Introduction to Mathematical Thinking
Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years.
Number Theory Real Analysis Mathematical Logic Language
Next cohort starts July 20. Accessible for free. Completion certificates are offered.
Affiliate disclosure: Please use the blue and green buttons to visit Coursera if you plan on enrolling in a course. Commissions Reddsera receives from using these links will keep this site online and ad-free. Reddsera will not receive commissions if you only use course links found in the below Reddit discussions.
Dr. Keith Devlin
Co-founder and Executive Director
Reddit Posts and Comments
0 posts • 109 mentions • top 88 shown below
6 points • SarcasmUndefined
I highly recommend starting with Intro to Mathematical Thinking. It'll give you the tools you need to better understand and do proofs. And general mathematics.
69 points • welshfargo
Susan Fowler also recommends undergraduate math books.
Problem Solving and Reasoning With Discrete Mathematics
10 points • qna1
ELI5: How are vacuous truths applied in computer science?
Currently taking this course in mathematical thinking, and several students and I are confused as to why statements such as these are automatically assigned to be true. Some searching lead to this document, which on page 18, has a footnote which explains that outside of computers science, there are no practical applications to vacuous truths, and would like to know how they are applied in computer science.
4 points • welshfargo
You might find this MOOC helpful.
4 points • Anirudh25
I am not sure but this might answer your question: Introduction to Mathematical Thinking
3 points • null_anecdote
No problem. Your inference there is probably right, I can't speak specifically to GMAT/LSAT but it makes sense. And yeah, the key here is validation -- especially for an established test, even the best question writers probably can't predict how good a question is until they get data on real students in test conditions.
Yeah, absolutely! AoPS is all about building up sets of concepts, NOT recipes, to solve generic problems. That's especially true for Probability & Number Theory, where that type of approach is necessary (and precludes them from appearing in standard ed). Definitely check the excerpts to make sure the material isn't way below your level though.
I like what I've seen from Khan academy, but can't personally recommend other materials since AoPS / math competitions got me from long division to calculus. MathCounts is extremely similar to GRE style and you can find 100 practice tests, FWIW.
I love this course for bridging the gap to rigorous and purely conceptual math (good for anytime after Calculus I).
3 points • Andrew_Tracey
Introduction to Mathematical Thinking, by Keith Devlin. It's a Coursera course, it's free, and it's what finally got me to understand and enjoy mathematics after decades of hating it thanks to consistently poor teachers growing up.
3 points • Duckanomics
Beginner Doubt In Logic
Hey Everyone. I am currently taking an online course, could not get my head around this one.
Consider the truth table analysis of Proof by Contradiction.
p is the initial assumption in this table, (a negation of some statement)
q is the conclusion that we arrive to be false in proof by contradiction
|p|q|p => q ( p implies q)|| |:-|:-|:-|:-| |T|T|T|(1)| |T|F|F|(2)| |F|T|T|(3)| |F|F|T|(4)|
"The instructor said since we carried out a proof of p => q , it means that p => q is true. "
What does the above statement mean? Can someone break down the intended meaning?
For anyone wandering what course I am referring to its :
2 points • KillMeFastOrSlow
De Morgan's laws and negations sound like Foundations. Many kids are not given a Foundations math class, in the state I live in, I saw logic and proofs briefly in 10th grade and then didn't see them again. Take an easy Foundations MOOCon Coursera to dip your toes in it.
2 points • l0gically
You can also try the the free MOOC on Coursera called introduction to mathematical thinking https://www.coursera.org/learn/mathematical-thinking It is also targeted at people just starting out college
2 points • beingisdoing
2 points • secret-nsa-account
Coursera has an intro to proofs type class that’s pretty good.
2 points • welshfargo
I am currently taking Keith Devlin's Coursera course (https://www.coursera.org/learn/mathematical-thinking), which should be the first course that math students take, in my opinion. He explains concepts very well.
2 points • holatuwol
Take Mathematical Thinking, offered by Stanford through Coursera, before you start college. It will make all the other math you need to take for Computer Science much easier.
21 points • welshfargo
Susan Fowler also recommends undergraduate math books.
