The fall semester was very busy for me. Unfortunately, I wrote 0 blogs during that time. I was teaching two classes that I had never taught before, so I had to do all of my course prep from scratch for those in addition to updating my material for my other two classes.

The first class was Discrete Math. I am obligated by my absurd sense of humor to inform you that this is not doing math in secret. It refers to the study of discrete mathematical objects (as opposed to continuous ones - think things that are separated from each other, like integers, instead of things that run together in a continuum, like real numbers). The real purpose of the class is to introduce mathematical logic and proof technique, but that's extremely dry (actually, very interesting! but only to me...) so I taught it using topics in discrete math as a vehicle. It's a very common first course in proof and logic. This was one of my favorite courses in undergrad, and one of the ones that convinced me to become a math major (I was a math minor before I took it). In the class, I taught how to prove things using pure logic starting with basic assumptions. We started with basic properties of integers (odd/even, prime/composite, divisibility) and moved on from there. We covered a wide variety of topics after that, from recursion to function theory to probability and counting to graph theory. It was a lot of fun for me.

The other class was called Foundations of Geometry. You might think the word foundations implies it is like high school geometry, but maybe easier (the foundations are the beginning, so they should be easy, right?). Really, it is a course on building all of geometry from a small set of axioms. We built Euclidean geometry (which is what you learn in high school) as well as much of hyperbolic geometry (which is a very strange place indeed). We also discussed other possible geometries. The strangest thing you might have learned in my class would be that the universe is probably not really Euclidean, but some kind of mix between Euclidean and hyperbolic geometries. This means that some of what you learned in high school geometry is actually a lie. Most surprisingly:

- Given a line and a point not on the line, there is more than one line through the point parallel to the original line (try drawing that to see that it's not what you expect).
- The angles of a triangle actually add up to less than 180 degrees (the larger the area of the triangle, the smaller the sum of the angles)
- There are no non-congruent similar triangles.
- There is no such thing as a rectangle.

I enjoyed teaching both of these classes. I was surprised how much I enjoyed the axiomatic approach to geometry. Basically, we started with the bare minimum assumptions, and then proved that the rest of what we know of geometry (and some stuff we didn't know) follows from those axioms using only pure logic. It was surprisingly interesting, and surprisingly difficult to teach. The main challenge was making it accessible. I spent countless hours on it, which is one reason I wrote exactly 0 blogs.

This spring marks the first semester in which I will not be teaching any classes that I've never taught before. All of my class prep will just be adapting what I did last time to make it better. This would have made it a very open semester for me, but we are starting a couple online classes that I will be teaching. You might think that it'd be hard to give an online class in math. Yep, it will be. We can't exactly just tell students to read the textbook and discuss it in an online forum. Instead, I'll be creating virtual lectures on my computer to closely approximate what you'd get in a classroom. I will make a video where I write on the screen like a whiteboard in the classroom, and narrate with roughly the same lecture I'd give in person. It will take quite a bit of time to create all the videos, but I'm hopeful that it'll go really well. Obviously in some ways it'll be harder for students - they can't interrupt me to ask questions - but it'll also be easier in that they can pause and rewind and re-watch anything they need to until they understand. And, of course, I will be available to answer questions online. This spring I'll be teaching both of the courses described above, but online. Let me know if you want to take one. Only partly joking.

So, enough about teaching. During all of this time, Lucas has gone from stumpy little guy saying 7 total words to basically-full-grown guy who says things everyday that I had no idea he knew. He's gotten really funny, too ("Mommy say wrong word," "Nap zero minutes!"), and to us he seems so smart. I have no frame of reference, but I'm constantly amazed by how much he understands and the complex thought patterns he displays.

One of my favorite things: whenever I'm playing anything with Lucas and he really starts to have fun, he gets all excited and giggly and shouts "Game!" One great game of note: he has me chase him around (or vice versa) and we each pretend to be something ("Dada 'tend piwate cat. Lukie 'tend monkey"). The game is called, simply, Chase. It is his favorite game in the whole world. In ideal situations, it is broken up by "Crash!"-es, which are: HUGS, after which I am assigned a new character to 'tend to be and we chase all over again.

To top it all off, Sharayah is cooking up another little marshmallow for me to play with, coming June 2017.

Yep, I have the best life in the world.

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