Wednesday, April 25, 2007

Memory Tools

Yesterday I had passed out a colored sheet of paper and asked my calculus class what topics they want to review as a class and also any suggestions on how we spend our class review time. They were to list a topic, and if their topic was already listed, they were to put a check mark by it, so I could gauge popularity. One of the topics they stressed was derivatives of trig and inverse trig functions.

Today, I listed the 12 trig and inverse trig functions on the overhead and discussed memory tools I had for the derivatives of the 6 trig functions (all the "c" ones are negative; there's always 2 secants and a tangent; cotangent is like the pesky little brother of tangent and wants to be just like him; cosecant ditto to secant). Then I gave them 3 minutes, and they were to come up with memory tools for the inverse trig functions. I didn't know how it was going to work, but there were several good discussions going.

As we met back as a class, the ideas were slow at first, and then I don't know what happened, but as a class they came up with several good ones to remember the derivatives of the inverse trig functions (sin^(-1), cos^(-1), ...):

* if the name has a "t" in it, then there's a "+" (1+xx) which looks like "t"
* if there's an "s" anywhere in the name of the function, it involves a "s"quare root
* anything starting with a "c" has a negative (think grade c-)
* we learn sine and cosine 1st, so those are "1-xx" (as opposed to "xx-1")

Whew! I think those may even help me remember the inverse trig derivatives :).

Wednesday, April 18, 2007


Yesterday I gave a mock AP calculus exam for my students while the rest of the school population was taking the state-mandated exit exam (yay TAKS). Three hours and 45 minutes or so of testing. Oh my goodness. For the most part (aside from doing some chores here and there), I took the BC exam with my students. Whew! I got exhausted from the extended mental challenge, so I can empathize with them when they come back from the test looking drained.

I can also empathize with them on another aspect. This is the first year I'm teaching BC, so it's the first year in a long time (and maybe ever for some topics) that I've seen various things: calculus on polar functions, on series, parametrics, logistic growth, etc. And in my treadmill type of year, I haven't revisited those topics since I taught them weeks and weeks ago. So. As I'm taking the exam and the topics come up, I felt like the students. I remember seeing the topics. I know I understood them when we were doing them. I'm not a stupid person. I just don't remember the details on some of them. I guess I must be human.

Here's to me not thinking of review in class as a "wasted day when we could be learning something new and why don't they review on their own what kind of students are they and I'm enabling them by hand holding them through a review oh my gosh the next thing you know I'm going to have to wipe their noses for them". Here's to me thinking of a review as a necessary part of the learning process that's worth "taking a day for". I could also think of it as money in the bank for the later part of the year when we take many days to review for the actual exam. Maybe it'll be an opportunity to think of memorizing tricks.

Aside: as a class we came up with a good one for memorizing the Taylor series expansion of sin x, cos x, and e^x. The sine function contains the odd numbers of the pattern, the cosine contains the even numbers, and the e function contains all the numbers in the pattern. So. The "i" in sine looks like 1, and that's odd. The "o" in cosine looks like zero, and that's even. And. "e" for everything. Woot woot. It worked as I was taking the test.

Thursday, April 12, 2007

Class Atmosphere

Last night in tap dance class, as sometimes happens, the teacher started teaching us a combination I swear at a revved up speed. She kept showing us the steps (say 8 sounds) quickly, and before I knew it it was over and I'd maybe got one sound. Then she'd do it again and I got the 1st and the 2nd sound. Then again, and I got the 1st, 2nd, and 3rd sound, etc. I was getting frustrated. She was getting frustrated. "Come on people. These are all steps you know. This class is about putting these steps to a different rhythm. If you're not getting it, then you're not practicing your basics at home. You've all seen this before."

I had a teaching epiphany. I eventually got the steps. But she was going too fast for my comprehension. She was chastising us for not practicing. It wasn't that I didn't want to learn and master the steps, it was just that I needed it taught in a different way. I started getting mad at her, and I didn't even want to hear the sound of her voice. It wasn't MY fault, it was HERS, I muttered to myself.

Then I thought about my teaching. I could see myself in class getting frustrated (hopefully internally) with the students as some of the them struggle with basic concepts that we've gone over in the past that I thought should be second nature to them, and they still didn't get it.

I hope to remember from last night's experience that everyone is human, people have a lot on their minds. It's not that they don't want to be successful in class, they just need a diplomatic reminder of past concepts and a helpful hand to guide them to understanding and mastery.

Thursday, April 05, 2007

Moment of Clarity

I'm going along, happily teaching logs and 10^x and e^x and ln(x) and making sure to put in cool examples and practice and descriptions. I especially was proud of my continual stressing to them to ask the question of themselves, "what is it asking for?" when faced with the equation y = "log base something of something".

I made sure to stress that log of a particular base can be rewritten in an exponential form and visa versa, and I said that they basically represent the same information. I also made sure to stress that logs of a base and that base to a power are inverse operations of each other. Can you see where I'm going with this one?

Well. Today when a student was reviewing for a test she casually mentioned that in the equation y = log a c you can just replace the right hand side with a^c because it means the same thing. Argh. Big miscommunication. I hopefully cured her of her nonunderstanding, but how many other kiddies are lurking out there thinking the same thing?

Monday, April 02, 2007

Random Things

I like when I individually tutor my kids after school, and in that 1-on-1 time, it's easier for me to probe their misunderstandings and ask the right guiding questions to get them on the correct track to solving a problem. Then I can use that information later in class because I'm sure that there are others having similar problems.

I like how being part of the public education system and interacting with kids every day forces me to take a deep breath and start anew every day even when I had a rough day yesterday and REALLY disliked a particular kid's behavior. I have learned (mostly :) ) to dislike the behavior and not (mainly) the kid (on a good day). (apparently with all my "(" and ")", I still have some learning to do).

I like how I constantly learn things from my kids. The other day 2 students were talking, and I didn't hear what the boy said, but I heard the girl's response, "wow, that was really gross. I don't know how to respond to that", and how she put it back on the boy instead of being embarassed.