Introduction to Mathematical Thinking

Learn how to think the way mathematicians do - a powerful cognitive process developed over thousands of years.

About the Course

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.

About the Instructor(s)

Dr. Keith Devlin is a co-founder and Executive Director of Stanford University's H-STAR institute and a co-founder of the Stanford Media X research network. He is a World Economic Forum Fellow, a Fellow of the American Mathematical Society, and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 32 books and over 80 published research articles. He is a recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. In 2003, he was recognized by the California State Assembly for his "innovative work and longtime service in the field of mathematics and its relation to logic and linguistics." He is "the Math Guy" on National Public Radio.

Course Syllabus

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 7. Proofs 8. Proofs involving quantifiers 9. Elements of number theory10. Beginning real analysis

Recommended Background

High school mathematics.

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.

Course Format

The course starts on Monday March 4 and lasts for ten weeks, eight weeks of lectures followed by two weeks of monitored discussion and group work, including an open book final exam to be completed in the penultimate week and graded by a calibrated peer review system in the final week. There are also separate tutorial sessions where the instructor will demonstrate solutions to some of the assignment problems from the previous week.

For those familiar with the first version of this course, offered last September, this is the "stretched" version many students asked for. The overall content is the same, but the pace is slower, to allow for students whose other commitments made it difficult to keep up. (The original-paced version will be offered again next September.)

ORIENTATION: The course website will go live at 10:00 AM US-PST on Saturday March 2.

FAQ

• Will I get a certificate after completing this class?

  The course does not carry Stanford credit. If you finish the course, you will get a Certificate of Completion, and for those who do well on the coursework and the final exam the certificate will indicate Completion with Distinction.

• What are the assignments for this class?

  At the end of each lecture, you will be given an assignment (as a downloadable PDF file, released at the same time as the lecture) that is intended to guide understanding of what you have learned. Worked solutions to problems from the assignments will be described the following week in a video tutorial session given by the instructor.

  Using the worked solutions as guidance, together with input from other students, you will self-grade your assignment work for correctness. The assignments are for understanding and development, not for grade points. You are strongly encouraged to discuss your work with others before, during, and after the self-grading process. These assignments (and the self-grading) are the real heart of the course. The only way to learn how to think mathematically is to keep trying to do so, comparing your performance to that of an expert and discussing the issues with fellow students.

• Is there a final exam for this course?

  At the start of the penultimate week, you will be given an open-book exam to be completed by the end of the week. Completed exams will have to be uploaded as either images (or scanned PDFs) though if you are sufficiently familiar with TeX you have an option of keyboard entry on the site. The exam will be graded during the final week by a calibrated peer review system. 

• How is this course graded?

  There are two final grades: “completion” and “completion with distinction”. Completion requires viewing all the lectures and completing all the quizzes (both in-lecture “progress quizzes” and weekly “credit quizzes”). Distinction depends on the scores in the weekly credit quizzes and the result of the final exam.

Provided by:

University: Stanford University

Instructor(s): Keith Devlin