Understanding algebraic notation from the students' perspective

Abstract

Kinzel looks at algebraic notation from the students' perspective in an effort to prevent future miscommunications about what is represented by algebraic notation.

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A few years ago, I noticed a student in an elementary algebra course having particular difficulty on an examination with such items as the following:

Find the equation of the line through the points (3, 2) and (-1, 4). (5 pts.)

Her difficulty surprised me, since I had thought that she was comfortable with the material before the examination. She continued to struggle with the test, turning in her paper at the last minute. When I asked her how she felt about the test, she replied that she had done what was asked on the problems but that she had no idea what to do with all those "pints." Her reply left me speechless. She had interpreted "5 pts."-that is, the value of each item, 5 points-as additional information-5 pints-to be used in the item!

Granted, the interpretation in this example was not essential to the mathematics involved and the miscommunication was ,easily corrected, but it opened my eyes to the possibility-perhaps the certainty-- that students do not see what I see, or intend, in my notations. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) advocates a decreased focus on symbol manipulation in high school algebra, encouraging instead a view of algebra as a "means of representation." If algebra is to be a means of representation, teachers must be explicit about what is being represented. I thought that I was representing the value of each test item; my student thought that I was representing additional information to be used in the task. How often do such miscommunications as one this occur in algebra classes? What do students think is being represented by algebraic notation?

Investigating students' use of algebraic notation is not a new idea. Perhaps the most familiar example is the "student-professor" problem:

Write an equation using the variables S and P to represent the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (Clement, Lochhead, and Monk 1981)

Although this question appears to be a straightforward one that is well-defined in the sense that the variables are identified and labeled, the "reversal error" of writing 6S = P is remarkably prevalent (Philipp 1992; Rosnick 1982). Interviews with students (Davis 1984; Philipp 1992; Rosnick 1982) reveal alternative interpretations for what is being represented. For example, one explanation for the equation 6S = P is the equivalence comparison "six students for every one professor," which is similar to the familiar comparison "12 inches = 1 foot." The comparison explanation requires that S and P be interpreted as labels for "student" and "professor," as opposed to the number of each, even though this interpretation was explicitly stated in the task. How can students who are attempting the task overlook the distinction between labels and quantities? Why would my student think that pints was an appropriate interpretation of my notation in the context of an algebra examination? In general, what do we know about how students interpret algebraic notation? What can teachers do to support appropriate interpretations?

THREE KEY FINDINGS ABOUT STUDENT VIEWS OF NOTATION

As illustrated by students' performance on the student-professor problem, students may not use symbols to represent the intended referent, yet a shift in referent occurs even when the students state the relationship themselves. In Rosnick's (1982) study, college students participated in a written examination, then follow-up interviews were conducted with ten students. On the basis of test results and interview data, Rosnick characterized students' use of symbols as labels for "broad, undifferentiated concepts." For example, a student may have written "b = books" but then shifted between interpreting b as the number of books, the value of a book, the value of the total number of books, and so on, while working on the task. Although a referent (b = books) had been at least nominally identified, this act of assigning a referent to b was not seen as binding, allowing the referent to shift within the activity.

Harper (1980) found a related phenomenon when interviewing elementary students. Students were shown two line segments labeled "a cm" and "b cm" and were then asked questions about the length of the segments, the values of a and b, and when one line would be longer than the other. Most students identified the concrete segment as the referent for the symbol, as opposed to connecting the symbol to the length of the segment, and they were not able to consider the possible effects of varying values for a and b.

More recently, Macgregor and Stacey (1997) found that prealgebra students had a tendency to assign. values to algebraic variables, often turning to other symbol systems to supply that meaning. For instance, when given h as a symbol for the height of a boy, some students assigned h the value of 8, since h is the eighth letter of the alphabet. Macgregor and Stacey and others (Booth 1988; Philipp 1992) found that using initial letters could confound the symbolizing issue. Booth (1988, 27) reported the response of fifteen-year-old Peter when asked what the y refers to in the task "add 3 to 5y." Peter responded that y "could be anything . . . a yacht. Could be yogurt. Or a yam." The symbol was seen as a label for something, the name of which begins with the letter y, not as a symbol representing a quantity that could be manipulated. This shift in attention from the letter as a label to the letter as an object on which to be operated does not seem to occur spontaneously; rather, it must be an explicit goal of instruction.