Problem Solving and Reasoning With Discrete Mathematics
16 points • [deleted]
Some things I found useful in self-studying/trying to obtain mathematical "maturity"
First 3 helped start the journey, 4 was my first dive into trying to do a proof-based class, 5 is a pretty good intro to analysis and proofs.
6, 7 have been pretty crucial in the past year or so of my self-study. 6 is really helping develop a problem solving mindset, 7 helping translate my intuitive problem solving/proof into something very rigorous.
Even proofs from just a few weeks ago seem like total garbage in comparison to where I'm at now and I'm sure in a few weeks I'll hate what I'm currently writing.
8 is good because there are a ton of valid proofs in different styles (induction, contradiction, contrapositive etc) for the same theorems.
So it's been good practice to apply techniques from 6 to prove theorems multiple ways and make them rigorous using style of 7 and then comparing the different proof techniques to understand why some methods are easier than other (eg one method requires a construction that might not be clear but the other might just required a counter example ie global vs local argument etc).
The main skill I'm trying to develop at the moment (other than problem solving -> proof) is being able to read less expository text and try to extract out the intuition/big picture.
1 points • Zorander22
Oh man, I have the perfect resource for you: Introduction to Mathematical Thinking. It is all about learning to think like a mathematician and starting to do proofs. It's free, with resources created by Dr. Keith Devlin, a Stanford math prof who has somewhat specialized into mathematical education. Also, for the first time in a few years, he's planning on getting involved in answering student questions. This is the perfect time to sign up for the course! It's not a book like you asked for, but I really recommend the course!
1 points • Zorander22
Great observations! It sounds like you actually care about concepts and understanding.
You may be interested in this free course on Coursera: Mathematical Thinking. The idea is to learn and train the way that mathematicians actually think, rather than learning set steps and just copying them. The course focuses on learning how to create proofs.
Math, the way it's often taught, is a set of rules to follow. Math, the way it is practiced by mathematicians, is a creative process of logical thinking. It doesn't really answer your question, but may be something you enjoy.
1 points • BlackFlatStanley
beginner to proofs, logic, and analysis. should i take this coursera course or should i read a book ('book of proof' or 'logic: the laws of truth'?)? i would prefer to take a course but if i absolutely should pick up a textbook i will
1 points • JoshuaZ1
> You took a predetermined sequence you know doesn’t have a specific number and equating it to another sequence which is much more random.
The point you seem to be missing here is that the example number I gave you shows that a decimal expansion can be non-repeating, non-terminating and still not have every finite decimal in it. Your intuition that somehow the digits of Pi look more random is a good intuition which is born out by the empirical data we have, but that's a very far claim from anything resembling a proof.
It may help for you to take some actual math classes. I recommend starting with Devlin's Introduction to Mathematical Thinking on Coursera.
1 points • Galois_Cohomology
I'm late here, but back in high school I used coursera's 'Introduction to Mathematical Thinking' by Keith Devlin which I'd definitely recommend
1 points • beingisdoing
Maybe start with that course for logic and problem solving.
Read How to Win Friends and Influence People by Dale Carnegie for social skills.
And for creativity I’d check out The War of Art by Steven Pressfield.
1 points • bluelite
Introduction to Mathematical Thinking: https://www.coursera.org/learn/mathematical-thinking
1 points • beingisdoing
1 points • sch77
Maybe you just need to step outside. Do something else, take a walk or you can relax and go through this course.
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.
1 points • control_09
I think maybe something like this: https://www.coursera.org/learn/mathematical-thinking? is what you are looking for.
1 points • Caderrific
I actually found a course on coursera that looks perfect:
They have discussion boards and chat rooms, peer-grading, and prof-grading for my proofs. Maybe I'll convince my friend from school to do it with me too.
1 points • _nefario_
(enroll in the free version.)
1 points • MrMehawk
I would like to very strongly urge you to take this course on an introduction to college-level mathematical thinking. It served me fantastically to bridge the gap to university when I was in a similar situation as you are now.
