Finally teaching makes sense in my life. Teaching is so much fun now that I know I only have a few more months of it left. (Of course, I’m sure I’ll be involved in teaching, tutoring, etc) in some regards forever.
I wake up in the morning and walk into my school bubbly happy because I got into Berkeley and am already having amazing conversations about education.
I’m just warming up for this post, lots and lots has been happening in life.
Big concepts…. emails with my possible future PhD adviser who is a brilliant and incredibly energetic math ed researcher. I wrote to him about my ideas about designing an interactive online math “book” that allowed kids to interact in ways that best suited their needs. He wrote to me that “perhaps you’ll be surprised to find out(or perhaps not) that researchers are investigating a lot of your ideas). Sentences like this make me feel like I’ve finally found my field. Of course, questions in math education are not as complicated to pose as questions in math. What type of textbook helps kids learn the most? Quiet rows or lively discussions about everything from math to what’s being served for lunch? What type of teacher training? What order should we put things in? How should we introduce integrals? What things can teachers do to be effective? Am I effective!?
Even though the questions are easy to pose, I’m not sure that everyone thinks like this. As I read papers on math education I can envision being able to write them. It’s a good feeling to think that a PhD is possible in this subject because you have to have a lot of good ideas to write a great dissertation. And I’m just so excited about it.
I’m reading about transfer of learning for Berkeley and thinking about how I set up lessons in the class. This week I’ve told my students that integrals were useful in a wide variety of subjects and showed them web pages from physics, statistics and photography that included integral signs before teaching them what the integral sign meant. One student said that he felt like he was learning one of the fancy symbols you see on blackboards in movies about genius children. And they are, it’s true. But it’s also just area.
The things that I’m saying to students are changing subtly as well. We are finding area under curves using a variety of methods and I have not really spent any time on direct instruction of Riemann sums. I gave students a problem asking them to find the total distance traveled by a car if you measure the velocity at a number of intervals. The next two questions involved finding the distance again but from a graph and from a function of velocity.
Kids were able to come up with all sorts of ways to solve the problem. They did left and right approximations, midpoint approximations, trapezoid approximations, counted squares, estimated the error and averaged various methods. When they were trying to find area under a linear function they used the area of triangle formulas and when the car was driving backwards they were able to interpret the area under the x-axis as a negative distance.
They could relate left and right hand sums to the graph.
And when I say, I mean that at least a few people in each class were able to figure these things out. I tried to select speakers based on their ideas and have them present on the board to get a picture of all of the different methods we’d come up with. I found myself having a hard time not stepping in while they explained because they struggled to explain concepts they’d just come up with. In both classes kids came up with the idea of making smaller rectangles to find a better approximation and connected that to limits. One said “I don’t like where this is going. Can we all drink a big glass of water before seeing where this leads?”
I purposefully didn’t introduce notation about Riemann Sums the first two days on integrals because I didn’t want them to lose their own agency. I want them to understand that they can make sense out of this without the formulas that describe it and without subscripts and sigmas. I am hoping that the conceptual understanding of what is happening will motivate and help them to learn the notation when we get down to the details next week.
I found myself saying “If that is logical, then it is correct” to a lot of students who’d come up with a variety of other ways to find area. I justified my focus on left and right hand sums because I said that they would be easier to write notation for later.
In Related Rates news, two students are trying to use the ideal gas law to relate the change in temperature, number of moles of gas, and pressure together. The Calculus part of the problem is easy in the sense that the formula is not hard to differentiate. Deciding which parts of PV=nRT are constants and variables is really tricky and required the help of the Chemisty teacher. She is also buying dry ice for class tomorrow so that they can conduct their experiment.
It’s amazing how difficult it was for me to use Calculus to solve a problem that wasn’t from a book. When there was no obvious way to decide which peices were constant and I had to look to the real world to decide the problems became much more difficult. It made me realize that I have a lot of science to learn about if I ever want to be really effective in creating problems with real context and real applications.
A lot of my students are not sure if they have a correct answer because the problems they created ended up so complicated and hard to verify based on the knowledge we have. I reassured them that I was impressed and asked them to reflect on their process. the group who collected data on the rate of change of a lunch line realized that doing a regression on their data was not going to work and is going to write about how the problem they invented is way too hard to solve using the tools they know. They ended up inventing a simpler formula to model their situation.
I didn’t expect the result of this lesson to be that students realized how difficult it was to actually solve real world problems. As a teacher, I still feel like I don’t really have a concept of what real scientists actually do with data and formulas. There are so many additional variables and considerations when a problem is not from a book. I find that my solution is to keep making things fit into a format that makes sense to me. An equation. An oversimplification. A cookie is a hemisphere. A line of people is modeled by a log graph.
In Geometry I’m struggling a bit with my introduction of proof. We are reviewing words from Geometry A and learning how to set up proofs. I don’t love class because it feels really teacher driven but I don’t want to take forever learning these words through exploration again. I honestly have no clear idea about how I’m going to teach proof. Some kids get what I’m saying about not using what we are trying to prove as a reason and some don’t. For the kids who don’t understand the basic logic I try to make examples that are simpler about cat’s and dogs and how many legs they have(if 4 legs then cat is false) but I’m not sure if that is effective. After getting the class to help me create a proof on the board I had them discuss all of the steps. I’m not sure if having a class help me create a proof is better or worse than telling it to them. Things seem more linear and clear when I’m the sole arbiter of what goes on the board. However, I don’t want kids to get the impression that they can’t actually figure these things out on their own. When they lose that sense of agency, I feel like I’m not accomplishing my biggest goals.
I keep wondering after I say something like “you can’t use the conclusion as part of your proof” if kids really understand. Often the statements they use are things that they know are true because a teacher told them. For example “If a shape is a parallelogram, then opposite sides are congruent.” They know this from Geometry A and struggle to decide which peices they can use and not use. And I havn’t even tried to touch on axiomatic systems(largely because I’ve read about the levels of Geometric thought and realize that my kids need to do proof first.)How do I explain that at the beginning you need to prove this property and then later you can use it in a way that actually makes sense. Prove them in this order because I handed out the packet in this way seems too arbitary.
The other thing that I’ve noticed lately that is probably occuring for a whole host of reasons, is that I’m actually running and bouncing around the classroom. The jokes are easier to tell. I do ridiculous things. I’m like an actor who is living in the moment and channeling my impluses into my body and words. I’m more me and less of a script than I’ve ever been before. I’m not trying to please my administrators, or think about how my actions today might affect my students behavior when I’m observed tomorrow. We went outside two times this week in 8th period because it was sunny and I wanted to go outside and the kids were fantastic. We look up random cool things(only momentarily) because I feel like it. I’m so open about my motivations mathematically and as an educator. I guess perhaps class is less planned and more planned all at once. I’m incorporating learning theories in what I say even as I come up with them on the spot. I think that my answers to some questions will help students see their own abilities in math in the way I want yet those answers are spontaneous. Perhaps the extra thought has given me the ability to ad lib easily and effectively.
The thing is, I have no real way of knowing if me feeling great about my job is actually helping my kids learn more. It might just be making my classroom crazier and less well managed. I’m certainly not being strict at all and I know that could get away from me.
I should wrap this up. To describe one day of teaching could take a book though. And I’m not going to have myself as an experiment for much longer. Who knows if we can generalize from me in any case.
I need to observe more classes, make videos of my class. I feel like I’m already starting my PhD. And I NEED to review my undergraduate math as well.
But, luckily, I have this new boost of energy and happiness and direction. It’s amazing how much positive reinforncement works for me.