About this Course

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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.

Flexible deadlines

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Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Intermediate Level

Approx. 39 hours to complete

English

Skills you will gain

Number Theory
Real Analysis
Mathematical Logic
Language

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Intermediate Level

Approx. 39 hours to complete

English

Instructor

Instructor rating

4.81/5 (357 Ratings)

Dr. Keith Devlin

Co-founder and Executive Director

H-STAR institute

427,015 Learners

1 Course

Syllabus - What you will learn from this course

Content Rating96%(19,028 ratings)

Week

1

Week 1

3 hours to complete

Week 1

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.

3 hours to complete

6 videos (Total 84 min)

Week

2

Week 2

4 hours to complete

Week 2

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.

4 hours to complete

6 videos (Total 131 min)

Week

3

Week 3

4 hours to complete

Week 3

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.

4 hours to complete

4 videos (Total 128 min)

Week

4

Week 4

3 hours to complete

Week 4

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!)

3 hours to complete

4 videos (Total 88 min)

Week

5

Week 5

4 hours to complete

Week 5

4 hours to complete

4 videos (Total 82 min)

Week

6

Week 6

4 hours to complete

Week 6

4 hours to complete

4 videos (Total 111 min)

Week

7

Week 7

5 hours to complete

Week 7

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.

5 hours to complete

4 videos (Total 101 min)

Week

8

Week 8

5 hours to complete

Week 8

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.

5 hours to complete

5 videos (Total 124 min)

Week

9

Week 9

5 hours to complete

Weeks 9 & 10: Test Flight

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.

5 hours to complete

4 videos (Total 83 min)

Reviews

4.8

624 reviews

5 stars
84.89%
4 stars
12.24%
3 stars
1.44%
2 stars
0.40%
1 star
1%

TOP REVIEWS FROM INTRODUCTION TO MATHEMATICAL THINKING

by PDJun 29, 2020

It has help evaluate what I put into decision in any applicable context. Since I've noted what decision I make and how outcome can be made to vary when I consider all components in a isolated manner.

by ADAug 16, 2020

This is an excellent course, which provides insights into how mathematicians think about proofs. The exercises are not hard, but they do require careful thought. This is a well constructed course.

by NRMar 4, 2018

An awesome course. Very easy to follow at the start, becomes more challenging at the end. I have a PhD in economics yet I struggled with the real analysis at the end. And that's just intro level! :-D

by ASNov 16, 2022

It was an excellent course and it is something that I love in the end. It teaches a lot about the idea of math is a subject that should not be feared. It is a subject that should be understood.

View all reviews

Frequently Asked Questions

When will I have access to the lectures and assignments?
Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I purchase the Certificate?
When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
Is financial aid available?
Yes. In select learning programs, you can apply for financial aid or a scholarship if you can’t afford the enrollment fee. If fin aid or scholarship is available for your learning program selection, you’ll find a link to apply on the description page.
What is the passing grade for this course?
Since the focus is to acquire a new way of thinking (as opposed to getting right answers), the passing grade for the weekly Problem Sets is 35%, and for the Test Flight Problem Sets 30%. Basically, this means that if you stick with the course and complete all the work diligently, you should get a passing grade.

More questions? Visit the Learner Help Center.

Introduction to Mathematical Thinking