Abstract

Discourse as a Problem Solving Strategy

Georgianna T. Klein Grand Valley State University, USA

Abstract

Drawing on research with students in a summer program for disadvantaged youth, this paper illustrates that students bring a variety of resources beyond mathematical expertise to the problem solving process and may be engaged in tasks quite different from and with different purposes from those intended by the researcher or teacher. In particular, the paper describes several routines jointly produced by Sam and the researcher that mimic traditional mathematics classroom routines, but which are not particularly productive to either solving mathematics problems or gaining insight into Sam's mathematical understanding. Failure to take into account these kinds of interactions between "expert" and student during teaching or research can lead not only to invalid judgments about what is occurring, but to misappropriation of responsibility for failures.

Introduction

Drawing on both social interactionism and radical constructivism, Bauersfeld (1995) suggests an orientation to analyzing the processes of communicating and their impact on personal development in the mathematics classroom. He argues that language cannot be thought of as a medium to be used or as an object. He maintains that a speaker's utterances can function for the listener only as directing the focus of attention, whereas the construction of what might be meant, that is, the construction of references, is with the listener. The speaker's utterances and intentions then can have no direct access into the listener's system. What the listener's senses receive, undergoes spontaneous interpretation. Bauersfeld further claims that such interpretations have emerged from many social interactions, encounters through which the person has tried to adapt to the culture by developing viable reactions and trying to act successfully. Bauersfeld describes how communication takes place:

The focus of attention, our "looking at," cuts something out of the perceived diffuse continuum and makes an object of it. This process does not necessarily result in the same consequences with all persons, because it is a historical, situated, and individual process. Only across social interaction and permanent negotiations of meaning can "consensual domains" emerge, so that these "learned orienting interactions" can lead an observer to the illusion of some transport of meaning or information among the members of the social group

(p. 275)

This position differs from that of traditional teaching, which is based on the assumption that the teacher teaches some objective version of mathematics, using language as a representing object and means. This latter view, which holds that language can exist on its own objectively and thus serve as a verbal transport of knowledge and direct learning and teaching, is in direct opposition to individual construction of meaning. In the face of such beliefs about teaching, what do students do? Bauersfeld (1995) claims that without simple transmission of meaning through language, the students often develop routines in which they learn to say what they are expected to say in certain defined situations. Students have learned, for example, certain routines that occur in mathematics classrooms that help them to get through the lessons successfully, sometimes without engaging with the mathematics.

This paper, using research interviews with students, gives evidence for the validity of Bauersfeld's arguments by exhibiting three such routines. One might argue that research interviews are not classrooms. They do, however, share the attribute that there is an "expert" in a power position who is directing the flow of the activity. The conversation between researcher and student is very similar to that of teacher and student in a traditional classroom, and in this case, the researcher was an experienced mathematics teacher.

The Context for Discussion

In a recent study, I sought to explore high school students' conceptions and representations of elementary function classes. Students were participants in a summer program for disadvantaged youth who were interested in science and mathematics. Data consisted of interviews of four focal students, pre- and posttests, written homework, and fieldnotes and videotapes of classroom observations. Students had five 45-60 minute interviews. Questions varied from simple tasks, such as translation of a linear function from tabular to symbolic form, to more complex tasks such as comparing two quadratic functions situated in an economic application context and critiquing the correctness of conjectured mathematics. As the interviews progressed, there were many instances where it appeared there was a mismatch between the orientation of students and the researcher. Both the classroom teacher and I, the researcher, were oriented by our views of the mathematics, setting tasks and looking for evidence of student understanding. Students, however, often seemed oriented less toward the mathematics per se, but more toward what it means to be in school and in the successful completion of the tasks. In particular, they evoked patterns of interaction with me that mimicked school routines, but could be described directly as problem solving strategies. The problem to be solved, however, was not simply mathematical, but rather, getting through the interview successfully.

Each routine involved questioning and answering. The strategies students used were: (1) responding to questions in a way that provoked a funneling pattern in the researcher's questioning (Bauersfeld, 1988), (2) determining how to proceed with a problem through questioning the researcher, and (3) gaining clues about how to proceed from the researcher's comments. All focal students participated in these kinds of interactions with the researcher, but one student, Sam, regularly used all three, so I use interviews with him to illustrate the points.

Sam is a boy who comes from an economically poor family in a small Midwestern city in the USA. At the time of data gathering, he had completed two years of algebra, receiving an A for the first year, and was reading above grade level. He had a strong sense of humor and reported a career interest in zoology. During an awards ceremony at end of the program, Sam was chosen by his teachers as outstanding student in his section in biology, physics, and anatomy. In class, Sam was often a lively participant, offering suggested solutions, asking questions, or perhaps commenting on difficulties with his calculator computations.

School Routines

Narrowing the Focus of the Questioning

One pattern in Sam's responses in interviews was that he often paid only lip service to my questions until they moved from being conceptual to requiring a highly specific piece of information. This interaction relied on a routine common in classrooms. In order to help a student who might be stuck to be successful and to arrive at the answer the teacher wants, the teacher continues to ask more and more specific questions or to give hints until a student succeeds in giving a "correct" response.

