Introduction to Mathematical Thinking

I'm doing Introduction to Mathematical Thinking on Coursera. It's quite enjoyable, the professor has a good humor about him -all is recorded but it still feels great and you can watch the short lectures over and over until you get it. It is much better than going straight for procedure, IMHO - as it is more about logic steps and basic language.

https://www.coursera.org/learn/mathematical-thinking

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/approved-textbooks/

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

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.

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.

I am not sure but this might answer your question: Introduction to Mathematical Thinking

check out intro to mathematical thinking on coursera it is free

https://www.coursera.org/learn/mathematical-thinking

Maybe for some parts of it, but you were asking for a big-picture view. The napkin gives you a comprehensive outline of almost every subject and a fairly unconventional path through it. I recommend the online course Introduction to Mathematical Thinking if you want an introductory view perhaps before you tackle the napkin.

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.

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).

I would do this to get an understanding of what proof based math can be like: https://www.coursera.org/learn/mathematical-thinking

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

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|p|q|p => q ( p implies q)|| |:-|:-|:-|:-| |T|T|T|(1)| |T|F|F|(2)| |F|T|T|(3)| |F|F|T|(4)|

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"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?

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For anyone wandering what course I am referring to its :

Intro to Mathematical Thinking

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.

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

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.

https://www.coursera.org/learn/mathematical-thinking/

Coursera has an intro to proofs type class that’s pretty good.

https://www.coursera.org/learn/mathematical-thinking

Some things I found useful in self-studying/trying to obtain mathematical "maturity"

1. A Mind for Numbers

2. How to Study as a Mathematics Major

3. Introduction to Mathematical Thinking

4. MIT's Mathematics for Computer Science

5. Understanding Analysis

6. The Art and Craft of Problem Solving

7. How to Write a 21st Century Proof

8. Intro to Graph Theory

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.

https://www.coursera.org/learn/mathematical-thinking

(enroll in the free version.)

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.

Introduction to Mathematical Thinking: https://www.coursera.org/learn/mathematical-thinking

1. Introduction to Mathematical Thinking, Coursera

2. Mathematics for Computer Science, MIT OCW

3. 15-251 Great Theoretical Ideas in Computer Science, CMU

I think maybe something like this: https://www.coursera.org/learn/mathematical-thinking? is what you are looking for.

I actually found a course on coursera that looks perfect:

https://www.coursera.org/learn/mathematical-thinking#

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.

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

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

I took a course from Coursera offered by Stanford "Introduction to Mathematical Thinking". It introduces general logic, symbols and methodology that is required in proofs.

If you're just starting off, these would be helpful:

• https://www.edx.org/course/effective-thinking-through-mathematics-utaustinx-ut-9-01x-0

• https://www.coursera.org/learn/mathematical-thinking

Check out Introduction to Mathematical Thinking

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.

> 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.

https://www.coursera.org/learn/mathematical-thinking Is a good one to give a glimpse about what happens in math after high school. I dont think you need algebra above a level you know. Know that it barely scratchen the surfaces though

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!

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!

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!

Precalculus:

Have a look at this precalculus book (which is available for free from the authors' website):

http://www.stitz-zeager.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.

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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:

https://www.coursera.org/learn/mathematical-thinking.

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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.

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.

" 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

mathe is an sich sehr spannend, das wird einem nur in der Schule so vermiesd. Also zeitlich haut das bei dir vielleicht nicht hin aber Mr Devlin ist sehr motivierend anzusehen in seinem kurs oder auf youtube zur historie der mathematik.

I'd suggest Keith Devlin's Introduction to Mathematical Thinking on Coursera. https://www.coursera.org/learn/mathematical-thinking

> However, I am absolutely AWFUL at math (I failed College Algebra three times during my first undergrad). It doesn't come natural to me; but I would be willing to buckle down and sweat over it to make it through the program.

Math doesn't come natural to 99.9999% of people. It's a learned skill like any other, that's perishable, and that with enough study and practice anyone that doesn't have some exceptional circumstance (such as a learning disability) can do.

Fear of mathematics is something that infects a lot of people - but as someone that used to outright abhor mathematics - I came to find that sitting down and taking it for what it is - just another subject to learn and apply to my life - was the easiest way to remove the mystique of mathematics and actually learn it.

If you're a reader, there's a short book called A Mathematician's Lament that will make you feel better about why you've probably had such a struggle with mathematics. It's written for the laymen and is a rumination (not a mathematics book) - it's also cheap, about $1.50 to read via a Kindle.

You may also find Keith Devlin's Introduction to Mathematical Thinking (Free) to be a rewarding investment for learning how to properly think about problems in a mathematical way.

And for instruction, practice and learning there's Khan Academy (Free)

1. 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.

2. 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.

3. A Mind for Numbers is more a of a pop-psych book but goes over learning techniques etc.

4. Probability 110 by blizstein https://projects.iq.harvard.edu/stat110/home/. Youtube lectures and a ton of solved problems.

5. 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.

Hey bud,
I have a few suggestions which helped master the math behind ML. I started out with the Coursera Course: Introduction to Mathematical Thinking by Dr. Keith Devlin.Link: https://www.coursera.org/learn/mathematical-thinking/home/welcome

I followed that up with a Specialization called Mathematics for Data Science.
Link: https://www.coursera.org/specializations/mathematics-for-data-science

These two took me around 5 weeks to complete. Post completion I was fairly confident with my Math skills. I also watched Videos on Youtube and referenced Khan Academy.

Hope it helps. :-)

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.

This is a great resource I used: https://www.coursera.org/learn/mathematical-thinking

Check out the book How to Prove It as well.

One thing that came to me after I was about halfway through a Theory of Computation course was the realization that I was not proving things to myself, first. I would try to start a proof by looking at the definitions, propositions, theorems, etc. and then try to guess my way into writing the proof using those. This was an awful way to go about it and was very stressful.

It wasn't until I realized that I needed to get an intuitive understanding of the thing I was trying to prove, and to reason/think about/diagram/play with the material to see exactly why something is the way it is. That shift in thinking, of first gaining an understanding of the material and proving to myself why something was true has made all the difference in the world for me.

That's a good start. I also like the course https://www.coursera.org/learn/mathematical-thinking.

I like Keith Devlin’s Intro to Mathematical Thinking on coursera.

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.

i like Hammack's Book of Proof for self-study. Velleman is good too though more advanced.

there's also a Coursera course on Mathematical Thinking you might find useful.

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 math-heavy subjects. They teach the course using visuals and they help you build intuition. It's currently closed (will reopen mid-August 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.

Online course called Introduction to Mathematical Thinking by Devlin is a great intro

https://www.coursera.org/learn/mathematical-thinking