As a symbol system, algebraic notation can be a powerful mathematical tool, a possibility that is directly related to the inherent ambiguity of the notation. We can think of 2x + 3 either as the process of multiplying a number by 2 and adding 3 or as the result of that process. The expression itself can indicate the operation to be performed on objects, that is, actual quantities; or it can be operated on as an object itself, that is, used as input into another operation, such as adding expressions. As Gray and Tall (1994,121) explain, the cognitive complexity of "process-concept duality" can be replaced by the notational convenience of "process-product ambiguity"; we can set aside the difficult task of specifying whether we intend the process or the concept through a flexible use and interpretation of the notation. However, this ambiguity is rarely discussed explicitly in mathematics classrooms, and it therefore remains an implicit awareness for those who are able to shift their attention and a mystery to those who are not.

When we speak of "interpreting notation," we are necessarily introducing a three-way relationship among the person, the notation, and an identified referent. (For a more detailed discussion of the triad notion, see Greeno and Hall [1997]; Hawkes [1977]; Kaput [1987]; Pimm [1995]). As Greeno and Hall (1997) put it, notations are "potential representations"; they become representations when someone interprets them, that is, when someone establishes a signifying connection between symbol and referent. In exploring this connection from the student's perspective, the emphasis is on how the student is perceiving the notation, what the student sees as being represented, and what role that representation plays within the mathematical activity. The nature of the symbol-referent connection shapes how the student uses the notation within problemsolving contexts. To illustrate this point, consider the treasure-hunt task in figure 1 (Kinzel 1997). ii as the distance traveled in the second step, with x representing the distance CD. This expression can be interpreted as the process of finding the distance, that is, add 7 to x and divide by 3; or the product of that calculation, which remains unknown at this point. Students interpreting the expression as a record of a potential process are unable to carry out the next step in the task; they are unable to proceed without having a specific value for this expression. This interpretation is evidenced by such comments as the following:

My problem is in deciding how I can get this

(...)

to equal where I am between C and D. For some reason I think this should equal 7.... But see, then how do you solve for x? Because you don't know what's over here. That's where I'm confused. That's why up here

(...)

I think it should equal something, so we have a value of how far you went. (Kinzel 1997, 51)

Viewing the expression only as a record of a particular process limited the student's ability to use the expression as an object available for further manipulation. The following discussion draws on empirical research results that address the interpretation of algebraic notation from the students' perspective.

Booth (1988) reports on data collected for the Strategies and Errors in Secondary Mathematics project, which was conducted in the United Kingdom from 1980 to 1983 with eighth- to tenth-grade students who had started studying algebra in seventh grade. Booth concludes that students tend to view the notation as a label for some concrete object and are thus unwilling to accept algebraic expressions as answers. For example, in one task, students were shown a path consisting of stages, each of which was 11 units long. Each student was told that the path had y stages. Wendy's response was given as representative; she was able to identify the operation-- that is, multiply 11 times y-but did not acknowledge that "11y" was an appropriate representation for the length of the path. Wendy had identified an appropriate referent for y, the number of stages in the path, and knew what she should do, that is, multiply; but her connection between symbol and referent did not allow her to view the result of that symbolic manipulation as an object. For her, "1ly" represented the process that she would carry out if she knew the value of y, not the result of that process. These results are consistent with Kieran's (1992) synthesis of research on the learning of algebra, in which she concludes that students' tendencies to interpret algebraic letters as specific unknowns contribute to their difficulties with viewing expressions as objects.

It is possible to imagine how loose interpretations may make sense from a student's perspective. Viewing the connection between symbols and referents as only a labeling relation allows for the following sequence: the student sets up a label, generates expressions, and performs manipulations. These manipulations change the objects under consideration-they certainly look different now, and so the student believes that it is reasonable to reinterpret the labels. Without a reflexive, symbolic connection between symbol and referent, the student has no reason to require that the interpretation be maintained throughout a task. In fact, we often do "change the rules" in algebra, for example, when we discard extraneous solutions to equations. So what influences students' interpretations of algebraic notations and manipulations on those notations?

