Conic Sections: a Visual Representation

Here's a video about how to slice a conic section and visually get a parabola, a circle, an ellipse and a hyperbola.
I was trying my best to visually describe the conic sections without the equations.
Music is "Pressure Drop" by The Stanton Warriors.

What do you think? Does it work for you?
Do you need the equations to identify each conic section?

Galois, Lie and Me

I'd love nothing better than to go back to a University and get my PhD in Mathematics. However, right now I can't afford it, both in time and money. So to keep my brain from atrophying, I've researched a few different branches of Math that are beyond the scope of my previous education. I figured that if I could just get my mind working through one idea in Mathematics that I've never pursued before, that maybe just maybe, I'd shake these cobwebs from my brain. And that, in turn, would make me a better tutor, and quite frankly, also a happier person in general.

Some subjects just didn't interest me. The way my brain works, I think in terms of circles, curves and continuity. Binary thinking is harder for me to gasp. But it's fascinating how you can feel compelled, drawn towards certain problems. I stumbled upon Galois Theory, and Lie Groups several times, but felt convinced that I could not understand, well, even the basics. Still, there was something compelling me to look deeper, even though it felt like I was stumbling around in complete darkness. 

I looked through my usual sources: Wikipedia, Khan Academy, journal papers and a variety of other material online. Alas, I just didn't get it, so I put it aside. I started reading about Benford's Law, and discovered Zilf's work, applying it to letters and words, and its applications to cryptography. I can imagine that finding anomalies in large data sets might be interesting and useful. But this didn't fascinate me.

Then, a few days ago, I joined twitter, as @toopretty4math. I started following some of the extraordinary Math Professors I'd found on twitter as @girlmeetsbike. Yesterday @Mathematicsprof tweeted something that changed my mind about the inaccessability of Lie Groups:

 "Robert Gilmore has a clearly written book on Lie Groups online at goo.gl/cphAI "

 I went to Professor Gilmore's website, expecting the same jargon that had been used in Wiki and every other source (it seemed) that I'd scanned. This was different.

Under the heading of the first chapter, Professor Gilmore states:

"Lie groups were initially introduced as a tool to solve or simplify ordinary and partial differential equations."

Thank you Robert Gilmore. I understand more about Lie Groups from this one statement than I have ever understood before. It's clear concise, and it gets me motivated to know more, without using jargon.

I downloaded the first chapter of "Lie Groups, Physics, and Geometry", and started reading. I found myself thinking, "OOOH Oh, that's cool", then I realized that he was using math that was easy for me to understand. I looked at the rest of the chapters (that were available for download), and indeed, he has incorporated both math and language that I understand. For the first time in a long while, I can sit back, sift through, and totally enjoy reading a math textbook. And yes, I'm even going to attempt the problem sets.

Are you interested in Lie groups? If you are currently reading "Lie Groups, Physics, and Geometry" by Prof. Robert Gilmore (or have before) and have some insights, please comment, and let's discuss.

Exploits of Tutoring "old math" in the "New Math" World

It's probably been 12 years since I've tutored anyone in Mathematics. Let me tell you, times have changed.


The good news, is that there are beautifully new ways to look at math, now that computers are there to do our bidding; that leaves us humans with more time to do solve problems by thinking creatively. 
Mathematicians now approach the same old problems in new ways that blend supreme levels of computing with ingenious techniques, but from completely new perspectives. 

There literally is, a new math for me to learn.

But, oh the bad news.


When I see the way that my beloved Mathematics is being taught, I am appalled. 

Teachers seem to be teaching the subject of Mathematics in a piecemeal way, as if you could teach the entire subject of English by giving each student only two books: one with only verbs, and one with only nouns.  In this way, Trigonometry and Geometry have been pulled from Algebra. But let's remember that it's not just the textbooks, it's the curriculum of the school system, that is currently in disrepair.  So it's not just two books, but two entire separate courses (one on verbs, the other on nouns) that are taught, and not concurrently. 
 In Community Colleges, students are given a course in nouns, and then told that if they want to study verbs, they'll have to go elsewhere.
But they can test out of adjectives completely.
BTW, in this scenario, grammar is an advanced course and rarely taken (much like Calculus).


And quite frankly, the language of the textbooks nowadays is about as exciting as a barren wasteland with no road. No wonder everybody hates Math nowadays. I would too.

These students are given no cohesive way of connecting the dots in Mathematics, in such way that it leads to logic, and critical thinking. A Math course should lead students along a certain mathematically logical journey that has a beginning, a middle, and an end.  If we're lucky, the teacher will make some real world connections, and give some motivation about why that's interesting. In other words, the teacher, as your tour guide, is responsible for making your Math journey interesting. Or not.

I've heard a few people say that they're not fond of Math, or that they're bad at math, or that math is hard.  I asked, in return "Who told you that?" Usually, the answer is, a Math teacher said that. Some Math-ey person in authority somewhere in their past, told them that "Math is hard" or that "some people just aren't cut out for Math."

What nonsense. What hubris on the teacher's part. 
Math is easy. 
Your Math teacher just didn't know how to teach well, or make it interesting.
Yeah, I just said that.


Proof to follow.

 Here are a few videos I made for a couple of my students. Yes, I realize that they need work, and I'm still nervous, so I make mistakes in fron t of the camera. (Note that when that happens, I will overlay text to clarify what I'm saying).


Here's a video where I describe how to very easily find the equation for a circle, using Pythagorean Theory. I then begin to describe how to relate a circle to an ellipse.

Here's the quick and dirty way to understand what the heck is the discriminant of a quadratic function.