This isn't by any means complete but I think it's better to study the bridging material now so that you can make the most out of your first year material rather than studying the first year material now already and then repeating it. Especially in physics, this stuff is unfortunately often swept under the rug a bit in first semesters but familiarity with this helps students a great deal.
5 points • gapry
1 points • constrito
Cursando matemática aqui.
Você não deu seu perfil então fica complicado falar em sugestões mas digo, quando comecei o meu curso não sabia quase nada, praticamente só o do ensino médio e bem que decorei as coisas para passar no vestibular. Matemática do ensino superior é bem diferente e tudo se renova: você vai aprender tudo do princípio só que com rigor das demonstrações e provar tudo o que faz, e é isso que faz melhorar seu raciocínio lógico e argumentativo, não é decorar fórmulas e resolver problemas encaminhados. Um curso muito bom é o https://www.coursera.org/learn/mathematical-thinking, estou até pensando em fazer novamente por ter curtido todo o conteúdo do curso.
3 points • another_josh
Coursera has a free class from Stanford on Intro to Mathematical Thinking, it sounds like it’s in line with what you’re looking for maybe. Read the description of the class: https://www.coursera.org/learn/mathematical-thinking
MIT has a free open courseware class on Discrete Math for computer scientists which also sounds applicable: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/
Coursera has more intro level classes on similar topics, as does MIT. Good luck!
3 points • toroawayy
If you're just starting off, these would be helpful:
1 points • -___-___-__-___-___-
I think this is a bit different from what you're asking, but generally speaking the "language of math" refers to notations. Kieth Devlin's Introduction to Mathematical Thinking covers quite a lot on this and is great for it.
For what you're asking though, I don't think it would be helpful to find a glossary of these words and study them, your method right now is perfect. It's better to stumble upon the words and Google the definition and understand how the word is being used relative to its context.
I'd focus more on studying the actual math and you'll slowly gain an intuition for its language. Good luck!
1 points • wsppan
Check out Introduction to Mathematical Thinking
6 points • vittujee
"Rank doesn't matter" is equivalent to saying "ELO doesn't matter" or "MMR doesn't matter". If you play a match against player who's rank is x and your rank is z, you gain/lose points based on algorithm and based on your points your match can either be demotion match, promotion match or just a match. Rank is result of what happened and it's not possible to manipulate freely without cheating.
I would assume it's entirely possible to get two tekken god primes using two accounts using ranked progress save states and network routing and few macro's to have it fully automated to run from beginner to highest rank available with enough time. Just like it's apparently possible to roll back demotions and losses using save states.
The only function matchmaking algorithms have is to attempt to find most suitable opponents to play against in order to make the game as interesting as possible for as many players as possible. The tekken's grind system doesn't let those who are skilled already to skip ranks, but in return it doesn't accidentally send you too far up in your placement matches from where you'll fall a long way down.
If the matchmaking algorithm is flawed, people will feel it's unjust and they'll either leave or try fix it their-selves, one way or another. Personally I think ELO would've been better for online than ranks, not because people would not value ELO as something as important as Ranks, but because I imagine it would do better job with matchmaking.
Ranks matter for matchmaking but the rank you are in does not define you, you being in a rank defines the rank among with everyone else in the rank. You're not the rank, but the rank is you and everyone else in that rank. If you think you are the rank, you need to study up on Set Theory;
2 points • LemmyEuler
I agree with the other comment, that Velleman's "How to prove it" is great.
If you want a video-based course, I can warmly recommend this free Coursera course from Stanford University:
If I were you, I would certainly get Velleman, and perhaps supplement it with the Coursera Course (depending on how difficult you find Velleman).
5 points • LemmyEuler
Have a look at this precalculus book (which is available for free from the authors' website):
Remember that in mathematics, it is very important to solve lots of exercises - just having read the relevant section of a textbook is only the beginning of mastering the material! :)
Reading and doing exercises from the book I linked would probably prepare you very well for calculus.
Proof techniques and mathematical thinking:
If you want an introduction to proof-based mathematics (at the introductory college level) a book I liked was Velleman's "How to Prove It".
