Real-World Contexts, Multiple Representations, Student-Invented ...

MATHEMATICAL THINKING AND LEARNING, 9(4), 387–418 Copyright © 2007, Lawrence Erlbaum Associates, Inc.

Real-World Contexts, Multiple Representations, Student-Invented Terminology, and Y-Intercept Jon D. Davis Department of Mathematics, Western Michigan University

One classroom using two units from a Standards-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly connected to the meaningfulness and implicit/explicit temporality of the contexts that students investigated as part of their classroom activities. This terminology held great promise for promoting the concept of y-intercept within a multiple representation environment. However, the teacher’s interpretation of different activities and his assumptions about the transparency of different representations, as well as students’ past experiences left the studentgenerated terminology and the concept of y-intercept disconnected from one another. This resulted in student-generated terminology that had limited applicability, a fragile understanding of y-intercept within different representations, and for some students, interference between their invented terminology and the concept of y-intercept itself.

INTRODUCTION The notion of a function is one of the most important ideas in mathematics (Harel & Dubinsky, 1992; Romberg, Fennema, & Carpenter, 1993). One of The author would like to thank Jim Fey, Jeremy Kahan, Jane-Jane Lo, and the anonymous reviewers for their helpful suggestions on earlier drafts of this article. Correspondence should be sent to Jon D. Davis, Department of Mathematics, Western Michigan University, 1903 W. Michigan Avenus, Kalamazoo, MI 49008-5248. E-mail: [email protected]

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students’ first formal learning experiences at the middle or high school level in this important domain occurs around linear functions. These functions are important because of their many real-world applications, but they are also important for the role they play in supporting conceptual understanding of more advanced mathematical topics such as the derivative in calculus. Studies have focused on students’ understanding of slope within abstract (Zaslavsky, Sela, & Leron, 2002) and real-world contexts (Stump, 2001). This study focused on the concept of y-intercept and its importance as a defining parameter of a linear function (Moschkovich, Schoenfeld, & Arcavi, 1993; Schoenfeld, Smith, & Arcavi, 1993). Although y-intercept may be seen as a less complex concept compared to slope, it can pose challenges for students when they must translate between different representations (Smith, Arcavi, & Schoenfeld, 1989). For example, when students are presented with a line in slope-intercept form, identification of the y-intercept is relatively straightforward. However, within a table and a graph, students must manage the value of the y-intercept with the x-coordinate as well.

Cartesian Connection The Cartesian connection is the realization that each point on a line represents an ordered pair that satisfies the equation representing the line (Moschkovich et al., 1993; Schoenfeld et al., 1993; Smith et al., 1989). The y-intercept holds the potential to promote the Cartesian connection because of its presence within the slope-intercept form of an equation. In other words, the y-intercept leads to the straightforward identification of a point on an oblique line since it is the y-coordinate when the x-coordinate is zero. Knuth (2000) researched students who had mathematical backgrounds ranging from first-year algebra to calculus and found that students did not apply the Cartesian connection to translate a graphical representation to an algebraic one. Instead, students preferred to move in the opposite direction, from algebraic to graphical representations, possibly because of the type of instruction they had received.

Standards-Based Curricula The assessments on linear functions above were situated within abstract mathematical settings. In the past two decades, national-level documents have appeared that emphasize the importance of connecting school mathematics and the real-world (Cockcroft, 1982; National Council of Teachers of Mathematics, 1989; 2000; National Research Council, 1989). One of the outcomes in the United States of the movement embodied in these documents was the creation

STUDENT-INVENTED TERMINOLOGY AND Y-INTERCEPT

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of K–12 Standards-based1 curricula designed to achieve this end, among others (Senk & Thompson, 2003). Although the five secondary mathematics curricula described as Standardsbased all differ in important ways such as mathematics content2 or how they are organized,3 they also share some similarities in how the activities that are designed to teach linear functions are structured. These include the use of realworld contexts as situations by which mathematics can be reinvented, the use of technology as a tool to solve problems, the application of different function representations (e.g., table, graph, equation) to solve problems, and a delay in formal mathematical terminology. There is some reason to believe that realworld contexts have the potential to facilitate the understanding of the Cartesian connection and to provide “cognitive hooks” for y-intercept in linear functions (J. T. Fey, personal communication, March 15, 2005). The following example, taken from the Standards-based curriculum, Contemporary Mathematics in Context: A Unified Approach (Core-Plus Mathematics) (Coxford et al., 1998), and shown in Figure 1 is helpful in illustrating this point. A meaningful question in this context is the initial length of the spring before compression. Students’ answers to this question have the potential to highlight y-intercept as the value of b in the point (0, b). Further, not only does Core-Plus Mathematics present the equation for a line in y = mx + b form, but it also places the terms in a slightly different order, as y = b + ax, to capitalize on this notion of y-intercept as a beginning point in a real-world context (J. T. Fey, personal communication, March 15, 2005). The delayed introduction of formal mathematics terminology and the increased role of students in fashioning their own definitions common to Standards-based mathematics programs have been advocated by Lakatos (1976) through his description of how Euler’s formula for convex polyhedra evolved. Lampert (1991) described the importance of using natural language to bridge the gap between what students already know and mathematical ideas to be learned. For example, she used the words “big” and “skinny pie pieces” (p. 135) as a means by which students could use their informal understandings to grasp the structure of decimal numbers. At the same time, students’ use of natural language may result in misunderstandings through a process of semantic contamination (Pimm, 1987). This can happen when the more common definition for a term contrasts with its use in the mathematics classroom. An example of this is the 1 These are curricula that were created because of a request for proposals from the National Science Foundation (1991). 2 For example, Voronoi diagrams appear in Mathematics: Modeling Our World (Garfunkel, Godbold, & Pollak, 1998) but not in other Standards-based curricula. 3 The Interactive Mathematics Program (Fendel & Resek, 1997) begins nearly every chapter with the introduction of a large, organizing problem, while Math Connections: A Secondary Mathematics Core Curriculum (Berlinghoff, Sloyer, & Hayden, 2000) does not.

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FIGURE 1 A sample graph from unit 3 of the first course of Core-Plus Mathematics (Coxford et al., 1998, p. 185). (Reprinted with permission of Glencoe McGraw-Hill publishing, © 1998.)

word limit, which students often think of as a boundary that cannot be crossed. This is not always true of a mathematical limit (Williams, 1991). Student understanding of terms within a Standards-based environment4 has been studied within geometry (e.g., Keiser, 2004). In other areas, such as number (Russell & Corwin, 1993) or measurement (Godfrey & O’Connor, 1995), accounts have focused on the teacher’s role in probing student-generated definitions, leaving student understanding unexamined. In this study, students experienced little difficulty connecting slope to contexts as a rate and to its more formal definitions. There were few instances in which informal student-generated terminology concerning slope conflicted with its formal definitions. On the other hand, the real-world contexts embedded throughout the curriculum constrained the action of the teacher and students to graphs whose independent values were nonnegative or positive. This led to the use of “start,” “starting point,” and other variations, which at times were similar to y-intercept, and at other times and within different representations, conflicted with its formal definition. Consequently, the purpose of this study was to determine how real-world contexts, multiple representations, natural 4 This is a classroom environment where instruction is similar to that advocated by National Council of Teachers of Mathematics Standards documents (1989; 1991; 2000).


