Introduction to Mathematical Thinking
Below are the top discussions from Reddit that mention this online Coursera course from Stanford University.
Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years.
Number Theory Real Analysis Mathematical Logic Language
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Taught by
Dr. Keith Devlin
Cofounder and Executive Director
and 11 more instructors
Offered by
Stanford University
Reddit Posts and Comments
1 posts • 180 mentions • top 50 shown below
69 points • welshfargo
https://www.coursera.org/learn/mathematicalthinking
https://www.reddit.com/r/learnmath/comments/7ta2we/psa_how_to_learn_math_from_scratch/
http://www.reddit.com/r/learnmath/comments/8p922p/list_of_websites_ebooks_downloads_etc_for_mobile/
https://openstax.org/
http://open.umn.edu/opentextbooks/
https://aimath.org/textbooks/approvedtextbooks/
http://mathworld.wolfram.com/
http://tutorial.math.lamar.edu/
http://patrickjmt.com/
YouTube: 3Blue1Brown
Susan Fowler also recommends undergraduate math books.
Problem Solving and Reasoning With Discrete Mathematics
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.
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 • Anirudh25
I am not sure but this might answer your question: Introduction to Mathematical Thinking
9 points • PublicRain
check out intro to mathematical thinking on coursera it is free
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
​
pqp => q ( p implies q) :::: TTT(1) TFF(2) FTT(3) FFT(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 :
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).
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 • secretnsaaccount
Coursera has an intro to proofs type class that’s pretty good.
https://www.coursera.org/learn/mathematicalthinking
2 points • beingisdoing
https://www.coursera.org/learn/mathematicalthinking/
2 points • l0gically
You can also try the the free MOOC on Coursera called introduction to mathematical thinking https://www.coursera.org/learn/mathematicalthinking It is also targeted at people just starting out college
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.
16 points • [deleted]
Some things I found useful in selfstudying/trying to obtain mathematical "maturity"
First 3 helped start the journey, 4 was my first dive into trying to do a proofbased 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 selfstudy. 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 • 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/mathematicalthinking, estou até pensando em fazer novamente por ter curtido todo o conteúdo do curso.
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!
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/mathematicalthinking
MIT has a free open courseware class on Discrete Math for computer scientists which also sounds applicable: https://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6042jmathematicsforcomputersciencefall2010/
Coursera has more intro level classes on similar topics, as does MIT. Good luck!
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 nonrepeating, nonterminating 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 • _nefario_
https://www.coursera.org/learn/mathematicalthinking
(enroll in the free version.)
3 points • toroawayy
If you're just starting off, these would be helpful:

https://www.edx.org/course/effectivethinkingthroughmathematicsutaustinxut901x0

https://www.coursera.org/learn/mathematicalthinking
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 • bluelite
Introduction to Mathematical Thinking: https://www.coursera.org/learn/mathematicalthinking
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.
1 points • Caderrific
I actually found a course on coursera that looks perfect:
https://www.coursera.org/learn/mathematicalthinking#
They have discussion boards and chat rooms, peergrading, and profgrading for my proofs. Maybe I'll convince my friend from school to do it with me too.
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 • MrMehawk
I would like to very strongly urge you to take this course on an introduction to collegelevel 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.
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 • wsppan
Check out Introduction to Mathematical Thinking
1 points • control_09
I think maybe something like this: https://www.coursera.org/learn/mathematicalthinking? is what you are looking for.
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 theirselves, 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;
https://en.wikipedia.org/wiki/Set_theory
http://matterhorn.dce.harvard.edu/engage/ui/index.html#/1999/01/82347
https://eliademy.com/catalog/physicalscience/fundamentalsofclassicalsettheory.html
https://www.coursera.org/learn/mathematicalthinking
2 points • timald
I'd suggest Keith Devlin's Introduction to Mathematical Thinking on Coursera. https://www.coursera.org/learn/mathematicalthinking
2 points • eucorri
Echoing the previous recommendation of Khan Academy. If you follow none of the rest of my advice, at least do 2030 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 • Fromthesewerr
" https://www.coursera.org/learn/mathematicalthinking "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
5 points • LemmyEuler
Precalculus:
Have a look at this precalculus book (which is available for free from the authors' website):
http://www.stitzzeager.com/szprecalculus07042013.pdf
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 proofbased 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:
https://www.coursera.org/learn/mathematicalthinking.
