Some Fundamental Theorems in Mathematics (2019) [pdf]
people.math.harvard.edu
Some Fundamental Theorems in Mathematics (2019) [pdf]
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Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#2Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#3Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
You cannot teach people--even people at a young age--to enjoy mathematics because mathematics is based on logic and almost everyone except for a few weirdos hates logic. Mr. Spock was designed to point out how unusual people who like logic are, but as a character from a minstrel show, not as a role model....
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#4Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
(Here I am focusing on math outside of the math taught in pure mathematics programs.)
If anything the focus on math being a tool should be even bigger than it is today if you ask me. Most if not all of the math you get taught in school is math that was invented to solve a problem, it was not created as art. Introducing this historical context would probably help students in understanding why learning math is useful instead of just teaching math without this context as is usually done today. The beauty, in theorems and their proofs, comes in my opinion from understanding the nitty-gritty details. Without understanding the logical steps it's hard, at least for me, to understand where the beauty lies. I cannot appreciate mathematical ingenuity without understanding it. This is a bit of an abstract question, but perhaps you could elaborate on how you see this beauty without understanding?
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#5Wow I really like this document. As someone who really like mathematics but didn't get to do this as my major in undergrad, I'm missing out on so much, but also there is not enough time to start with undergrad books and do three years of basics... This document with all the clear statements at least allows me to see what is there to learn. Yes it's long, but compare that to dozens of the other books I would have to read to encounter these theorems, it's actually very short!
There is some useful "about this document" info near the end:
> The motivation to try such a project came through teaching a course called Math E 320 at the Harvard extension school. This math-multi-disciplinary course is part of the “math for teaching program”, and tries to map out the major parts of mathematics and visit some selected placed on 12 continents. [...] A goal of this project is also to get back up to speed up to the level of a first year grad student (one forgets a lot of things over the years) and maybe pass the quals (with some luck). via http://people.math.harvard.edu/~knill/graphgeometry/papers/f...
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#6This is awesome, although a bit in the old sense of the word, since it's scary to see entire courses I've taken distilled into one result that I probably couldn't even attempt to prove now.
aside: wonder why for group theory (51) Lagrange's theorem is only stated for cyclic subgroups, instead of any subgroup.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#7Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
I'm convinced that if Book of Proof [1] was taught in high school, most people would (a) understand math, and (b) not hate it.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#8Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
You cannot teach people--even people at a young age--to enjoy mathematics because mathematics is based on logic and almost everyone except for a few weirdos hates logic. Mr. Spock was designed to point out how unusual people who like logic are, but as a character from a minstrel show, not as a role model....
Joan Pearce and Peter Sloman of De Forest Research, Inc. were probably unusual people who like logic, having provided the science feedback for TOS.
Some other people certainly found Spock to be a role model in 1968:
https://ca-times.brightspotcdn.com/dims4/default/6870952/214...
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#9Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with MULTICOLORED (whoa!) text. There is no communication between modern mathematicians and grade school teachers in America; they may as well live on separate planets. Math classes in American grade school are little more than busy work meant to weed out people who hate busy work. Infer from that what you will.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#10Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.
I can't agree more. Speaking as someone who never enjoyed math at school, but now have an avid interest in it - the thing that converted me was building stuff with math. Much like you talk about.
When we learn woodworking we learn some basic tools and how they work, and then we build stuff. In my experience we don't typically get to build stuff with math. At least not the type of stuff that most people find remotely interesting.
I'm now building stuff with math all the time and enjoy it alot. The set of projects to choose from must be diverse though, some math isn't for everyone, just like not all kids will find working on building a baseball bat rewarding.