Objective: Students will be able to use the array method to represent and compute one-digit by two-digit and two-digit by two-digit multiplication problems.
Although I enjoyed teaching a lesson to a group of students, I found teaching one isolated math lesson to be particularly challenging. Currently, Ms. H’s class is learning both the lattice method and the standard algorithm for solving multiplication problems. However, many of the students either seem confused by these methods or do not fully understand why they work. As a result, the goal of my lesson was to teach students the array method in order to make the other multiplication algorithms more transparent. Unfortunately, we did not have enough time to connect the array method to other multiplication strategies, and I am not entirely sure where the students are in terms of their understanding.
Moreover, I found I struggled to appropriately differentiate the lesson to the needs of all the students. Even though Ms. H gave me a group of students who were all “in the middle of the class,” their abilities still differed greatly. For example, most of the students understood that the array we drew represented the Thanksgiving problem, but one student struggled with this concept. As a result, I decided to do a turn and talk in hopes that talking with her peers would clear up her confusion. While she and her partner were discussing the array, the other group of students had already come to a consensus. Because I wanted them to keep thinking, I asked them how they thought we could break up the array by place value. However, looking back, I realize that this probably was not a good additional question because this was something that we would talk about as a group anyway. Moving forward, I will try to think of more situations in which students have different levels of understanding and account for these situations as I plan my lesson.
If I were the classroom teacher for these students, I would follow up this lesson with other lessons regarding the array method until it seemed that students fully understood what they were doing. Because place value seemed to cause confusion, we would first revisit its meaning, perhaps even modeling with objects that the tens place represents groups of ten and the ones place represents groups of one and so on. I would also allow students to explore making arrays until they developed a relational understanding of the tool. I could ask them to draw a variety of different arrays, including ones they thought of on their own, and explain how their array represents a specific multiplication problem. Allowing students to become more familiar with arrays themselves before moving on to use them as a multiplication technique most likely would improve their understanding. Eventually, we would discuss how the different multiplication methods—the standard algorithm, the lattice method, and the array method—are related and the role that place value plays in each one.
As Fosnot and Dolk (2001) explain, “[u]sing algorithms, the same series of steps with all problems, is antithetical to calculating with number sense. Calculating with number sense means that one must look at the numbers first and then decide on a strategy that is fitting—and efficient.” Thus, I would like to expose students to numerous multiplication strategies so they continue to develop number sense and explore the meaning of multiplication. However, having students calculate with number sense is not always an easy task: “evidence suggests that once students have memorized and practiced procedures without understanding, they have difficulty learning to bring meaning to their work” (Russell, 2000, p. 156). As a result, I wonder what is the best way to teach students who have been conditioned to use algorithms—how can we break them of this habit? Furthermore, although the goal of teaching the array method was to increase students’ number sense and “relational understanding” of multiplication, to many of the students it may have seemed like just another algorithm. I feel that if I had given students more time to explore and understand arrays in general the method may have made more sense.
Similarly, while I plan to continue to work on selecting and using representations to make mathematics meaningful and draw connections between mathematical concepts, I now realize that this pedagogical focus most likely cannot be addressed in an isolated lesson. Instead, for students to have a relational understanding of the mathematical concept as well as make connections to other mathematical concepts, they must have ample time to explore the ideas. Achieving mathematical understanding is a process that cannot be rushed.
Fosnot, C. T. & Dolk, M. (2001). Algorithms Versus Number Sense. In Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, ME: Heinemann.
Kazemi, E., & Hintz, A. (2014). Intentional Talk: How to Structure and Lead Productive Mathematical Discussions. Portland, ME: Stenhouse.
Russell, S. J. (2000). Developing Computational Fluency with Number. Teaching Children Mathematics, November.
Skemp, R.R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77, 20–26.