I would like to take a small detour for this entry and use the start of the new year reflect on the previous year. Reflecting on the past as a teacher can help us to think about what we might consider for the future. So what were the top lessons learned at AIMS in 2017? I should mention, these are just my own top lessons. I am sure my colleagues would have a different list.
4: Students build from what they know to learn something new.
I have always believed this, but it became apparent this year as we worked with students that the steps between what they know and the math which we engage them in is often much greater than students can connect. This can lead them to blindly follow procedures to get an answer, but not make sense of the math.
3: Student behavior is just a clue to what students know, not what they actually know.
I think we often confuse student behavior for the inferences we make about the source of student behavior. It is only later, when the context of the math changes, that what we thought students knew is not what further evidence might suggest they, in fact, do not know. Read that again. I had to.
2: Our understanding of how students might learn math influences what we observe in students.
As teachers, the more deeply we understand how children learn mathematics, the more refined our observations and the complimentary inferences we make about their understanding become. When I think of math as just following steps, I look for process behaviors and then assume process understanding. What I miss is what they do or do not understand about the process that might limit the use of that process in another context. What I also miss is the limitations on my own selection of tasks for students to do that are outside of the realm of just following a process.
1: We first learn by reflecting on something concrete.
There are two parts to this that are critical. First, we have to have concrete examples that relate to the more abstract ideas we want to develop. In other entries on this blog, we have been talking about tasks that promote multiplicative thinking. The description of the tasks is the concrete example. Second, we have to reflect on that experience. We have to sit back and let our minds think about our representation of those experiences. We need enough of these experiences to be able to remember them clearly enough to then reflect on them and come to conclusions about how they relate and what we might understand from those experiences. This is also true for our students. Concrete experiences and time to reflect are keys to students making sense of the mathematics in which you are asking them to engage.
If you were to make a list, what would be on your list?
I will continue to share our reflections as we work with classrooms and look for ways to engage students in tasks that help them construct ideas about math that will transcend the math of the moment.
Next time I will continue with the lessons learned from our making a game out of our multiplication tasks.