States》[1]，就能深入了解美国（或者更广泛地说，整个英语世界）在小学数学教学方面究竟缺失了什么。基础数学里有很多东西绝非易事，如果你追求的是深层理解……
…In general, any reader of Hacker News could learn a lot about what is missing in the United States (or, more generally, in the English-speaking world) in elementary mathematics instruction by reading Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.[1] There is a lot to basic mathematics that is far from easy, if deep understanding is what you are looking…
……运算以及它们为什么管用，因为我们都是以此为基石建构起来的。”关于理解算术——真正地理解它——以及这如何与算术之外的学习产生关联的更详尽内容，可以在马立平的《Knowing and Teaching Elementary Mathematics》一书中找到 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... （两位研究数学教育的数学家对此书评价极高）http://www.ams.org/notices/199908/rev-howe.pdf http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf……
…operations and why they work because WE BUILD FROM THERE." The longer story about understanding arithmetic--REALLY understanding it--and how that relates to learning beyond arithmetic can be found in the book Knowing and Teaching Elementary Mathematics by Liping Ma http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... (well reviewed by two mathematicians who study mathematics education) http://www.ams.org/notices/199908/rev-howe.pdf http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf…
……在开始微积分课程之前具备“对基础数学的深刻理解（PUFM）”。促使我进一步思考学生如何能把数学学得更好的一个链接，是 Richard Askey 对马立平著作《Knowing and Teaching Elementary Mathematics》的书评：http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf。数学家 Roger Howe 对这本佳作的另一篇书评 http://www.ams.org/notices/199908/rev-howe.pdf 同样非常发人深省……
…of fundamental mathematics (PUFM)" before starting the calculus course. A link that furthered my process of pondering how students might learn mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma. http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf Another review of that excellent book by mathematician Roger Howe http://www.ams.org/notices/199908/rev-howe.pdf is also food…
……去教。分数只要教得好，其实并没有那么难于上青天，但在美国的各级数学课堂上却极少有人能教好。如果你想了解更多这方面的背景，可以看看马立平写的《Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States》http://www.ams.org/notices/199908/rev-howe.pdf https://webstorage.worcester.edu/sites/rbisk/web/marlboro%20...……
…teach. Fractions are not insuperably hard if they are well taught, but they are rarely well taught at any level of mathematics classes in the United States. If you'd like more background on this topic, see Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma http://www.ams.org/notices/199908/rev-howe.pdf https://webstorage.worcester.edu/sites/rbisk/web/marlboro%20...…
……在中国长大。从本文所指的意义来看，中国的数学教学根本不是死记硬背。关于中美数学教学这一话题有一本好书，是马立平写的《Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States》[1]。在她这项由美国联邦政府资助的实地研究中，明确指出了小学……
…up in China. Mathematics instruction in China is not at all about memorization, in the sense used in this article. A good book on the topic of mathematics instruction in the United States and China is Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.[1] Her field study, with sponsorship by the United States federal government, made clear that elementary…