Math: Core Decisions
What:
I am planning to teach a lesson on analyzing and solving story problems involving multiplication and division to a small group of fourth grade students.
To understand when to multiply or divide, students must understand the relationships that these operations represent. While procedural competency is important, “simply being able to perform calculations does not necessarily mean that students understand these operations (Chapin and Johnson, 76)”. Rather it is necessary for students “to develop deep conceptual knowledge of multiplication and division in order to apply and use these operations to solve problems (76)”. Sorting problems can be challenging for students “since it requires a deep understanding of problem structure,” which “develops gradually over time after extensive involvement with problems (Chapin and Johnson, 83)”.
How:
Tasks:
The lesson will be composed of two tasks. First, students will work independently to write a series of story problems, illustrating their thinking for each through a drawing or the use of manipulatives. For the second task, I will provide each student with a sheet with 4-5 multiplication/division story problems. I will ask the students to analyze these problems, write a number sentence for each, and to illustrate their thinking by drawing a picture or using manipulatives.
My goal is to help students develop deeper conceptual understanding of problem structure and the relationships multiplication and division represent, through solving problems. As Hiebert explains, “it is only through solving problems that their concepts and procedures develop together and remain connected in a natural and productive way (25).”
Discourse:
Following independent work, periods of group activity will be structured to allow students to lead the discussion. I will act as a facilitator, and focus on having students explain their thinking and revoice the thinking of other students to focus on the underlying meaning of multiplication and division relationships.
Tools:
I will provide each student with sheets of partial number sentences, scratch paper and pencils, and a variety of manipulatives for the first task. For the second task, I will also provide a sheet of story problems. All of these materials are familiar to the students. My intention is to help support the students’ thinking by providing a variety of differentiated options with which to approach the problems.
Norms:
I will review and rely on established class norms. The students often work independently before sharing and discussing their work with the group.
Why:
Following distinct units on multiplication and division, the students are now being presented with multiplication and division scenarios simultaneously, requiring them to analyze each problem to select the appropriate mathematical operation before solving. I have noticed the students struggling to determine whether to multiply or divide when presented with a sheet of mixed problems. As I recently reviewed a student’s work, I noticed that she had used division to solve a multiplication story problem. When I inquired as to why she had chosen to divide, she responded that: “That’s what we’ve been doing.” I asked her to attempt the following problem, and to explain her reasoning to me. She proceeded to scan the text for the numbers within the story problem, and then wrote a number sentence using multiplication. Ironically, it was a division problem.
This observed difficulty indicates that while the students may have developed procedural competency with multiplication and division, they have yet to develop deep conceptual understanding of problem structure and what each operation represents. I have designed this lesson with the goal of strengthening the students’ conceptual knowledge of multiplication and division, by allowing them to generate and solve mathematical problems. As Hiebert explained: “Teachers can point out relationships, but they become meaningful as students use them for solving problems (22)”.
As such, I have chosen to begin by having the students write their own story problems before presenting them with prewritten story problems. It is my hope that scaffolding the tasks in this order will help the students to connect the logic behind multiplication and division scenarios to problem structure.
As the lesson begins, each student will be provided with scratch paper, a pencil, and access to a variety of manipulatives. After modeling/creating a story problem as a group, the first task I will pose is for the students to write a series of story problems, illustrating their thinking for each through a drawing or the use of manipulatives. Each student will work independently to write at least one problem for each operation. I will provide a sheet with incomplete number sentences to aid this process, and streamline discussion once the group comes together for discussion at the end of the task.
For the next task, I will provide each student with a sheet with 4-5 multiplication/division story problems. I will ask the students to analyze these problems, write a number sentence for each, and to illustrate their thinking by drawing a picture or using manipulatives.