LEARNERS' DIFFICULTIES WITH QUANTITATIVE UNITS IN ALGEBRAIC WORD PROBLEMS AND THE TEACHER'S INTERPRETATION OF THOSE DIFFICULTIES
Abstract
This study examines 8th grade students' coordination of quantitative units arising from word problems that can be solved via a set of equations that are reducible to a single equation with a single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as a necessary construct in dealing with such problems. We base our analysis within a framework of quantitative reasoning (Thompson, 1988, 1989, 1993, 1995) and a theory of children’s units-coordination with different levels of units (Steffe, 1994) that both encompass and are extended by these two constructs. Our data consist of videotaped classroom lessons, student interviews and teacher interviews. On-going analyses of these data were conducted during the teaching sequence. A retrospective analysis, using constant comparison methodology, was then undertaken during which the classroom video, related student interviews and teacher interviews were revisited many times in order to generate a thematic analysis. Our results indicate that the identification and coordination of the units involved in the problem situation are critical aspects of quantitative reasoning and need to be emphasized in the teaching-learning process. We also concluded that unit coordinationandunit conservation are cognitive prerequisites for constructing a meaningful algebraic equation when reasoning quantitatively about a situation.
Introduction
The research presented in this paper indicates that the ability to coordinate different units in a quantitative situation is an important skill for students to develop in order to be successful in both representing and solving algebraic word problems. Whether this be coordinating different levels of units in a whole number multiplicative situation (e.g. Steffe, 1994) or in a fraction situation (e.g. Lamon, 1994; Olive, 1999; Olive and Steffe, 2002; Steffe, 2002) or in dealing with intensive (e.g., miles per hour) as well as extensive (e.g., number of hours) quantities (Schwartz, 1988) the crucial point is to understand what is being done with the varying quantities in these situations and how the units involved can be related (Thompson, 1988, 1989, 1993, 1995).
Theoretical Framework
This study is part of Project CoSTAR (Coordinating Students’ and Teachers’ Algebraic Reasoning)1 that has as its main purpose the coordination of research on students’ understandings and teachers’ practices and interpretations of students’ actions relative to algebraic reasoning. This particular study is informed by recent research on students’ understanding of algebraic symbols (Kieran and Sfard, 1999), and students’ construction and coordination of quantitative units (Lamon, 1994; Olive, 1999; Smith and Thompson, (in press); Steffe, 1994, 2002; Thompson, 1988, 1989, 1993, 1995).
The main theoretical framework for this study is based on the construct of quantitative reasoning. Thompson (1995) states that “Quantitative reasoning is not reasoning about numbers, it is about reasoning about objects and their measurements (i.e., quantities) and relationships among quantities” (p. 206). Thompson (1988) defined a quantity as follows: “A quantity is a measurable quality of something. A magnitude of a quantity is the quantity’s measure in some unit.” The important distinction that Thompson makes in this definition is between the quality of something and the magnitude of that same thing. A quantity has to be both named (by its quality) and measured by some identified unit. We concur with Thompson on this necessity for forming or identifying a unit by which the named quality may be measured or quantified. We need to delve further into the nature of these quantities in order to uncover the units associated with them. An ordering of the form (name, unit) is helpful for the sake of proper coordination. For instance, the coordination (dime, number of dimes) is not the same as (dime, value of a dime) or (dimes, value per dime). Smith and Thompson (in press) formalize this “name—unit” coordination in their representation of quantitative relationships throughout their article. They “use ovals to represent quantities and place inside those ovals all relevant information about those quantities—their ‘name,’ their units of measure, and any numerical value or expression that is given or can be inferred.” (footnote 4 on p. 19)
Schwartz (1988) used the term referent in a way similar to how we are using identified unit of measure and called such quantities adjectival quantities. (p. 41) He stated that all quantities have referents and that the “composing of two mathematical quantities to yield a third derived quantity can take either of two forms, referent preserving composition or referent transforming composition.” (p. 41) The referent transforming composition, Schwartz claims, forces us to distinguish between two different kinds of quantity: extensive quantity and intensive quantity. An extensive quantity can be counted or measured directly, whereas an intensive quantity is derived from the multiplicative combination of two like or unlike quantities, and is usually recognized by the use of “per” in its referent unit (e.g., miles per hour, price per pound). Schwartz (1988) also distinguished the name of the quantity from its referent unit. For instance, the intensive quantity speed could have referent miles per hour or feet per second.
In word problems involving extensive and intensive quantities, one further step is needed, beyond coordination of each one of those quantities. We somehow would need to reconcile all these quantities, each of which can be coordinated in the form (name of quantity, unit of the quantity). In other words, we not only look at each coordinated quantity separately, but also look at all these quantities together as a whole. This coherence of the whole requires that we meaningfully combine each coordinated quantity: A coordination of coordinated quantities. We refer to this second level of coordination as “quantitative unit conservation”.