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language, and a delay in the introduction of formal mathematics terminology influenced students’ understanding of the y-intercept concept and their abilities to use it. THEORETICAL FRAMEWORK Multiple Representations Lesh (1979) developed a multiple-representation framework that consists of realworld situations, manipulative models, pictures, written symbols, and spoken symbols to more clearly elucidate Bruner’s (1966) theory of learning. Translations within a node, such as from one manipulative (e.g., fraction circles) to another (e.g., chip model), are just as important as those between nodes (e.g., creating written symbols to represent a real-world situation). In a similar vein, research in the area of functions has concentrated on students’ use of functions within different representations such as tables, graphs, and equations (Confrey & Smith, 1991; Janvier, 1987; Kaput, 1989; Moschkovich et al., 1993). The framework used in this study is shown in Figure 2 and involves a combination of these three function representations with those described by Lesh. Tables and algebraic5 representations appear as separate nodes within the framework. At the center of this framework are real-world contexts, not because this is a privileged representation, but, rather, because of its constant presence in Standards-based curricular materials. Students’ investigations are dominated by real-world contexts and students are frequently translating between real-world situations and tables, graphs, and equations, and vice versa. The spoken-language node is similar to the spoken-symbol node in the Lesh (1979) model. This node was included in this framework, as students in classrooms using Standards-based materials often engage in classroom discussions with the teacher or peers and are routinely asked to justify their explanations (Trafton, Reys, & Waseman, 2001). Although in other multirepresentational frameworks (e.g., Moschkovich et al., 1993) graphs are considered a monolithic category, the use of real-world contexts and the less-frequently appearing abstract mathematical contexts led to two different graphs that students produced or that appeared in Standards-based materials and are distinguished by the domain of the functions they represent. The first, shown in Figure 3, will be described as a limited domain graph (LD), and the second, in Figure 4, will be referred to as a full domain graph (FD). The domain of an LD graph consists of values that are either nonnegative or positive. This results in a graph that resides in either the first quadrant only or the first 5

Algebraic and equation representations will be used interchangeably throughout this article.

Tables Tickets 0 20 40 60

Profit –450 –400 –350. –300

Spoken Language

Algebraic y = –4x + 7

“start” “starting point” “y-intercept”

Real-World Contexts The distance that a bungee jumper falls before bouncing back …

LD Graphs

FD Graphs

FIGURE 2 Theoretical framework guiding the study.

FIGURE 3 Example of a student-generated limited domain (LD) graph.

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FIGURE 4 Example of a student-generated full domain (FD) graph.

and fourth quadrants. An FD graph, on the other hand, is valid for negative and non-negative values and can appear in any two or three of the four quadrants. As is true of the Lesh model, this framework also includes translations within and between nodes. Further, the movement between nodes is bidirectional and is represented by solid lines that connect any two nodes. The only exception to this translation property is between LD and FD graphs. Unlike other representations, which provide a unique lens to view any function, the choice of a LD or FD graphical representation is determined by the domain of the function itself. In other words, while students could conceivably translate from LD to FD graphs without loss of information, the opposite is not true. Thus, these two representation nodes are connected by a unidirectional arrow in the framework. The focus of the analyses here will be primarily on between-node translations involving the spoken language, tables, LD/FD graphs, and algebraic representations, and within-node translations involving spoken language. Students will understand the concept of y-intercept when they are able to correctly translate between spoken-language referents and FD/LD graphs, equations, and tables. An important aspect of this model is the students’ ability to use representations to facilitate translations from one node to another (Behr, Lesh, Post, & Silver, 1983; Davis, 2004; Dufour-Janvier, Bednarz, & Belanger, 1987; Lesh, 1979). For example, if students are struggling to find an equation from a table, they may draw on the graphical representation to plot the points, fit a line to the points, and determine the slope and y-intercept from the line once it has been drawn. There is a potential for ambiguity in this article in spoken-language referents and the concepts that they denote. Thus, spoken language will be distinguished from a particular concept through the use of quotation marks. For example, “y-intercept” will denote the spoken-language representation within the framework, while y-intercept denotes the concept as seen in its connections within and between the six nodes of the framework.

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This study is informed by analyses of the written curriculum and the developer’s intent in designing these activities. Attention is also paid to how the teacher interprets the written curriculum. This is of special significance, given the disparate ways that teachers enact lessons drawn from Standards-based curricula (Collopy, 2003; Lambdin & Preston, 1995; Remillard & Bryans, 2004; Wilson & Lloyd, 2000). At the same time, teachers’ work is constrained and enhanced by students as has been shown by Henningsen and Stein (1997). These factors will be addressed in developing an account of student understanding within this environment. METHODOLOGY Research Site and Participants This study took place at Jefferson High School,6 which is located in a large, urban, Midwestern setting in the United States. Jefferson was originally a private school but had become a public charter school in the summer of 2003. Mathematics courses at Jefferson followed a traditional mathematics sequence of Algebra I, Geometry, Algebra II, Precalculus, and Advanced Placement Calculus. Mr. Grant7 began teaching at Jefferson in the fall of 2001 after graduating with a Bachelor’s degree in mathematics. In the summer of 2004, after having taught at Jefferson for three years, he received his state-issued teaching certificate. During his preparation for this certification, he learned about the Core-Plus Mathematics instructional model and worked through Core-Plus Mathematics activities in a similar manner to students. While the study was being conducted, we conversed at least once a week by telephone to discuss issues on his use of the materials. Mr. Grant was in his fourth year as a mathematics teacher at Jefferson when this study began. Eight students (five boys and three girls) of different grade levels from one second-year algebra classroom participated in the study. All students were of middle-socioeconomic status. Carl, Christine, and Bart were juniors and had just begun at Jefferson when the study began. Ann and Joel were the only two sophomores in the class, but had come to Jefferson as freshmen, enrolling in geometry. Jim and Darla were seniors. Jim was an exchange student from Iceland and had taken several mathematics courses there. Darla first came to Jefferson as a freshmen and had enrolled in algebra at that time. Six of these eight students learned from a more traditional algebra textbook such as Merrill Algebra One (Foster, Rath, & Winters, 1990). In this book, students encounter 6

Pseudonym. This name is a pseudonym that matches the individual’s gender as are all others used throughout this article. 7


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functions in abstract settings and principally translate between tables and FD graphs to equation representations in slope-intercept form. Steve and Christine had learned from several units of Core-Plus Mathematics as freshmen in two separate school districts. All the students’ algebra experiences had occurred two to three years before this study had begun. Standards-based activities. Because of dissatisfaction with the mathematics curriculum that had been in place at Jefferson, Mr. Grant created his own curriculum that drew heavily from Algebra and Trigonometry, Functions and Applications (Foerster, 1994). In the past, he supplemented this textbook with other activities, such as a unit on modeling real-world data with linear functions. In the summer of 2004, Mr. Grant agreed to replace this unit with activities drawn from units 2 and 3 of the first course of the Core-Plus Mathematics program. A sample context and question from unit 2 of the first course is shown in Figure 5. Each Core-Plus Mathematics lesson consists of launch, exploration, share and summarize, and apply. Students are introduced to the context in which the lesson is embedded through the launch activity, which usually involves a reading selection followed by several student questions. Students begin inquiring into the mathematics of the lesson in the exploration phase through carefully planned investigations. The teacher facilitates the sharing of student strategies used to solve problems within the investigation and summarizes the main ideas of the lessons with the help of student answers to specific questions in the checkpoint section. In the last phase, students begin working individually or with others on problems related to the lesson. Mr. Grant generally followed these four phases of instruction, sometimes changing the size of groups or extending investigations across several days because of a 40-minute class period.

