Introduction to Mathematical Thinking
持续时间: 10 weeks
课程评论: 2 个评论
NOTE: For the Spring 2013 session, the course website will go live at 10:00 AM US-PST on Saturday March 2, two days before the course begins, so you have time to familiarize yourself with the website structure, watch some short introductory videos, and look at some preliminary material.
The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.
Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
The primary audience is first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. They will need mathematical thinking to succeed in their major. Because mathematical thinking is a valuable life skill, however, anyone over the age of 17 could benefit from taking the course.
Instructor’s welcome and introduction
1. Introductory material
2. Analysis of language – the logical combinators
3. Analysis of language – implication
4. Analysis of language – equivalence
5. Analysis of language – quantifiers
6. Working with quantifiers
8. Proofs involving quantifiers
9. Elements of number theory
10. Beginning real analysis
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Disclaimer: the review is more or less subjective. I have absolutely no intention to make it objective. You have been informed.
这门课是高中数学到大学数学的一个过度。高中数学一般重计算不太注重证明，这门课讲了基本的逻辑，数学语言（两个 quantifier，there exists， for all）和证明的几个基本方法，比如证明充要条件要从两个方向证、证伪只需要举个反例，原命题不好证的时候可以证等价的逆否命题以及很常用的数学归纳法。课程讲了数论里一些基本定理，然后通过让你证一些看起来显然而不需要证明的证明题来训练你证明的技能和逻辑思考的能力，看起来显然的命题也是要证明才能说服人的，课程最后简略的讲了下数学分析里面实数的引入，但这部分讲的不完整。Keith Devlin 是个 old school 的讲师，上课只用纸和笔，也是属于比较热情的讲师，他每周都会录几个答疑的视频。这门比较适合大一的新生上，开得也比较频繁。
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