SUPPORTING STUDENTS’ CONCEPTUALIZATION OF ALGEBRAIC EXPRESSIONS AND OPERATIONS USING COMPOSITE UNITS

Diana Underwood Gregg Purdue University Calumet diana@calumet.purdue.edu

Erna Yackel Purdue University Calumet yackeleb@ calumet.purdue.edu

Abstract

In this paper we attempt to share understandings that we have developed by investigating how students' notions of algebraic expressions and operations might be fostered in a conceptually sound way. To that end, we describe the generalized candy factory instructional scenario that was used in a classroom investigation in a university developmental-level mathematics class. The analysis emphasizes the utility of the notion of composite units as a guiding construct in understanding and further developing students’ conceptualization of operations with algebraic expressions.

Purpose

The developmental mathematics program at our university is designed to help students whose background in algebra is deemed inadequate. Students enrolled in these courses typically have had more than one year of algebra during high school. However, their understanding of algebra consists of an isolated repertoire of proceduralized skills, which they often perform inaccurately. Over the past six years, rather than simply addressing the symbol manipulation difficulties of these students, we have worked toward identifying and developing certain concepts and notions that are foundational to algebraic thinking and reasoning, as well as creating materials and methods to help students develop a conceptual understanding of basic algebra. In this paper we attempt to share understandings that we have developed by investigating how students' notions of binomial multiplication might be fostered in a conceptually sound way.

Methods of Inquiry and Data Sources

The data for this study was collected in a zero-credit, developmental-level mathematics classes for entering university students. The class was comprised of 14 female and seven male students. One of the researchers conducted the class sessions during each classroom investigation. Data for the study consist of a graduate assistant’s field notes, videotape data, and a documentation of the development and revisions of the instructional sequence.

Theoretical Framework

The overarching theoretical framework used to guide the study is a version of social constructivism, called the emergent perspective (Cobb & Bauersfeld, 1995). According to this perspective, interactionism and psychological constructivism are coordinated to account for learning and teaching. Interactionism is the social perspective that is taken from communal or collective processes, while psychological constructivism is the perspective that is taken from an individual’s activity as he or she participates in and contributes to the development of communal processes (Cobb & Yackel, 1996). This theory is compatible with the Realistic Mathematics Education (RME) instructional design approach (Freudenthal, 1991; Treffers, 1987). Using this approach, students are supported in the “guided reinvention” of mathematical concepts through a process of mathematizing activity in problem situations that are experientially-real to the students. Reasoning with conventional mathematical symbols is an end-point of this process.

Composite Units

Our past research has indicated that young children develop increasingly sophisticated concepts of a unit as they participate in classroom situations in which the instruction has been designed to foster this development. We have also noted the difficulties that students experience in constructing composite units when the nature of the instruction is centered on teaching students how to operate procedurally on numbers. It is within this frame that we began to view students’ conceptual difficulties with algebraic expressions and operations on algebraic expressions as struggles with constructing and operating on composite units in an unknown base.

Steffe’s notion of a composite unit (1992 and personal communication) is useful for making sense of an individual’s activity from a psychological point of view. According to Steffe, when a child has developed a notion of composite unit, she can coordinate units of different rank. For example, she can treat a number, such as 23, as a single unit comprised of 23 ones, or as a group of 23 individual units and can move back and forth between these conceptions and coordinate them in flexible ways. She might think of a group of 23 combined with a group of 25 more as two groups of 20 and 8 more, or possibly think of it as 2 fewer than two groups of 25. This is the type of flexibility needed to have an understanding of place value numeration.

In an analogous way, conceptualizing an unknown as a composite makes it possible to think of X+X as 2X where the 2X is the result of counting X units, followed by counting X units again. That is, 1, 2, 3,. . .,X, X+1, X+2, . . .X+X, where X+X is now seen as 2X. In the same way, an algebraic expression such as 3X-2 can, depending on one’s current needs, be viewed as a single composite unit, as 3X-2 individual units, as 2 fewer than 3 units of (size) X, or as 2 units of (size) X and 2 fewer than another unit of (size) X. In this way, this conceptualization of algebraic expressions is analogous to conceptions of place value numeration. With this in mind, our teaching-learning goal was to develop an instructional sequence following the RME instructional design theory that would support students’ notion of an algebraic expression as a composite unit and that would facilitate their operations on these units.

