# Teaching as a Primary Instructor. Day 1.

Today was designed to be an introduction / icebreaker sort of day. I passed around a prop, and had the students introduce themselves to the class when the prop got to them. Once I got the prop back, I asked them if they knew what it was, knowing that none of them would. It was a slide rule. I told them it was a calculator, watching for a moment as the blank looks on their faces turn into dumbfounded looks before telling them a little bit about how slide rules were the calculators used before electronic calculators came on the scene in the 1970s. Of course I brought half a dozen slide rules with me today as part of what was probably the geekiest icebreaker ever. I showed them how to use the C and D scales to multiply 2 by 3, and split the class into groups with a slide rule in each group and gave them the task of multiplying 4 by 3. That was a trick, of course, because the opposite 1 is used for those two multiplications. Then I had them look at the A and D scales, using the hairline to line up with the 5 on the D scale. And I had them do the same with 3 on the D scale. They were pretty quick to notice that the operation there is squaring, despite the initial appearance that 5 takes you to 2.5. After asking what else they could expect to be able to do with a slide rule, given that you can multiply and square numbers, I asked them to find the square root of 2, using the slide rules I handed out. I hope they had fun with that. I know I did.

Still in their groups, I had them talk to each other for about ten minutes to round out the icebreaking in a more… normal? way. I wandered around for a bit, answering questions and addressing concerns, and after the ten minutes was up, I handed them their Problem of the Day. I wanted to make it very light for a very light day, so the questions were: What is my name? What are the names of the people in your group? and What is the name of the prop we used today? I had my name and the words “slide rules” written on the board, so it was mostly a matter of, “Are you paying any attention?” After I collected the PotDs, I pointed out the reason I decided to use slide rules as a prop for the icebreakers – logarithms are what make slide rules work, and they will be learning about logarithms later on in the Precalc class portion of their Summer Bridge ensemble. The point of the geeky icebreaker: I whetted their appetites for things to come. I also explained that the multiplication on a slide rule would be addition if the scales were linear instead of logarithmic, which means logarithms effectively turn a multiplication problem into an addition problem.

By then, they started asking questions, like why I like mathematics. I answered that “mathematics is fun (interruption: frustrating) – yes, it can be frustrating at times, but it is still fun – and beautiful. No, I’m serious.” I think any subject, if you go deep enough into it, has some kind of beauty. Math has lots of really cool things you can look at. Some examples I gave were that we can show $\sqrt{2}$ is irrational… primes are 0% of all whole numbers, but there are just as many primes as whole numbers… just some snippets of cool stuff that would be accessible at the precalc level.

There was another question about what’s the hardest math I’ve dealt with. I told them the most abstract I’ve dealt with is Category theory, but the one with most nitpicky detaily nuissances that I’ve dealt with is Analysis. I don’t think I explained very well. I mean, how do you start to outline what’s so cool about Categories or any branch of Analysis when they’re taking precalc? For the sake of having somewhere to start, I reduced Categories down to manipulating arrows, and I reduced Analysis down to the details for why you can do what you will do in calculus. That is a gross oversimplification, and I don’t think I did a very good job of explaining that much. They seemed kind of bored with my answer to that question. Oh well. The remainder of the time I spent with them was answering questions about course materials and the like.

I suppose I should also note some notable things that happened last week.

Last week the other class I was supposed to teach today got cancelled because there was insufficient enrollment. That kind of sucks because it means my paycheck will be half as much. In anticipation of something like that, along with finances already being tight, I filed the FAFSA the weekend before so I could be eligible for student loans. I’ve never taken out a student loan before, so that’s kind of new territory for me. Don’t ask me how I managed to get to the beginning of my fourth year in a Ph.D. program without taking out any student loans; I would just say it’s a miracle. Turns out, even if I get offered student loans, the earliest they could possibly be disbursed would be somewhere in the middle of September. I was able to get an emergency loan through Grad Division, though. That should at least help tide me over. Basically, finances are stressful.