Profits and Losses Typical businesses watch patterns of change in their costs, income, and profit from operations. For example, the Palace Theater shows only second-run movies and charges a single low price of $2.50 for all shows all day. The income from ticket sales depends on the number of tickets sold. Ignoring costs for operating the concessions stand, the operating expenses for the Palace Theater average $450 per day. Assume there are no other expenses nor other sources of income. How could you determine the theater’s daily profit? Use this relation to make a table of (number of tickets sold, profit in dollars) data like the one below. FIGURE 5 A sample context and problem from unit 2 of the first course of Core-Plus Mathematics (Coxford et al., 1998, p. 122–123). (Reprinted with permission of Glencoe McGraw-Hill publishing, ©1998.)

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DATA COLLECTION This study followed an instrumental case study design as described by Stake (1995). Thirty-two lessons over a ten-week time period were videotaped by senior students enrolled at Jefferson. The camera was sometimes stationed in a corner of the classroom so that the students’ interactions with Mr. Grant at the front of the room could be recorded. At other times the camera followed Mr. Grant throughout the classroom as he interacted with groups of students. Occasionally, the individual operating the camera would roam the classroom, spending several minutes recording the conversation of different groups of students. Each week’s videotaped classroom episodes were sent to the author, who transcribed them in chronological order. Phone conversations with the teacher were held several times during each of the first three weeks of videotaped lessons and once a week thereafter, until four weeks after the lessons were implemented in the classroom. Initially, these focused on questions that Mr. Grant had about implementing the Core-Plus Mathematics lessons. As the videotaped lessons were analyzed, questions arose concerning the decisions Mr. Grant made in the classroom. Phone conversations were held with Mr. Grant to ascertain his decision-making regarding these episodes. The notes from these conversations became another source of data to help understand the reasons behind Mr. Grant’s pedagogical moves in the classroom. Prior to the beginning of the study, students were given paper and pencil tests to determine their abilities to translate between different representations. This assessment covered translations within an algebraic representation and translations from tables to equations, equations to a real-world context, real-world context to equation, table to LD graph, and LD graph to real-world context. During the ten-week duration of the study, students invented several different terminologies to depict the first point in an LD graph. Three weeks after the conclusion of the study, the term that had been used most often, “starting point,” was used to assess student understanding. Students were given paper and pencil tests in which they were asked to describe their own definition of “starting point.” These tests also included translation tasks between “starting point” and FD graphs, LD graphs, graphs of their own creation, tables, and equations. These translation tasks asked students to identify or locate the “starting point” in a graph, table, or equation. These assessments also shed light on students’ abilities to transfer this knowledge to abstract contexts. Students were also asked if they thought starting point was the same as y-intercept. A similar set of questions were given to students, asking them to make translations between “y-intercept” and FD graphs, LD graphs, equations, and tables. In addition, Mr. Grant was given a paper and pencil test in which he was asked to construct graphs that contained a “starting point” and a “y-intercept.” He was also asked to define y-intercept. Finally, through interviews and e-mail conversations, data were collected on the


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Core-Plus Mathematics developers’ intent in specific activities as well as overall design of the first-course units involving linear functions.

DATA ANALYSIS All transcriptions of classroom lessons were coded for student use of invented terminology regarding y-intercept. First, all utterances of student-generated terminology related to y-intercept in the classroom transcriptions were coded by individual and representation (table, graph, equation, and real-world situation). If the representation was a table, graph, or equation, the selection was also given a code for the context (e.g., Palace Theater) of the problem in which it occurred. Second, all utterances of the phrase “y-intercept” were coded in a similar manner. Last, the utterances made by the teacher, as found in the first and second stages of the analysis, were further investigated by examining their role in relation to students’ use of terminology. “Start” and its variants were also examined for whether they were student or teacher initiated within the transcripts. Nvivo qualitative research software (QSR International, 2002) was used to organize and facilitate the analysis of the transcript data. The nature of students’ connections between formal mathematical terminology and student-generated terms vis-à-vis different translations was first analyzed at the classroom level through the coding procedure described above. Second, analyses were conducted using data that were ascertained from paper and pencil tests (see the Appendix for sample items from these tests). This second set of analyses was used to verify conjectures about student understanding that were formed from analysis of classroom transcripts. These paper and pencil tests provided much information on the stability of students’ translations between spoken language and the other representations. In some instances, students were asked several questions that involved the same type of translation. If student translations were similar to y-intercept in all instances, the author reasoned that this translation was stable, and so it was coded as “S” for similar to y-intercept. On the other hand, if students identified the “y-intercept” in a table representation as the value b in the ordered pair (0, b) in one instance but as both the x-coordinate and y-coordinate in another, this translation was coded as unstable for this student and consequently given a code of “D” for dissimilar to y-intercept.

RESULTS AND DISCUSSION Classroom Transcripts Summary use of “start” and “y-intercept.” Table 1 displays the student and teacher utterances of “start” and “y-intercept” by type, frequency, context,

TABLE 1 Utterance by Individual by Representation by Abstract or Real-World Context

Individual Ann

Bart

Carl

Christine

Darla

Jim

Joel

Steve

Mr. Grant

Total

Type of Utterancea Start Start y-int Start Start y-int Start Start y-int Start Start y-int Start Start y-int Start Start y-int Start Start y-int Start Start y-int Start Start y-int Start Start y-int

–S –D –S –D –S –D –S –D –S –D –S –D –S –D –S –D –S –D –S –D

Frequency of Utterance

RW

A

C

E

T

LD

FD

3 0 8 2 0 3 4 1 2 5 2 0 2 0 0 1 1 1 1 4 1 27 3 6 44 11 23 88 23 44

3 0 1 1 0 0 4 1 0 4 1 0 2 0 0 1 0 1 1 3 1 25 3 1 41 8 11 81 17 15

0 0 7 1 0 3 0 0 2 1 1 0 0 0 0 0 1 0 0 1 0 2 0 5 3 3 12 7 6 29

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 6 0 0 9 0 0

1 0 2 1 0 0 3 0 1 3 0 0 2 0 0 1 1 0 0 1 0 13 0 0 21 0 3 45 2 6

0 0 2 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 2 0 0 3 0 1 9 3 1 16 5

2 0 1 1 0 0 1 0 0 2 1 0 0 0 0 0 0 1 0 1 1 8 0 1 16 1 9 30 3 13

0 0 3 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 3 0 5 0 1 8 4 1 20

Contextb

Representationc

a Start—S refers to uses of temporal-based language that is used in a similar fashion to the concept of y-intercept. Start—D includes uses of temporal-based language that is used differently than the concept of y-intercept. “Y-int” are those utterances in which individuals used the terminology “y-intercept.” b RW refers to a real-world context, while A denotes an abstract or purely mathematical context. c C—represents those situations in which students connect “start” or “y-intercept” to a context; E—equation or algebraic representation; T—table representation; LD— limited domain graph; and FD—full domain graph.

