probabilitytheory
proofsareart:

Linear Algebra. Linear algebra is the study of spaces which are in some sense “flat” like points, lines, planes, and their higher-dimensional analogues. The tools of linear algebra often help to formalize geometric intuitions and give a clean picture into the behavior of spaces which may have literally millions of dimensions (or more!).
The proposition statement references “the action” of Gamma (the half-T thing). Gamma is a group, whose elements can be thought of as symmetries; the action is the mechanism by which those elements are “decoded” from their abstract forms into physical symmetries such as reflections and rotations.
To finish the rest of that clause: a point in space is called a fixed point of Gamma if the action of each element leaves a point alone. For example, in two dimensions a reflection leaves an entire line alone, and so if Gamma consists of two reflections whose lines are not parallel* then Gamma has only one fixed point — where the two lines meet.
( * and the identity and their products, technically. )

proofsareart:

Linear Algebra. Linear algebra is the study of spaces which are in some sense “flat” like points, lines, planes, and their higher-dimensional analogues. The tools of linear algebra often help to formalize geometric intuitions and give a clean picture into the behavior of spaces which may have literally millions of dimensions (or more!).

The proposition statement references “the action” of Gamma (the half-T thing). Gamma is a group, whose elements can be thought of as symmetries; the action is the mechanism by which those elements are “decoded” from their abstract forms into physical symmetries such as reflections and rotations.

To finish the rest of that clause: a point in space is called a fixed point of Gamma if the action of each element leaves a point alone. For example, in two dimensions a reflection leaves an entire line alone, and so if Gamma consists of two reflections whose lines are not parallel* then Gamma has only one fixed point — where the two lines meet.

( * and the identity and their products, technically. )

nobel-mathematician-deactivated
lthmath:

Sergei Lvovich Sobolev born in 1908 was a Soviet mathematician working in mathematical analysis and partial differential equations.Sobolev spaces are used for growth conditions on the Fourier transforms and are an important subject in functional analysis. Generalized functions (later known as distributions) were first introduced by Sobolev in 1935 for weak solutions. The theory of distribution is considered now as the calculus of the modern epoch.He was a Moscow State University professor from 1935 to 1957 and also a deputy director of the Institute for Atomic Energy 1943-57 where he participated in the A-bomb project of the USSR. He was the founder and first director of the Institute of Mathematics at Akademgorodok near Novosibirsk, which was later to bear his name. In 1962 he called for a reform of the Soviet education system. 

lthmath:

Sergei Lvovich Sobolev born in 1908 was a Soviet mathematician working in mathematical analysis and partial differential equations.
Sobolev spaces are used for growth conditions on the Fourier transforms and are an important subject in functional analysis. 
Generalized functions (later known as distributions) were first introduced by Sobolev in 1935 for weak solutions. The theory of distribution is considered now as the calculus of the modern epoch.
He was a Moscow State University professor from 1935 to 1957 and also a deputy director of the Institute for Atomic Energy 1943-57 where he participated in the A-bomb project of the USSR. 
He was the founder and first director of the Institute of Mathematics at Akademgorodok near Novosibirsk, which was later to bear his name. In 1962 he called for a reform of the Soviet education system

nobel-mathematician-deactivated

jacksophone asked:

Why do people hate math?

imathematicus answered:

That’s both a good and difficult question.

Here’s a TL:DR; 

  • it’s more demanding of students than other subjects
  • it gets harder more quickly than other subjects
  • lack of practice makes it difficult to catch up 
  • when students don’t see the purpose, they have no desire to try

Full answer below.

Read More

imathematicus:

cofinaldestination:

ryanandmath:

I don’t think I quite agree with where this answer is coming from. There seems to be a lot of emphasis on the student’s lack of ability to comprehend / practice mathematics and less on how mathematics is taught. Certainly the student has to take some responsibility but I think that’s less so in grade school math, and “math hate” seems to primarily stem from how students are taught it when they are young.

It’s very easy to make math be boring from the start. If you’re not making activities related to addition or division or geometry interesting (and more importantly, thought-provoking and beyond rote calculations) in the 2nd and 3rd grade, you’re gonna start losing kids. So by the time they’re even just in 4th or 5th grade (where maybe you can expect some more personal responsibility), you’ve already lost the battle, they already hate math. And this attitude will follow to high school. And high school mathematics won’t save them because, honestly, algebra and geometry and how they’re taught suck too. So you’re correct in saying that at some point it will be entirely overwhelming to even try and catch up. But the reason they fell behind in the first place is probably because the teacher did not do a good job introducing them to math years ago, not because they should have been practicing this whole time. Who wants to work at something that is taught in such an atrocious way?

I guess what I’m trying to get at is that answers like “it’s more demanding”, “it gets harder”, “lack of practice” all seem very student focused and I think it should be less aimed at the student who has a good reason for hating math, and more at the practices which are producing math hate. Kids are willing to work for the answer, but only when they feel like they can engage with a question. Your last answer “when students don’t see the purpose” seems to address this, but I think that should be the biggest part of the answer.

