I could continue teaching if I really wanted to, but it has been ridiculously difficult for me especially amidst the pandemic.
I've mentioned this a few times here now, but basically, I went into this position with nothing other than a set of learning outcomes. None of the textbooks I've seen were sufficient for teaching what I deemed necessary, so I spent a few hundred hours creating my own source material, on top of maintenance and catching up with technological changes every semester. Additionally, enrollment for this class has been historically low, so I've had to promote it as well.
I'm proud of what I've done though: I've had first-semester community college students place in contests against graduate-level students. The relationships I've built and the things I've accomplished - especially since this was the first time I taught anything professionally - are what I'll leave with me.
that sounds cool. teaching is really a labor of love, there's not a lot of day-to-day thanks in it. i can only imagine what it's been like with new tech all the time.
i think there's something to be said for it. it's more useful than calculus, and everyone says teach that in high school. but it's easier to teach formulas in a cookie-cutter way than to teach programming.
It doesn't take deep knowledge to learn how to manipulate numbers at scale in a spreadsheet. Honestly, handheld calculators are outdated and IMO people are kidding themselves if they act like calculators are how computations are done at scale.
my daughter actually did subtraction today. she wanted five little crackers. my wife gave her three. she said she wanted more crackers. we asked how many more to get five. we had to ask it five or six times in various ways before she understood. then she said: "two!"
@Astyx The course is a first course and these students haven't done any amount of set theory (so no cardinality). But I prefer this inverse argument. It brings together all the pivot machinery they have been using in their problem sets.
not sure if it's literally a vertical tilde or just a visual equivalent. in graphics packages (i dunno about mathjax) they also have general tools to rotate symbols. i like \sim horizontally, but \wr seems to do an OK rotated job.
at least they preserve the orientation. i dunno what to tell you if you want a left-handed version of one of those.
@Clarinetist i think probability theory can't really be taught in high school, but statistical methods can, as long as the emphasis is put less on proving the CLT and more observing the CLT holds, it seems to work fine
it does devolve into memorizing procedures to do significance testing when students have to sit exams, but at least some appreciation for statistics can be developed
Let $G$ be a finite group and $S$ generate $G$. Let $s \in S$ such that $<s>$ is a normal subgroup of $G$. Now $<s>$ is abelian subgroup. I have been reading a text and there for an element $x$ in a coset of $<s>$, they have talked about an $s^x$. What can $s^x$ mean? Can it be $s*x$, where $*$ is the group operation? They have not mentioned regarding such notation..
sometimes i have seen superscripts used for conjugation (e.g. s^x = x^{-1} s x or xsx^{-1} depending on whether you want to act on left or right), or other for other group actions that are somehow fixed by context.
Hmm, ok. Probably it is the conjugation in this case then. There, they say "since $<s>$ is abelian, if $x,y$ are elements in the same coset of $<s>$, then $s^x=s^y$. I do not get what they mean though.
Maybe they mean by $s^x$ the map $S \to S$ given by $z \mapsto x z x^{-1}$, and not the element $x s x^{-1}$. That would make some sense.
With this notation I think you have $(s^x) (s^y)^{-1} = s^{xy^{-1}}$, as an equality of maps on $S$. If $x$ and $y$ are in the same coset, this map is "conjugation by something in $S$," which will fix all of $S$ because $S$ is abelian. So $s^x (s^y)^{-1}$ will be the identity map on $S$.
i think that's weird notation to say the least, if $s$ is also a generator of $S$. but it makes sense if you ignore that.
Thank you very much @leslietownes Another small question. How did you determine it is going to be "conjugation by something in $S$,"? $x,y$ are elements in the same coset of $<s>$. but does $xy^{-1}$ necessarily be in the generating set $S$?
Is it because the action should be by an element in the generating set?
a good earl grey in the late morning or early afternoon is nice, not too late so the caffeine won't be flowing through the system late into the night....
I could imagine that. I've recently changed my "math study" time to start a little later in the afternoon and not wrap up until around 3-4 am. I find the silence of the night allows for better thinking.....as a result I may sip on a yerba mate late at night