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and representation during the ten weeks of the study. Overall, “start” was used 111 times and “y-intercept” was used 44 times. Only 13 of these uses of “start” came within abstract representations, reflecting the connections between this language and the real-world contexts used in classroom lessons. On the other hand, y-intercept was used nearly twice as often in abstract (29 times) than in real-world contexts (15 times). Table 1 also shows much variance in the frequency of “start” used by students throughout the study. Steve used “start” most often (30 times), but the next most frequent occurrence was Christine, who used it 7 times in classroom conversations. Three students only used “start” on 2 occasions. Nonetheless, this informal terminology was a constant presence in the classroom, as seen in the teacher’s use of this terminology (55 times) when compared to the more formal “y-intercept” (23 times). Students used “start” and its variants in 56 instances; 52 of these were initiated by students. In the case of the teacher, 21 of 55 of his uses of “start” were in response to the students’ use of the “start” terminology. In the sections that follow, these results are explored in more detail through examinations of the classroom transcripts. Conditions leading to student-generated terminology for y-intercept. A sample of the contexts that students encountered during the ten weeks of videotaped instruction appears in Table 2. Many of the contexts that students worked with in units 2 and 3 of Core-Plus Mathematics contained an explicit temporal quality. In other words, time appeared explicitly as the independent variable. This was seen in the installment plan in which students envisioned themselves paying back a certain amount of money over a period or the cost of making phone calls of differing lengths of time. Some contexts, on the other hand, did not have time as an independent variable. This was the case in the Palace Theater, which involved number of tickets sold and profit. Indeed, these were only meaningful when the independent variable was nonnegative or positive. This led students to believe that they had a clear beginning. Moreover, the format of the tables in which the independent variable was in ascending order strongly suggested that the actions occurred in a specific sequence. The order in which points were graphed both with and without technology also suggested temporality. For example, LD and FD graphs were constructed by hand and with the use of technology from left to right by substituting steadily increasing values of the independent variable into the linear equation. The independent variable of these contexts also had an implicit or implied temporality. For example, the independent variable in the Palace Theater context was the number of tickets sold. Since it was not possible to sell five tickets before selling one, there was an implicit ordering, or temporality, embedded in the independent variable. These factors led to an imagined or implicit temporality, which became evident through students’ terminology for this beginning location. A specific example of this is the installment context in

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TABLE 2 A Sample of Contexts, Variables, Temporality and Meaningfulness

Contexta Bungee Experiment Earnings Installment Palace Theater Phone Company Remaining Fuel Ice Cream Projector Ratings Game Springs Male and Female Medical Doctors Telephone Companies Walking

Independent Variable

Dependent Variable

Temporalityb

Meaningfulc y-intercept

Jumper Weight

Cord Stretch

I

Y

Hours Worked Number of Payments Number of Tickets Length of Calls Time Number of Scoops Distance from screen Ranking of Show

Money Earned Balance

E I

Y Y

Profit

I

Y

Cost Fuel Price

E E I

Y Y N

Size of projected image Households Watching Length Percent of All Doctors

I

Y

I

N

I E

Y Y

Weight Time

Number of Calls

Price

I

Y

Time

Distance

E

Y

a These situations are only a sample of those that students worked on during classroom activities. b I represents implicit temporality; E represents explicit temporality. c The y-intercepts have a meaningful interpretation within the context.

which the number of payments is the independent variable and the balance is the dependent variable. Mr. Grant: I want you to explain What does that 1100 minus 130x mean? Carl: You like start out at 1100 and then you figure out like wherever you want you just you know you have 130 because that’s what you’re subtracting by times x. (Day 4, September 24, 20048 ) Thus, the equation comes about by reenacting the implicit or explicit temporality of the situation. There is a clear beginning to the situation, and new values are generated by subtracting a constant repeatedly. The two principal pieces 8 This notation represents the day of the study, out of 32, and the date that this data was drawn from.


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of a linear equation at this point were not slope and y-intercept, but instead a constant value that was either repeatedly added or subtracted and a beginning location, respectively. Students used these pieces to find the equation, given a table of nonconsecutive points. This is seen in the Palace Theater context involving number of tickets and profit. Number of Tickets 80 140

Profit in dollars $15 $270

Steve: I took 270 – 15 and that equals 255. Then I saw there was 60 difference between there and I divided it by 6 and that should give me every 10 tickets equals 42.5 dollars, and then I had to get every single ticket, so I divided it by 10 and that equals 4.25 and that’s my equation right there. [I]t goes up by 4.25 every x and then I had to get the starting point so I just took 80 and then I minused [sic] it 42.5 over and over again 8 times for the number of tickets, and that got my starting point of negative 325. (Day 7, September 30, 2004) During the second week of the study, Steve introduced “starting point” to the teacher and other students. Mr. Grant repeated this terminology in classroom conversations after Steve used it. Steve continued to be the only student who used this terminology for several days. Up to this point, Christine, Darla, Carl, and Joel had used “start” in their classroom conversations. During the third week, however, Mr. Grant began introducing “starting point” and using it in his class discussions. This move appeared to grant the terminology a status that it had not had before. Consequently, other students began incorporating “starting point” into their conversations. Promoting connections with start. As described earlier, Core-Plus Mathematics activities in the algebra and functions strand frequently involve students in working within and between the six representations of the function framework. The spoken language representation, as exemplified with the “start” and “starting point” descriptions, had an important function within this environment as a way in which students could connect their work to other representations. For example, in the excerpt above, the student used the phrase “start out at 1100” to justify this number in the equation y = 1100 – 130x. At this point, because of the temporality/meaningfulness of the contexts students worked with, “start” referred to the beginning of the context or a physical location9 on the LD graph. Thus, the 9 The physical location is a place on the graph as opposed to the ordered pair or coordinates that represent the position mathematically.

