Good article. I have a different opinion on how to Read/Study/Teach Mathematics. Contrary to popular belief, Maths can be a "spectator" sport. There is a very

Good article. I have a different opinion on how to Read/Study/Teach Mathematics. Contrary to popular belief, Maths can be a "spectator" sport. There is a very big difference between Reading/Understanding (to a certain extent) and Doing Maths. The latter is not a necessity unless you plan to be a Mathematician while the former is a necessity for every educated individual in our Scientific Society. To paraphrase Richard Hamming's quote; "The purpose of Mathematics is Insight, not Numbers". Mathematics is a Language. Thus it is important to become fluent in the Notation of the Language first. The Notation is used to express Concepts (eg. Sets), Relationships(eg. Functions) and then build Complex Edifices(eg. Linear Algebra) using them. The metalanguage of Maths is Logic. The purpose of Maths is to Describe and Explain Nature (notwithstanding "Pure Maths"). As V.I.Arnold said in his essay "On Teaching Mathematics" (https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html) - "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap". Hence always look for Applications of Maths in various fields. Abstraction should only happen after studying a lot of Concrete Examples else it becomes incomprehensible. Thus Read lots of Maths books, get comfortable with the notation, follow the worked out examples closely making sure that you comprehend every step and finally if you want to, do some exercises. PS: I love this dig at 'Murica in the article :-) P: For example, the author is not looking for a solution like this: everyone lives in Independence Land and is born on the 4th of July, so the chance of two or more people with the same birthday is 100%.