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| Breaking counters into stacks to think about factors |
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| Using geoboards to create rectangles and think about factors. At this point it had devolved into a bit of free time. |
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| Playing multiplication memory |
This Saturday we are hosting a conference called "Free School 101" where we will share with participants the ins and outs of how our school works. I am *not* giving a workshop about math at the free school, which is great because I feel like I am just getting a handle on all the possibilities that are available in teaching and learning math in a Free School setting.
I *am* setting up a table display with posters and some samples of kids work. So I have to figure out what to say about our math curriculum.
It says something about the way math has been prioritized here that I, an intern, am the de facto math teacher. The two teachers who work with younger elementary grades do math with their home room classes but for grades 4 and up I am it. It seems a little shocking to me, coming from a school where kids did math for about two hours per day, that math was not necessarily a regular activity for most kids in previous years. On the other hand, kids here learn all kinds of things that kids at my old school did not - cooking, music, art - many of which include math.
For the most part I have undertaken to plan and execute my lessons on my own. I have a lot of autonomy and that is ok with me. But it does mean no one gave me any outlines for what teaching math at the Free School should look like.
At the beginning of the year
I thought about what I value when I teach math.
- I want students to learn to solve problems that take a variety of forms and types - I want them to learn to think flexibly.
- I want them to use logical reasoning when they solve problems. Especially at the Free School where students naturally want to understand they why of everything, I want students to learn to do math based on their own thinking, not recipes.
- I want students to develop the habit of perseverance.
- I want students to get in the habit of looking for patterns and testing conjectures.
- I want students to learn to connect concrete representations of numbers with abstract ones and work within systems of abstractions.
Okay, that's a long list of goals. Where to begin?
I started all three of my classes of older students doing similar things - multiplication and multiples. I tried to get a sense through informal assessment of students' abilities.
In fourth grade, it seemed like we needed to keep practicing multiplication, so we did times tables, multiplication memory games, multiplication array pictures, and word problems. After a couple weeks of that I also started to teach kids how to find factors by making arrays, drawing rectangles on dot paper or creating them with geoboards, and using skip-counting. On any given day we will do two or three activities, a mix of word problems, hands-on explorations, games, drills, and worksheets
(Aside - I am not anti-worksheet as such. I think sometimes kids just need to practice a skill. I think it's a mistake to do worksheets to the exclusion of other, richer forms of teaching.)
The menu of options is similar in my 5th/6th grade class but we are covering topics at a quicker pace and have already done least common multiple and greatest common factor. The mix of activities skews more toward word problems and cooperative problem-solving.
I am finding that the students complain just as much as those at my old school when things are "too hard" but they have a greater ability to improvise strategies and try a variety of methods - a more robust bag of conceptual tricks. These may have been taught to them but I also think Free School students tend to be flexible thinkers in general because they spend so much time in self-directed play.
In my junior high class we have now begun a unit on fractions. We do hands-on activities less frequently - once a week instead of nearly every day. The students in that group have much more well-developed number sense and can understand new concepts abstractly rather than always needing a concrete grounding with manipulatives.
So how is Free School math different than good math teaching anywhere else? Sometimes it is not. I am re-reading "Teaching Problems and the Problems of Teaching," a UTEP favorite, and finding it to be a great guide. I do feel that there are fewer limits on my teaching here - my classes are small, the students are generally pretty motivated, and I do not need to lay out my scope and sequence with the goal of getting through as much as possible before the state tests.
We are taking longer than I expected with certain topics because the students need more time to master them. There have also been moments where I teach and topic and know that it has not been mastered but we move on anyway because I can tell the topic is not compelling to the kids and I don't believe it is one of the most important things for them to learn. An example of a topic we gave up on was greatest common factor in 5th/6th grade. I know it will come up again when we do fractions and it will be a lot more useful to them then, so we don't need to master it now. I have a lot of flexibility in what I am teaching. But I don't ditch a topic just because students are struggling a bit - if I
feel that the struggle is interesting and productive I encourage
students to keep on.
That flexibility is a bit worrying to me too, though. No one at the Free School ever talks about accountability, which is refreshing, but in practice it means that there is no formal system by which we track students' progress or communicate with them or their parents about that progress. Some parents and teachers take a very relaxed attitude about learning math - they believe it is important but they accept that every student learns on their own timetable. This is awesome but sometimes I worry that students could go through school without getting the basic skills that they need. I don't know whether it is my place, or whether it is possible for me, to try to create systems of accountability. I do know that it is my place to care that students are learning, but how much learning is enough?
Ultimately, I do not have total control over what my students learn, because at the Free School kids have the option of not participating in a given class or activity. I have been thinking a lot about "opting out" and so I want to write a full post about it later this week.
Readers, please feel free to ask me questions about teaching math at the Free School, either in the comments or via e-mail. It will help me clarify my own thinking.