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spoken-language node represented by “start” in this case connected students’ translation work to an LD graph. Other translations between equations and different contexts were aided by spoken-language representations that consisted of the word “starting” and a noun related to the context. This happened within the context of the profit of a theater after selling a number of tickets. Mr. Grant: Profit at the Palace Theater is a function of number of tickets sold according to the rule P = –450 + 2.5T. Without making a table or graph, before you start doing this one, I want you to explain the equation P = –450 + 2.5T. What is –450 in this case? Steve: The starting price. (Day 27, November 18, 2004) Here the student interpreted the daily operating costs in terms of the price that the theater would have to pay for these services. At this point, start and its linguistic variations consistently represented the constant in the slope-intercept form of an equation. This constant also happened to be where the contexts were meaningful in many cases and the place in which the graphical representations of these contexts began. However, at this point it did not necessarily represent the concept of y-intercept. Influence of everyday experiences. At first glance, the context represented in Table 3 does not appear to have a meaningful y-intercept, since the table begins when the independent variable is one. However, students would often tap their everyday experiences to translate between a table and equation. In this phone context, the concept of y-intercept was meaningful since it was possible to have a monthly charge for phone service without making any phone calls. One student described the situation in the following manner: “You need to pay starting 2.50 for just picking it [the handset] up” (Darla, September 30, 2004). They used their experiences with these settings outside school to make sense of the missing values in this table, such as the point (0, b). In other cases, such as the ratings game situation shown in Table 4, the y-intercept was not connected to the context; it was not possible to have a show with a rank of zero. Nevertheless, students’ experiences with Core-Plus Mathematics up to this point in class had emphasized connections between TABLE 3 Price of a Phone Call Depending on Length of Call in Minutes Call Length, L, in minutes Cost, C, in dollars

1 3.00

2 3.50

3 4.00

4 4.50

5 5.00

10 7.50

15 10.00

20 12.50

Source: Coxford et al., 1998, p. 148. (Reprinted with permission of Glencoe McGraw-Hill publishing, ©1998.)


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TABLE 4 Rank, Name of Show, and Average Weekly Audience

Rank 10 20 30 40 49 59

Show Friends, NBC 60 Minutes, CBS The Simpsons, FOX Dateline, NBC The Drew Carey Show, ABC Ed, NBC

Average Weekly Audience (millions of households) 109 91 84 76 67 58

Source: Coxford et al., 1998, p. 162. (Reprinted with permission of Glencoe McGraw-Hill publishing, ©1998.)

equations and the spoken-language representation. As a result, students attempted to make a connection where one did not exist. This is seen in the conversation below. Darla: Negative 0.1x + 11.7. Mr. Grant: Alright, now these are all very close, aren’t they? And they should all be very close shouldn’t they? What is that 11.5, 11.6? Steve: Starting point. Mr. Grant: Our starting point would be the show that was ranked what? Steve: Zero. Mr. Grant: Zero that would be the best you could ever do. Actually, the show that is ranked best is ranked what? Steve: One. (Day 16, October 19, 2004) In this example, students were willing to translate between the “starting point” and the context even when it did not make sense to do so. In other cases, students refused to connect the “initial value” to a context because their experiences outside class conflicted with this interpretation. This happened in one of the three contexts that Mr. Grant had created involving number of scoops and the price of an ice cream cone. Jim: Christine: Mr. Grant: Jim: Mr. Grant: Jim:

The cone costs $1.00. Which makes a lot of sense then. Is $1.00 our initial cost then? No, 1.2 is our initial value. Why? Because you can’t buy an ice cream that doesn’t have a scoop in it. (Day 8, October 5, 2004)

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The students had determined the price per scoop was $0.20, but, based on their experiences, it was not possible to buy just a cone. The “starting value” was $1.20 instead of the $1.00 that they had calculated. Thus, in some cases, realworld contexts do not aid students in developing an understanding of concepts such as y-intercept. Table to spoken language translations. Table representations have many different beginning or starting values, which are tied to the form of the representation. This led to different referents for the spoken-language representation within a table setting, as seen in classroom transcripts. Indeed, as seen in Table 1, 16 of 17 utterances of “start” within a table representation were not used in a manner similar to the concept of y-intercept. The only correct use came from the teacher. Thus, the ability of real-world contexts and their spoken-language artifacts to play the role of an anchor for student understanding of y-intercept within this representation is limited. An analysis of these uses of “start” by students within a table representation led the author to make two different interpretations that contrasted with its meaning in graphs, equations, and real-world contexts. First, in some contexts, the initial value of the dependent variable or “start” of the table did not occur when the independent variable was equal to zero. This is shown in Table 4. Second, “start” was used by students as any value of the independent variable. Textbook exercises frequently asked students to find either the independent or dependent value of a function when given one of the two values. For example, Joel was provided with Table 3 and asked to find the cost of a phone call that lasted eight minutes. Joel: Ahh, right here, it starts at 5 so it’s just 5 dollars so I built onto the 5 and of course I erased my answer, but I just built on to 50 cents until I got to 8. (Day 7, September 30, 2004) Notice that Joel stated, “it [the table or the context] starts at 5,” which happened to be the independent value of the function instead of the dependent value when x = 0. In another example, Christine was working with a table representation of the Palace Theater context that contained the ordered pair (80, 15). As she described how she created the equation representation, she stated, “we start at 80 and get 15” (Day 7, September 30, 2004). This connection between “start” and the value of the independent variable was further strengthened when students used graphing calculators to solve problems similar to the one that Joel answered. Students simply entered the equation in slope-intercept form (y = mx + b) into the calculator’s list of equations. The first row of the displayed table is determined by setting the “tblstart” to the independent value of the function with which students want to begin. Consequently, the use of technology reinforced


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the connection between “start” and the independent value of a function, as seen in the first row of the calculator’s table representation. Connecting formal and informal terminology. As mentioned earlier, the developers intend for students to connect “y-intercept” to the contexts in which they work. As the formal terminology of “y-intercept” is introduced midway through unit 3, opportunities arise for it to become connected with the informal terminology that students have been using up to this point. The teacher echoed this curricular organization in his classroom interpretation of the text since he used “y-intercept” only once in week 2 and once in week 5. After its introduction in the text during week 7, it played a greater role in classroom discussions. In week 7, students worked through investigation 1 of lesson 2, unit 3, involving the real-world representation of the amount of stretch or compression of springs when different weights are attached. When students translated from tables, graphs, equations, and real-world contexts to spoken language, they did so through the use of “starting point.” The checkpoint or summary at the end of this lesson asks students how the “slope” and “y-intercept” can be identified in tables, graphs, and equations. At this point, the Core-Plus Mathematics developers intend for students to generalize “slope” and “y-intercept” to any table, graph, and equation (C. R. Hirsch, personal communication, March 14, 2005). The teacher made a similar interpretation, but chose to accomplish that generalization in two ways, which did not take students’ prior work into account. First, instead of beginning with the LD graphs and real-world contexts that students had been working with, the teacher situated his examples within FD graphical representations. Second, he focused students on translating between tables, graphs, and equations to spoken language via “y-intercept” rather than beginning with the “start” terminology, which students had previously been using in these situations. Week 8 included an activity in which students investigated FD graphs by altering the parameters a and b in the equation y = a + bx and noting the effects on their graphing calculators when domain and range values varied from –10 to 10. This activity was situated within an FD graphical representation, but the text surrounding this problem in the Core-Plus Mathematics written materials included no mention of the phrase “y-intercept.” It appeared that the teacher initiated the use of y-intercept in this abstract setting because his past experiences in which translations were made from FD graphs and equations to spoken language through the use of “y-intercept.” Only one student, Steve, did not use the phrase “y-intercept” in this activity. He experienced no inhibitions in translating between FD graphs and spoken language using “starting point,” even though the graphs did not contain such a point in everyday parlance. Translations around y-intercept. Within the classroom lessons, students and teacher made distinct translations from tables, graphs, and equations to