Introducing a second level of coordination of quantitative units necessitates a view of different levels of quantitative units. Behr, Harel, Post, & Lesh, (1994) conducted a conceptual analysis of different levels of quantitative units. Steffe (1994) provided a psychological analysis of children’s construction of composite units at different levels of composition (a singleton unit, a unit of units, and a unit of units of units) involved in multiplicative reasoning. Olive (1999), Olive and Steffe (2002), and Steffe (2002) extended the idea of multiple levels of units to children’s reasoning with fractional quantities. We shall show, in the analysis of the coins problem in this study, that students need to operate with three levels of units in order to successfully make a coordination of coordinated quantities necessary for quantitative unit conservation.
Thompson (1988, 1995) provides another way of thinking about this second level of coordination through his description of quantitative operations and relationships. Thompson (1995) states: “quantitative operations and numerical operations should not be thought as being the same.” (p. 212) In his 1988 paper he describes four types of quantitative operations: combining quantities either additively or multiplicatively, and comparing quantities either additively or multiplicatively. He goes on to state that:
Complex quantitative reasoning entails relating groups of quantitative mental operations, such as in forming a multiplicative comparison of an additive comparison and an additive combination (i.e., “How many times bigger is this difference than is this combination?”). Quantitative reasoning also entails reasoning relationally about quantitative structures, entails the constitutive mental operations for comprehending a quantity situationally, and entails the constitutive mental operations which allow one to recognize a quantity as one whose value varies or can vary. (p. 165)
We consider this elaboration to be supportive of our construct of quantitative unit conservation. With this construct, we are opening one more theoretical perspective that covers a range of mathematical practices associated with solving word problems. Such mathematical practices include, but are not limited to, taking care of priority of operations, using parentheses appropriately (in order to combine quantities), and substituting literal expressions for other literal symbols. All these mathematical practices serve one crucial idea, and that is to maintain the equality of expressions on both sides of an equation (Chazan and Yerulshalmy, 2003), while being aware of what's happening on both sides: Things we are adding or subtracting have to be like terms while those we multiply or divide do not necessarily have to be so. The simplified expressions on both sides of the equation must be “like-terms” in the sense that they both have the same “simplified” unit. The simplified unit throughout the process of obtaining equivalent equations must be conserved.
In this paper we explore the units coordination arising from situations that can be represented by linear equations involving more than one unknown or variable but that can be reduced to an equation in a single variable; that is, a system of linear equations that can be solved by substitution. We consider these situations to be ones that require complex quantitative reasoning (Thompson, 1988) in that they entail relating groups of quantitative relationships; they also involve the use of algebraic notation that adds another layer of symbolic complexity to students’ quantitative reasoning. Students often associate the algebraic symbol with the name of the quantity rather than its magnitude, as Thompson (1995) pointed out:
When we reasoned symbolically, we needed to remind ourselves continually that W stood for the number of women and that M stood for the number of men. When students fail to keep in mind that letters represent numerical values, they will think of an expression like W=8/9M as saying “one woman is eight-ninths of a man” instead of thinking “the number of women is eight-ninths the number of men.” Also, students will often read the (equivalent) equation 9W=8M as “There are 9 women for every 8 men” instead of as “9 times the number of women equals 8 times the number of men.” Students’ thinking of letters as standing for objects is well researched, and it has pernicious consequences for students’ understanding of algebra... (p. 209)
Several students in this study made similar associations between the letters in an algebraic expression as standing for objects (names of quantities) rather than numbers (magnitudes of the quantities). Through our analysis of the classroom discussions, students’ explanations and responses to interview tasks, along with interviews with the classroom teacher, we have come to realize that the identification and coordination of the units involved in the problem situation (the name-unit coordination) are critical aspects of quantitative reasoning that need to be emphasized in the teaching-learning process.
Context and Methodology
This study took place in an 8th-grade classroom in a rural middle school in the southeastern United States. The 24 students were between 13 and 14 years old and had been placed in the algebra class based on their success in 7th-grade mathematics. The students were racially, socially and economically diverse, with an approximately equal distribution of gender. All eight class lessons on a unit that focused on writing and solving algebraic equations from word problems were videotaped using two cameras, one focused on the teacher and the other on the students. Four students were interviewed twice in pairs (a pair of girls and a pair of boys) during the three weeks of the study. The classroom teacher was also interviewed twice during the three weeks. All interviews were videotaped. Excerpts from the classroom videotapes were used during both student and teacher interviews to initiate discussion of the learning that was taking place in the classroom. Excerpts from the videotapes of student interviews were also used in the teacher interviews. The first author conducted all of the interviews.
This paper focuses on problems arising from a particular contextual situation (the Coins Problem). The data for the Coins Problem were collected during the two class lessons and subsequent student and teacher interviews that dealt with the following word problem from UNIT 4 of College Preparatory Mathematics (CPM) Algebra 1, 2nd edition (2002):
Mrs. Speedy keeps coins for paying the toll crossing on her commute to and from work. She presently has three more dimes than nickels and two fewer quarters than nickels. The total value of the coins is $5.40. Find the number of each type of coin that she has.