This free Coursera course is good for that as well:
If you already master the precalculus material and have taken an introduction to proofs as mentioned above (and perhaps read a book on the basics of calculus) then my favorite calculus text is "Calculus" by Spivak. It is very rigorous compared to introductory calculus text, in the sense that every result is proven, and he starts from the axioms of the real number line. If you read Spivak and do a lot of exercises (preferably after doing precalculus, an introduction to calculus and an introduction to proofs) you will be very well set to study mathematics in college. Do not start on Spivak until you have a firm grasp of the material in the precalculus book and Vellemans "How to Prove it". Even after those, it would perhaps be best to start with a gentle introduction to calculus.
2 points • Fromthesewerr
" https://www.coursera.org/learn/mathematical-thinking "What is it starting from?, will it start from the basics of the basics and assumes you have little to know experience?, what are the pre requisites
2 points • Andrew_Tracey
Yes, I got into discrete mathematics. Start with Keith Devlin's Coursera.org course, Introduction to Mathematical Thinking (anybody can do this, you need only the most basic of highschool alegebra; it's almost entirely logic with very little numbers/algebra involved), then move onto Susanna Epp's Discrete Mathematics with Applications. The textbook by Epp was recommended to me by an assistant on Devlin's course when I asked where I could go from there.
2 points • eucorri
Echoing the previous recommendation of Khan Academy. If you follow none of the rest of my advice, at least do 20-30 minutes a day there. Don't miss a day; if you are short on time, 10 minutes of exposure beats none at all.
I'd also suggest you check out /r/learnmath and their stickied thread of resources. PatrickJMT saved my ass in calc.
If you have already completed a high school level algebra course and want to progress into trig, calculus (or even further!), I'd also suggest the Coursera MOOC Introduction to Mathematical Thinking, which will teach you how to approach problems in the same way a mathematician would.
2 points • timald
I'd suggest Keith Devlin's Introduction to Mathematical Thinking on Coursera. https://www.coursera.org/learn/mathematical-thinking
2 points • Zorander22
You already have good advice here, but I'm going to go a different route. I hope that you look into Khan Academy, and do learn calculus - that's wonderful!
If, as seems likely, you go on and learn calculus, you may wish to watch the lectures and try the exercises from this course: https://www.coursera.org/learn/mathematical-thinking?
In schools you won't learn how to do math as a mathematician does it, which is about creating new math, discovering (or creating, depending on your view) new proofs.
The course I linked to will start to teach you how to think like a mathematician. It starts off slowly, and gradually ramps up. However, keep in mind that mathematical thinking is one very powerful type of thinking. If you only study mathematics, it is easy to only view the world through one particular lens.
I know some mathematicians, so if you decide you want a mentor who is a university professor (especially after you get the hang of calculus), let me know and I can put you in touch.
3 points • welshfargo
Susan Fowler's story should encourage you to try.
And this: https://hbpms.blogspot.com/
There is also this course: https://www.coursera.org/learn/mathematical-thinking
3 points • NonwoodyPenguin
Understanding Analysis by Abbott is a very friendly introduction to analysis and its proofs. I can't quite remember if it goes through the proof types, but it's very readable.
https://www.coursera.org/learn/mathematical-thinking. Goes through the very basics of high level math including notation, logic rules and then the basic proof types.
A Mind for Numbers is more a of a pop-psych book but goes over learning techniques etc.
Probability 110 by blizstein https://projects.iq.harvard.edu/stat110/home/. Youtube lectures and a ton of solved problems.
How to Study as a Math Major https://smile.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316?sa-no-redirect=1
IIRC , lara alcock does her research of math education and transition to higher math.
1 points • sovietcableguy
1 points • gazorpazorpazorpazor
Wife started taking this: https://www.coursera.org/learn/mathematical-thinking
Hopefully that could help you. More about thinking like a Mathematician and demystifying math than a particular deep dive into Math.
0 points • mrpalmer16
I like Keith Devlin’s Intro to Mathematical Thinking on coursera.
1 points • bullpup1337
That's a good start. I also like the course https://www.coursera.org/learn/mathematical-thinking.