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“y-intercept.” For example, in an FD graph to spoken-language translation, “y-intercept” was defined by Ann as “where the line intercepts the y-line” (Day 21, November 3, 2004). Within a spoken language representation, Mr. Grant linked “y-intercept” in the last week of class to “where the line intercepts y” (Day 32, December 3, 2004). In the equation to spoken-language translation, “y-intercept” was defined as the “plus whatever” by Steve (Day 21, November 3, 2004). In the table to spoken-language translation, it was defined as the y value when x was zero. The students performed many of these translations between “y-intercept” and different representations, but they were not provided with opportunities to reflect on the similarities across different translations. Had this consolidation occurred, it may have helped them develop representational transparency so that the concept of y-intercept could become identified with the Cartesian connection. Overall, the students and the teacher mentioned “y-intercept” 44 times in the 10 weeks of this study. As was shown, “y-intercept” was used predominately in translations involving equations and FD graphs. In fact, 59%, or 26 of 44 utterances of “y-intercept” occurred within translations emanating from FD graphs or equation representations. Alternatively, “start” terminology was only used twice, once by Steve and once by the teacher when translating from FD graphs to the spoken-language representation. Given these classroom experiences, it was possible for students to consider translations involving “start” as being different from translations involving “y-intercept.” This issue was explored in more detail through the paper and pencil tests. Paper and Pencil Tests Students’ initial understandings. Recall that students had taken an assessment focusing on their abilities to translate within algebraic representations and across a number of different representations prior to the beginning of the study. Students’ scores on this test ranged from a low of 7 out of 45 points for Joel to a high score of 22 for Ann. Seven of the 8 students had scores ranging from 15 to 22. Thus, with the exception of Joel, student scores were very similar. Christine and Steve, who had previous experience with Core-Plus Mathematics, struggled with translating between different representations and real-world contexts, as did the other students. Overall, the students were most successful solving equations using formal symbolic manipulation techniques. Transfer of learning. Table 5 examines students’ translations involving informal language within real-world contexts and was assembled from classroom transcripts. The paper and pencil test results shown in Table 6, on the other hand, show students’ translations around informal language within an abstract context.


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TABLE 5 Student Translations Surrounding Informal Language in Real-World Contexts Type of Utterancea

Equation

Table

24 1 96%

0 6 0%

Start – Similar Start – Dissimilar Percentages Similar

LD Graph 12 1 92%

FD Graph 2 0 100%

a Start – Similar are those translations that are similar to “y-intercept,” while Start – Dissimilar are those that involve “start” but are different from “y-intercept.”

TABLE 6 Student Translations Surrounding Informal Language in an Abstract Context Graph Student

Table

FD

LD

Equation

Percent Correct Translations

Ann Christine Darla Jim Carl Steve Joel Bart Average

D S D D S D D D 25%

D S D D S S D D 38%

S S D D D S D D 38%

D S D D S S S S 63%

25% 100% 0% 0% 75% 75% 25% 25% 41%

Note. “S” represents that the student translated “starting point” in a manner similar to “y-intercept,” while “D” represents translations dissimilar to “y-intercept.”

These results provide data on students’ abilities to transfer10 their informal terminology to an abstract context. In Table 6, students translated to equations in a manner similar to “y-intercept” 63% of the time, to tables 25% of the time, to LD graphs 38% of the time, and to FD graphs 38% of the time. Table 5 shows that students were able to translate from “start” or “starting point” to equations in a manner similar to “y-intercept” 96% of the time, to LD graphs correctly in 12 of 13 situations (92%), and to FD graphs similarly in all occasions (100%). Students did not make any translations between “start” and table representations within real-world contexts in a manner similar to “y-intercept.” Looking across these 10

The author thanks the anonymous reviewer for pointing out this implication.

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translations, students did much better in classroom activities that were tightly connected to real-world contexts. Hence, it appears that students’ success within equations, LD graphs, and FD graphs did not transfer completely to an abstract context—that is, an important element of students’ successful translations during classroom lessons was the real-world contexts within which these activities were embedded. The one exception to this finding was on table representations in which students did better within abstract settings. This may have been due to the ways that “start” was used by students within table representations that were tied to real-world contexts. In other words, tables were used to solve problems that did not include or highlight the “y-intercept.” Students’ translations surrounding starting point. Students’ definitions of “starting point” fell into one of three categories. First, Ann, Bart, Darla, and Jim believed that the “starting point” was simply the place in which a line begins, as seen in Darla’s definition: “The starting point is the point in which you put the first mark for the line on the graph.” Second, Carl and Christine gave a definition of “starting point” that was tied to the formal definition of yintercept: “Starting point is the y value when x is 0.” Third, Steve and Joel, on the other hand, believed that “starting point” was connected to either the “x-” or the “y-intercept” in a graphical representation. Joel remarked, “The starting point is the point where it encounters the x-axis or the y-axis.” Thus, only two students identified the spoken-language referents “y-intercept” and “starting point” as equivalent. Stable connections. As seen in Table 6, only Christine viewed starting point as similar to y-intercept when translating across tables, FD/LD graphs, and the algebraic representation. Recall that she was a junior in this class and had studied Core-Plus Mathematics as a freshman at another school. This past experience with the curriculum may have contributed to her understanding, but, unlike Steve, she did not use the “starting point” terminology until the last week of class. Within the paper and pencil test, she defined “starting point” as the y-coordinate when x was zero. Her definition of “starting point” in terms of the more formal definition may have helped her in connecting these different representations. Moreover, of all the students, with the exception of Steve, Christine used “start” as “y-intercept” most often in classroom discussions. In fact, the nature of these translations involving “start,” as seen in this conversation, may have helped her to connect it to the coordinates of that point. Mr. Grant: We can put lines in on the bottom [Mr. Grant puts hash marks on the x-axis.]. But the problem with that is we don’t know how long it will take. It seems that we are going at intervals of 50 hours here so I’ll make this one 50, then 100, then 150, so now we have a scale here and a scale there. What would the first point be? Joel: Zero and a 100.


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Mr. Grant: Zero and a 100, right? So that’s right here and that point is known as Christine: Starting point. Mr. Grant: This is the starting point. It’s got a name. What is it? What axis is it? Steve: Y-axis. Joel: Y-intercept. (Day 30, November 19, 2004)

Unstable connections. Other students such as Carl and Steve had unstable or incomplete connections between “starting point” and “y-intercept” within the spoken-language representation. The ambiguity of “start” when translating from table representations may have caused Steve’s understanding of starting point to be separated from y-intercept. He clearly indicated that the “y-intercept” was the same as the “starting point” when translating from spoken-language representations to LD/FD graphs and algebraic representations, but in tables that contained the point (0, b), he was unable to identify the value of b as the “starting point.” Of all the students, Steve more frequently referred to “start” in a table as its beginning value, regardless of whether it was the ordered pair (0, b). A lack of emphasis on the formal definition may have also contributed to this disconnection for Steve, since he defined “starting point” as “where the equation’s line intercepts with the y-axis.” Although Carl also defined “starting point” in terms of the formal definition of y-intercept, his understanding was not as stable as Christine’s. For Carl, the everyday connotations of “starting point” were too difficult to shake after being removed from the real-world contexts in which it had been created. He identified the beginning location as the “starting point” in three LD graphs regardless of whether they represented the “y-intercept.” Thus, he reverted to this everyday connotation within a graphical setting where it was most likely to appear. This vignette provides additional evidence for the lack of transfer of “starting point” from real-world contexts to those that were more abstract. Carl’s formal definition could be applied in equation, table, and FD graph representations, but it appeared that real-world contexts were a necessary ingredient by which the definition became functional in LD graphs. Students’ translations surrounding y-intercept. Students’ translations from “y-intercept” to the other representations are shown in Table 7. Overall, the percent of correct translations between “y-intercept” and the other four representations was higher than for “starting point.” This may have been the result of students’ prior knowledge with this concept in their previous algebra classes, and it may reflect the limits of natural language as shaped by the contexts and the representations that students used. Despite this improved performance with the use of the word “y-intercept,” students still did not have a high success rate.

