Ambitions for teaching

In a month and a half, I will be teaching a class for the first time ever. I’ve taken a teaching seminar, and spent the past semester designing a curriculum myself, and listening to experts tell me about teaching gifted students, and I’m terrified. I’m teaching a summer class for gifted high school students that I’m calling Much Ado About Knotting, because nothing is better than math puns. I get to design the entire syllabus myself, write the lecture notes, prepare the TA, and then I will teach, for 6 hours a day, for 3 weeks. Did I mention I’m terrified? Talk about jumping head first in a pool of cold water.

I have a lot of fun knot theory planned out for my students to learn. They will prove Reidemeister’s theorem, study some interesting knot invariants, look at some applications to areas like biochemistry, medicine, and cryptography, and finally we’ll look at my favourite thing: graphs of knots. There will be problem solving, and theorem proving, and some playing with a computer program called KnotPlot. (You should check out the website, it has some of the most amazing pictures of knots I have ever seen). Along the way, students will learn about prepositional logic, equivalence relations, some group theory, some number theory, some mathematical cryptography, and tons of other fun things.

I’m planning on teaching the class using the Moore method modified with think-pair-share. With the Moore method, the instructor only provides students with definitions, and then they get to work out all problems and examples themselves, and present them in class. Since I have the students for full days instead of 3 hours a week, I’m modifying the method to let them work in class. One of the teaching methods we have learned is effective, is think-pair-share. Students get to think about a problem for a few minutes, then pair up and talk about their idea, and finally share them with the whole class.

Everything I shouldn’t do first day of class, from spikedmath.com

I want to use this class to teach them not just a little knot theory, but how to think like a mathematician, and a little about what math is really about. So many of the students I’ve seen in tutoring think that there is just one correct way to solve a math problem, and their job as math students is to memorize that way. This of course leads to major problems when they come across a problem that looks different from the sample examples in the book. I want to show them that math is a set of rules to structure your thoughts, not a set of algorithms. I expect knot theory with me will be quite a shock for students who have spent their lives in an educational system that punishes students for using a different, but equally correct, method than the teacher. Instead, they need to realize that there are many different ways, most of them correct, and each has advantages and disadvantages. I want to teach them to figure out for themselves what is best in a given situation.

Did I say I’m terrified to teach this class? I am, but I’m also excited, and curious to see how many adjustments to my careful plan will be necessary when it meets real life humans. I hear teenagers can destroy the most thought out of plans. I’m also very exited to see what happens this fall when I start TAing a calculus class. I won’t have to jump headfirst into that without teaching experience. And I’ll tell you more at some point about all the exciting ways I want to change my recitations from traditional recitations.

Mindful practice

Oh impostor syndrome. I still feel like everyone else in my program is just gliding through the classes, and will ace the prelims in August, whereas I am struggling with most of it. And then I have a sudden breakthrough when I finally figure out a homework question on semidirect products that no one could figure out during the semester, and all the feelings of inadequacy go away. For a few hours, or maybe a few days, only to come back again. So I’ve been reading a good book on what connects successful people over all disciplines: math, music, sports, business, and anything else you can think of. It’s called “Talent is overrated” by Geoff Colvin. From the title, I imagine you have some idea what it might say. The author goes through the research on successful people from various disciplines, and discusses some very interesting findings. For example, there is no evidence that what we call “talent” even exists. Mozart started preforming at a very young age, but he had already been trained by his father, one of the top music teachers in Vienna, for years before he started. Those of his works that are still preformed today were composed when he was around 20, after 17 years of training.

The same pattern is found for all other successful people. Tiger Woods trained from a very young age, much harder than anyone else, and thus became a golf prodigy. Simply because he had more practice than anyone else that young. Not because he had magic golf playing genes. Simple practice. Benjamin Franklin spent years practicing how to write, and emulating famous writers, before he wrote any of the things he is now famous for.

This is where you object “but write 3000 word for my blog every week, and I’m not a professional writer”. That objection get at the heart of the problem: what sets successful people apart from average performers is what is called “conscious practice”. Playing the same piece over and over again on the piano will not help you improve, no matter how many hours you spend with it. Sitting down for an hour or two every day with a specific goal in mind: “today I will practice the transition between chord X and chord Y” will help you improve. Spending hour every day mindlessly playing chess will not make you a chess grad master. Spending hours every day analyzing games played by masters and your own games for advantages and weaknesses will help you get better. This is the reason ballerinas spend hours every day at the barre, practicing the same developpe they have been doing for 20 years.

The idea is that being mindful and practicing the basic, underlying skills of your trade will make you better at it: much, much better than simply mindlessly trying to dance that pas de deux, or playing chess over and over again. What separates the experts from average performers is their absolute mastery of the basic skills of their trade.

I sure hope that’s conscious practice (curtesy of SMBC).

So, how does this translate to math research? The basic skills needed are a deep knowledge of your particular area of interest, as well as any closely related fields, and a knowledge of the current research. I’m a graph theorist, so for the basic knowledge of my field, I have a few books to work through: currently “Graph theory” by Reinhard Diestel. Every day, I spend at least 30 minutes working through the book. I don’t focus on exercises: I focus on parts where I can get immediate feedback. I make sure I understand the text, try to predict what would be the next natural development, and try my hand at all of the proofs. I read the theorem or lemma, try to prove it, and then compare how my (sometimes only partial) proof compares to the one in the book. Which one is more elegant? easy to understand? This gives me immediate feedback on how I’m improving as a mathematician, and how well I’m internalizing the key methods used in my field. To supplement this, I also keep up to date with current research. Every week, I pick a newly submitted paper from ArXiV to read. I spend a half hour every night on this paper, with special focus on the proof technique(s) they used. After all, proof techniques are the tool kit of a mathematician.

