Though I don't write about it much, I've been teaching College Algebra this year. The kids aren't earning college credit for it, it's more of an alternative to Pre Cal we're offering for students that need another year of algebra. The Pre Cal we teach isn't like some places where you're doing Algebra 2 all over again, so our version of College Algebra makes a nice option.
Interestingly enough, I have two very small sections, 18 and 16 kids in each. It's been a long time since I've had classes this small. We had a bit of a population boom since I started and some quirks happened to bring class sizes down this year despite a very high overall population. The biggest lesson I learned from those early years with small classes is that they were totally wasted on early teaching me. I always wanted a chance to do a better job with that kind of environment.
The first challenge was their work pace. These kids are great, but they're all over the place. I developed a technique for dealing with that last semester.
The new challenge became holding their attention. For whatever reason, they aren't interesting in listening to me speak in front of the group in a traditional sense. I noticed this back in 2013 when I reframed Algebra II around lots and lots and lots of classwork. The kids got so used to knowing math class would require them to do something, that listening became problematic. They just wanted the classwork and it was crazy. Same thing here, this College Algebra crew eats up their classwork. But lecturing? Forget it.
How did I fix it? Let's look at a multi-part assignment I gave for inverses:
On the surface, this looks like bad textbook worksheet moves. Steps are labeled. Kids follow the steps. Repeat.
The Mini Lesson
As mentioned, these groups work at varying paces. As the semester has gone on, they've gravitated to sitting with kids who move at their same pace. Generally, I know which table is going to finish first and which takes some time. This frees me to wander, stop, and initiate a discussion.
This multi-part assignment came with three discussion phases. As a group finished a part, we discussed what they learned. In Part 1, we made some observations about the data points from the table. We ironed out the relationship I asked them to observe, many noticing right away that it looked like some kind of reflection. I floated around and had the Part 1 discussion several times, with audiences of 3-6 kids each time.
For Part 2, we expanded the idea. Can entire equations be reflections of one another? Do the observations about points still hold true? Are the points of one equation just reflections of another? Could we come up with a counter example?
Part 3 is a culmination of the first two discussions. Now that we seem to have a definition of inverses in general, can you determine them on your own and check your work with a graph.
My classroom is uniquely designed for this kind of set up. Near each table is a screen replicating my teaching computer. I can carry a keyboard and trackpad with me to manipulate stuff from wherever. I can also make use of their work as we talk since it's right there with us at their table.
I think it went really well. It was easy to get the kids to focus because there's no hiding when it's a group of three, or even six. The kids were eager to share their observations and made some good ones, all catching on quickly. The pacing allowed kids to work independently and have a piece of my attention. In this classwork heavy setting, I have been very free to offer more face time than these kids could ever want.
From an outside perspective it probably looks chaotic. From a breakthrough perspective it's not that special. It's just some scripted tasks leading kids towards an observation. Kids do some work, practice some things, get tested over those things. The key part I think is that the teaching element hasn't gone away. I'm still their main source of information, rather than videos or whatever. I'm still facilitating discussion and offering new ideas even if the tasks aren't particularly exciting, such is the nitty gritty algebra sometimes. And the kids are more than willing to give everything a try, mistakes are no big deal when it's just a few people sitting around a table.
I like where this is going.