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TABLE 7 Student Translations Surrounding “Y-Intercept” Graph Student

Table

FD

LD

Equation

Percent Correct Translations

Ann Christine Darla Jim Carl Steve Joel Bart Average

S S D S S D D S 63%

S S S S S S S S 100%

S S D D S S S S 75%

S S D S S S S D 75%

100% 100% 25% 75% 100% 75% 75% 75% 78%

Note. “S” denotes that the student translated “y-intercept” and represented it in a manner similar to the concept of y-intercept. “D” denotes translations dissimilar from y-intercept.

For example, one student translated correctly 25% of the time, while four other students missed one translation, each resulting in a classroom average of 78%. Students were most successful at translating between FD graphs and “y-intercept,” which may have been a result of the frequency of these types of translations during classroom activities and Mr. Grant’s definition of y-intercept as “the place where the line intercepts the y-axis.” Similar to “starting point,” table to “y-intercept” translations were the most difficult for students to perform. The difficulty experienced by two students when translating between LD graphs and “y-intercept” may have been because of decisions of the curriculum designers and the teacher’s choice of using “y-intercept” when working with FD graphs. Similar to translations around “starting point,” students more often correctly translated between “y-intercept” and algebraic representations. This may have been a result of students’ prior experiences with a traditional algebra curriculum, which works predominately with graphical representations, equations, and the slope-intercept form for a line. Christine, Carl, Ann, and Jim correctly translated from table representations to the spoken language representative, “y-intercept,” by identifying it as the dependent variable when its accompanying independent value was zero. Joel and Steve, on the other hand, did not have the same conception of y-intercept across different representations. For example, they identified “y-intercept” in a table representation as both the x- and the y-coordinates, but in the equation representation, they all identified “y-intercept” correctly. Interference between y-intercept and starting point. Ann, Darla, Joel, and Jim all stated that “y-intercept” involved a line crossing the y-axis. However, this


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metaphor appeared to influence students differently. Although Ann defined “yintercept” as crossing the y-axis, she correctly translated between “y-intercept” and LD graphs in which the linear function touched the y-axis. However, for Darla, the crossing metaphor interfered with her ability to translate between the spoken-language referent “y-intercept” and table and algebraic representations. For example, when asked to identify the “y-intercept” in the equation y = –4x + 7 she underlined the 7 explaining, “Because there is a positive and negative number . And they eventually will pass threw [sic] each other.” When asked to identify the “y-intercept” in an equation representation she circled the slope. In Jim’s case, the crossing metaphor only appeared within LD graphs. This was seen in the context in which a student named Lori bought a computer for $1,000 and paid off $200 each month. Jim said that the situation did not have a “y-intercept”: “Because it is a gradual decrease beginning at (0, 1000). Lori cannot owe money for her computer before the plan begins.” Hence, he felt that a “y-intercept” only occurred when there were points on the negative side of the axis as seen in his statement “before the plan begins.” His graphs of these situations clearly touched the y-axis, but he stated that they did not have a “y-intercept” because they did not cross the axis at the point (0, b). Teacher’s connections between starting point and y-intercept. Of all individuals in the classroom, Mr. Grant used “start” and its different variations in his verbal communication most often. He made 55 separate translations between “start” and the other four representations. Of these, 11 were not similar to mathematically acceptable translations involving “y-intercept.” Furthermore, he stated that “y-intercept” was equivalent to “starting point” during classroom lessons on two occasions, thus translating between these two within a spokenlanguage representation. This led to the conjecture that when Mr. Grant heard “starting point” in classroom conversations this was akin to y-intercept. His paper and pencil test results lend credence to this conjecture.

IMPLICATIONS AND CONCLUSION Benefits and Liabilities The student-invented terminology in this study had benefits for students. First, since it was familiar to students, it connected their actions in the mathematics classroom to their everyday understandings in a world suffused with temporality. This is in contrast to “y-intercept,” which has fewer connections to students’ lives and the vocabulary to which they are accustomed. Second, it had the unique characteristic that it could be modified so that its original intent was not lost, but instead, its meaning could be expanded to focus on its real-world connections. This was seen in the students’ use of “starting price.” This terminology still

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evoked the notion of a beginning point on an LD graph or equation, but it also connected students’ work to the real-world context within which they had been working. At the same time, however, this terminology had limits. For instance, as seen in Table 1, it was most often used in a manner different from y-intercept when it was used in a table. This was because of the presence of the independent and dependent variables in the table, the table’s flexibility in beginning at any ordered pair for the function, and the use of technology in which the independent value of the function denoted the starting value. In addition, “start” was rarely used by students within FD graphs since the activities focused on real-world contexts and the LD graphs they promoted. Moreover, there were few connections between “start” terminology, as it was thought about in an everyday sense and FD graphs—that is, an FD graph has no real beginning or end. The terminology “starting point” may also become problematic for students because they may think about it as representing an ordered pair because of the use of the word “point” instead of solely using the dependent value when the independent value is zero. Because of these liabilities, the informal language used by students in this study did not make a good foundation for creating a network of connections across different representations. This was seen in the uneven performance of students when translating between “starting point” and other representations in Table 5, and the decrease in overall performance of students in abstract contexts when compared to similar translations in real-world contexts (see Table 6). Past Experiences In terms of the students’ past experiences, six out of eight students in this study had made translations from “y-intercept” to tables, equations, and FD graphs through activities in their former algebra textbooks. Moreover, inattention by the curriculum and specific teacher moves reinforced these past experiences by focusing the use of “y-intercept” within abstract contexts or without explicitly connecting it to students’ informal terminology. Thus, there were already strong expectations in place that “y-intercept” was linked to FD graphs. This led some students to believe that “y-intercept” was the place at which a line crosses the y-axis, and since this did not occur in LD graphs, these did not have a “y-intercept,” even though they did contain the point (0, b). Two aspects of the teacher’s experiences shaped students’ understandings of “starting point.” First, Mr. Grant was able to abstract the important idea of y-intercept from the class’s use of “starting point” because of his past experiences within the terrain of linear functions. He may have assumed that the students also made this connection. Evidence for this assertion comes from an analysis of classroom transcripts. Out of 44 instances of “y-intercept” in classroom