I’m with ryanandmath here - students get introduced to “math” as a series of rote calculations to be memorized, and it’s a losing battle from there.

He hit the nail on the head, so I’d like to delve into a related problem - I say “math” because I’d wager these students don’t actually hate mathematics, they hate arithmetic, computation, hate rote calculation, etc. (I do too!) They - and most of the population - have no idea what mathematics, as a field of study, actually is. Maybe I’m playing semantics here, but it always blows non-mathematicians’ minds that I could spend a day doing math and never see a single number (this is an especially delightful conversation to have with kids). I spend my time looking at rotation groups and metric spaces and homeomorphisms and ultraproducts and all manner of fascinating things, and the tools I’m using to do that are so far removed from what most people think “math” is that it isn’t even funny.

Show someone a bit of actual math - graph theory via the Bridges of Köngisburg, or topology via a Möbius strip, or just logical thinking via any number of interesting puzzles - and then tell me they hate it. 

Cofinal, I agree with you that the higher level maths is the heart of it all, and that if people looked at graph theory and such, they would better appreciate math as a whole.

Ryan, Cofinal, as for the idea that math hate stems from the teacher, I am on the fence with that. I don’t think an engaging teacher can make every student appreciate math, but they can definitely make more of them appreciate math. Perhaps if we consider the base case - yes, this is getting tautological - that the first math teacher a student has, has to be engaging in order to start the student on the right foot. So when the student doesn’t yet have an opinion of math, if they are influenced in the right direction, they might stay on the right path. Whereas, if a student didn’t like math in elementary school - for whatever reason - might still not like it regardless of having an engaging teacher in junior high. This is a crude example, no doubt.

I do see where you are coming from, but my main problem with math hate being teacher-based is that it’s easier to blame other people for your problems. 

nobel-mathematician-deactivated
sixpenceee:

The Tesseract is a fourth dimensional cube. As you may know, the 1st dimension is a line, the 2nd dimension adds width to the line (square) , and the 3rd dimension adds depth (cube). The 4th dimension is impossible for us to imagine because we live in a 3D world, but mathematically it exists.  In his theory of special relativity, Einstein called the fourth dimension time, but noted that time is inseparable from space.
Imagine how confusing a drawing of a cube would look like to someone who lives in a 2D world and has never experienced a 3D world. To them it would be overlapping squares. That’s exactly how we perceive the 4th dimension. We don’t understand how it looks but we can represent it on a 3D world. 
If anyone is interested here are some cool articles on this topic. (Can our brains perceive the 4th dimension?) (The 4th Dimension) 

sixpenceee:

The Tesseract is a fourth dimensional cube. As you may know, the 1st dimension is a line, the 2nd dimension adds width to the line (square) , and the 3rd dimension adds depth (cube). The 4th dimension is impossible for us to imagine because we live in a 3D world, but mathematically it exists.  In his theory of special relativity, Einstein called the fourth dimension time, but noted that time is inseparable from space.

Imagine how confusing a drawing of a cube would look like to someone who lives in a 2D world and has never experienced a 3D world. To them it would be overlapping squares. That’s exactly how we perceive the 4th dimension. We don’t understand how it looks but we can represent it on a 3D world. 

If anyone is interested here are some cool articles on this topic. (Can our brains perceive the 4th dimension?) (The 4th Dimension) 

spring-of-mathematics

spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.

  • A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
  • The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.

See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.

Figure 3: Chess and goose game board at The Metropolitan Museum of Art.

spatialtopiary
ryanandmath:

darkuncle:

flowchart of the relationship of various branches of mathematics - found via twitter (IIRC) but source unknown and unfortunately not listed. If you know the author, please let me know; I’d love to give credit.

I love this, I was just looking for something like this last night. I think it’d be nice to see a couple more branches in here, and have it organized more outwardly so that the center is the core and everything coming out of it builds on that core.

ryanandmath:

darkuncle:

flowchart of the relationship of various branches of mathematics - found via twitter (IIRC) but source unknown and unfortunately not listed. If you know the author, please let me know; I’d love to give credit.

I love this, I was just looking for something like this last night. I think it’d be nice to see a couple more branches in here, and have it organized more outwardly so that the center is the core and everything coming out of it builds on that core.

lthmath
lthmath:

Pierre Deligne is a Belgian mathematician born in 1944. He is known for work on the Weil conjectures (on generating functions derived from counting the number of points on algebraic varieties over finite fields), leading to a complete proof in 1973. What he did is considered the geometric analogue of the Riemann hypothesis. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, and 1978 Fields Medal, making him one of four mathematicians to achieve this.

lthmath:

Pierre Deligne is a Belgian mathematician born in 1944. He is known for work on the Weil conjectures (on generating functions derived from counting the number of points on algebraic varieties over finite fields), leading to a complete proof in 1973. What he did is considered the geometric analogue of the Riemann hypothesis. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, and 1978 Fields Medal, making him one of four mathematicians to achieve this.

Fluttershy - Working In Background