# Good vs. Evil

Dudes, this post has layers. Like an onion. Or an ogre.

Also apologies in advance for formatting, I’m too lazy to figure out math type.

First layer. When we got back from winter break we started our Rational Functions unit. We talked about how to simplify rational expressions and we talked about restrictions on the domain, and then we started to talk about asymptotes. When we first introduced the idea, we showed it through transformations with simple reciprocal functions. 1/(x+4) – 3 has a vertical asymptote at x = -4 and a horizontal asymptote at y = -3.  If you’re doing (h, k) form (a la Glenn) then it turns out the asymptote is usually y = k.

Our mistake here was that we weren’t talking about WHY the horizontal asymptote was y = -3, we just asked students to look for patterns. I think the patterns are useful as shortcuts, but students didn’t understand why any of that was actually true. Which I realized the second we showed them real rational functions and they all wanted to tell me the horizontal asymptote of (x – 3)/(x – 5) was y = 5.

We followed up a lesson with a sort-of discovery lesson about finding the horizontal asymptotes of different rational functions and students generating those stupid “Degree rules”.  You know the ones, if the degree of the thing is greater than the thing then the asymptote is this other thing. The only reason I remember them is because I just taught them. And I know when I was in school I could never keep the rules straight and really didn’t understand it (which is just one of the many reasons I wrote off Algebra 2 as pointless when I was 14).

What should come as a surprise to no one who actually thinks about how students understand math, my kids all bombed the test. Over half of my students failed (although in their defense, not just because of the one question on asymptotes).  So in a panic, I decided to offer test corrections. It was a huge mistake that I will blog about later, but long story short, I wound up having to explain asymptotes one-on-one to about 30 different students. They all knew that HAs tell us end-behavior, but they’re not clear what that means other than “When x is really big.” When I first taught it I had described “fuzzy math” where we replace x with infinity, but that seems too abstract in hindsight. This time around I started to say x = 1,000,000. It’s a sufficiently big number that kids are in awe of it, but also fairly easy to work with. I go through some examples and they kind of get the idea I think.

My big concern, though, is that it won’t stick. They have a week to their midterm and 5 months until the State Test, so I was trying to figure out a way to make it memorable, and I came up with the analogy of Good vs. Evil.

This picture is so freaking cool.

The numerator is the forces of good, the denominator is the forces of evil, and they’re fighting for control.  Kids are pretty good at understanding if the numerator is bigger than the denominator, but they lose track of what that entails. So I phrase it like this: If the numerator is more powerful than the denominator, then good beats evil, and we all live happily ever after without any restrictions on our lives (No Horizontal Asymptote).  If the denominator is more powerful than the denominator, then evil wins and burns everything to the ground and we’re left with nothing (HA: y = 0).

It gets a little trickier when the numerator and denominator have the same degree. If I have something like 20x/5x, and you ask “Which is bigger, the numerator or the denominator?” I always get “Numerator” but then I have to say “But is it much much stronger, or only a little bit stronger?” Because they’re fairly well balanced, I always say that they fight to a standstill.  The place of the standstill is decided by the leading coefficients.  I think next year I’ll have little chibi images of Yoda and Darth Vader in the numerator/denominator to make it more visual. It could also work with Harry Potter or anything else where there’s good and evil.

I hang a lantern on the whole thing at the beginning, noting how cheesy it is. But I say that if it works then that’s awesome. And some kids tell me it has. I’ll let you know after I grade their midterms this week.

But here comes the second layer, which is more pedagogical: I’ve been conflicted the last week trying to figure out if this counts as a trick or not. It’s been my goal this year to nix as many tricks as possible. I’ve trained my kids to the point that when I said “FOIL” this week they all said “NO MR. B”. So now I’m trying to figure out if this counts. My gut tells me if I still present the topic with large numbers then my method becomes more of a mnemonic. Yet here I am stressing out about it. Because obviously as a second year teacher I need more to stress out about.

Being reflective on teaching practice is great, usually. It helps me to improve my method and better reach my kids. But sometimes it’s so exhausting. I tell myself you can’t fix everything in a day. Bleh.

Anyway, tomorrow is another day, and I have to go grade some quizzes.

