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Introduction to Mathematical Thinking
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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.