Several studies have empirically investigated specific influences. Philipp's (1992) study, in a context similar to the student-professor problem, manipulated two factors: the physical resemblance of the notation to the quantities and the explicitness of the relationship between quantities. Each student in a large sample (N = 295) of high school mathematics students enrolled in first-year algebra, geometry, or second-year algebra worked on one of four problems that were similar in structure and that varied across factors. Using letters related to the quantities, for example, S for the number of students, did not focus the students' attention on the quantity being represented. Leaving the relationship between variables implicit, for example, using pennies and dimes as opposed to students and professors, also interfered with success. To further investigate the implications of this finding, seven adults were interviewed while they worked on two of the four problems used in the first study. The participants justified the meaning of an equation by changing the meaning of the variables, applied dynamic meanings to variables within a single problem, and used or referred to multiple meaningful referents for variables. As in Rosnick's (1982) study, participants' interpretations of the notation shifted as they worked on the task, reflecting loose, vague connections between symbol and referent. Even when efforts were made to make the intended connections explicit, students seem to have operated with these loose connections and interpretations.

Macgregor and Stacey (1997) report on data from written tests given to approximately two thousand students, ages 11 to 15, in twenty-four Australian secondary schools. They identify such origins for students' difficulties as intuitive assumptions and pragmatic reasoning, analogies with other symbol systems, interference from new mathematical learning, and poorly designed teaching materials. In one task, students were told that David is 10 cm taller than Con and that Con is h cm tall. When asked what they could write for David's height, some students applied a form of pragmatic reasoning to the task-either assigning a reasonable value for Con's height or assuming that another letter, perhaps an adjacent letter, such as g, should be used to represent David's height. This second interpretation seems to ignore the information about the relationship between the boys' heights. As mentioned previously, some students used other symbol systems, as in a response of 18 (h is the eighth letter of the alphabet; 10 more is 18). Some students evaluated a letter as 1 unless otherwise specified (so that 10 + h = 11) or allowed the letter to have a general referent (h means "height," so it means both "David's height" and "Con's height" in the expression h = h + 10). Demby (1997) tested students in seventh grade and at the end of eighth grade to explore the procedures that they used to simplify algebraic expressions. Students explained their individually constructed rules in terms of actions taken, as shown in figure 2. Demby distinguishes between "rules" and "quasi-rules" on the basis of consistency as opposed to accuracy. The two examples given in the table were both classified as rules, since the procedure was applied consistently, even though the second procedure is not correct. The application of a quasi-rule can be seen in the following example: in simplifying -2x^sup 2^ + 8 - 8x - 4x^sup 2^, a student stated that first you must "raise to the power," applied that rule to the first term, but failed to apply it to the last term: -2x^sup 2^ + 8 - 8x-4x^sup 2^=4x^sup 2^. A significant result of Demby's findings was that students using consistent rules, even if inaccurate, showed greater progress than students applying ad hoc, haphazard quasi-rules. This result has at least two important implications: students should be encouraged to articulate the rules that they are applying, and students should be encouraged to reflect on the consistency of those rules.

In short, failing to focus explicit attention on the interpretation of algebraic notation forces students to operate with the notation as objects without the cognitive foundation necessary for such operation. Research evidence supports the hypothesis that students tend to focus on the labeling aspect of symbols and that viewing the symbols in this way inhibits the potential power of the notation as a symbolic system.

TWO RESEARCH-BASED STRATEGIES

Make symbolizing an explicit aspect of students' problem-solving activity

Teaching experiments at the elementary level have been successful in developing students' ability to use symbols (Bednarz et al. 1993; Steffe and Olive 1996). The primary notion is creating genuine situations that provoke the use of notation. Meira (1995) applied this notion to students' work with functions. Students worked in pairs to answer a series of questions using one of three function machines: winch, springs, or computer (see fig. 3).

Each machine represented two linear functions; for example, the springs machine had two springs attached to the top of a numbered track; small weights could be attached to the springs, and the length of the stretched spring could be read off the track. Students were asked to respond to such questions as "What will happen with each spring as you hang weights on it?" for the springs device or "What will happen with each output as you type in inputs?" for the computer device. Extrapolation questions went beyond the physical limitations of the machines, for example, questions asked about weights heavier than those supplied. No method of recording was prescribed or required. Students encountered the need to record results and conjectures as part of their activity. Meira analyzed in detail how the students' constructed representations shaped their continued problem-solving activity. Students most often created tables, as shown in figure 4, and used these tables to display the information as well as to support elaborate computational activity.

As an example, consider the work of a student pair using the springs device and investigating the question, "Will any of the springs be longer after 15 pounds?" The lengths of the springs are given by the equations y = 2x + 8 and y = 2x + 2. The students had observed that each spring stretches two inches for each additional pound added. In the process of thinking about the question, the students began recording their data in the table shown in figure 5.