Instructional Sequence

The instructional sequence that we employ is adapted from a sequence developed for children to facilitate young children’s development of place value numeration. For children, we use the scenario of a candy shop, in which pieces of candy are packaged in rolls of ten, ten rolls of ten pieces are packaged in a box, and ten boxes of ten rolls are packaged into a case. In our work, the scenario of the shop is also used, but now the candies are packaged into rolls of some specific, but unknown, quantity. Similarly, the same (specific but unknown) quantity of rolls are packaged into a box, the same (specific but unknown) quantity of boxes are packaged into a case. The purpose of using the scenario is that students’ mathematical activity is grounded in real-world imagery. For example, to figure out the result of (3X+5) + (2X-1), students might imagine the activity of combining 3 rolls and 5 pieces of candy with 1 roll and a roll missing 1 piece of candy. (An earlier investigation of students' development of an algebraic expression as a composite unit within additive settings can be found in Underwood and Yackel, 1998)

To promote a notion of arrays, the following scenario is used. When candies are produced in the candy factory they come out of the machine onto a conveyor belt. The number of candies on each row on the conveyor belt is exactly the number needed to make a roll, that is X candies are in each row. Further, the conveyor belt stops after making X rows so that the candies can be packaged. Each row is packaged into a roll and the rolls are packaged into a box. An alternative description is that the machine makes an X by X array of candies before the conveyor belt stops so that the pieces may be packaged. The critical feature in this scenario is that the quantity X now takes on two different roles. On the one hand, X represents the quantity in a row (the number of candies in a roll), and, on the other hand, X represents the number of rows (the number of rolls in a box). Using the language of composite units, we would say that in thinking of the X by X array, the student would need to think of iterating a composite unit (consisting of X ones) X times. The conveyor scenario was further extended to situations in which the machine breaks down. The difficulties that are encountered may be of two types. First, the machine may make too many or too few candies in each row. Second, the machine may make too many or too few rows before it stops so the candies can be packaged. In some cases, the machine may make both types of errors at once. For example, the machine may make two extra candies per row and may make three fewer rows than it should before stopping. From our perspective, the situation just described can be thought of as an X-3 by X+ 2 array.

An Example

Below, we describe the small group activity of one group of four students during their work on the broken machine tasks. Although the students in this group were among the least conceptually advanced students in the class, they were among the most active participants in small-group and whole-class discussions. The analysis of their activity gives us an opportunity to consider the nature of the difficulties that students encountered in this instructional sequence. As can be seen below, the group is attempting to figure out the number of rows, the number of pieces per row, and the total number of pieces which were made by the broken machine that produced X+2 pieces per row and X+1 rows. Just prior to this episode, a research assistant had guided the group through a similar task. Therefore, this particular episode illustrates that the students’ difficulties were not due to a lack of understanding of the conventions of the task. The analysis suggests that the students were unable to think about simultaneously as the number of pieces in a roll of candy and the number of rows and, therefore, move flexibly between these two ideas.

Marla: For the number of rows, you got. No, .

Christine: No. (the other group members agree with Christine). That’s the number of rows. Now we got to figure out the number of pieces per row, which is ,…

Jesse: The number of pieces per row? X.

Marla: X. It’s -- I mean, plus X.

Christine: No. These are saying the same thing. How many pieces per row? ? Because we got—this (pointing to the two columns of X) is our 2X from our X pieces. (Several minutes later.) Okay is the number of rows. is the number of pieces per row, then is the number of rows.

This vignette illustrates the group’s inability to keep track of the units that they were counting. For instance, at the beginning of the episode, Marla correctly labeled the number of rows as X+1 but immediately changed it to . While there are pieces in the square array, there is 1 extra row below this array. She was mixing the unit pieces with the unit of rows. Christine and Jesse had the same problem later in the episode. Christine stated that there were pieces per row and rows.

While in subsequent tasks that involved the candy factory sequence there was evidence illustrating that this group had made significant progress in the coordination of units, the group’s initial inability to keep track of the units that they were counting is similar to the difficulty that young children have in coordinating units of different rank in base 10 situations. For example, when asked to figure out the number of tens in 230, many second- and third-graders will count 10, 20, 30, … up to 200, while keeping track that they have counted 20 tens. Then, realizing that they have 30 remaining, they add 20 and 30 to get 50 and report that there are 50 tens. The children's activity indicates that they are not able to coordinate units of different rank. That is, they could not keep track of which unit they were operating in as they solved the task, in the same way that the group in the vignette could not coordinate the units that they were counting.

Analyses of the developmental-level students' activity in our study indicate that to distinguish between X as the quantity in a row (the number of candies in a roll) and X as the number of rows is an essential first step in conceptualizing the quantity of candies as an array. The activity of making this distinction is what Steffe (1992) refers to when he says, "For a situation to be established as multiplicative, it is always necessary at least to coordinate two composite units in such a way that one of the composite units is distributed over the elements of the other composite unit.”

Final Thoughts

The significance of our research extends beyond the analysis of this classroom data. Using the construct of composite units has provided us with a lens for interpreting students’ mathematical activity and a framework for developing conceptually sound instruction. We believe that other algebraic constructs may also be investigated by considering students’ understanding of related arithmetical operations.

References

Cobb, P., & Bauersfeld, H. (1995). The coordination of psychological and sociological perspectives in mathematics education. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 1-16). Hillsdale, NJ: Erlbaum.

Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist. 31, (3/4), 175-190.

Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer.

Steffe, L. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4, 259-309.

Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics education: The Wiskobas project. Dordrecht: Kluwer.

Underwood, D., & Yackel, E. (1998, April). Developing a concept of an algebraic expression as a composite unit. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.