I’m still working on learning Hebrew, and I’m supposed to get a new apartmentmate some time this week. Things feel like they are falling apart at times, but other times I know they are falling into place. Where things land may not be comfortable all the time, but there are definite opportunities for growth from the experiences. Also! A few little notes. The Associator animation is not yet in its final state. It works, but there are things that could be better. The other little note – I finally found the spamomatic filter and noticed it deleted and removed 45 comments already. I don’t know what those comments said, so if you made some comments and they never posted, I now know one more place I have to look, when approving comments.

# Associator

After more delays that I would like to admit, I finally got around to making this. Perhaps some explanation of what’s going on in the picture is in order. We begin with a category with (all) pushouts, starting with some objects and morphisms arranged as chained spans. A pushout is used to compose the two spans on the left. This gives a new span, $L \rightarrow \Gamma'' \Gamma' \leftarrow J$, which in turn can be composed with the span to its right via pushout. This is just one way that we can compose the chain of three spans. If we had built up from right to left instead of left to right, we would have $\Gamma'' (\Gamma' \Gamma)$ at the top instead of $(\Gamma'' \Gamma') \Gamma$.

The associative property is really nice, so it would be good to show those two ways of composing spans are “the same” in some appropriate sense. Spoiler: that sense is “up to isomorphism.” Instead of directly finding them isomorphic to each other, it is easier to show each is isomorphic to some other thing – the colimit of the entire diagram we started with. This colimit will here be called an associator; it is something that both compositions of all three spans will be isomorphic to. I’m just going to sweep under the rug the question of whether that colimit will exist and just assume it does in this post.

So we’ve got the groundwork set, now we need to take advantage of some universal properties to give unique maps that make sub-diagrams commute. The first universal property taken advantage of is that of the first pushout. The associator has arrows going to it from $\Gamma''$ and $\Gamma'$, so there will exist a unique morphism, $\Gamma'' \Gamma' \rightarrow \Gamma'' \Gamma' \Gamma$. such that everything commutes. Going up a step and using the arrow we just made, we can play the same game to get a unique morphism $(\Gamma'' \Gamma') \Gamma \rightarrow \Gamma'' \Gamma' \Gamma$ that still makes everything commute.

To go in the reverse direction, we need morphisms from each of the Gammas to $(\Gamma'' \Gamma') \Gamma$. We already have one from $\Gamma$, and the other two can be built by composing morphisms. We can join two arrows together, tail to tip, to get a single arrow regardless of what category we are in. Now the universal property of the associator kicks in, giving a unique morphism $\Gamma'' \Gamma' \Gamma \rightarrow (\Gamma'' \Gamma') \Gamma$, but there’s a catch. We only know it makes some of the diagram commute. We have to put in a bit more effort to show the arrows involved in the pushouts will commute, too. The first step of this isn’t too bad — the three arrows that don’t point directly at $\Gamma'' \Gamma' \Gamma$ will commute because the morphisms from $\Gamma''$ and $\Gamma'$ to $(\Gamma'' \Gamma') \Gamma$ commute with them by construction.

The universal pushout maps are a bit trickier, but since both legs of the pushouts commute with everything else, the universal maps will commute as well. Without the picture, that sentence might be a bit hard to understand. Look at the picture. They say those things are worth a thousand words. As parts of the diagram are shown to commute with the blue arrow on top, those part turn dark green. The arrows that already commute with everything get highlighted when they are used to show other arrows commute with everything. That should clarify what the picture is doing. Once everything commutes, the arrows turn black again, and we are left with unique maps in both directions at the top of the diagram that commute with each other. Long story short, this gives an isomorphism between the composition of spans by pushouts and the composition of spans “all at once” by associator. Watching the animation in a mirror gives the other way of composing spans by pushouts as isomorphic to this as well.

So this picture together with its mirror image shows $(\Gamma'' \Gamma') \Gamma \simeq \Gamma'' (\Gamma' \Gamma)$, which is really the best we could possibly hope for when pushouts themselves are only unique up to isomorphism.