Observing their table, the students made the following comments:

AL: So it's just a difference of six. "Will any of the springs be longer than-" . . . oh, yeah!

CC: Yeah, because they'll go down at the same rate . . . and that one is always biggest.

The episode cited here is significant for two reasons: first, the table is nonstandard and thus represented a reconstruction of a technique within the problem-solving activity as opposed to the rote application of a procedure; and second, the role of the table shifts from a record of the students' observations to the object of their reasoning-the students reflected on the values in their table and began to notice relationships. Although Meira does not report on students' construction of symbolic rules to describe the behavior of the devices, it is easy to suppose how such activity could be supported. Asking this pair to justify their conclusion about which spring would be longer could provoke both a verbal and a symbolic description of their observations. Student-constructed notations became both a means of communicating mathematical ideas and a support for mathematical reasoning. External representations constructed by students in the course of problem solving not only offer windows into students' thinking processes but also contribute to the problem-solving process itself.

Attend to the development of the concept of variable

Much of students' difficulty with algebraic notation reflects a limited conception of algebraic variables. Traditional instruction supplies the student with the notation but does not attend to developing appropriate mental referents for the notation. Thus, the student may have only vague mental images of variable to serve as a referent. As a result, students' interpretations reach for other, more familiar referents. Sutherland and Rojano (1993) have experimented with using spreadsheets to support the development of a concept of variable. In the spreadsheet environment, students can refer to a cell by using the mouse or arrows and can operate on the cell before using explicit symbolic reference. The following activities were implemented with ten- to eleven-year-old students in Britain and Mexico. Initially, students were introduced to the spreadsheet activities through a rule-exploration task such as "undoing a formula" (see fig. 6).

Teacher interventions encouraged students to explore a range of input values as opposed to focusing on the initial given value. Away from the computer, students were asked to write the rule that they would use for a table of values. Pretest and posttest results showed an increase in students' ability to identify inverse operations and to write rules in algebraic code, for example, writing "= x . 4" to represent y = 4x. Between one-fourth and onethird of the students spontaneously used algebraic code to write rules. At least half the students were able to write algebraic code when asked.

The second block of instructional activities involved using a spreadsheet to solve such word problems as the following (Sutherland and Rojano 1993, 369):

Measurement of a field task: The perimeter of a rectangular field measures 102 meters. The length of the field is twice as much as the width of the field. How much does the length of the field measure? How much does the width of the field measure? Students often use an informal approach to solve such tasks as this one, working from the known back to the unknown. The spreadsheet environment supports the development of working from the unknown. Students can enter the rules to calculate values from the unknown values represented by cells, as shown in figure 7. Once these rules are in place, the initial unknown is varied to produce the desired result. In this way, students can learn to operate on unknown quantities, thus supporting work with algebraic notation.

CONCLUSION

Traditional algebra instruction addresses the interpretation of algebraic notation only implicitly. Despite explicit instructions to "identify and label variables," the actual act of symbolizing has received little or no attention. As a result, students have no reason to establish a reflexive relationship between symbol and referent that will support mathematical activity. As we strive to implement the vision of the NCTM's Standards and to move toward algebra as a means of representation, symbolizing and interpreting must receive more explicit attention in our classrooms.

In the face of reform and technological advances, finding a definition for symbol sense takes on added significance. What do students need to understand about, and be able to do with, symbols? Arcavi (1994) proposes a set of desired behaviors, ways of operating with symbols that seem to be indicative of symbolic thought. He talks about encouraging the development of a "feel" for symbols-an understanding of both the usefulness and applications of symbols, as well as their limitations. Knowing when to abandon symbols is as crucial an aspect of symbol sense as knowing when they will support reasoning. He also refers to the ability to "read" symbols, an ability that reflects the nature of the connection between symbol and referent as the individual's ability to flexibly shift her or his focus of attention between the symbol and its referent as needed within a task.

An example of this ability to shift is given in the work of a student solving a linear equation: the student arrived at 3x + 5 = 4x. Instead of proceeding mechanically, the student switched modes, observing that to obtain 4x from the 3x on the left, one must add x. Thus 5 must be the value of x. In developing the ability to shift the focus of attention in this way, the signifying connection itself can be viewed from a new perspective, as an object that can be manipulated mentally. Explicitly attending to the establishment of reflexive relationships between symbols and referents has the potential to support the development of this vision of symbol sense.

References

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