Introduction to Mathematical Thinking
HSTARY0001
Description
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 insidethebox. In contrast, a key feature of mathematical thinking is thinking outsidethebox – a valuable ability in today's world. This course helps to develop that crucial way of thinking.
The course is offered in two versions. The eightweeklong Basic Course is designed for people who want to develop or improve mathematicsbased, analytic thinking for professional or general life purposes. The tenweeklong Extended Course is aimed primarily at firstyear students at college or university who are thinking of majoring in mathematics or a mathematicallydependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.
Subtitles for all video lectures available in: Portuguese (provided by The Lemann Foundation), English
Course Syllabus
Instructor's welcome and introduction
 Introductory material
 Analysis of language – the logical combinators
 Analysis of language – implication
 Analysis of language – equivalence
 Analysis of language – quantifiers
 Working with quantifiers
 Proofs
 Proofs involving quantifiers
 Elements of number theory
 Beginning real analysis
Recommended Background
High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.
A good way to assess if your basic school background is adequate (even if currently rusty) is to glance at the topics in the book Adding It Up: Helping Children Learn Mathematics (free download), published by the US National Academies Press in 2001. Though aimed at K8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the TwentyFirst Century (which was the National Academies' aim in producing it).
Instructor
Dr Keith Devlin, Cofounder and Executive Director HSTAR Institute
Delivery Option: 
Online

Online Course  $0.00 
Notes
Course Format
The Basic Course lasts for ten weeks, comprising ten lectures, each with a problembased work assignment (ungraded, designed for group work), a weekly Problem Set (machine graded), and weekly tutorials in which the instructor will go over some of the assignment and Problem Set questions from the previous week.
The Extended Course consists of the Basic Course followed by a more intense two weeks exercise called Test Flight. Whereas the focus in the Basic Course is the development of mathematicallybased thinking skills for everyday life, the focus in Test Flight is on applying those skills to mathematics itself.
Suggested Readings
There is one reading assignment at the start, providing some motivational background. There is a supplemental reading unit describing elementary set theory for students who are not familiar with the material. There is a course textbook, Introduction to Mathematical Thinking , by Keith Devlin, available at low cost (under $10) from Amazon, in hard copy and Kindle versions, but it is not required in order to complete the course.
For general background on mathematics and its role in the modern world, take a look at the five week survey course on mathematics ("Mathematics: Making the Invisible Visible") Devlin gave at Stanford in fall 2012, available for free download from iTunes University (Stanford) , and on YouTube (1 , 2 , 3 , 4 ,5 ), particularly the first halves of lectures 1 and 4.
Coursera Course Certificate Available for a Fee
You can earn a Course Certificate for this Coursera course. A Course Certificate is proof that you completed and passed the course.
In order to get a Course Certificate, you must:
 Complete ID Verification , which includes submitting a photo and verified ID
 Pass all required assignments in the course
 Pay the Course Certificate fee , or apply and be approved for Coursera Financial Aid