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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/mathematical-thinking
课程评论：没有评论
课程名称：数学思维导论 课程概述：本课程旨在帮助学员学习数学家独特的思考方式，这是一种经历了数千年发展的强大认知过程。数学思维与常规的学校数学学习截然不同，学校数学通常注重学习解决高度规范化问题的程序，而职业数学家则通过特定的思维方式解决来自日常生活、科学或数学内部的真实问题。成功的学校数学学习往往是在“盒子”内思考，而数学思维的关键特征则是“盒子”外的思考，这在当今社会中是一项极为重要的能力。 课程大纲： 第一周：课程介绍，了解课程内容及结构，并进行背景阅读。注意不要急于跳过，建议组建学习小组进行讨论。 第二周：继续学习数学语言的形式化部分，掌握课程基本结构，做好作业和视频辅导。 第三周：继续分析数学语言，关注如何将日常生活中的概念形式化，建议找到学习伙伴以便更好理解材料。 第四周：完成语言分析，为19世纪数学家提供的处理无穷大的语言工具打下基础。 第五周：首次接触数学证明，现代数学的基础。 第六周：进一步学习数学证明的内容。 第七周：探索数论，古希腊数学的分支，对数学、物理及现代技术有广泛影响。 第八周：学习实分析的基础，深入研究实数系统，为微积分奠定严谨基础。 第九与第十周：测试飞行阶段，体验评估他人的数学论证，分为三个阶段，需顺序进行，强调过程重要性。 本课程不仅提供数学思维的理论指导，还通过具体的实例和练习帮助学员提高处理复杂问题的能力。
Name:Week 1
Description:START with the Welcome lecture. It explains what this course is about. (It comes with a short Background Reading assignment, to read before you start the course, and a Reading Supplement on Set Theory for use later in the course, both in downloadable PDF format.) This initial orientation lecture is important, since this course is probably not like any math course you have taken before – even if in places it might look like one! AFTER THAT, Lecture 1 prepares the groundwork for the course; then in Lecture 2 we dive into the first topic. This may all look like easy stuff, but tens of thousands of former students found they had trouble later by skipping through Week 1 too quickly! Be warned. If possible, form or join a study group and discuss everything with them. BY THE WAY, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.
Name:Week 2
Description:In Week 2 we continue our discussion of formalized parts of language for use in mathematics. By now you should have familiarized yourself with the basic structure of the course: 1. Watch the first lecture and answer the in-lecture quizzes; tackle each of the problems in the associated Assignment sheet; THEN watch the tutorial video for the Assignment sheet. 2. REPEAT sequence for the second lecture. 3. THEN do the Problem Set, after which you can view the Problem Set tutorial. REMEMBER, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.
Name:Week 3
Description:This week we continue our analysis of language for use in mathematics. Remember, while the parts of language we are focusing have particular importance in mathematics, our main interest is in the analytic process itself: How do we formalize concepts from everyday life? Because the topics become more challenging, starting this week we have just one basic lecture cycle (Lecture -> Assignment -> Tutorial -> Problem Set -> Tutorial) each week. If you have not yet found one or more people to work with, please try to do so. It is so easy to misunderstand this material.
Name:Week 4
Description:This week we complete our analysis of language, putting into place the linguistic apparatus that enabled, mathematicians in the 19th Century to develop a formal mathematical treatment of infinity, thereby finally putting Calculus onto a firm footing, three hundred years after its invention. (You do not need to know calculus for this course.) It is all about being precise and unambiguous. (But only where it counts. We are trying to extend our fruitfully-flexible human language and reasoning, not replace them with a rule-based straightjacket!)
Name:Week 5
Description:This week we take our first look at mathematical proofs, the bedrock of modern mathematics.
Name:Week 6
Description:This week we complete our brief look at mathematical proofs
Name:Week 7
Description:The topic this week is the branch of mathematics known as Number Theory. Number Theory, which goes back to the Ancient Greek mathematicians, is a hugely important subject within mathematics, having ramifications throughout mathematics, in physics, and in some of today's most important technologies. In this course, however, we consider only some very elementary parts of the subject, using them primarily to illustrate mathematical thinking.
Name:Week 8
Description:In this final week of instruction, we look at the beginnings of the important subject known as Real Analysis, where we closely examine the real number system and develop a rigorous foundation for calculus. This is where we really benefit from our earlier analysis of language. University math majors generally regard Real Analysis as extremely difficult, but most of the problems they encounter in the early days stem from not having made a prior study of language use, as we have here.
Name:Weeks 9 & 10: Test Flight
Description:Test Flight provides an opportunity to experience an important aspect of "being a mathematician": evaluating real mathematical arguments produced by others. There are three stages. It is important to do them in order, and to not miss any steps. STAGE 1: You complete the Test Flight Problem Set (available as a downloadable PDF with the introductory video), entering your solutions in the Peer Evaluation module. STAGE 2: You complete three Evaluation Exercises, where you evaluate solutions to the Problem Set specially designed to highlight different kinds of errors. The format is just like the weekly Problem Sets, with machine grading. You should view the Tutorial video for each Exercise after you submit your solutions, but BEFORE you start the next Exercise. STAGE 3: You evaluate three Problem Set solutions submitted by other students. (This process is anonymous.) This final stage takes place in the Peer Evaluation module. After you are done peer reviewing, you may want to evaluate your own solution. It can be very informative to see how you rate your own attempt after looking at the work of others.
Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of 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.
