Math in groups
St. Joseph's School | 5th Grade | Math | 3/11/09 and 3/12/09 | Present: 24
students, teacher and myself | 3.5 hours field experience including
reflection time
The lessons on Fractions continued this week. I couldn't get enough of what I have been observing so I requested to go in on two days this week. Fractions are intriguing to me. Watching fifth graders try and grasp fractions has made me realize that even today, after all my years of training in math, I still don't get fractions completely. A simple example brought this home to me. A student described 10/3 as "ten-thirds", that it is "ten of those one-third parts". I have not thought of 10/3 that way in many years or ever. I just think of it as "ten divided by three" or "ten by three". Realizing this subtle difference was monumental for me. There is a big difference in thinking of 10/3 as "ten one-thirds" versus "ten divided by three". There is a conceptual difference. In the former, you see that one-third is a part and that there are ten of them in play. The focus is on the one-third part. In the second, you think of ten as a whole number being divided into three parts. The focus is on the ten. Quite different, I think, especially when you are working with fractions.
This and other realizations I had as I watched this fifth graders work so hard to understand fractions has made me realize that I need to go back and do some work on my fractions. I plan to do this by designing a unit plan in Math that makes fractions come alive, at least to me.
When I stepped into the classroom on Wednesday, I noticed the class split in close to what looked like a half. One half sat towards the front being instructed by the teacher while the second half was near the back of the class working on their own. One the teacher's request, I moved to help the half in the back. The group was working on simplifying fractions to their simplest form. Soon enough, I realized this is not easy to do. To begin with, one wonders why one needs to do this. Then there are all these steps that you have to keep in mind, including running divisibility rules to find common factors and so on. Then there is all this new jargon. At once, I was struck by how foreign all this can see to a child. We give our children 2 years or more of 24×7 exposure to language before they are expected to speak it. While expectations are out of place in my view of education, even if I allowed them, surely we would give our children more than a week of an hour of math instruction every day to start speaking this foreign language of fractions. Especially when it isn't meaningful to their existence. I worked with the students, breaking down the steps to get to simplest form, finding myself focused on the mechanics of the process instead of getting to the core of the ideas, wondering why we need the simplest form and how every child can arrive at his/her own understanding of it. It is so easy to focus on the procedures in math and lose sight of the concepts and how each child will internalize a concept in his/her own way.
Soon after, the class halves swapped. The teacher focused on revising the concepts of number equivalency. I myself remained intrigued by the idea of these groups. So, I spent a good chunk of time that day and the next observing what it means to do math in groups. These groups were based on student readiness. The students were encouraged to put themselves in the group they felt most ready to be in. Each group was focused on a different level concept of fractions. You could tell that some students really struggled putting themselves in a group that was at a lower level of understanding than the one they wanted to be in, either because they thought they should know better or because they did not want to lose face in front of their friends. I completely appreciate the teacher's rationale in creating learning groups. It is hard to have every child keep pace with the class. So, the teacher is trying to respect individual learning paces. That said, having groups still doesn't keep the focus on the individual. There is the pressure and all the dynamics of belonging to a group, keeping pace with the group and comparing with a different group. In some ways, it is worse than being in the whole class. This is probably why, Math is done on an individual basis in progressive schools. This is hard to do in a class of 25 students and much easier in a class of 3 students working with one teacher. I walked away with the feeling that about half of each students brain was occupied with thoughts of which group they belonged to. This was more apparent in those students that were in groups of lower understanding but I cannot help imagine it was any different for those who were in groups of higher understanding. They were likely feeling proud of where they were and fearful of losing their place when the next new concept was introduced. Oh, I felt the pain of this teacher. We talked about it forever. She is trying so hard and so well. If only she could work with 3 students at a time!
The next day, the class started out together revising number equivalency concepts. I thought this was a good strategy for bringing everyone together and defusing some of the strong feelings that arise because of the groups. This was followed by a number mixer game in which each student was given a card with a number on it. They were tasked to find the one other number in the class that was equivalent to the one they carried in hand. I liked how this placed everyone at the same level and allowed for partner learning as the students worked to determine equivalency. Examples were shared at the end of the mixer. Then, the students moved into groups. I loved how the teacher worked to de-emphasize the obvious prestige issues associated with being in a particular group. She did this by promising that eventually they would all get to the same destination, just in different ways, at different paces. That this was like walking. Just like it was not fair to expect all of them to walk on the same day, it was not fair that they all learn fractions concepts on the same day. I think the students appreciated this analogy.
The teacher then spent time with the group that was struggling the most to redefine and revise some of the vocabulary they had been learning. She had the sense they a lot of them understood concepts but didn't know how to name them. This seemed meaningful to the group.
I also thought it might have been useful to create the opportunity for breaking down the procedural steps so students could make note of them for themselves. This would especially serve those students who struggle with memorizing procedures.
I have learnt a lot watching a single Math concept being explored continuously. I get to see more of the same this coming week. I am closing with a list of the teaching methods I have seen in practice this week:
Group revisions and practices in which the students teach the concepts learnt
Group games
Partner learning
Worksheet usage
Vocabulary practice and notes
A walk about that included solving problems posted at different places in the hallway — nothing like getting students to move to learn
Individual work on worksheets and otherwise
Writing a fractions story to explore all the concepts learn
I love this teacher and her efforts. She is working so hard!