I’ve only been adhering to the principle of mindful practice for a couple of months, but I already feel more confident in my weekly meetings with my adviser, and during the department graph theory seminar, because I have so much more background knowledge. As for how it will work out over the long term, I will keep you updated!

Hall’s Marriage Theorem

Let’s talk matchings! A perfect matching of a graph is a set of independent edges such that they cover the entire vertex set of the graph.

Now, let be a bipartite graph with vertex partitions and . The question Hall’s marriage theorem answers is: what condition(s) on are sufficient and necessary for to have a perfect matching? First of all, it is obvious that for any , a subset of say A, we must have that |S| is at most equal to the number of neighbours of the vertices of S, denoted N(S). It turns out that this criteria is also sufficient. I’m going to present my favourite inductive proof of this theorem (all credit to Graph Theory by Reinhard Diestel).

Borrowed from Dinosaur Comics, my favourite way to waste time

Theorem. (Hall, 1935) Let G be a bipartite graph with bipartition (A, B). For any $\latex S \subseteq A$, denote by N(S) the number of neighbours of the vertices in S. Then G has a perfect matching if and only if for each , we have .

We already have that the condition of the Marriage theorem is necessary. So in each proof, we only need to show that given a graph that satisfies the marriage criterion, we can find a perfect matching.

Proof. We apply induction on |A|. For |A| = !, the assertion is true. Now let , and assume that the marriage condition is sufficient for the existence of a matching of A whenever |A| is smaller.

If for every non-empty set , we pick an edge and consider the graph . Then every non-empty set satisfies:

$|N_{G’}(S)| \geq |N_G(S)| – 1 \geq |S|$

so by the induction hypothesis G’ contains a matching of . Together with the edge ab, this constitutes a matching of G.

Suppose now that A has a nonempty proper subset A’ such that |B’| = |A’| for B’ := N(A’). By the induction hypothesis, the graph contains a matching of A’. But G\G’ satisfies the marriage condition too. So by the inductive hypothesis, G\G’ contains a matching of A\A’. Putting the two matchings together, we obtain a matching of A in G. Q.E.D.

Mathematics, poetry and beauty

Peter Cameron's Blog

Comparing mathematics with poetry is an infinitely rich game. For every opinion you express, there is an equally valid counter-opinion. Contrasted to Hilbert’s dismissal of a student who had left mathematics for poetry, “I always thought he didn’t have enough imagination for mathematics”, someone said to me recently that the early death of Schubert was a greater tragedy than that of Galois, since what Galois could have achieved would sooner or later be done by someone else, whereas Schubert’s potential was lost forever.

So it isn’t so surprising that a book by Ron Aharoni, newly translated into English, doesn’t come to a definite conclusion one way or the other. The best we can do in a book entitled Mathematics, Poetry and Beauty is to give many examples of beautiful mathematics and beautiful poetry and discuss what the similarities and differences are.

Ron Aharoni is a mathematician whose field is combinatorics…

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Normal spanning trees and swirl cupcakes

Today I had my last official class of my first year, a topology class. We talked about homology invariants and ate the chocolate vanilla swirl cupcakes I made for the occasion. It’s weird that I don’t have any more classes left. I almost even miss analysis homeworks. I have plenty of things to keep myself occupied until August though. Tomorrow I have a mandatory research ethics course to attend. 8 hours of listening to someone talk about how we shouldn’t experiment on people without the permission of the ethics review board. I imagine it will be moderately interesting, at best. Well, unless I come up with some experiment like these two mathematicians, courtesy of Spiked Math.

Today I want to share some interesting graph theory. Given a connected graph G, we can find a spanning tree T. This can be proved by simple induction. However, it is also true that we can find a normal spanning tree in any connected graph.

Given a tree fix a vertex called the root, r, of the tree. Next, we define a partial order on the vertex set of G by if x is in the unique path from r to y. This is the partial tree-order associated with r.

We call a rooted tree with root r normal in G if every pair of vertices that are adjacent in G is comparable in the tree-order associated with r.

Theorem. Every connected graph G has a normal spanning tree.

The reason I wanted to share this result with you is because it has a beautiful inductive proof where we induct on the number of edges. Think about it for a few minues before you scroll down to see.

Proof. For the base case, consider a connected graph on one edge. Clearly, this graph is , which is its own normal spanning tree. Now suppose every graph on n edges has a normal spanning tree. Then there are three cases for the inductive step: if we add another vertex, then the new normal spanning tree will be the old one joined with the new edge. Since there are no more edges connected to the new edge, this is still a normal spanning tree.

In the second case, we add the new edge to the old graph, but it does not affect the partial order on the normal spanning tree, so we are done.

The final case is the tricky one: suppose that the new edge connects two vertices that are not comparable in the tree order. In this case, we create a new spanning tree by adding in the new edge, connecting two branches of the spanning tree. Then we pick one of the original branches and delete the edge in it closest to the root of the tree. Then we have a spanning tree again, and it is normal. Q.E.D.

Oh, and those cupcakes I mentioned? I used a recipe I found on JoyOfBaking. This is one of the best sites I have ever found for basic recipes, like vanilla cupcakes, brownies, red velvet cake, peanut butter cookies, and so on. I highly recommend it. I separated the frosting in two bowls, melted a handful of dark chocolate chips and whipped into one of them to make half chocolate and half vanilla frosting. And here’s the end result:

Place a teaspoon of each frosting on the cupcakes like this

Use the tip of a butterknife to swirl the colours into each other

A closeup of the result