# Exorcising Teacher Demons (or “I’m going to vent for 600 words and you don’t have to read it”)

My kids are lazy, selfish, and possibly stupid and they don’t appreciate anything I do.

They’re also hilarious, brilliant, and weird in a good way, and I’m completely failing them.

Which of those I believe at any given moment depends on which demons have been whispering in my ear at any particular point during the day.  It’s like the whole cast of Inside Out is whispering in my ear all day long.

From left: “You can’t teach” “You’re not doing any good” “Teaching is amazing!” “STOP TALKING SO HELP ME GOD” “Ugh these kids”

Obviously the truth is that my kids are people, and people are complicated. Especially people who aren’t quite done cooking yet.  Teaching undercooked people is tough. Teaching them a subject that it’s apparently socially acceptable to be bad at is more tough. And actually giving a shit makes the whole thing extra tough.

I’ve been having a rough couple of weeks. Despite my almost week and a half off for winter break, I’m exhausted and stressed out. Assessment results tell me my kids aren’t learning, and inter-visitations have been more constructive than supportive. Which is good, on the one hand, because I know I have a lot of things to work on. But it’s exhausting, too.  So I weave back and forth between “I’m doing my job, the kids are just too dense to get it” and “I’m awful at this job, these kids should have someone else.”

Neither of those things are true, of course. Or at least not completely true. There are kids that are difficult and should care more, and there are definitely things I could be doing better. It just gets harder to realize that after your fourth class in a row of blank stares and off-topic conversations. Why are you not working? Why are you not getting this? Why are you completing the classwork and homework but failing my tests and quizzes? Why are you telling me you can’t make-up my quiz today because you have to make-up your History quiz? Why didn’t I remember to follow-up with the four students who were supposed to come for help and didn’t?  Why did I think this explanation would make sense to anyone?  Why do these kids ever listen to me?

It’s really awful sometimes. I love my job, and I love my career even more, but it’s still awful. Especially when it’s cold and I’ve put on weight that I don’t have the energy to lose and blah blah blah. I know it sounds like I’m whining, but I’m really just venting. Today we did a Consultancy protocol in our new-teacher meeting, and I was on stage. My dilemma was “I want to create a safe space in my classroom” and the conversation turned to “Well it sounds like he just hasn’t set up a good classroom culture”.  Thanks team.

Part of me keeps telling myself “You’re doing the best you can, and it’s only your second year, and your first year was unspeakably bad, so don’t expect the world.” Then the aforementioned demons grab the voice of reason and wrestle him back into a closet so he can’t spout any more of his nonsense.

But my lovely co-planner had a point: “If you were asked to come up with 10 things that you’re doing well this year, you could do it”. So that’s what I’m going to do to exorcise some of these demons.

10 Good Things:

1) My kids are now so afraid of killing puppies or kittens that they don’t fuck up distributing exponents or simplifying rationals. Seriously, only two students out of 112 incorrectly tried to simplify part of a polynomial on the last test.

2) Several students came to me to ask for recommendations, telling me I was the best teacher they had this year because I explain things in an engaging and fun way.

3) I can get the class under control by counting to 3. And when I make the universal “Shhhh” gesture they actually stop talking.

4) I’ve made some very good lessons for the Algebra/Geometry hybrid class I have.

5) I’ve been told by several members of my department that I’m a good addition to the department and a good fit for the school.

6)  Kids have started saying “Divide to one” instead of cancel, and they know that if they say FOIL or “Plug In” I will jump on them.

7)  My “Cards of Destiny” cold-calling system has improved who I hear from. I’m not getting 100% participation but it keeps me honest about who I call on.

8)  This year planning takes me maybe two hours if I’m being super lazy, rather than 6 hours of stress that doesn’t even pay off anyway.

9)  I have much more of a balance of “I’ve had a long day, so I’m just going to sit and watch TV, and this grading will have to wait until tomorrow” than I ever had last year. Granted I’m always behind on grading, but I’m at peace with it.

10) I’ve developed a reputation in my department as being super well connected with different resources thanks to the awesome Math Twitter Blogosphere (and my Professional Learning Community has increased dramatically)

This doesn’t kill the demons, just shuts them up for a while. I have to grade some quizzes tonight which will wake them up again, but it’s the little things.

Until next time.