A few points with pure math in mind (notes to my past self, perhaps):
Mathematics is a single discipline, knowing each of its basic topics helps in understanding the other topics. Don’t omit anything on undergraduate level, include some topology, set theory, logic, geometry, number theory, category theory, complex analysis, differential equations, differential geometry where some courses skip them (in addition to the more reliably standard linear algebra, analysis, abstract algebra, etc.).
The goal is fluency, as in learning a language, not mere ability to parse the arguments and definitions. It’s possible to follow a text that’s much too advanced for your level, but you won’t learn nearly as much as if you were ready to read it.
Reading unfamiliar mathematics is difficult, while familiar material can be rapidly scanned. As a result, reading partially redundant supplementary texts comes at a relatively modest cost, but improves understanding of the material. In particular, some books can be included primarily to connect topics that are already known. Other books can be included as preliminary texts that precede other ostensibly introductory books that you could parse, but would learn less from without the preliminary text.
Learn every topic multiple times, at increasing levels of sophistication, taking advantage of the improving knowledge of other topics learned in the meantime. The rule of thumb is to read 1-2 books on undergraduate level, and 1-2 books on graduate level.
Don’t be overly obsessive, it’s not necessary to repeat all proofs in writing or solve all exercises.
Don’t shy away from revisiting elementary material. It’s not there just as a stepping stone to more advanced material, to be forgotten once you’re through, it should remain comfortably familiar in itself.
Don’t shy away from revisiting elementary material. It’s not there just as a stepping stone to more advanced material, to be forgotten once you’re through, it should remain comfortably familiar in itself.
This is in regards to low-level math (the higher maths are beyond me), but I thought some of you might find my story inspiring:
I was never fond of math. I got through Calculus in HS, but literally spent most of my math classes sitting in the back coloring in rainbows on graph paper. After getting a BA in history though, I decided I actually wanted a useful degree, and decided to get a second one in engineering. Of course, first thing you had to do was take a Math Placement Test.
It had been over 8 years since I took my last math class, which had been “Math for Elementary Teachers”, and I had pretty much forgotten everything past very basic algebra...I could remember the quadratic equation, because my teacher had taught it to the tune of “Pop Goes the Weasel”, but I couldn’t remember what it did, or what a,b, or c was supposed to represent.
Anyway, NOT being willing to pay thousands of dollars, and take a bunch of boring intro classes to work my way back up to mathematical literacy, I decided to spend a summer reviewing math on my own. I got a bunch of “For Dummies” and “Demystified” books from the library, and started with basic arithmetic and Algebra 1. Over the course of the summer I worked my way back through everything I learned in jr. high and high school, and managed to test into Calculus (as high as the math placement test would go), saving myself thousands of dollars.
I have never learned anything so well, as the I did during review work I did that summer. Those 3 months of self-motivated study are probably the best investment I ever made in my learning. In high school, I only ever understood one concept at a time, which I promptly forgot after the test. Studying them in one fell swoop allowed me to understand it all as a whole.
From then on, with a firm foundation from which to build, math seemed easy. (well, except for Calc 3, but that’s a whole nother story...)
I’ll admit that its more like algebra-to-half-an-engineering-degree. I got divorced a while back and don’t have the resources to finish grad school.
Also, I would say that it’s not so much that I was ever inherently bad at math. I never had to study at it or anything. I personally just think I was socialized to not like it.
I’ve read somewhere (but of course I can’t find it now, as usual) that when males standardize test really high in math, they are more likely to be worse at language arts. However when females test really high in math, they are more likely to test even better at language arts.
I know this was the case for me. Math was what I was “bad” at. The argument went on to say that this was perhaps one reason why females were more likely to choose to go into liberal arts than STEM fields.
A few points with pure math in mind (notes to my past self, perhaps):
Mathematics is a single discipline, knowing each of its basic topics helps in understanding the other topics. Don’t omit anything on undergraduate level, include some topology, set theory, logic, geometry, number theory, category theory, complex analysis, differential equations, differential geometry where some courses skip them (in addition to the more reliably standard linear algebra, analysis, abstract algebra, etc.).
The goal is fluency, as in learning a language, not mere ability to parse the arguments and definitions. It’s possible to follow a text that’s much too advanced for your level, but you won’t learn nearly as much as if you were ready to read it.
Reading unfamiliar mathematics is difficult, while familiar material can be rapidly scanned. As a result, reading partially redundant supplementary texts comes at a relatively modest cost, but improves understanding of the material. In particular, some books can be included primarily to connect topics that are already known. Other books can be included as preliminary texts that precede other ostensibly introductory books that you could parse, but would learn less from without the preliminary text.
Learn every topic multiple times, at increasing levels of sophistication, taking advantage of the improving knowledge of other topics learned in the meantime. The rule of thumb is to read 1-2 books on undergraduate level, and 1-2 books on graduate level.
Don’t be overly obsessive, it’s not necessary to repeat all proofs in writing or solve all exercises.
Don’t shy away from revisiting elementary material. It’s not there just as a stepping stone to more advanced material, to be forgotten once you’re through, it should remain comfortably familiar in itself.
This is in regards to low-level math (the higher maths are beyond me), but I thought some of you might find my story inspiring:
I was never fond of math. I got through Calculus in HS, but literally spent most of my math classes sitting in the back coloring in rainbows on graph paper. After getting a BA in history though, I decided I actually wanted a useful degree, and decided to get a second one in engineering. Of course, first thing you had to do was take a Math Placement Test.
It had been over 8 years since I took my last math class, which had been “Math for Elementary Teachers”, and I had pretty much forgotten everything past very basic algebra...I could remember the quadratic equation, because my teacher had taught it to the tune of “Pop Goes the Weasel”, but I couldn’t remember what it did, or what a,b, or c was supposed to represent.
Anyway, NOT being willing to pay thousands of dollars, and take a bunch of boring intro classes to work my way back up to mathematical literacy, I decided to spend a summer reviewing math on my own. I got a bunch of “For Dummies” and “Demystified” books from the library, and started with basic arithmetic and Algebra 1. Over the course of the summer I worked my way back through everything I learned in jr. high and high school, and managed to test into Calculus (as high as the math placement test would go), saving myself thousands of dollars.
I have never learned anything so well, as the I did during review work I did that summer. Those 3 months of self-motivated study are probably the best investment I ever made in my learning. In high school, I only ever understood one concept at a time, which I promptly forgot after the test. Studying them in one fell swoop allowed me to understand it all as a whole.
From then on, with a firm foundation from which to build, math seemed easy. (well, except for Calc 3, but that’s a whole nother story...)
.
I’ll admit that its more like algebra-to-half-an-engineering-degree. I got divorced a while back and don’t have the resources to finish grad school.
Also, I would say that it’s not so much that I was ever inherently bad at math. I never had to study at it or anything. I personally just think I was socialized to not like it.
I’ve read somewhere (but of course I can’t find it now, as usual) that when males standardize test really high in math, they are more likely to be worse at language arts. However when females test really high in math, they are more likely to test even better at language arts.
I know this was the case for me. Math was what I was “bad” at. The argument went on to say that this was perhaps one reason why females were more likely to choose to go into liberal arts than STEM fields.