Tasks: The activities that students engage in and the problems that they solve. What students are asked to do, think about, or work on during mathematics instruction.


What kind of problem is it?
To begin the lesson, I gave the students the following question:
Thanksgiving Problem However, instead of asking the students to solve this problem, I asked them to think about what kind of problem this story represented.

Mathematical Focus: “Building an understanding of the concept of multiplication requires developing a language for thinking about and describing multiplicative situations and their qualities” (Smith and Smith, 2006, p. 41). Thus, by asking students to discuss what type of problem my story represented, I wanted to get the students thinking about what kind of situations lead to multiplication problems. By giving a “real life” example, I tried to make it something that the students could visualize, which I hoped would help when setting up the problem later.

Appropriateness for Students: Since Ms. H’s fourth grade class has not yet mastered multiplication, many students are still unsure of what types of problems lend themselves to which operations. This task requires students to both identify the story as a multiplicative situation and use the appropriate vocabulary. Also, students are accustomed to immediately finding the solution to a problem, so this task helped them take a step back—even though several students immediately told me the answer anyway. Furthermore, since Thanksgiving had just passed, I hoped they would be able to relate to the problem, but I did not consider that some of the students in my group might not celebrate Thanksgiving.

Fit with Learning Goals: Because the cumulative goal of the lesson was to solve multiplication problems, I wanted students to begin thinking about characteristics of multiplicative situations. Although I did not do this during the lesson, I could have further discussed what in particular made this problem a multiplicative scenario and distinguished it “from other situations suggesting addition, subtraction, or division operations” (Smith and Smith, 2006, p. 42). Furthermore, this word problem became the basis for the multiplication problem we did together. 

Array Introduction
Getting out a piece of ½ inch graph paper and reinforcing that our problem talked about groups, I asked students how we could represent the Thanksgiving problem on the graph paper. After setting up the array, I had students evaluate whether or not it represented our problem. Then, we divided 15 into its place value components and represented this on our array. After we went through a guided example, the students were instructed to set up a separate multiplication problem on their own. 

We set up an array to represent the Thanksgiving problem.  The first column illustrates the pieces of pie the first person ate, but for the sake of time I did not draw the rest of the pieces of pie.
We set up an array to represent the Thanksgiving problem. The first column illustrates the pieces of pie the first person ate, but for the sake of time I did not draw the rest of the pieces of pie.

Setting up arrays to represent a multiplication problem
Mathematical Focus: Representing a multiplication problem as an array requires students to think about the meaning of multiplication. Thinking critically about multiplication is important because many students memorize different algorithms and do not consider what they are actually doing and why what they are doing makes sense.

Appropriateness for Students: Because many of the students are accustomed to immediately finding the answer when presented with a math problem, a few of them had trouble thinking about the process. Furthermore, students became hesitant when asked if the array represents our problem and if so, how, so I decided to do a turn and talk.

Here, J shows that he had a lot of difficulty drawing arrays to represent multiplication problems.  In the top array, by drawing the box around his labels, he actually had an array that was 4 x 14.  [He attempted to fix the number but then crossed it out any tried again.]  In the second array, J had the correct number of rows, but his labeling still had him end up with 14, instead of 13, columns.
In the top array, by drawing the box around the labels, J actually had an array that was 4 x 14. In the second array, J showed improvement but was still incorrect.

For the most part, the students seemed interested in the conversation, but one student in particular became frustrated when one of his peers struggled to understand why the array represented the problem. When working independently, one aspect of the arrays that seemed to hinder student understanding was labeling the grids. Originally, I labeled the grid boxes to show students exactly what I was doing, but in making their own arrays, many students labeled the boxes incorrectly and thus drew the arrays inaccurately. I wonder if it would have made more sense had I just focused on the number of boxes inside the array rather than labeling them. Moreover, I wish I had given students more time to explore making arrays.

Fit with Learning Goals: Having students represent the multiplication problem as an array had multiple purposes. First, I thought seeing a pictorial representation would be helpful to students, particularly visual learners, in making multiplication more transparent. Additionally, this is the first step in the array method, which was the goal of this lesson.

After discussing the place value components of 15, we represented this on the array.
After discussing the place value components of 15, we represented this on the array.

Dividing arrays into their place value components
Mathematical Focus: The goal of dividing the array by place value was to break multiplication involving two-digit numbers into simpler problems. Furthermore, in showing how place value plays a role in multiplication using the array method, I hoped students would able to make a connection to the role place value plays when using the standard algorithm or the lattice method. However, we did not have time to discuss how the array method connects to other methods for solving multiplication during this lesson.

Appropriateness for Students: Although I did a quick review of what place value means, we should have done a deeper analysis to assure students really understood its meaning. As Van de Walle (2004) explains, “[a]n idea fully understood in mathematics is more easily extended to learn a new idea” (p. 26). As a result, I worry that students did not fully understand place value and consequently had trouble extending this concept to the array method.

Z drew the place value line on the incorrect line of the the 10th column, which later affected his calculations.
Z drew the place value line on the incorrect line of the the 10th column, which later affected his calculations.

Furthermore, some students drew their place value line on the incorrect side of the tenth column. I think explicitly showing the number of boxes on each side of the place value line, rather than just labeling it, would have make this more transparent.

Fit with Learning Goals: In dividing the array into its place value components, I hoped students would be able to see the role place value plays in multiplication.

We put all the pieces together to solve a multiplication problem using the array method.
We put all the pieces together to solve a multiplication problem using the array method.

Using Arrays to Solve Multiplication Problems
The final step of the array method is using the array to solve the multiplication problem. First, I modeled what I wanted students to do using the 5 x 10 box, showing them where the five and the ten came from and why we should multiply them. Then, through guided instruction, we discussed how to determine how many squares are in the remaining box. Once we determined the area of the place value rectangles, we discussed how to find how many grid blocks were in the entire rectangle.

Mathematical Focus: Using arrays to solve multiplication problems is the most complex task of the lesson, requiring students to analyze what it means to multiply numbers and why you can break them into their place value components.

Appropriateness for Students: In working through the guided example, students seemed to understand this step. However, I planned to wrap up the lesson here, but once students tried this step independently, many were more confused than I had realized—perhaps this task was not as accessible as I had originally thought. I think giving students more time to explore arrays in general would have made this step more transparent.

Fit with Learning Goals: This task was the culmination of the lesson. The hope was that students would be able to put all the steps together to solve problems using the array method. By solving problems this way, I hoped multiplication would become more meaningful.

 

References:
Class notes.
Kazemi, E., & Hintz, A. (2014). Intentional Talk: How to Structure and Lead Productive Mathematical Discussions. Portland, ME: Stenhouse.
Skemp, R.R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77, 20–26.
Smith, S.Z. & Smith, M.E. (2006). Assessing elementary understanding of multiplication concepts. School Science and Mathematics, 106(3), 140-149.
Van de Walle, J. A. (2004). Developing Understanding in Mathematics, Ch. 3 in Elementary and Middle School Mathematics: Teaching Developmentally.

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