​
Calculus:
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.
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/mathematicalthinking. 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 poppsych 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/HowStudyasMathematicsMajor/dp/0199661316?sanoredirect=1
IIRC , lara alcock does her research of math education and transition to higher 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/mathematicalthinking.
1 points • booleankey
I would recommend you should read or take some basic course on Logic. But I don't your educational background and age to recommend specific book.
There is a good course on coursera.org for beginners. Although I haven't taken that course myself.
2 points • nouveem
If you're looking for a (very good) refresher, I'd suggest you take Linear Algebra: Foundations to Frontiers by UT Austin. The course is comprehensive and especially helpful for people going into mathheavy subjects. They teach the course using visuals and they help you build intuition. It's currently closed (will reopen midAugust I think), but you can start at your own pace now.
Alternatively, you can watch Professor Gilbert Strang's Lectures on Linear Algebra from MIT OpenCourseWare. Like the course I mentioned, he starts at the very basics and you're guaranteed to understand, since his style of teaching is beyond great.
Check both and choose which fits you better, or take the two if you have time.
Good luck! It's not as hard as you might think.
Edit:
• I can't exactly recommend one over another, since I don't know how much you've forgotten. But Prof. Strang starts at the very basics, unlike LAFF, which assumes that you have graduated recently.
• I'd highly recommend you take Introduction to Mathematical Thinking from Stanford University. It doesn't teach you mathematical methods as much as it gives you the intuition of mathematics and provides you with a mathematical mindset. It's about understanding math, not doing it.
1 points • gymtanlibrary
https://www.coursera.org/learn/mathematicalthinking
The book for the course is
https://www.amazon.com/gp/aw/d/0615653634/ref=mp_s_a_1_1?ie=UTF8&qid=1513886056&sr=81&pi=AC_SX236_SY340_QL65&keywords=introduction+to+mathematical+thinking&dpPl=1&dpID=41LEgIHzvjL&ref=plSrch
1 points • gazorpazorpazorpazor
Wife started taking this: https://www.coursera.org/learn/mathematicalthinking
Hopefully that could help you. More about thinking like a Mathematician and demystifying math than a particular deep dive into Math.
1 points • sovietcableguy
i like Hammack's Book of Proof for selfstudy. Velleman is good too though more advanced.
there's also a Coursera course on Mathematical Thinking you might find useful.
1 points • chrislit
It sounds like you're enrolled in a session that hasn't started yet. Until the session start date, you can't see some areas of the course in the menu. But they're still there, and you can directly navigate to the discussions, grades, etc.
13 points • Mauss22
Free, general logic resources:
Stanford's Intro to logic and Mathematical Thinking w/ Free online tools for completing exercises & interactive community;
J. Ehrlich's "Carnap Book"  w/ free exercises & tools;
Paul Teller's Modern formal logic primer  w/ free tools for completing exercises;
Peter Smith’s Teach Yourself Logic and other materials, like his reading guide;
~~Katarzyna Paprzycka Logic SelfTaught~~ ~~ w/ free workbook;~~ (edit: link dead)
Free Modal Logic: http://cgi.csc.liv.ac.uk/\~frank/MLHandbook//
Free Software: Carnap!, Truth Table Gen, Taut (prop/pred)
Not Free: Gensler's Introduction to Logic, Howard Pospesel's Introductions to Formal Logic (prop and pred, which I used in my undergrad). If you buy Pospesel's, see this prof's page. She uploads videos using the software, solving proofs, etc. It would help with self study
1 points • wbowers
I'm plugging away at my own curriculum. My main goal is to learn Machine Learning and eventually take a series of courses on selfdriving cars.
I'm currently studying Calculus, taking a higher maths transition course, and reading How to Solve It. This week I wrapped up differential calculus and have moved on to integral calculus.
I started reading A Walk Through Combinatorics last week but quickly realized I was in over my head. With no experience with proofs I wasn't getting much out of the book. I've moved my study of Combinatorics until after Abstract Algebra and bumped Mathematical Thinking (transition courses, proofs, logic, etc) up my list. I believe this will help with Linear Algebra as well.
