Real Math People Study the Appendix

In this episode of Office Hours with The Math Sorcerer I answer a question I received from a viewer. He is trying to learn mathematics.
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[jamieg]

learning tips:

1. review
if you do a problem today, then don't touch it for five days, you may forget how to do the problem. for each book, keep a list of problems that exemplify each concept. if you find a juicy problem that bundles multiple concepts together, then you only need that one problem instead of several for each concept.

2. time
difficult courses may take 10--15 hours (sometimes more) a week of study outside of class. don't overload yourself, so you have enough time for the class.

3. spread
if possible, only do three to ten problems from a book section per day. once you hit your quota, move on. return tomorrow. this spreads out the learning, allowing you to absorb material slowly, while also moving quickly.

4. repetition
when you finally get a tough problem, do it again once or twice. treat math like an instrument. repetition is helpful.

5. speed
it is important to write out all your steps, but there are shortcuts too. even if you don't use these shortcuts on an exam, they can help you repeat a problem more quickly. here's an abbreviated difference quotient example from calculus, done the typical way:
lim . . . /h
= lim . . . /h (algebra)
= lim . . . /h (algebra)
= lim . . . /h (algebra)
= answer

that's a lot of repetition. instead of carrying the entire problem onto each line, use variables to name the pieces of your problem. this way, you can work on each piece, which is faster, then pull them back together at the end.
L = lim Q (L for limit; Q for quotient)
Q = . . . /h
= . . . /h (algebra)
= . . . /h (algebra)
= . . . /h (algebra)
L = answer

6. illusion of competence
reading a book or your notes makes you FEEL like you know something, but it might be an illusion. you have to reproduce the knowledge with pencil and paper or voice. think of notes as a way to help you make quiz material for yourself: flash cards, problem descriptions, or similar things to QUIZ yourself with. you do not know something unless you ask yourself a question on it and give the correct answer without help.

7. solutions
download the solutions manual. only use the solutions manual to nudge you in the right direction. you have to be able to reproduce the correct answer by yourself and understand why it works.

8. lectures
lectures have been shown to give students false confidence and poor competence. a teacher only has a couple hours of class per week, so their treatment of the material will be superficial at best. you MUST fill in the rest of the details yourself outside of class with the textbook and other materials. this is not extra.

9. work linearly
do a problem straight down a page. if you write a bunch of calculations on the side of your paper or jump around, it can make your logic hard to follow. in other words, keep your math solutions skinny. also, you can use indentation and headers to give you work some structure:
slope?
y = mx + b
y - b = mx
■ m = (y - b)/x

10. books and notes
extensive notes are often overkill.
1. look through a chapter's headings to see what's going on
2. read through and work the examples along with the text as if they were hookworm problems. definitions or equations you need for those examples can go on a separate sheet. this sheet will be notes for the chapter, and will contain only the most important ideas.
3. work problems. again, when you find you need an equation or definition, copy that equation to a separate sheet. after a bunch of problems, you can go back through the text to add any loose ends to your notes.

[MikeSpinoza]

It's actually gastroenterologists who study the appendix.

[christopherallan]

The idea that a person can improve their proof writing by using proper "etiquette" resonates with me as someone who writes computer code for a living. Even if you work in a highly-technical field (as opposed to working in the arts or humanities), there's a big difference between building a thing that merely "works", and building something whose design reveals itself to an observer in a natural way (writing "self-documenting" code, or writing a proof or design specification with a high degree of simplicity and elegance), and that is easy (and maybe even pleasant) to understand, modify, or build upon.

[christressler]

When I started to review my calculus book by Larson, Hostetler (edition is old enough to include Edwards), I too started with the appendix, which is preliminary work. This book also has a chapter 0 that also includes pre-cal algebra material.

When I got into Ueno's Introduction to Algebraic Geometry (NOT his other three volumes), I started with its appendix first also.

Finally, I just started David Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and again started with its appendix first. Lol but while it is useful to know before delving into the main material, it have a feeling it's going to be harder!

[xhivo]

Huge congratulations to Michael! Good luck on your studies!!!

