I've written about how long it takes to become a master teacher and that even after 9 or 10 years most teachers are really just advanced beginners. A big reason for that is our long feedback loop. You do something and you can't do it again for a year.
I was listening to a podcast on my morning run by the Hudson River and something came up about differences between quantities of items. The difference between having n versus n+1 of something changes based on the value of n. If, for example, you have 1 versus 2 shoes - that's a BIG difference. On the other hand, if you have 100 vs 101 shoes, not so much.
This got me thinking about how I framed things when teaching big-O notation. In big-O you generally just look at the dominant term. If you have an algorithm that runs in \(O(n^2+n)\) time you usually just say \(O(n^2)\) because \(n^2\) is the dominant term. If you n is small, let's say 2, the difference between 2 and 4 is big so the n part of \(n^2+n\) seems important but as n gets large, say 10,000, then the difference between 100,000,000 and 100,010,000 is negligible for our purposes.
I try to introduce this concept when teaching big-O but listening to the podcast and reflecting gave me some thoughts as to how I can do it better.
The trouble is, I won't be able to for a while. I introduced big-O to our teachers in our summer intensive. My next opportunity to cover the topic will be sometime this Spring with my first year undergrads. Same topic, but different audience so it won't be quite the same. The next time I get to do this with teachers is next summer. All this supposes that I remember the changes I'm thinking about making and work them in. I'm writing myself a few notes but even then, who's to say if I'm going to remember the nuance that I'm thinking now?
That's the teaching feedback loop - get an idea, try it, and then can't iterate for a year. Get a new idea and you can't try it until the topics comes up which could be a long time from now. True, some things like a new pedagogical trick can be tried out more quickly but even then you can't just do it. You've got to find where it works within your current classes and still have to account for the countless differences that make up your classes when you're ready for the next iteration.
None of this is good or bad, just the nature of our profession. Just something that I was thinking about this morning.