Introduction to Mathematical Thinking に戻る

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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....

Mar 05, 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

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.

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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 Колесник П П

•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 Jimmy

•Apr 16, 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 Christopher B

•Jan 17, 2019

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

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 Jorge D P

•Apr 08, 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 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 Gerard D L D

•May 10, 2017

Great course ! Provided new skills on how to think about maths and improved my overall confidence with the subject. I feel like I can go further in the study of mathematics thanks to this.

by Yuliya S

•Jun 12, 2017

Some work is involved and probably a good ideas to start on assignments earlier then later, but it's really thought through material that is being presented and great concepts.

by Alexis H

•Feb 12, 2019

Touches the foundamental parts of mathematics. Really helped with my mathematical thinking.

by Fighting F

•Aug 30, 2019

Good Course, A must take for college math majors. While the course covers most of the contents, Professor should have given more attention to Proofs and Real Analysis which are assigned just two weeks in this course.

by Shangyan Z

•Oct 19, 2018

The content of this course is very helpful. It explains the fundamentals of college mathematics, and the possible confusions about symbols. I'm not sure if it can clear all my confusions about symbols, but the light weight of the content has captured the gist of college mathematics. Even after I have graduated from college for 7 years and have rarely used something like partial differential equations, this course brings all the math I learnt back to life in a few hours. I also recommend following Prof. Keith Devlin's twitter. Although there's a bunch of political content in his tweets, reading them can help you get closer to the mind of the author, and get the sense of reality besides this MOOC.

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 Victor O

•Sep 03, 2018

Keith did really great course for those who missed formal proofs method at school or at university. I wish I had this course in high school or in my first college year. Math, unlike to other STEM disciplines, is based not on experimental confirmations, but on the chain of logical steps (aka proofs). Keith explains and teaches you how it all works. I strongly reccommend to everybody who wants to establish truly solid background in maths and then be able to study it further in his required directions. Really enjoyed. Universe is very beautiful and math is such an astonishing side of it. Keith is amazing.

by Dr. A S M

•Aug 17, 2017

This is a very well designed course which is essential for mathematicians as well as for those who want to use mathematical way of thinking (which is needed in most of the subjects ) in their life. It prepares a sound base for the ones who have mathematical aptitude but could not study mathematics in their life . In fact thinking mathematically is the modern and most effective way of dealing with the situations at hand and this course is most suited for this purpose. Thanks to the entire team of professors and others who have developed this course.

by Dana W C

•Aug 25, 2017

The one thing that was a problem for me was the last week, the scoring of peers was presented to me PRIOR to the videos working through solutions and "training" me to score those particular questions. Therefore, I did those tasks out of order. I believe for the second and third peers, their scores were fine, but the initial peer was rooked. Reflecting, they should have earned an overall score closer to 46 than the 33 attributed. If there was someway to remedy that...

Thank you.

by Edwin C v E

•Nov 03, 2018

Excellent course. Many important mathematical concepts are covered during the course, such as proof techniques, quantifier handling, logical and mathematical thinking and objectively assessing other people’s work. The pace of the lectures as well as the variety of topics is good. The final chapter might be a little out of some students’ comfort zone, especially since the prof takes a deep dive quite quickly, but remains interesting nevertheless. Highly recommended.

by GBVN K

•May 17, 2017

Extremely interesting with nice and lucid explanation. This 9 weeks' learning journey is thoroughly knowledgeable and mathematics has become my favorite subject after completion of this course. Advise my fellow students to make best use of this course and come out of fear in order to learn mathematics. Extremely grateful for helping me earning first online course from the reputed Stanford University.

Thanks and Regards,

G. SAI YASHWANTH,

INDIA

by Nickolas F

•Aug 31, 2017

A truly amazing course that gives the student an excellent taste of mathematical thinking and makes even those who hate mathematics understand its depth. A special thank you to Dr. Keith Devlin for teaching this course and giving me a true glimpse into university level mathematics, but also to Coursera for giving me the ability to be an active student in such amazing courses. I hope that more courses like this are to follow in the future.

by Harold L M M

•Aug 23, 2019

This is a great course on mathematical thinking. It covers important topics such as: the mathematical language, logic, number theory, real analysis.

These topics are presented in a clean and concise way by Dr. Devlin, and I highly recommend this course to any person interested in mathematical thinking, how to read and understand mathematical proofs, and for those interested in pursuing a mathematical career.

Thank you professor Devlin.

by Christopher C

•Mar 31, 2018

What anyone could ask for in a basic proofs course: logic, proofs, induction, some application of what they learned to higher mathematics. I like the idea of teaching a bit of abstract algebra right away, but this course goes with some basic results of analysis (probably for the low-hanging surprising results in analysis). Nice job!

Grading proofs was subjective, but keep in mind the threshold to pass a quiz is very low.

by Justin L

•Feb 19, 2017

With consistent effort, I have found this course to re-ignite my passion for mathematics. Keith Devlin, the instructor, does an excellent job of helping us to develop our ability to think in mathematical and logical ways. In short, I have found a new joy and excitement for doing mathematics, even stuff I consider hard. I have a new-found passion and new "eyes" for maths that I didn't have before. Excellent course!

by George P

•Mar 19, 2017

It is very good course that teaches simple at first glance concepts that later evolve to something more complex. Mr. Keith Devlin is very charismatic professor who knows his material very well and passes his energy and enthusiasm to students. One minor notice about the real numbers material, in my opinion it is too brief and needs slightly more time to be understood by many students.