connecting students' informal language to more formal definitions

Connecting Students' Informal Language to More Formal DefinitionsAuthor(s): Jon D. DavisSource: The Mathematics Teacher, Vol. 101, No. 6 (FEBRUARY 2008), pp. 446-450Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20876175 .

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Definitions

Jon D. Davis

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ommunication is an essential part of mathematics and mathematics education,' (NCTM 2000, p. 60), and an indispensable component of this communication is formal

mathematical terminology. Many mathematical terms have been appropriated from everyday language and

transported to the mathematics classroom (Halliday 1978). The mathematical definitions of these words

may be analogous to their meanings outside the classroom. For example, the word similar is defined in Webster's New World College Dictionary (1999) as "nearly but not exactly the same or alike; having a resemblance." This definition is akin to the math ematical definition of similar triangles?triangles that are alike in some properties (corresponding angles) but different in others (corresponding sides).

Still, there are some words, such as odd, whose mathematical definitions bear no resemblance to their everyday counterparts. Although some stu dents may think odd numbers are indeed strange, this perception is unrelated to the term's mathemat ical definition. This difference in the use of terms inside and outside the mathematics classroom may confuse students, as seen in the example (Pimm 1987) shown in figure 1, in which a student inter

preted the word right with regard to its orientation rather than to the measure of its largest angle.

446 MATHEMATICS TEACHER | Vol. 101, No. 6 February 2008



Left-Angled Triangle Right-Angled Triangle

Fiq. 1 Confusing the everyday meaning of right with its use in the mathematics classroom

STANDARDS-BASED MATHEMATICS PROGRAMS The issue of how language is used inside and out side mathematics classrooms has grown in impor tance recently as the number of schools choosing to

implement Standards-based mathematics programs increases (Senk and Thompson 2003). Although these programs may differ from one another in orga nization or content (Martin et al. 2001), they often

delay the introduction of formal mathematics ter

minology, involve student-to-student and student to-teacher communication, draw on real-world con

texts, and use several representations (e.g., tables, graphs, and equations). These features allow stu dents to engage in the practices of mathematicians

by coining and defining new terms, but, as I intend to show in this article, these opportunities must be

carefully monitored and nurtured by the teacher. The study described here occurred in one secondary

mathematics classroom that used Contemporary Math ematics in Context, a series of books in the Core-Plus

Mathematics Project (Coxford et al. 1998). Students were videotaped as they worked through activities within the algebra and functions strand of the first course of this series. These videotaped classes were

later transcribed and analyzed to understand how stu dents created their own informal mathematical terms and how they used the terms within different function

representations (tables, graphs, and equations). At the end of the classroom instruction, paper-and-pencil tests were given to determine whether students viewed their own terminology as similar to more formal mathemati cal words and to measure their ability to transfer their invented terniinology to an abstract context.

HOW INFORMAL TERMINOLOGY DEVELOPED IN THE CLASSROOM The students worked with problems set in many differ ent types of real-world contexts. Some involved time as an independent variable, such as when students inves

tigated the relationship between a phone call's cost and its length in minutes. In others, the use of time was not explicit, but the functions themselves had a clear

beginning. For example, in the Palace Theater context, students needed to determine the revenue when given

Table 1

Number of Tickets Sold and Profit for the Palace Theater

Number of Tickets Sold Profit in Dollars

80 $15

140 $270

Fig. 2 Student-generated limited-domain graph

the number of tickets sold (see table 1). Students see that there is a clear begirrning location for the context,

graph, and equation when zero tickets are sold. They also have a sense that the events are governed by time, because the theater will sell 50 tickets before it sells 75. Contexts such as this led to limited-domain graphs that involved positive real numbers or nonnegative integers and remained in the first quadrant or in the first and fourth quadrants. An example of one such graph con

structed by a student is shown in figure 2. The curriculum and the teacher frequently gave

students opportunities to explain their thinking, which led to the use of informal terminology to refer to the beginning value of the function.

Teacher. 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 know you have 130 because that's what you're subtracting by times x.

Vol. 101, No. 6 February 2008 | MATHEMATICS TEACHER 447



During the second week, another student began using starting point in his mathematical explana tions. In this example, the student was asked to find the equation for the information in table 1.

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... I had to get every single ticket so I divided it by ten and that equals 4.25 and that's my equation right there. It goes up by 4.25

every x... Then I had to get the starting point so I

just took 80 and then I minused it 42.50 over and over again eight times for the number of tickets and that got my starting point of negative 325.

