Do multiplication arrays need to be introduced as a ready-made convention, or can they be “constructed” by children using concrete objects while engaged in an open-ended task?

(An action research collaborative effort by second grade lead teachers, Tasha Hernandez, Bill Miller, and Lower School Math Coordinator, Debra Rawlins)

We began this project as a way to build math relevance into the annual second grade canned food drive. The students visit Saint John’s Food Pantry and spend a morning learning about the needs of the community. Then they pitch in and help out the staff for a few hours. When they return to school, they create signs advertising the canned food drive, and place collection boxes at various locations around the school.

Over the course of several weeks, the cans are collected and brought into the classroom, where they begin to stack up…and stack up. Before long, there are cans everywhere! We thought that this would be a good opportunity for a “hands-on” approach to a math problem.

The concept of arrays was not one of our goals when we first started thinking about using cans to design a worthwhile task for our students. There were several other math content areas we wanted to concentrate on as we framed a problem for this investigation.

We wanted our students to develop the capacity to move from counting objects by 1’s to understanding and counting sets. Counting sets is listed in NCTM’s Principles and Standards for School Mathematics (2000) as, “Count with understanding and recognize ‘how many’ in sets of objects” (p. 78). Counting sets can be described as understanding that when counting by 5’s, the child comprehends that each number in the count represents five objects, and as the child says each number in the skip-count sequence, they are adding another group of five. The Common Core State Standards for Mathematics (2010) emphasizes that second grade students need experience working with equal groups of objects to lay a foundation for multiplication.

Over a two-week period, the students were given two open problems. The first one was:

“You have 12 cans. How many ways can you arrange them to make them easier to count?”

Show your work. Write an equation for each way.

We intentionally posed an open problem because we wanted to use it as a formative assessment. We planned to observe students closely as they worked, and to notice if they counted the cans one by one, or whether they would place them into equal groups and count them as sets.

We assigned four students at random to work collaboratively at each table. Each group was given twelve cans to arrange, but they were individually responsible for recording their own thinking on the assignment paper provided. We watched carefully to see how they counted and recorded their work.

Many students organized the cans into equal groups and then counted the groups as sets out-loud in sequence while pointing to each group (2, 4, 6. . .).  Most students recorded their equal groups in the form of repeated addition equations (example: 2 + 2 + 2 + 2 + 2 + 2 = 12).

A couple of students numbered each can by ones, consecutively, even though they had drawn the cans in equal groups. We felt that these students were not yet counting sets and needed more experience with grouping and counting concrete objects. Other students placed the cans into unequal groups and recorded their thinking as addition equations (10 + 2 = 12; 8 + 4 =12).

As students continued to find more than one way to arrange twelve cans, we noticed that at some tables, they were creating equal groups that looked like clusters, and at other tables, they were placing the cans into equal rows (example: 4 rows of 3).

After class we gathered together to look over the students’ recorded work. There were a broad range of approaches and solutions to this problem. We couldn’t help but notice that most of the drawings fell into two separate and distinctly different categories; equal groups of cans versus equal rows of cans. We wondered if students were making a connection between equal groups and equal rows.

We also questioned whether the students who drew equal rows of cans were deliberately and intentionally placing the cans in rows, rather than in groups. It was difficult to know for sure because twelve cans were not many objects to group together. This limited number of 12 might also have been the reason that some students solved the problem by adding unequal groups. This was reflected in their two-addend equations using combinations that they were familiar with (example: 10 + 2 = 12).

We wondered if students might create equal rows that resembled arrays if they were given a similar task that required them to work with a larger number of cans.

This exploration concentrated on canned goods used as concrete manipulatives. Much of the prior research on arrays has focused on pictorial representations. Two different array models are described by Outhred et al, (2002). The first model is shown in the form of rows and columns of discrete dots. This type of discrete object array can also be seen in real world examples such as muffins in a baker’s pan showing equal rows, or egg cartons showing two equal rows of six.

The second model that Outhred describes is an array of “contiguous squares”. In this latter version, unit squares are aligned in equal rows and columns in an array grid.

 Model 1: discrete dots                                                    Model 2: contiguous squares

 

 

 

 

 

 

(From Outred, L. and Mitchelmore, M. (2002), pg. 51).

Outhred discusses whether students can readily “generalize” from the discrete object model to the array grid model. Since the grid array model is used to develop the understanding of area measurement, and multiplication in later years, it would be important for teachers to relate the discrete object model to this square grid array model in order to help students make this conceptual transition.

