Back to Introduction to Mathematical Thinking

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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....

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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.

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Although I'm a 9th grade student and found many concepts hard, the course was very informative and interesting. Thank you coursera for offering me the opportunity to take this course with stanford.

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By Ho X V

•Jun 21, 2018

My love for Mathematics in the past few years is like the electrocardiogram (Điện tâm đồ) of a near-death patient, which is peaceful as the surface of a calm lake. This does not annoy me at all, despite for the fact that I used to be called the smartest one in class, in this specific topic particularly. Then, how did I end up like this? Fortunately, Professor Keith Devlin points out that this is not exceptional. It is not true that people are born to be good or bad at math by default, and keep up the trend till they die. In fact, there is always a transformation in mathematical thinking from high school to university level, that almost no one was told when they were students. Moving from being taught a fixed set of techniques to solve a problem, now you are asked to be able to abstract things and reason about it, which naturally confuse learners. To prepare for such transformation, this is the exact course you need. My personal opinion: I wish that I have taken this course 5 years ago, so I can learn a bit more about Algebra. First year undergraduate students, mark my words: take it!. Course review: In the first half of the course, Professor teaches the importance of understanding the primitive mathematical notations, how careful you should be when formulate a problem, and do not rush when attack it. The quizzes successfully demonstrate the second and third points. In the second half, it's amazing to see how one can derive mathematical reasoning solely from the truth tables, and apply it to produce appropriate techniques to solve problems. Only now that I know it's not black magic when somebody choose an arbitrary value that I have no idea where it comes from in a proof. In the last 2 weeks, he gives us a few touch on set theory and real analysis, the two subjects that I suddenly feel attracted to, not to mention his sense of humour when introducing them ;) Notably, throughout the whole course, Professor shows us how to evaluate a proof, and I do learn a thing or two about it. As he always says, the point of a proof is 1) establising the truth and 2) explaining it for the readers, math will be interesting once you learn its way of thinking. I now feel more ready than ever to read a mathematical problem; specifically, where to look for in the beginning of a proof, and the abstract idea of how the authors come up with their chain of deduction. Thank you so much Professor.

By Will C

•Apr 25, 2020

this course is given by a rock star in math (prof at Stanford and behind the interesting BrainQuake). Also some interesting concepts are shared. And finally: I love math! So why such a low rate? Because this is a very badly done MOOC from the pedagogical angle: you can't do 27-minutes video when we know that the average focus of a student is lower than that. Also don't ask for a lecture to read separately if you can't at least share the url where we can safely read it online. Finally locking the next weeks is ridiculous: let the student advancing at their pace. So very good in terms of pure math (not a surprise since Keith Devlin is pretty famous in the math world), but very frustrating in terms of pedagogy.

By Chris B

•Jan 17, 2019

I found this course to be incomprehensible. The professor was rude and dismissive, when I asked for help.

By David P

•Apr 8, 2018

I love this course.

I cried of the emotion when Dr. Keith explained the meaning of the Implication. I'm 35 and I did not know that. After week 2, all things that I have learned before of this course became make sense.

You should take this course. It’s a new way of thinking about things.

This course is like one of the greatest pieces of classical music if you like the classical music.

If you want to take the course, I recommend that you take it in advance. I recommend seeing the material slowly. Some parts require some effort. Please do the exercises and review the forum group.

By 柳子豪

•Jul 1, 2020

This course is great because it teaches you the foundations of mathematical thinking, namely how to write rigorous and concise proofs. I really enjoyed the course and would recommend it to a friend.

By Nathaniel R

•Mar 5, 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 Ching K R L

•Apr 5, 2020

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 clearly and concisely explained, I am sure even for people who are learning university mathematics for the first time, the lectures are very easy to understand.

That being said though, as in any university math courses, solely listening and taking notes in the lecture could not make you a decent grade. The most important material provided by this course is the assignment, which is provided at the ‘download’ tab at the bottom of the video. The assignment deliberately included confusing examples and counterintuitive facts to trick people who have misunderstandings in their concepts, and detailed explanations for harder questions were provided as well. I love how professor Devlin reason as well: he proceeds very carefully, writing down what he speaks in every argument, and then translate it to symbols in mathematics.

I found that technique very efficient as I tackle the assignment problem. If you want to get out most from the course, remember to complete all the assignment and check your answer by asking your friends / in the forum!

The test flight peer-assessment is also a very enjoyable experience, looking at other MOOC-mates attempting to prove the same questions enables me to learn more on how to improve presentation style when writing as a mathematician.

Overall, this is a very interesting course for people who loves mathematics, it serves as a transition from high school mathematics to university mathematics as well. Highly recommended to people who wants to know more about what university mathematics looks like!

By Ali M

•Nov 24, 2019

It was an amazing course! Lots of interesting content. The content is also explained really well, i found it really easy to understand. The assessments are a little challenging, but reasonably sized.

By Phumlani D

•Jun 30, 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 Jimmy

•Apr 15, 2019

In the last lecture, based on what he previous taught, the professor give us the definition of limitation which is the beginning of the Calculus. I wish I taken this course before university.

By Randall G T

•Feb 19, 2018

Good overall, but Dr. Devlin tends to ramble in lectures rather than getting to the point. He also seems somewhat inconsistent in how he evaluates proofs. Since proofs and their evaluating are the core of the course I found it frustrating that there isn't more care given to clarifying what constitutes a good proof.

