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Stanford University

Introduction to Mathematical Thinking

Stanford University via Coursera


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.


  • 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.
  • 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.
  • 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.
  • 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!)
  • Week 5
    • This week we take our first look at mathematical proofs, the bedrock of modern mathematics.
  • Week 6
    • This week we complete our brief look at mathematical proofs
  • 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.
  • 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.
  • 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.

Taught by

Keith Devlin



4.4 rating, based on 53 Class Central reviews

4.8 rating at Coursera based on 2391 ratings

Start your review of Introduction to Mathematical Thinking

  • Ryan Lam is taking this course right now, spending 5 hours a week on it and found the course difficulty to be medium.

    This is by far the best course of mathematics on Coursera! The content was spread evenly, it includes everything that is required for a student to read and understand mathematical theorems. Concepts such as implications, equivalence, quantifiers are...
  • I have just finished this course. I treated it just like a real course and did all the assignments, watched the videos, and took the final assessment. In the beginning, this class is billed as a great class for anyone including those not good at or...
  • Profile image for Sami Laine
    Sami Laine

    Sami Laine completed this course, spending 4 hours a week on it and found the course difficulty to be medium.

    An excellent introductory course to mathematical thinking or a companion course to follow while shuffling through your first book about mathematical proofs. Professor Devlin's way of putting out lots of exercises and going through them meticulously afterwards...
  • Rishika
    I really enjoyed the online course. I thought it was well planned and layed out, easy for me to follow. The work load(h.w. & test)was just enough, so i could finish everything with enough time, learn about the topics and not feel over loaded and rushed....
  • Change Nexus

    Change Nexus completed this course and found the course difficulty to be hard.

    I think this is one of the most useful, real-world applicable courses on Coursera or the entire Internet, and I would recommend it to anyone. It’s so rare that a course teaches a new way of thinking, rather than new concepts or new data. Prof. Devlin has a clear, friendly style that makes difficult concepts easy to understand. The teaching method where he hand-writes proofs while narrating is the next best thing to actually seeing a brain work. And I’m no longer afraid of a jumble of math symbols when I see them on a page. I can dive in, sort out the meaning, and understand the point the author is trying to make. Which is amazing.
  • Seb Schmoller

    Seb Schmoller completed this course, spending 10 hours a week on it.

    I am finding the course considerably less discussive than I did Peter Norvig and Sebastian Thrun’s AI course. This is partly because the video lectures are longer and more formal; and partly because there seems to be less active discussion in the course-provided discussion forums, possibly on account of the way in which students have been encouraged to make their own arrangements, which was far less the case with the AI course. But what this course shares with the AI course is the feature that struck me so forcefully in 2011: the feeling that you are getting one-to-one personal tuition from a very skilled and interesting teacher.
  • Anonymous

    Anonymous completed this course.

    Thank the stars that I bought and read his book, before the course. It's most likely the only reason that I was able to "hang on" and finish the course. The pace was "Stanford full speed ahead."

    Professor Devlin paid attention and saw that speed was an issue to some of us. To his credit I note that he plans to now offer the MOOC at varying speeds.

    If you really want to understand such thinking buy the book (its small and cheap), read it and then jump into the course with both feet.
  • Anonymous

    Anonymous completed this course.

    I would say this class is more for people who want to major in mathematics than anything else. It begins with lessons on logic and how to think about the language in a logical fashion. It eventually moves into mathematical proofs. I feel that more of the class should have been spent on proofs, mostly because I have always struggled with them. Professor Keith Devlin seems like someone I would want to have lunch with while discussing math, philosophy, and science in general.
  • Profile image for Gulnur Makulbekova
    Gulnur Makulbekova

    Gulnur Makulbekova is taking this course right now, spending 2 hours a week on it and found the course difficulty to be medium.

    I absolutely love First in Math. I haven’t come across another website or app that is as engaging as First in Math. Students love logging on and trying to earn as many stars as possible. I like how it is aligned to the Common Core standards. Students...
  • Wei En

    Wei En completed this course, spending 5 hours a week on it and found the course difficulty to be medium.

    An excellent course on mathematical thinking. Prof. Keith Devlin does a great job of going through the material and explaining solutions to the problems. There's also loads of student interaction in this course, though unfortunately, student activity dropped off towards the end as many people dropped out. Going through this course was definitely worth it though -- I have a much better understanding of proofs now.
  • Elazar Newman completed this course, spending 2 hours a week on it and found the course difficulty to be hard.

    As a mathematics teacher and a major in mathematics I found this course mostly a waste of time (from the point of view of a student intending to study mathematics) and a far cry from the description given in Mr. Devlin's introduction. Mr. Devlin seems...
  • Anonymous
    As a computer science, engineering graduate learning at your university will help me with my future plan of becoming an IS specialist. For this, I believe UCD is the starting place for my career. In Ethiopia, IS is new to the tech industry and I want...
  • Anonymous

    Anonymous completed this course.

    Instructor is very engaging and thorough, does a very good job of helping to ensure that difficult concepts are understood. Assignments (both graded and practice) are very helpful, instructive, and quite fun, though often challenging. Since the course is essentially pass/fail based on completing the assignments, the challenge is definitely worth it. Overall, a very interesting course taught by a very good professor that would be highly worthwhile for anyone in STEM fields, not to mention the general public.
  • Anonymous

    Anonymous is taking this course right now.

    This course has, in my opinion, a huge weakness, because, after week three, there is no answer key to all but a few of the ungraded exercises. Different people do try working them out online, but there is no way to tell if their answers are correct, unless a mentor weighs in. The instructor assumes that people will form study groups and work out answers together, but this isn't an option for some people.
  • Achei o workshop muito bom. A demonstraçâo grande conhecimento. O diferencial foi a capacidade de visão critica em cada trabalho, me impressionou que além da didática incrível em relação ao método para escrever o artigo, ela ainda demonstrou grande conhecimento científico, que auxiliou bastante para o enriquecimento do artigo. Agradeço muito a oportunidade,
  • QZ

    QZ completed this course, spending 10 hours a week on it and found the course difficulty to be hard.

    This was an interesting course. It assumes no understanding of advanced mathematics. It teaches you how to think which is an important skill for possibly every job out there. The time you spend every week depends on your own goals. Sure, you can just do the quizzes and still get a certificate of accomplishment; but to get the most out of it you'll need to utilise study groups and the course discussion forums.
  • Matthew Philip

    Matthew Philip completed this course, spending 8 hours a week on it and found the course difficulty to be medium.

    This was a great course. The instructor Dr. Devlin was able to make an otherwise difficult topic easy to understand. His explanations were crystal clear. There is a fair amount of assignments to do. This can be challenging, both qualitatively and quantitatively. In the end, I learned to write mathematical proofs.
  • MAMartin

    MAMartin completed this course, spending 5 hours a week on it and found the course difficulty to be hard.

    Very good and worth doing but hard work. My education has been in history - reading texts from strange times and cultures and weighing half-formed, messy evidence. This was technical; it has real world application but, as with any course in logic, do not expect human color. I will give it another, effortful try at a later time.
  • Anonymous

    Anonymous completed this course.

    I enjoyed this course, I am glad I took it. The Professor was very enthusiastic, and the lectures were well done. All the course material was available on time. The material does not lend itself as well to the online format as say the programming courses. I thought the course was well done in spite of that. I think more peer review on the assignments instead of just the final would help.
  • Profile image for Miftahus Salamah
    Miftahus Salamah
    Fun science math

    Don't be afraid to learn Math

    Let's learn Math happily

    If you practice you can do it

    Mathematics is not just memorizing formulas

    But have to practice

Never Stop Learning.

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