Additional research (Battista et. al 1998) has shown that pictorial grid arrays are difficult for students to understand from a spatial-structuring viewpoint. In contrast to this, “In the traditional view of learning, it is assumed that row-by-column structuring resides in 2D rectangular arrays of squares and can be automatically apprehended by all” (pg. 531). Battista’s research found that “consistent with a constructivist view of the operation in mind, such structuring is not ‘in’ the arrays-it must be personally constructed by each individual.” They describe the ability to see and understand the row-by-column array structure as the result of a process that requires the learner to go through several spatial-structuring transitions.

The previous research underscores the complexity and challenges students have in their ability to understand arrays. It also reinforces the need to provide more teaching time and opportunities for students to explore these concepts deeply. By incorporating physical concrete objects, such as cans, and giving them extra time and space to work, our students were able to find many ways of counting 30 objects. The configuration of arrays emerged as a natural outcome while students explored the different ways they could group the cans.

By the second week of the food drive, we had collected a lot of cans and they were overflowing off the shelves! We framed our next open problem using 30 cans, and changed the wording from “Show your work”, to “Draw a math diagram of each way”. We hoped that this language was more explicit and would encourage students to record an accurate representation of their can configurations.

The next problem we posed to the class was:

“You have 30 cans.

How many ways can you arrange them to make them easier to count?”

Draw a math diagram of each way. Write an equation for each way.

All students approached this task by creating either equal groups or equal rows.

We concluded that the magnitude of 30 cans necessitated students to organize them into equal amounts to make them easier to count, rather than arranging them into unequal groups. In the process of doing this task, we anticipated that students would arrange and rearrange this larger number of cans in a variety of ways. We expected that this would reinforce students’ understanding that the total number of items would remain the same, regardless of how they were physically arranged. This represents Piaget’s research on Conservation of Number (1965).      

Students recorded their work in the form of drawings and multiple-addend equations representing either groups or rows. One student created equal groups of five, and then also created equal rows of five, recording both ways alongside each other on the paper.

Across the room, another student counted five rows of six cans, while a student at the same table, looking from a different angle, counted six rows of five cans each. Both students recorded what they saw from their point of view.

After the class, we looked over the student work for the purpose of selecting examples to use for a class discussion the next day. For the first two samples, we concentrated on a drawing that showed groups of five, and another drawing that showed rows of five. By limiting the work to the same number of rows and groups, we planned to help students focus-in on the similarities and differences between the two visual representations.

Each student displayed their work using a document camera and then led the class in a discussion with support from the classroom teacher. Equal groups of five were presented and discussed first, then an example of equal rows of five were shown and discussed. During the class discussion, the term “row” and “column” were introduced by the teacher as a need to identify and differentiate the student language of “across” or “sideways” and “up-and-down”.

The last student to present had drawn three rows of tens and he also drew ten rows of three adjacent to it. He counted the first array’s rows by tens, and then pointed to the second array and counted each row by threes. To emphasize his thinking, the student rotated the drawing ninety degrees under the document camera to demonstrate how the rows could become columns.

A student explained,

“The rows turned into columns, and the columns turned into rows.”

This represented a commutative way of thinking about an array (3 x 10 = 10 x 3).By looking at the array from these two angles, this student was exploring the spatial-structuring properties of a rectangular array.

Watch  this video that shows how students arranged cans to make them easier to count, and also how they presented their thinking to their peers:

This mathematical exploration started out as a way to creatively make use of canned goods as math manipulatives in order to give our students “hands-on” experience with organizing objects in different ways, and counting equal groups. Once our students started working on these problems, it became clear that there was more mathematics for them to discover than we realized. The openness of the problems allowed for deeper mathematical thought and exploration. The problems also provided differentiation within the class because all students were successful at creating and counting equal groups, but many students could approach the problems at a higher level by starting to think about equal rows, and columns.  By working with cans as concrete manipulatives, our second grade students began their journey toward constructing and understanding arrays.

References

Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. (1998). Students’     spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29, 503-532.

Common Core State Standards Initiative (CCSSI), 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and Council of Chief State School Officers. http://www.corestandards.org

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Outhred, L. and Mitchelmore, M. (2002). Across the Great Divide: from process to structure in students’ representations of In: Rectangular arrays. Nata, R., Progress in Education, Volume 5; Chapter 4, 51-66. Macquarie University, Department of Education, Australia.

Piaget, Jean. The Child’s Conception of Number. New York: W. W. Norton & Company, Inc., 1965