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conversations, only 3 utterances were connected specifically to “starting point.” Hence, there appeared to be an implicit assumption on the part of the teacher that “start,” or its other variants, served as a window through which the concept of y-intercept could be viewed. This may have played a role in students’ inabilities to connect “starting point” to “y-intercept” across different representations. Second, Mr. Grant’s own experiences with the concept of y-intercept shaped his instruction and played a role in students’ understanding. For instance, during week 8, as students worked on FD graphs, the teacher introduced “y-intercept” into classroom discussions because of his past experiences teaching the concept. The students followed his lead and used “y-intercept” predominately during this activity, which reinforced their own past experiences in the domain. This resulted in “starting point” becoming separated from “y-intercept” within a spoken-language representation. Curricular Implications Although teachers are active designers of their classroom lessons, there is some evidence to suggest that they tend to follow the textbook closely when implementing a Standards-based curriculum for the first time (Hetherington, 2000). Therefore, it is important that the curriculum show what is possible through student activities and notes for the teacher. In that light, this study recommends the following curricular changes. The curriculum should include spoken-language and real-world situations as representations akin to tables, graphs, and equations. Moreover, graphical representations should no longer be thought of as monolithic. Specific activities should involve translations among real-world contexts, spoken language, tables, equations, LD graphs, and FD graphs. Activities that involve translations between different representations can easily confuse the trees for the forest. That is, movements between different representations are important in their own right as methods to solve problems, but they can also lead to a deep understanding of a concept through the systematic comparison of one translation against another. The curriculum should provide students with activities that promote discussion of the different translations between “starting point” and other representations. This may lead to an examination of the inconsistencies between “starting point” to equation and “starting point” to table translations. These discussions have the potential to encourage students to construct an understanding of starting point similar to y-intercept, its disuse due to the ambiguities of “start” within a table representation, or the creation of a new term that still retains features of the contexts from which it arises, yet reduces ambiguity. The delay in formal terminology, the postponement of formal definitions, and the use of multiple representations can lead students to create multiple informal

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definitions for y-intercept that, on the surface, appear to be different as seen in the descriptions of “y-intercept” as the “plus whatever” in an equation, or as “where the line intercepts the y.” The curriculum itself defines y-intercept as “the point where the graph intersects the y-axis” (Coxford et al., 1998, p. 186). The curriculum needs to be aware of this possibility and should contain specific student activities in which informal definitions for formal terms across representations are compared, thereby leading to the development of the y-intercept as the value b of the point (0, b). This may, in turn, lead to the Cartesian connection. Six of the eight students in this study all had prior experiences with algebra as it is traditionally approached. In the United States, this mirrors the experiences of a growing majority of students at the beginning of high school, who may have already experienced a traditional algebra class in middle school. The curriculum must take these prior experiences into account and seek to bridge the gap between this traditional background and activities that include real-world contexts, technology, multiple representations, and a delay in formal mathematics terminology. Such work may avoid the separation of informal from formal terminology, as was seen in the students in this study who had different definitions for “starting point” and “y-intercept.” This case also suggests that teachers using these curricula may view students’ informal terminology around y-intercept, as well as other concepts that are similar to their own more advanced understanding, even if students do not. Therefore, teachers may not bring comparisons between informal and formal language to the attention of students so that they can be examined explicitly. Thus, it is important that the curriculum include these activities for students, as well as notes in the teacher’s edition to draw attention to this possibility. Although real-world contexts hold potential for making students’ mathematical investigations more meaningful, they must be used carefully in the classroom. Indeed, these contexts and a delay in formal terminology may encourage students to bring informal or colloquial terminology into the mathematics classroom. The results of this study suggest that students’ may not be able to perform as well on tasks involving this terminology when they are set within abstract contexts. The curriculum must be cognizant of the potential for informal terminology to become disconnected from formal mathematical terms and for success to decrease when tasks are set within abstract contexts. It must address these possibilities through notes to the teacher and specific activities that establish and nurture these connections between informal and formal terminology and real-world and abstract contexts. In doing so, the curriculum can become a site for teacher and student learning. Future research needs to be conducted in other classrooms learning from curricula with real-world contexts, delayed introduction of formal terminology, and multiple representations to determine the prevalence of temporal-based


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student-generated terminology and the understandings that students develop from these environments to determine the generalizability of these findings. As this case illustrates, it is especially important to document the actions of the teacher in such studies of student understanding. Further work in this area can help delineate the conditions under which students can more tightly connect informal and formal mathematics terminology. Because of the everyday underpinnings of student-invented terms, research of this nature may also help us better understand the factors with which students are able to generate mathematical definitions for commonplace terms such as limit. Ultimately, this work can help teachers and students find and capitalize on learning opportunities that were once lost.

REFERENCES Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91–125). New York: Academic Press. Berlinghoff, W. P., Sloyer, C., & Hayden, R. W. (2000). Math connections: A secondary core curriculum. Armonk, NY: It’s About Time. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press. Cockcroft, W. H. (1982). Mathematics counts: Report of the committee of inquiry into the teaching of mathematics in schools. London: HMSO. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers’ learning. The Elementary School Journal, 103, 287–311. Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes, multiple representations, and transformations. In R. Underhill & C. Brown (Eds.), Proceedings of the 13th annual meeting of the North America chapter of the international group for the psychology of mathematics education (pp. 57–63). Blacksburg, VA: Division of Curriculum and Instruction, Virginia Polytechnic Institute. Coxford, A. F., Fey, J. T., Hirsch, C. R., Schoen, H. L., Burrill, G., Hart, E. W., & Watkins, A. E. (1998). Contemporary mathematics in context: A unified approach. New York: Glencoe/McGrawHill. Davis, J. D. (2004). Supplementation, justification, and student understanding: A tale of two contemporary mathematics in context classrooms. Unpublished doctoral dissertation, University of Minnesota, Minneapolis. Dufour-Janvier, B., Bednarz, N., & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 109–122). Hillsdale, NJ: Erlbaum. Fendel, D., & Resek, D. (1997). Interactive mathematics program. Berkeley, CA: Key Curriculum Press. Foerster, P. A. (1994). Algebra and trigonometry: Functions and applications (2nd ed.). Menlo Park, CA: Addison-Wesley. Foster, J. G., Rath, J. N., Winters, L. J. (1990). Merrill algebra one. Columbus, OH: Merrill. Garfunkel, S., Godbold, L., & Pollak, H. (1998). Mathematics: Modeling our world. Cincinnati, OH: South-Western Educational Publishing. Godfrey, L., & O’Connor, M. C. (1995). The vertical handspan: Nonstandard units, expressions, and symbols in the classroom. Journal of Mathematical Behavior, 14, 327–345. Harel, G., & Dubinsky, E. (Eds.). (1992). The concept of function: Aspects of epistemology and pedagogy. Washington, DC: Mathematical Association of America.

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APPENDIX Sample Paper and Pencil Test Items 1. Describe the meaning of starting point in your own words. 2. Graph a line on the axes below. Does the line have a starting point? Yes No (Circle One). If so, label it.

3. Place numbers in the table below for a linear function. Does it have a starting point? Yes No (Circle One). If so, circle it.

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x

y

4. Does the equation below have a starting point? Yes No (Circle One). If so, circle it. y = −4x + 7 5. Is the y-intercept the same as the starting point? Yes No (Circle One) Please explain your answer.