The term starting point was first used by a few students during the first and second week of instruction. During the second week and thereafter, the teacher incorporated this student-invented term into his classroom communication when address

ing the whole class. For instance, during the second week of class, the teacher summarized one student's statements about which parts of a graph were

important to see in a graphing calculator's window.

Teacher. Okay. Steve said that these are break-even

points and these are starting points and we want to see both of them. How much more do we

want to see?

In this way, the teacher appeared to validate these terms for other students, and those who were

reluctant to use start began appropriating it into their classroom discourse more

often after the second week of instruction. During the ten weeks of the study, students used the term 56 times while the teacher used it 55 times. The over

whelming majority of these student uses (93%) were

initiated by the students themselves. In contrast, 21 out of 55 (38%) of the teacher's uses of start were in

response to the students' use.

STUDENTS' MEANING FOR INFORMAL TERMINOLOGY

Within real-world contexts, the start, or starting point, of graphs and equations became identified with the concept of the ̂-intercept. Students began to understand the concept of the ̂-intercept by con

necting it with their experiences of events, both inside and outside school, that have a clear beginning point. This student-generated term also possessed

Thereafter, the teacher incorporated this student

invented term into his classroom communication

when addressing the whole class

this important characteristic: It could be connected to different contexts while still being identified with the beginning value of a limited-domain graph. The

following conversation between the teacher and a student validates the flexibility of the term:

Teacher. 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

negative 450 in this case? Steve. The starting price. (Day 27, November 18,2004)

Ensuing questions by the teacher revealed that the student interpreted the daily operating costs of the theater in terms of the price the theater would have to pay for those services. The teacher also used the informal terminology starting by using starting distance when discussing a context involv

ing two students walking to school. When students were working with an equation representation, the

^-intercept was called the starting number. Thus, the invented terminology also enabled connections between different representations of a function.

Start, or starting point, had an unequivocal mean

ing as it was used within real-world contexts, limited domain graphs, and equations. In all three of these

representations, student-invented terms represented

the ̂-intercept. However, there was a difference when students used a table representation. A table repre sentation is a window onto a possibly limitless set of values for the function. This window can view any set of function values that the user wishes to explore. As

such, the first value in the table may be anywhere in the domain and range of the function. Within a real

world context, a limited-domain graph, and an equa tion, the use of start, or starting point, draws students' attention to the dependent value. The independent value of the function is zero here and, as such, fades into the background. However, with the table repre sentation, both the x- and the ̂-coordinates are present

within each row of the table. This type of representa tion has benefits for students in that it may lead them to understand better that each point on a graph is

composed of an ordered pair that satisfies the equation representation of the line. However, it also introduces

ambiguity into students' use of start, because the start of a table could refer to the x-coordinate, the ̂/-coordi nate, or both values in the first row of the table.

In the ten weeks of videotaped classroom instruc

tion, the term start or starting point was used fifteen times by students and the teacher when referring to table representations. Fourteen times the term was used in a manner different from the ̂-intercept. For

example, students were given the information in table 2 and asked to find the cost for a call that was 8 minutes long. One student's explanation follows:




Table 2

Price of a Phone Call Depending on Length of Call in Minutes

Call Length L in Minutes 10 15 20

Cost C in Dollars 3.00 3.50 4.00 4.50 5.00 7.50 10.00 12.50

Source: Coxford et al., 1998, p. 148. Reprinted with permission of Glencoe McGraw-Hill Publishing, ? 1998.

Joel Ah, right here, it starts at 5 so it's just 5 dollars so I built onto the five 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 for Joel, "it" (the table or the con

text) "starts at 5," which happened to be the inde

pendent value of the function instead of the depen dent value when x = 0.

STUDENT SUCCESS ON TRANSLATIONS SET WITHIN REAL-WORLD AND ABSTRACT CONTEXTS Starting point was used to mean something similar to the ̂-intercept in equations 96 percent of the

time, limited-domain graphs 92 percent of the time, and full-domain graphs 100 percent of the time when these representations were set within real world contexts. In an abstract context, students'

success in using starting point to mean something similar to the ̂-intercept dropped to 63 percent for

equations, 38 percent for limited-domain graphs, and 38 percent for full-domain graphs when these

representations were set within abstract contexts.

In sum, these data suggest that a substantial portion of students' success in using their own informal ter

minology was tied to the real-world contexts from which it arose.

