学习美国数学
为什么要学习美国数学?
在很多人的印象中,美国的中小学数学很简单,中国学生可以"秒杀"美国学生数学。 其实美国的中小学教育并不像传说中的那么简单,美国有很多世界一流的大学, 众多诺贝尔奖、菲尔茨奖(数学最高奖)获得者,以及众多科技及人文的创新。 诺贝尔奖没有数学奖,数学界的最高奖是菲尔茨奖(Fields Award), 菲尔茨奖每四年颁发一次,颁奖给全世界最优秀的40岁以下的数学家。我们来看看过去30年来获得菲尔茨奖的美国数学家的情况。

获奖年份 获奖者 获奖者获奖时所在的大学、研究院
2018年 Akshay Venkatesh 美国斯坦福大学 (Stanford University)
2014年 Maryam Mirzakhani 美国斯坦福大学 (Stanford University)
2014年 Manjul Bhargava 美国普林斯顿大学 (Princeton Univeristy)
2010年 Ngô Bảo Châu 美国高等研究院 (Institute for Advanced Study)
2010年 Elon Lindenstrauss 美国普林斯顿大学 (Princeton Univeristy)
2006年 Terence Tao 美国加州大学 (University of California, Los Angeles)
2006年 Andrei Okounkov 美国普林斯顿大学 (Princeton Univeristy)
2002年 Vladimir Voevodsky 美国高等研究院 (Institute for Advanced Study)
1998年 Maxim Kontsevich 美国 Rutgers University
1998年 Curtis T. McMullen 美国哈佛大学 (Harvard University)
1998年 Richard Borcherds 美国加州大学 (University of California, Berkeley)
1994年 Efim Zelmanov 美国芝加哥大学 (University of Chicago)
1990年 Edward Witten 美国高等研究院 (Institute for Advanced Study)
1990年 Vaughan Jones 美国加州大学 (University of California, Berkeley)
注: 以上数据来自维基百科

以上数据可以从一个侧面反映美国数学研究的实力。美国的中小学数学其实一点也不简单,作为参考,以下是美国的一套数学教材的目录, 可以看到,6年级的教材有600多页,7年级的教材也是600多页, 几何的教材有900多页,代数的教材有800多页。 (注: 美国没有全国统编教材,各个学区制定自己的教学计划,以下这套教材是美国加州使用的一套教材。)

Mathematics Course 1: Numbers to Algebra (Grade 6)
Mathematics Course 2: Pre-Algebra (Grade 7)
Mathematics Geometry
Mathematics Algebra 1

这是美国小学3年级的一次数学作业,这个作业是一个小project, 共30页,要求在1个月内完成。 注意这是一道题目,这一道题目共30页, 所以这是个小project、大题目。 这个project培养学生如何完整地、从头到尾地、系统地运营一个项目,培养学生的规划能力,同时为学生提供了自我设计的空间,可以让学生展示自己的个性。 最后的结果不是最重要的,重要的是学生在整个过程中各个阶段的体验和收获。 可以看到美国中小学数学的阅读量很大,需要较强的阅读理解能力。 美国学生从小就做各种各样的project, 比如这个作业就是一个project, 30页纸就是一道题目,这是一道大题目,1个月之内完成。 所以美国学生长大后科研能力相对比较强,从上面菲尔茨奖(Fields Award)的获奖情况也可以看出。


美国的中小学数学教学重视启发、探索,重视应用、轻松、有趣、与实际生活紧密相关, 启发学生如何 thinking mathematically,培养学生 creative thinking, critical thinking, and problem solving skills, 即探索如何用数学解决实际生活中的问题,培养学生创造性思维、批判性思维、和实际解决问题的能力。
What is “Thinking Mathematically?”
  • Many people associate mathematics with tedious computation, meaningless algebraic procedures, and intimidating sets of equations.
  • The truth is that mathematics is the most powerful means we have of exploring our world and describing how it works.
  • To be mathematical literally means to be inquisitive, open-minded, and interested in a lifetime of pursuing knowledge.
Creative Thinking, Critical Thinking, and Problem Solving Skills:
  • Creative thinking that focuses on the skills of fluency, flexibility, elaboration, and originality coupled with the affective characteristics associated with creativity such as curiosity and risk-taking.
  • Critical thinking as the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and evaluating information to reach an answer or conclusion; a reasonable, reflective thinking focused on deciding what to believe or do.
  • Inductive and deductive reasoning skills such as analysis, evaluation, and predicting.
  • Problem-solving skills, using a math heuristic to outline the process.
Logic and Analytical Skills, Logic Relations, Inductive Reasoning, and Deductive Reasoning:
Learning is an interactive process. The goal of education should be to provide the settings and opportunities for the student to become actively involved in the learning process. In a general sense, learning and intellectual development are not passive, sporadic activities, but dynamic, ongoing processes. The ability to acquire knowledge is built upon the capacity to organize and structure a concept’s key components. Furthermore, this process is based upon the development of logical relationships. Thus, it is necessary, first of all, to identify those logical relationships that serve as the foundation of intellectual development, then provide the settings within an academic discipline that will enable the student to acquire proficiency with these relationships.

The application of logic and analytical skills to numerical and spatial concepts is introduced in the questions through activities that are designed to focus student attention on the tasks of examining, discussing, and describing numerical and geometric relationships in terms of logical relations.
These logic relations include:
  • analyzing similarities and differences
  • recognizing sequences and patterns
  • using numerical and spatial concepts and functions
  • applying the concept of analogies to relations and functions

In addition, many of the activities stress using inductive reasoning to extend patterns, make predictions based upon available data, and formulate inferences. The role of deductive reasoning is introduced to students through the use of logical connectives, counterexamples, and the application of the process of elimination to derive solutions to numerical and geometric problems. Students should realize that mathematics does not necessarily restrict itself to a single simple solution or a single strategy to arrive at a solution. Such analysis and verbalizing results in students developing an appreciation that mathematics is indeed a logical discipline with recognizable patterns, order, and structure.
“Tell me, I will forget. Teach me, I will remember. Involve me, I will learn.” - Benjamin Franklin (富兰克林)




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