Back to Introduction to Mathematical Thinking

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527 reviews

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....

PD

Jun 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.

AM

Nov 23, 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 Victor O

•Sep 3, 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 Edwin C v E

•Nov 3, 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 Alexis H

•Feb 12, 2019

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

By Mudita S

•Jul 7, 2020

I work in publishing and have absolutely no day-to-day brush with Mathematics but since I had a high school background in the subject, I wanted to make contact again. This course served my purpose perfectly. The instructor, Dr. Devlin, has provided very detailed lectures and he never makes the subject boring. The in-lecture quizzes are a big help to keep your attention in check.

I was a bit apprehensive of the course content as I'm not from the United States and wondered if it would make sense to me, given my limited knowledge. But I stand corrected; it was easy to follow.

From structure to progression, this is indeed value for money. I'd highly recommend it!

By Ken L

•Jul 24, 2020

This was a return to mathematics for me after many years (I am in my late 60s). I completed modules in mathematics during my psychology degree and throughout my career had to use statistics for research purposes. The course has reawakened my interest and revealed something that I have overlooked, that deep down I have a love for mathematics!

The course tutor is outstanding, both in clarity, sense of humour and in having a kindly approach that inspires confidence. It is not an easy course and it certainly requires effort, but the reward for the effort is that what seems initially obscure gets quite a bit clearer and appreciation grows.

By Bruce S

•Jun 30, 2020

The course was interesting and challenging. Professor Devlin did a very good job of explaining the material and, while I appreciated that he did not want to lead us by the hand, it would have been helpful to have all the assignment and problem set answers available somewhere. I went into the class knowing nothing of higher level mathematics or proofs and am interested in learning more. I would be interested in a followup course focusing on proofs.

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 James H

•Jul 18, 2020

This is an excellent preparation for higher pure maths study. It's essentially an introduction to proofs - building them and assessing them.

It's worth looking at, even if you think you know this stuff. If it's 10 years since you graduated, it's still worth working through, to get your mind in order; if you think you are comfortable with the material, watch the videos at high speed to save time, until you start failing the quizzes :)

By Christopher C

•Mar 30, 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 Marcus K

•Mar 17, 2018

I work in the digital visual effects industry as a developer and wanted to understand scientific papers better. Searching the the for solutions, I came across this course. It is very well structured and offers tons of materials to work with. You can test yourself with quizzes, assignments and problem sets throughout the course.

I can only highly recommend it .

By Yuliya A 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 Wakash K

•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 Wesley T

•Dec 5, 2017

I really want to learn this material. I believe the module may be more improved if the videos were subdivided into smaller chunks. Passionate speaker though, theres a lot of promise here.

By CA G N

•Jun 26, 2018

think differently....

By Mohammed A S

•Sep 26, 2017

This Course is only good if one has a group to study with. Online , whtatsapp, or social networking groups are not that effective. If you want to do this course make sure that you have friend or two planning to enroll with you.

By zarghonaHussainzada

•Aug 30, 2020

all the videos and quiz are lock

By S M R R

•Oct 8, 2020

This course has a great introduction to why you should take this course. Professor Keith Devlin introduced us to why high school mathematics is different from university mathematics. High school mathematics deals with math which invented kind of before the 18^{th} century. But what happens to the mathematics that was invented in the 19^{th} century. Because of your kind information many modern mathematics techniques invented in the 19^{th} century. How to deal with those problems?

In varsity level mathematics you just don’t use or apply formula that was predetermined but you think about the problem. You brainstorm and make the reasoning behind the problem. A mathematical proof needs to be dealt with great care. Because you have to deal with its uniqueness universality. Of course, you have some certain predefined tools that you learned in high school, but you need to think before applying, not as it was done as a template set in high school.

So, Coursera offers this course to introduce you to a glimpse of how mathematical thinking procedure works. This course introduces us to Logic and Logic Combinators, Logic Implication, and Equivalence, Quantifiers, several proof techniques such as proof by contradiction and proof by Induction, proofs involving Quantifiers, introductory Number Theory, and some proves associated with it. The hardest part of this course is Real Analysis. Each proof associated with Real Analysis involves so much thinking and reasoning. At the very last week, i.e. at the 10th week, you need to submit some assignments involving proof techniques that will be peer-reviewed before you get your certificate.

