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Introduction to Mathematical Thinking
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If you are into Coursera / video lectures this lecture series fits the bill for "exceedingly gentle introduction to writing proofs": https://www.coursera.org/learn/mathematical-thinking
Wow, so, so much venom on this thread. For those of you who enjoyed the article (like myself), you'll see that his class ("Introduction to Mathematical Thinking") starts today as a Coursera Stanford MOOC here: https://www.coursera.org/learn/mathematical-thinking
⬐ lordgrenvilleDon't Coursera classes always start $TODAY?⬐ sidpatilSome do. Others have fixed starting dates.⬐ diegoperiniSome classes are browsable everyday but the classroom experience had a schedule last time I checked.
Introduction to Mathematical Thinking (https://www.coursera.org/learn/mathematical-thinking/home/we...). A foundational course on mathematical language and thinking, a great refresher course, at least for me after leaving school a long time ago
there's also Intro to mathematical thinking: https://www.coursera.org/learn/mathematical-thinking/home/we...
and mathematics for machine learning: https://mml-book.github.io/
You internalise mathematical notation by using it to solve mathematical problems and express mathematical ideas.
Two excellent resources are:
1. Introduction to Mathematical Thinking (if you prefer moocs) - https://www.coursera.org/learn/mathematical-thinking?
2. How to think Like a Mathematican - https://www.amazon.co.uk/How-Think-Like-Mathematician-Underg...
Think it's this course: https://www.coursera.org/learn/mathematical-thinking
Introduction to Mathematical Thinking (https://www.coursera.org/learn/mathematical-thinking). Aims to teach you what it is like to think like a mathematician. Covers the elements of topics that you probably encounter in the first semester of an undergraduate maths degree: logic, induction, proof construction, real analysis, etc.
Machine Learning (https://www.coursera.org/learn/machine-learning). I'm still working through this course but am finding it extremely interesting. I find that having to implement things in matlab/octave gives you a deeper understanding than using a framework like tensorflow or keras.
Both of the above courses have good instructors, which I think is the main factor that makes a good mooc.
Coursera's 'Introduction to Mathematical Thinking' is a great starter course for real Analysis https://www.coursera.org/learn/mathematical-thinking
A couple of years ago I did the Introduction to Mathematical Thinking course on Coursera . Even though I found it hard, I enjoyed it and learned a lot, and I feel I got some insight into mathematical though processes. Recommended.
Earlier this year I did the Stanford Introduction to Mathematical Thinking course on Coursera . I found it fairly challenging but managed to finish with a distinction. The instructor was particularly good.
I'm now working through UCSD Interaction Design specialisation , which is a series of courses followed by a project. So far its been very good, although the short course format (3-4 weeks) means that there isn't time for much of a community to form among the participants. I've learned a lot though.
I'd recommend both courses.
⬐ codexjourneysI also loved the Mathematical Thinking course, it was the first MOOC I completed and still one of my favorites!
"Mathematical thinking", on this course: https://www.coursera.org/course/maththink
A related "Ask HN" from a couple of months ago: How or where to begin learning mathematics from first principles? 
I'm at the very start of what I hope might be a similar journey, and have signed-up for a Coursera "Introduction to Mathematical Thinking" course. I'm hoping it might give me some insight to build on. The course starts in about ten days, so apprehension hasn't kicked-in yet.
Some I can recommend that are still available on Coursera:
- Introduction to mathematical thinking 
- Introduction to Mathematical Philosophy 
- Machine Learning (actually a CS course, but involves linear algebra and some calculus) 
- Calculus: Single Variable 
⬐ mitochondrionMany thanks!
Have you seen the coursera introduction to mathematical thinking ? That may be a good starting point.
I know he mentions people doing their own googling, but I'd like to recommend this coursera course: https://www.coursera.org/course/maththink
It's one of my favorite MOOCs of all time, a fantastic intro to mathematical thinking.
The author of this article teaches a course on coursera that I can't recommend enough, he put so much effort into that course. https://www.coursera.org/course/maththink As you might have guessed its not really about 'math' in traditional sense.
I think this skill is something all hackers should learn, as they implicitly use it in most of their ventures. Once mathematical thinking (abstract/outside the box thinking) is learned, it will be far easier to learn new math techniques and concepts in the future.