I have come to understand how important it is that students understand the relationships between number operations and how they are all inter-connected – addition, subtraction, multiplication and division. I believe that with only a strong understanding of these connections that our students will have real fluency with number. I find myself making this more and more explicit when I am working with students. I don’t want them thinking that we are “doing multiplication” and then moving on to “doing division”. I want them to be inter-woven and for the students to seek out the connections and relationships.
I visited Anna Nachbar’s grades 2&3 class at McNeely and they had been learning about multiplication and beginning to make the connection to division. I began with a mini-lesson modelling the two types of division. Although when presented with a purely number-based division fact, it doesn’t really matter which type of division a student uses to figure out the answer, being aware of both approaches to division again builds fluency. Contextual problems that have a story behind them often lead students to solving a division problem using one way over the other.
When working with primary students I use the terms sharing and grouping division because I think they are clearer in meaning than the more mathematical terms of partitive and measurement. When looking for clarity around number operations, a key resource I go to is the recently updated Children’s Mathematics: Cognitively Guided Instruction by Thomas Carpenter, Elizabeth Fennema et al. Most school libraries will have this excellent book and CD. Here is how the two processes of division are defined:
Partitive Division [sharing equally or dealing out]: Gene has 4 tomato plants. There are the same number of tomatoes on each plant. Altogether, there are 20 tomatoes. How many tomatoes are there on each tomato plant?
Measurement Division [making groups of a certain size]: Gene has some tomato plants. There are 6 tomatoes on each plant. Altogether there are 24 tomatoes. How many tomato plants does Gene have?
The example I used with Anna’s students was the sharing of “treats” I used wooden beads, a mat and some serving bowls. I began by saying I had 12 treats and I wanted to share them equally between myself and two friends. I put out three bowls to represent myself and my two friends and one by one, I “dealt” out the treats to each bowl. Students often make connections to dealing out cards when they see this…one for you, one for you, one for you. We recorded 12 divided into 3 equals 4 treats each. Next, I used the same materials and explained I had 12 treats and 4 were able to fit in a bag and I wondered how many bags I could make. I took 4 treats and put them in one bowl, then another 4 and put them in the second. We recorded 12 divided into 4s equals 3. One student noticed, “you keep taking four away until you have none left” and we were able to make the connection between repeated subtraction and division.
We had several materials out on tables for the students and provided them with three division questions to investigate. The students chose what materials they would work with and how they would record what they found out.
I also should comment on the serving bowls – this were from my own home but when I knew I was going to this classroom and being aware of the community, I thought some students might enjoy using these bowls. One of the students was beside himself with excitement exclaiming that they used similar bowls in his family’s restaurant. When it was time for choosing materials, he moved quickly to the table with the bowls (which he taught me how to say correctly using his first language) and didn’t leave.
Here are some links to short videos of the students engaged in the process of division:
After investigating division with the materials, we came together to share our findings and to discuss the question that I had posed to them at the beginning of our time together: What is the relationship between multiplication and division? The students listened carefully to each other and built on each others’ ideas getting to the essence of the question is that both operations were focused on equal groupings.
I think sometimes we can get off-track a little as teachers and focus too much on what students are “doing” instead of what they are thinking and learning. By asking students questions that uncover their understanding such as What is <division>? (or any other concept) and What relationships do you notice? we are helping students connect math to math and deepen their mathematical fluency.