The main thing I did today was watch the Wildberger video--below, and then thought about how to prove the communitive law to someone who doesn't understand modern language or numbers. I'm not sure the way I came up with works without the explanations around it.

I don't really understand the distrubutive law or why we are using k, m, or n. I'll have to read an explanation of it before I can try to prove it in the same way I did the communitive law.

I'm not really sure how to structure these daily updates, but for now they will probably take the form of a small end of day update--above, and then random thoughts and stuff I'm encountering during the day--below.

This Will Make You Better at Math Tests, But You Probably are Not Doing It

To get better at doing tests the reccomendation is to do the test twice and then compare answers. Being able to solve the problems quickly will allow answering them twice rather than just looking over the answers.

Arithmetic with numbers / Arithmetic and Geometry Math Foundations 2 / N J Wildberger

Give reasons why these rules are true:

Prove them true in a way that a 10 year old 100 thousand years ago would understand. Up until this point Wildberger has used lines for counting and has introduced the notation of a succesor function. I should be able to essentially create drawings that I could show to someone who doesn't understand the concept of numbers or speaks a language I understand.

Here is my attempt at the communitive property:

I'm going to show that nm=mn using 4 and 3.

first we show that m(4) groups of n(3) is equal to 12

Then we show that n(3) groups of m(4) is equal to 12

Finally, we show that m(4) groups of n(3) is equal to n(3) groups of m(5) using the method of drawing connections between our lines that represent a unit of 1.

someone who doesn't know the number 12 should be able to compare m groups of n compared to n groups of m and determine it's the same.

Collection of thinking tools and frameworks to help you solve problems, make decisions and understand systems

This looks interesting

How to Understand Math Intuitively

"
3:22

you should stick to your simple toolbox initially

Start with simple problems, and then work your way up to very hard problems

using these exact same simple tools

You would be very surprised how advanced problems can be solved

using just a hammer and a screwdriver of mathematics

The third step, after you have really exhausted all the tools in your toolbox

is going on and adding a new tool to your toolbox

The problem with most education systems in the world is

that this step happens way too early!

Typically, you've just barely learned how to solve like three or five

different types of problems using a given tool

And then you move on to learning the next one

And the word "learn" may be an overstatement here

because many students just memorize the steps to solving a problem
"

Reccomended books:

A Concise Introduction to Pure Mathematics

Art of Problem Solving - good introductory content for different competency levels.

Problem-Solving Strategies

Love and Math: The Heart of Hidden Reality

Don Knuth - Interview 2006

In the back of Thomas’s book he had supplementary problems, the teacher didn’t assign the supplementary problems; I worked the supplementary problems. I was, you know, I was scared I wouldn’t learn calculus, so I worked hard on it, and it turned out that of course it took me longer to solve all these problems than the kids who were only working on what was assigned, at first. But after a year, I could do all of those problems in the same time as my classmates were doing the assigned problems, and after that I could just coast in mathematics, because I’d learned how to solve problems. So it was good that I was scared, in a way that I, you know, that made me start strong, and then I could coast afterwards, rather than always climbing and being on a lower part of the learning curve.

Someone improved my code by 40,832,277,770%

Stand-up Maths wrote code that ran in 31 days, and then his viewers wrote code to solve the problem faster using both math and computer programmer techniques. It seems like the fastest improvements came from switching from python sets to binary, and other improvements that were programming details and not maths.