Topology Notes

I’ve been talking about writing a topology textbook introductory notes on topology for years. Basically since I wrote my Rethinking Topology (or a Personal Topologodicy) post 2 years ago — it’s hard to believe it’s been that long!

In any case, I finally started writing it. I’ve done a mild review of existing introductions to general topology (ie. I skimmed through the first few chapters of a dozen topology textbooks), so I feel somewhat comfortable contrasting my work to existing literature. It’s quite a different approach.

Topological Anatomy: Closure, Interior, and Boundary

I initially develop topology based on closures and adherant points. Kuratowski’s closure axioms are then built up with natural explanations. Emphasis is given to the variety of possible definitions (along the lines of Lakatos et al’s Proofs and Refutations) and exercises encourage the reader to explore the variety of possible definitions. I attempt to justify the axiomatic approach in a manner similar to Pinter’s wonderful A Book of Abstract Algebra, though I may fall very short. From here, we build intuition for closure, boundary, and interior with some diagrams and proofs of identities. Finally, we wrap up the first chapter with a visual interpretation of the closure axioms.

The indiscrete closure operator on {1,2}

(You can find the most recent version of the book on github.)

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Rethinking Grade School Algebra

There’s a question I like to ask random people: where is the flaw in the argument that because ? I very rarely get a satisfactory response. Usually the answer is that “you’re not allowed to multiply both sides by zero.” But we can come up with a slightly subtler argument: because . Some just don’t answer, other will insist that its not allowed… To me it suggests something is deeply wrong with how most people understand algebra.

They don’t know mathematics, they know voodoo-mathematics, a series of mysterious steps that result in their test being returned with a checkmark beside the question.

Now it may seem that I’m being a pedant. After all, they know it isn’t true; what does it matter if they can’t tell me why? But even if we set aside the fact that it simply feels wrong to not understand why the math works, it has practical implications because there are cases where the mistake won’t be as overt as above. And then these people won’t see the mistake.

So I’d like to use this essay to go over grade school algebra from a different perspective.

Drumbeat Toronto

Yesterday I went to Mozilla Drumbeat Toronto, an event designed to promote a open Internet.

The Modern Demonic Parody of Education

The Surface of the Problem

I’m angry. Well, actually, I’ve calmed down. But I was angry. I’ll probably get angry again as I write.

Why? Because I was, once again, reminded of the depravity of modern mathematics education and by extension all education.

My sister had a friend over and while we were eating dinner my sister mentioned they were going to do homework, she music and her friend math. Her friend groaned that she hated math. I was, of course, scandalised and asked why.

To begin with, her math teacher last year thought math wasn’t important and decided to just give them a pile of worksheets. This year, well, one question she was working on was turning into a `mixed number’ (a number of the form that must be written in the notational monstrosity ). Well, that’s fairly easy once we know what hoops we need to jump through: it factors to which is equal to which can be turned which is (, *shuders*). Simple right?

Well, she didn’t factoring was, or that you were allowed to cancel things multiplied on the top or bottom. She just removed twelfths from the top until she couldn’t. So, besides doing stupid, meaningless exercises she was doing them with a huge handicap. No wonder she thought math is boring.(Side note: when she asked a teacher for help, she was laughed at.)

This sort of `teaching’ is not unique to mathematics. My sister’s classes for the two previous years learned nothing (or so one might infer from the fact that she could miss a quarter or so of the year and not need to to any catch up. And not have any homework, whatsoever.) in all subjects. This sort of teaching ruins subjects for the student. It isn’t just wasted time, it is damaging.

These miserable excuses of teachers disgust me. They receive one of the most important roles in society: educating children. They are put in a position of power over children and they use it to damage them. They teach them not to be curious, not to explore, not to care about learning, not to care about that subject…. It makes me angry.

I showed her fractals, 3d plots, topology (cup turning into doughnut), knots, transcendental numbers, and natural numbers/arithmetic from sets. We probably talked for about half an hour. At several points she made comments like, “That’s so cool!”

That’s the response every student should have every day in every class.

I refuse to believe that isn’t possible. If it can’t be done for a subject, that subject isn’t worth teaching.

Examples of Education Done Right

The obvious question is `how do we do this?’ I’d like to look at the some of the best educational experiences I’ve had.

• In grade 9,  I had an amazing science class. Part of it was that it was taught well and was my first real introduction to science, but part of it was something else. I had a spare at the same time as the teacher, Mr. Maharaj, and he would come by every 15 minutes and answer any question (except “what is a photon?”) that I had, both during the spare and lunch. So I’d get the basics in class and then read wikipedia articles, intermittently asking questions. I learned a ton.
• hacklab.to Unpatched Tuesdays. Surrounded by a bunch of really knowledgeable people working on really cool projects. Enough said.
• hacklab.to workshops. If someone decides to teach a workshop at hacklab, its fairly safe to that they’re really interested in the subject. In particular, getting taught art (block press printing (1, 2, 3) and knife sharpening) by someone who keeps a jar of human teeth, made a robot to tear up essays, and otherwise embodies the insane(ly awesome) artist is very different than learning it at school.
• University lectures can be very good. But it helps if you’re there of your own choosing and are being exposed to fascinating ideas for the first time.

Common themes:

• Optional.
• Knowledgeable teachers.

Interpret that how you will. I’ll write more later.

Auditing University Courses as a High School Student

As a high school student who is somewhat advanced in certain subjects, I’ve been enjoying auditing university courses at the University of Toronto. Not only have I learnt a lot about the subjects I’ve audited, I’ve also met lots of interesting people and learned a lot about University life.

Up to this point, no professor has refused to let me audit their course. In fact, they’ve all been very friendly, although the ones that teach higher level courses are often concerned that I’ll “confuse myself.” (The professor of 237 was very adamant that math is like a house and needs firm foundations before it can be built higher. He’s probably right… But I think that a lot of people have a firm (enough) foundation before the powers that be notice this.)

So far, that hasn’t been a problem. In fact, I’ve found that most of the content is fairly straight-forward, and the lectures are much more interesting than high-school ones (probably because I don’t already know the content…). The one thing that did throw me off a little were some unfamiliar notations I ran into when I started MAT237 (multi-variable calculus).

I would encourage any fellow `advanced’ high-school students to do this as well.

Math Textbook Update

The math textbook is undergoing a huge revision. The more I look, the more I see that the way I understand things has changed and thus the book no longer reflects it…

I hope to release the next version around Christmas. Of course, this will hardly be a finished version but I’ve come to the realization that the math textbook won’t ever be finished. I’m trying to put everything I know about math in it and I keep learning!