[nuggamcnugget]

Michael been through some sh!t, damn. Coming up again and working your way to the top. Great! I been drinking and was worried I damaged my brain way too much in the process. I realized I lose noting on trying going for science courses. Good to hear someone have been going through the same thing I have been through. You are not alone :)

[leelwx]

Memorization first and question later resonates deeply with my learning experience. I was one of those slowest to grasp new ideas. I thought I was dumb. Repetition helps me a lot to catch up.

[jsween]

I like what you said around the 8 minute mark. Memorise first, understand after. Sometimes people get stuck in the trap where they feel they need to understand something before learning it. If you just imbibe the information first, organise it somewhat in your head, and then apply it to routine problems in the textbook, the understanding will usually fall put naturally

[jbbentley]

I just finished my first semester with a 4.0!!
Let’s go to summer classes to grad faster!

[KMMOS]

First of all, good on you for seeking focus and clarity away from drugs. Second, Stewart and co-authors have a textbook about algebra and trigonometry that is much longer and better than the appendices in the calculus book. Get this book, work through the problems, and save your work in a three-ring binder as you progress so you can refer to it as you do more advanced math. Analogously, get Schaum's outline series books about your math subjects, and work through those problems, saving your work papers in three ring binders. This may seem like a lot of extra work, but it will instill personal discipline and form an excellent reference record for future math studies and homework.

[davidgrenier]

A few things: First, either have good handwriting or take the time to learn LaTeX. An overleaf subscription is expensive but worth it if you're in school. Second, when solving an equation, include every single step of the process. Only change one or two things per step. This will help identify where anything goes wrong. Thirdly, same with proofs. LEarning to write proofs is tough and a lot of math books have REALLY BAD proofs that just make a bunch of implicit assumptions. State everything explicitly and explain each logical step you make. I tend to break my proofs into a bunch of individual lemmas, each only stating one thing and each having its own proof. Then the final "main" proof is a lot of "By Lemma 1 we know X, By Lemma 2 we know Y, Therefore Z". It's similar to how I write code when I program. Every function should do one thing and be testable. Then I can write unit tests to make sure everything works and a main function that just calls the other functions in order.

[douglasstrother]

Look through the whole book cover-to-cover; a lot of interesting stuff can be packed into the appendices. "Oh, I wish I knew that while studying Chapter 5."

Prayers to you to stay clean, and keep on keepin' on!

[helioliskfire]

I've never heard of "etiquette" in writing math solutions. That's interesting. Although, in a complex analysis class, I remember the professor striking out the "explanations" in my proof. Lolz. It was in such a way that I felt his annoyance if not rage at my submission. In a linear algebra class, where I used a theorem from the assigned reading but was not discussed in class, the professor refused to give me full points because I should have explained the theorem in my submission. So I guess, there's different kinds of professors out there.

[Verbal]

After self studying mathematics for a a couple years now, I naturally gravitated towards reading the appendix. Incredible video 🔥

[budgarner]

Nice advice. I do this for all books that I read and study. It's a nice way to see the trend on what the author thought was important and how they organize their thoughts. You are more than just a Math Sorcerer. Keep it going.

[martinhawrylkiewicz]

Love reading the appendix of the more advanced Tao's Analysis 1 has a great appendix on Logic (introduction). Love it.

[niteman]

This is an idea that just occurred to me, but you could consider setting aside some of your proofs and come back to them days or weeks later to see if they still make sense to you once they're no longer fresh in your working memory. If you realize there are details that would be good to add, you can make a conscious effort on any proofs you're currently working on to add them in.

[suredeydo]

This is such a good YouTube channel 🥰 I'm so glad I found you.

[bvds]

Really interesting question and answer. I did not expect the answer, as I feel math papers or even math in applied fields (e.g. financial engineering) are often very terse and very brief, with almost no wordy explanation. Great video.

[ValenceFlux]

I knew an ac theory instructor that once shouted at an apprentice 'if you can't stay sober for this career go do something else with your life. If you can't uphold the purpose of the code, go be a plumber. At least there your mistakes won't get someone killed.' I heard that instructor was also a helicopter pilot who knew how to shout for a living and that experience was hard to forget.