161–170 of 215 posts
It’s a revisionist history of mathematics aimed to demonstrate how ideas flow from one to another. He eliminates a lot of dead-ends and takes a perspective as to one “ought” to move from one subject or discovery to another. OTOH, the historical perspective is both readable and often a missing piece which makes other math tougher.
Earlier quoted context omitted.
Depends what your goal is. I cant imagine anything worse than High School style teaching of mathematics. Its awful. That repetitive calculation, entirely unnecessary for almost any real world situation. Almost every time you are doing repetitive calculation what you should have been taught was a second or third conceptualisation of whatever maths you are looking at. So linear algebra I was only ever taught numericall…
My high school calculus and physics teachers synchronized their classes, so shortly after we learned something in calculus, we would use it in physics. Since then I’ve learned this is not as common as it should be! Perhaps it works especially well in that case because calculus was literally invented to do the kind of physics you learn in high school, so it’s a very natural combination. Linear algebra (e.g.) is so gen…
>My current proficiency level is pre-high school mathematics as I didn't pay much attention in high school, learning effectively nothing. Advice from someone who was in the same position: Take a class. Multiple classes. Go sign up for a Mathematics AS at your nearest community college right now. You will never know enough of what you don't know to learn this stuff on your own. A lot of it is just doing the painful re…
Depends what your goal is. I cant imagine anything worse than High School style teaching of mathematics. Its awful. That repetitive calculation, entirely unnecessary for almost any real world situation. Almost every time you are doing repetitive calculation what you should have been taught was a second or third conceptualisation of whatever maths you are looking at. So linear algebra I was only ever taught numericall…
Adult brains, with less plasticity, are able to synthesise new concepts that build on what they already know.
https://j2kun.svbtle.com/mathematicians-are-chronically-lost...
It begins (and there is far more of value in the post):
Many people who are in this position, trying to learn mathematics on their own, have roughly two approaches. The first is to learn only the things that you need for the applications you’re interested in. There’s nothing wrong with it, but it’s akin to learning just enough vocabulary to fill out your tax forms. It’s often too specialized to give you a good understanding of how to apply the key ideas elsewhere. A common example is learning very specific linear algebra facts in order to understand a particular machine learning algorithm. It’s a commendable effort and undoubtedly useful, but in my experience this route makes you start from scratch in every new application.
The second approach is to try to understand everything so thoroughly as to become a part of it. In technical terms, they try to grok mathematics. For example, I often hear of people going through some foundational (and truly good) mathematics textbook forcing themselves to solve every exercise and prove every claim “left for the reader” before moving on.
This is again commendable, but it often results in insurmountable frustrations and quitting before the best part of the subject. And for all one’s desire to grok mathematics, mathematicians don’t work like this! The truth is that mathematicians are chronically lost and confused. It’s our natural state of being, and I mean that in a good way. ...
So I believe the best way to learn math is by finding areas where you are forced to apply it. And it is never too late. I learnt most of it after turning 30.
Earlier quoted context omitted.
I find there are things I’m missing. Does the precalculus stuff cover geometry and polynomials?
Precalc basically covers algebra and trigonometry. From my experience as a CompSci major, euclidean geometry is not really necessary.
Earlier quoted context omitted.
To the above I would add: How to Think Like a Mathematician - Kevin Houston (an excellent book to read before starting) How to Read and Do Proofs - Solow The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning - Solow Calculus - Spivak (Actually a Real Analysis book, not a Calculus book, see e.g. https://math.stackexchange.com/questions/1811325/spivaks-cal... ) Linear Algebra Done Right - Axler (Inte…
There's also How To Solve It by Polya. Small, accessible, casually written.
I had read it many years ago. It may be influential beyond what people know. There is a version inspired by it, for programming, called How to Solve it by Computer [3], by R. G. Dromey, who, IIRC, was/is a professor at an Australian university (Wollongong?).
I had the Dromey book. It is not exactly parallel to the Polya book, because it shows the details of how to come up with a solution, either in pseudocode or in a Pascal-like language, while the Polya book, IIRC, is more about principles and techniques for general problem-solving.
[1] https://press.princeton.edu/titles/669.html
[2] https://en.wikipedia.org/wiki/How_to_Solve_It
[3] https://en.wikipedia.org/wiki/How_to_Solve_it_by_Computer
I'm pasting below the first few paragraphs from the URL [1] above:
[ A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out—from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft—indeed, brilliant—instructions on stripping away irrelevancies and going straight to the heart of the problem.
First published in 1945.
George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety. John H. Conway is professor emeritus of mathematics at Princeton University. He was awarded the London Mathematical Society's Polya Prize in 1987. Like Polya, he is interested in many branches of mathematics, and in particular, has invented a successor to Polya's notation for crystallographic groups. ]
The John Conway mentioned is the one who invented the Game of Life.
Nothing beats having a (good) teacher. Self-learning, no matter how smart you are, is pitifully slow without a teacher. Half an hour with a good teacher can save you weeks of table head-butting. (But obviously you can't rely only on the teacher.) As for books, it's not a spectator sport: you gotta do it yourself. Read a sentence, then work it out yourself with pen & paper. You can't get it just from reading alone. Fi…
I disagree with the general statement that leaning maths is "pitifully slow without a teacher". So long as one has structure (e.g. some kind of syllabus) and has access to google, then learning can proceed very efficiently indeed. In addition, by being able to work through difficulties independently you can "be your own master" so to speak, earning the confidence to solve new and difficult problems without assistance…
There are even a lot of math teachers who feel anxiety about basic math. Why don't they simply self-study?