By rajat j

•Sep 27, 2018

I was interested in mathematics till high school. But after that in engineering we have to learn Applied Mathematics(AM) and which is quiet boring for me. Due to these type of mathematics , I decided not to learn math anymore. But this course was very intuitive and focuses more on thinking and logical steps rather than computational parts. Now I love math more than I used to love before Engineering. I will definitely learn more mathematics because of enormous interest in this field. THANK YOU Prof. Keith Devlin for this course. Since, you're the reason why I love Mathematics. Sorry for English because it is not my native language. Sir please reply to my review if you have time :) .

By Joseph J

•May 10, 2018

Really great course. It's aimed at high school students, but I found it valuable even as a college graduate who has taken several math classes but never had a formal introduction to how to think about proofs. I wish I was able to take this class when I was an undergrad.

By Michał F

•Aug 26, 2017

Great course, it's really shows the way how to think in mathematical rigor. It gave me an intuition about what Mathematics is itself.

Thank you Profesor Keith Devlin, you are a great teacher !

By maria

•Sep 9, 2017

The detailed explanation of proving certain claims in the course are very ambiguous, which is the reason I discontinued the course.

By Pavel K

•Feb 15, 2019

Очень сложно сказать, что курс был полезен для меня в карьере или учебе (или как-то крайне интересен), однако благодаря курсу я стал лучше понимать язык математиков и освежил память по некоторым математическим законам. Огромное спасибо создателю курса за отдельные turorial-видео после assignments!

By Sean R

•Nov 22, 2017

Overall I thought it was pretty good. Explanations are straight forward, and easy to follow. Small little quizzes are given throughout videos to ensure that people are paying attention during the lessons.

The only negative part is that it's clearly a class that's designed around you working with multiple people. If you don't, you won't be able to study as well. Fortunately, there are forums, but with the nature of forums, responses can range from "immediate" to "never."

By Deleted A

•Jan 29, 2017

I am really not sure if you need 20-30 minutes videos to explain the material - I think maybe 7-10 minutes clear with some good examples would be more helpful, I fell sleep during some of the lectures, boring, long and not sure if necessary - on the other hand I guess 20-30 minutes is good because you learn to put up with arguments and clear them up yourself - also maybe this course was directed towards high school students thus the long explanations -Overall, as I said on the beginning, it would work better with clear definitions and some examples and discussion board -

By E

•Jun 21, 2017

I am not familiar with the material and consider this course as a baby step into a completely new language, a journey in a world of thinking that shows our education system fails to inform and develop logic and rational thinking at the basic levels. At 67, I should have the basic knowledge to complete this course with ease. It reflects on the substance, or lack thereof, of the education system and the quality of teachers. Given the time, the effort and the patience of good teachers, every reasonable thinking person ought to be able to manage these basic skills. Thank you, Professor, for sharing your knowledge, it has been a great discovery for which I am immensely grateful.

By David S

•Jul 2, 2020

An excellent course from an outstanding educator. The first few lectures should be of interest to anyone who wishes to think and communicate in a logical manner —including anyone alienated by high-school mathematics. The remainder of the course is a solid preparatory foundation for anyone considering a university mathematics degree but is also valuable more widely for the sciences, engineering and emerging technology-based subjects like data science. I took a physics degree (and later a PhD) nearly forty years ago, and while I had a good high-school mathematics background, I wish I'd had something like this in addition.

By Mathula M

•Jun 6, 2020

The skills learned in this course are useful for someone who hasn't yet reached college math (like myself) but from what other people are saying, it may be basic for someone who has passed beyond this point (ask others about this). I gave this 4 stars because though the professor is engaging and his concepts are memorable, his lectures are slightly longer than I can hold attention for and the peer review process at the end is tedious, especially for someone who doesn't know how to use Latex format.

By steve g

•Feb 9, 2020

Professor Keith Devlin wrote and participated in this course for some years. He has stepped away from it since (he explains why in a post on https://mooctalk.org/ ) but the course remains a valid educational experience. I return to it every so often, mainly to have a go at answering other learners' questions on the discussion forums - which is a learning experience in itself- and try out different approaches to proof-writing. I can do this because the course has a peer-review element at the end, where your work will be read and reviewed by other students. If a given approach is faulty, difficult to follow or contains a mistake that you have over-looked, one of your fellow students will pick up on it. This might sound daunting but, ultimately, it is a much better deal than just satisfying yourself. A third aspect of this course which is challenging is that you are also assigned the task of reviewing other people's work, which is a surprisingly difficult thing to do well - your target is complete and constructive feedback, with each of your grading decisions explained and sound advice given - that demands a thorough understanding of the course material and also of the traps and pit-falls which are all too easy to fall into and which can, ultimately, demotivate a student if they fall foul of one and you, as the reviewer, are not 'on their side' .

By Juan A

•Aug 20, 2021

last test is unnecesary and annoying

By Paul O

•Jun 30, 2020

This was a great course. I've studied Mathematics independently for most of my career (I am a Software Engineer) but I always struggled with the less formulaic aspects of the field. I could memorize the quadratic formula, but I couldn't derive it. I could remember properties of real numbers, but couldn't provide an original proof to save my life. After taking this course, I'm now confident enough to reason about numbers, write effective proofs, and generally think more critically and abstractly about Mathematics. I'm looking forward to continuing to study the field, now with more confidence and practice!

By Andrew D

•Aug 17, 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.