2 points • OutOfNiceUsernames
MOOCs on logic, math, statistics and probability, neuroscience (brain 101,^[1]^[2] its vulnerabilities to addiction, etc), metalearning, etc. If you want to give them a try, and if your priority is selfimprovement and disclipline, start with the metalearning one. Which will also help with the process of gradual introduction of MOOCbased learning into your daily (monthly, yearly) schedule.
Also Duolingo for languages; Khan Academy, artofproblemsolving.com (too expensive, unfortunately), and brilliant.org for math (still looking for additional good math learning platforms that would have nonpaywalled interaction with human tutors). And /r/booksuggestions/ when I’ve already determined what specifically am I looking for.
There should also be a good CBTrelated selfhelp resource listed here, but I haven’t found any yet.
2 points • ISeePumpkins
I started off by recapping matrices and vectors from Khan Academy and moved to watching the videos of two precalculus courses:
https://www.coursera.org/learn/precalculus
https://www.coursera.org/learn/trigonometry
A really useful resource for trigonometry is this article, which finally boiled me down what the heck these things are and why they are used. During this time, I took also a linear algebra course from my local summer uni, but that course was very difficult because it was rigorous and I didn't have the required approach for universitylevel mathematics. Thus, I did Stanford's Introduction to Mathematical Thinking which I highly suggest:
https://www.coursera.org/learn/mathematicalthinking
After that, I did Calculus One from Ohio State University (also known as MOOCulus One) which I can't find from Coursera anymore. It went into great detail of the basics of differentials and integrals and had me practice much of the algebra, too. The textbook can be found from their website and the professor Jim Fowler has the lecture videos in his YouTube channel.
After this, I finished MIT OpenCourseWare's Single Variable Calculus' parts that were not covered in Calculus One. Those are probably very similar to Calculus 2 in many universities. Finally, I did the multivariable calculus from Khan Academy and some exercises from MIT OCW's Multivariable calculus concentrating on double and triple integrals, because I've understood that those are the most important things for probability.
E: the trigonometry article
1 points • cb_hanson_III
You are starting an undergrad programme at Bocconi, right? I know many people who went there for undergrad and then went to a top school for masters/PhD in US/UK.
I'm not really sure why you are worrying about masters and PhD textbooks at this point. The math that you need to know is no different from the basic math that any engineering/science/math undergrad student needs when starting university.
It sounds like you are lacking practice and confidence in your math. In my opinion it would be a mistake to tackle something too ambitious in the beginning. (For example there are many excellent, free online courses from MIT Open Courseware in calculus, linear algebra, differential equations, probability/statistics, but they might be too hard for you to start. Or not; by all means, check out these courses for yourself.) From my experience, math is something that is cumulative. When you skip ahead without having firm foundations at each step, it is a recipe for disaster. You'll actually make faster progress in the long run by going a step or two back and going slow.
One suggestion is to start with some high school level books aimed at ambitious students. Take a look at the brilliant set of books by AOPS: https://artofproblemsolving.com/store. They require far deeper understanding than your usual high school calculus books where the emphasis is on routine problem solving. Take a look at the precalculus and calculus books or even the algebra book. Even if it appears too basic, the time you spend making sure you know the basics well is never wasted, particularly for someone like you who has PhD goals. (They also have solutions manuals which make them good for self study.) These books are deeper than Khan Academy courses, which are fairly superficial. After these books, then you'll be ready for the MIT undergrad courses.
Additional suggestions:
This Stanford course called Intro to Mathematical Thinking (https://www.coursera.org/learn/mathematicalthinking), which is aimed at entering undergrads to teach how proofs are done and the basic language of maths, is worth a look. This knowledge will save you hours and hours later if you learn the rules of the game early on.
When you start your undergrad courses (or the MIT OCW courses), you might consider getting and using Mathematica on your computer. It's a great tool for visualizing functions, doing calculus, and a whole lot more. (You can probably get a student license; home edition is also quite reasonably priced.) It's not intended to replace pen and paper skills, but I found that for myself it helped to build intuition when I was learning various concepts. (Also in actual practice and research, i.e. outside your math class, you usually will be using computer tools to do your work. May as well get used to them early.)
For philosophy and motivation, see this first lecture from an undergrad math for econ course. http://ocw.uci.edu/lectures/math_4_lec_01_math_for_economists_introduction_to_the_course.html
TL;DR:
(1) AOPS books to get the high school math basics (through calculus) down cold at a conceptual level
(2) MIT OCW  excellent courses that give you a very sold foundations