Students were first asked to create a “guess and check” table to find possible solutions to the problem. They were then challenged to write an equation to represent the problem. This Coins Problem gave rise to student difficulties that can be explained in terms of unit identification and coordination.
Analysis Process
Each day the classroom video data from the two cameras were viewed and digitally mixed using a picture-in-picture technology. A written summary of the lesson with time-stamps for video reference was created from the mixed video. This written summary also contained comments about any significant events and screen shots from the video when needed for clarification or highlight. These written “lesson graphs” were then used to select excerpts from the classroom video to be used in the student or teacher interviews, and to plan questions and related problems to pose to the interviewees in an effort to understand how the students (and teacher) had interpreted the problem and the classroom discussions that followed from different students’ attempts to address the problem.
CPM ALGEBRA I UNIT 4, Choosing a Phone Plan (CP)
CPM UNIT 4 / CLASS / Students P&M / Students B&G / TEACHERCP 0,1 / 10/25/04
CP 1,4-8 / 10/26/04
CP 15,16 / 10/27/04
10/28/04
10/29/04
CP 17,18 / 11/01/04
CP 27, 28, 38 / 11/02/04
CP 45, 40, 48(a), 39(c), 65(b) / 11/03/04 / 11/03/04 / 11/03/04
11/05/04
CP 92 / 11/08/04
CP 92 / 11/09/04
11/10/04
Figure 1: Connections among class lessons, student interviews & teacher interviews
After the end of the three weeks of data collection, the corpus of classroom video data was reviewed, along with the associated lesson graphs to generate possible themes for a more detailed analysis. All student and teacher interviews were transcribed from audio files created from the videotapes of the interviews. A chart of relationships among class lessons, student interviews and teacher interviews was then created. This chart indicated which class lessons (including the specific activities from the CPM unit) were used or referenced in which student and teacher interviews, and which student interviews were used or referenced in the teacher interviews (see figure 1). A retrospective analysis, using constant comparison methodology, was then undertaken during which the classroom video, related student interviews and teacher interviews were revisited many times in order to generate a thematic analysis from which the results emerged.
Results of the Analysis
The protocols we are presenting below came from important events in class lessons, student and teacher interviews. By important events, we mean critical instances that are meaningfully related to our two theoretical constructs (quantitative unit coordination and conservation). For the coins problem, we started with a detailed description of the situation from the related class lesson. We then looked for where else the coins problem was used among class videos, student interviews, and teacher interviews. We revisited the transcripts of these video-episodes many times, and refined them further. The protocols presented below do not necessarily follow a chronological order; rather, critical incidents that occurred during class lessons are followed up through excerpts from both student and teacher interviews before returning to the protocols excerpted from the class lessons. This sequence of evidence follows the thematic analysis that emerged from the retrospective analysis of the total set of video data.
The Coins Problem
Mrs. Speedy keeps coins for paying the toll crossing on her commute to and from work. She presently has three more dimes than nickels and two fewer quarters than nickels. The total value is $5.40. Find the number of each type of coin she has. (from CPM Algebra 1, UNIT 4, CP-16, 2002)
This problem was introduced during the third class period in Unit 4 on 10/27/04 (see Figure 1). Problem CP-3 in the Unit was very similar, also dealing with combinations of nickels, dimes and quarters, but the teacher had skipped over that problem in a previous class period, thus problem CP-16 was the first one of this type that the students had encountered. In this problem situation, when trying to calculate the total value of all coins, the monetary values of specific coins are intensive quantities (they are the values per coin) and the numbers of each coin and total value are extensive quantities. Distinguishing between these two different types of quantities surfaced as a problem during the classroom discussions. Associating appropriate units with the different quantities and combining unknown quantities emerged as further problems during the student interviews.
A major confusion arose during the class lesson on 10/27/04 in naming the quantities in the situation. Students had chosen the letter N to represent the nickels in the problem, however, it became apparent from the discussion that, while N stood for the number of nickels for the teacher and for some of the students, for others it either represented the value of all the nickels together or just stood for the coin (a nickel). When the teacher, Ms. Jennings2 asked the students “What are we gonna call dimes?” (immediately after writing “n=nickels” on the classroom board) some students answered “two N”, and this could be a corroboration that those students saw N as the value, and not the number of coins under consideration. The following dialogue between Ms. Jennings and a student, Cathy, taken from the classroom video illustrates the confusion:
Protocol I: Student's confusion about naming coins (from classroom video on 10/27/04)
Ms. Jennings: We are not done... We are just naming our variables right now. We haven't begun to make an equation yet. We have to know what we are naming, before we put in an equation.
Cathy: So why can't we just put them all with their first letter? Like N equals nickels, just keep doing, D dimes, Q quarters.
Ms. Jennings: Let me ask you this question and see if you can solve it: “N plus D plus Q equals 5 dollars and forty cents. How many of each one do I have?”