By 徐致远

•Feb 14, 2020

First of all, thank you very much for this masterpiece. This is the most challenging course I've ever had on the Coursera.

One more comment on the final peer review assignment: the question about whether 0 is a natural number.

Only after taking this course and doing some research on the Internet do I realise that there's a lot of argument on this question. I am a college student from China, and I major in physics, rather than mathematics. For me, the statement "0 is the least natural number" is a fact that was written in my primary school's textbook. So I was quite surprised when I was told that 0 is not a natural number. After searching on the Internet, I finally got more knowledge on this.

Fortunately I was not supposed to lose too much on that. However, I do recommend that you make this clear in the course content so that everyone can discuss under a common assumption, regardless of their background education.

Above all, this course experience is a lot of fun. There's no standard answers here, only remarkable thinking and a space of infinity to explore. Thank you for all of it! I'm giving 23 out of 24 marks for this course, taking away 1 point for clarity about whether 0 is a natural number.

By Eray C

•Feb 2, 2021

This is one of the best and most useful courses I made so far.

But the content also quite sophisticated! So be ready for struggles. Actually there is a problematic fact, that should be considered by the course provider:

Not everything is covered in the video lessons that is asked in the problem sets. And in the final test you have to work on some tasks for which you most probably will need to consult the internet for getting the required skills.

Nonetheless, by working on this course one really improves the way of thinking and understanding. Be ready for logic, number theory, mathematical proofs and some hard times ;)

The professor is very very likable and really wants you to improve your way of solving problems. And you instantly know that he really loves math and wants to share his enthusiasm. That's just so nice!

So really fantastic course!

By Avanish M

•Jul 3, 2020

It really was a different perspective to mathematics, and I enjoyed working with proofs and entering the world of mathematicians where even simple problems may have rigorous proofs. It was novel to view how these fundamental proofs have shaped the current state of mathematics today. Dr Keith Devlin was very efficient at teaching and I understood everything in the course and trusted his method even when the beginning weeks were very simple because he felt really knowledgeable and reliable. I would recommend this course to anyone who is interested in mathematics or wants to explore another side of mathematics that we wouldn't ever see in school.

By Cedric V L

•Oct 14, 2020

I used this course prior to enrolling at university for a software engineering degree. I always was bad at mathematics and was extremely worried I would never manage to get through the foundational courses which are very much focused on discrete mathematics. This course literally made me think different. It showed me that math is nothing to be afraid of, and literally reset my thinking about mathematical thinking. In fact, when I had to write my first proofs, I noticed it was very similar to what is taught here. I passed the courses with confidence and am now finishing my Master's thesis. Thank you!

By Dr. A S M

•Aug 16, 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 Shubham V S

•Jun 30, 2020

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.

Good to learn this course as it really helps me to open up my mind to opt for different approaches of an answer rather then getting the actual answer....

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

This is by far the best course of mathematics on Coursera!

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

By Charles K

•Aug 2, 2021

I thought the course delivered everything it promised and then some. It was well paced, well taught and the emphasis was correctly placed on learning to think mathematically as opposed to chasing grades or regurgitating formulas. Setting the pass mark at 30% allowed us to take chances and fully explore the material. I took a similar class 50 years ag and this time I decided to accompany my sister who had never takin this kind of course. We both had a great time discussing the concepts and examples. I highly recommend it.

By Gilbert Y

•Aug 20, 2020

What a humbling course. I learned I'm not nearly as intelligent as I thought I was. Most of the content was doable with some effort. The most problematic was the content near the end about real analysis.

The course is all about mathematical proofs: expressing thoughts precisely, how to write proofs, how to reason, and grading proofs written by others. I didn't think that's what I'd be doing when I enrolled.

My one suggestion for the course is chop up the videos (averaging 30m each) into a bunch of shorter videos.

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