STUDENT DEFINITIONS OF STARTING POINT As part of the paper-and-pencil tests, students were asked to create their own definitions of starting point. These student-generated definitions fell into three categories: formal, beginning mark, and axis

intercepts. Two students held a definition of starting point that was equivalent to the formal definition of the ̂-intercept when they wrote that "starting point is the g value when jc is 0.'' Four other students

wrote that starting point is "the point in which you put the first mark for the line on the graph." The

remaining two students connected starting point to the place where a line intercepts the axis, but they

made no distinction between the two axes, as seen in the following statement: "The starting point is the

point where it encounters the x-axis or the #-axis."

Thus, at the end of the study, only two students had constructed a written definition of starting point that was equivalent to the formal definition of the

^-intercept.

IMPLICATIONS FOR TEACHING In conclusion, the real-world contexts that students

engaged in and the explanations that the curricu lum and teacher elicited from them stimulated the creation of informal language, specifically, start. This informal language had benefits for students in that the terminology was connected to their under

standing of time and thus provided a way for them to make sense of the ̂-intercept. This terminology could also be combined with different nouns, such as point, price, and distance, thus enhancing the

meaning of this concept by connecting it to a graph ical representation or a real-world context.

However, the benefits of student-generated ter

minology did not extend to tables because of the nature of this type of representation. Students can connect their use of starting point across different

representations through activities in which they translate between graphical representations, real

world contexts, equations, and tabular representa

tions that contain the ̂-intercept and then identify the starting point within each. Later, students can

generate tables from the same function that do not contain the ̂-intercept. Students can then be asked to locate the start, starting value, and starting point

within each representation. Further, the students in this study had a dif

ficult time using starting point as the ̂-intercept in abstract contexts because the graphical repre sentations of the real-world contexts they worked

with did not have clear beginning points. Stu dents should explore these limitations by working through activities where they are asked to identify the starting point in tables, graphs, and equations generated from abstract contexts.

In the study reported here, students worked

through these activities as part of the paper-and pencil assessments. Some students correctly identi fied the starting point as the ̂-intercept, but several did not. Such divergent responses pave the way for discussions that help students see the importance of mathematically precise definitions. The conver sations generated from these activities could lead students to create a definition for starting point that is consistent across different representations or to invent a different term that eliminates the ambiguity caused by table representations and abstract contexts.

Informal language is not the final destination as students venture into real-world contexts and

Vol. 101, No. 6 February 2008 | MATHEMATICS TEACHER 449



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multiple representations. After students have constructed definitions for their own informal terminology that are simi lar across different representations and

contexts, they can be introduced to the term y-intercept and its formal defini tion. This step is essential for students as they become acculturated into the wider mathematical community. It will also help students engage in communica tion with mathematics textbook authors as they "learn to read increasingly tech nical text" (NCTM 2000, p. 351).

REFERENCES Coxford, Arthur F, James T. Fey, Christian

R. Hirsch, Harold L. Schoen, Gail Bur

rill, Eric W. Hart, and Ann E. Watkins.

Contemporary Mathematics in Context: A

Unified Approach (Course 1). New York: Glencoe McGraw-Hill, 1998.

Halliday, Michael A. K. Language as Social Semiotic: The Social Interpretation of Language and Meaning. Baltimore, MD:

University Park Press, 1978.

Martin, Tami S., Cheryl A. Hunt, John Lan

nin, William Leonard Jr., Gerald L. Mar

shall, and Arsalan Wares. "How Reform

Secondary Mathematics Textbooks Stack Up against NCTM's Principles and Standards." Mathematics Teacher 94, no.

7 (October 2001): 540-45, 589. National Council of Teachers of Math

ematics (NCTM). Principles and Stan dards for School Mathematics. Reston, VA: NCTM, 2000.

Pimm, David. Speaking Mathematically: Communication in Mathematics Class

rooms. London: Routledge, 1987.

Senk, Sharon L., and Denisse R. Thomp

son, eds. Standards-based School Math

ematics Curricula: What Are They? What Do Students Learn? Mahwah, NJ: Lawrence Erlbaum Associates, 2003.

Webster's New World College Dictionary. 4th ed. Edited by Michael Agnes. New York:

Macmillan, 1999. ?>

^JON D. DAVIS, jon.davis? wmich.edu, teaches at ^^^H Western Michigan Uni- ^^^H versityr Kalamazoo, Ml

49008-5248. He is interested in ^^^H technology and how teachers use and ̂ ^^H learn from reform-based curricula.

^^^H PHOTOGRAPH BY LORI DAVIS; ALL RIGHTS RESERVED

^^^^H



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