For example, in the last interview Sam was engaged in a card sorting task and had decided that the graph of a rule on card B, Profit = -300p² + 3600p - 1500, was a parabola, but when he graphed the function on his graphics calculator, he got a vertical line because the y-dimensions for the viewing window were too small relative to the specific x-dimensions chosen. (Note that: "Trace" moves the cursor along the points on a graph (both within and outside the viewing window) and displays the values of the x and y coordinates of the current point. If the point is off the screen, the cursor itself is not displayed. The image scrolls left or right if one traces beyond the x-values or the window currently displayed.)

S: Is it a parabola? [turns on TRACE and moves the cursor along the graph] It don't look like it's ever going to go down.

G: You're on TRACE and looking at (values of) the x and y? [S: Um-huh.] What are the numbers doing?

S: Keep going up.

Sam correctly thought that this graph was an inverted parabola, and he expected to find the vertex by finding a place where the y-values reversed directions. This strategy could have helped him find an appropriate viewing window on his calculator by locating a maximum function value or, through the pattern in the data, confirmed his thinking that the graph was a parabola. However, since the vertex was to the right of the cursor and Sam was moving the cursor to the left, the cursor failed to reach the vertex. As he moved the cursor down the left-hand branch of the parabola, both the x- and y-values continued to go down. Sam thought because the magnitude (absolute value) of his negative numbers was going up, the cursor on the calculator ought to go up as well and thus ultimately reach the vertex of the parabola. He was trying to coordinate a change in magnitude (size) with an upward direction along the y-axis and thus with an upward change in position of the cursor along the graph. Although these three kinds of change are ordered (a math convention) in the same way for positive numbers, the direction of change in magnitude for negative numbers is exactly backwards from that of both y-values on the axis and the relative vertical position of the cursor.

As we continued this problem, my efforts to straighten out the number confusion led to continuing difficulties with the viewing window, in part because of Sam's style of responding. I started by deciding to help Sam discover his error and asked if the numbers were positive or negative. He said they were both negative. Then I tried to call his attention to the distinction between the magnitude (absolute value) of a number and the relative order of two signed numbers:

G: Are they going up in absolute value or are they going down? Are they going up if it's negative?

S: If it's absolute value, it's goin' to be positive all the time.

G: I meant the size of the number, is the size of the number going up?

S: Down. [He continues tracing.]

G: It's going down. If I said -200 is going to -300, is it going up or going down?

S: Say that again.

G: If I went from -200 to -300, would you say that's going up or going down?

S: Up.

G: It's going up in size, but the number itself is actually going down, isn't it? Isn't -300 less than -200?

S: Um-huh.

During this exchange, Sam was very busy tracing on the calculator. He appeared to be paying only cursory attention to me. When I raised absolute value, he gave a "textbook" answer about absolute value being positive all the time, not attempting to answer either of the questions I had asked. Thinking perhaps that the difficulty was my poor questioning, I reconsidered and asked only about the size of the number. I had hoped he would understand this informal description that is sometimes used with absolute value, but he answered my question as if it were about the order of signed numbers when I had asked about unsigned numbers. I echoed that he had said "down," thinking perhaps he had figured out the discrepancy between magnitude and direction. To confirm which question he was answering, I asked a more focused question about how he perceived change between two specific numbers, from -200 to -300. His answer confirmed that he had not sorted out the order concept, but perceived these numbers as increasing, rather than decreasing.

The question Sam avoided was a conceptual question about the magnitude of numbers and how numbers are ordered. In the passage above on questioning whether numbers were increasing or decreasing, Sam continued to give me partial or misleading answers. His responses to my questions had an effect on the interviewing process. They caused me to narrow the questions from one about a general concept to one involving the relationship between a single pair of numbers. Although Sam might have been ignoring me somewhat as he concentrated on the problem, the net effect of the interaction was that ultimately I asked a question that he could readily answer and finally told him a "correct answer." This style of delaying or deflecting responses was a reasonable problem solving strategy on his part that shaped my behavior as an interviewer to simplify the task and move the interview forward. It did not, however, really help Sam, nor did it give me much insight into his understanding of the task at hand It simply confirmed the point that he viewed -300 as larger than -200.

Seeking Clues

Another style of interacting with me that seemed to advance Sam's progress in the interview was his use of questions to gain clues for how to proceed. Sam began 7 of the 17 problems he worked on during interviews with a sequence of questions that sought additional information about the problem. On two of these occasions, his questions were only to find out what I expected of him, much as a cooperative student would ask a teacher in class. For example, at the beginning of the fourth interview he asked if I wanted him to think aloud and whether I wanted him to write his response. Sometimes Sam wanted clarification of terms or of the problem statement that might be necessary to solve the problem, or after I had explained their meaning, to check if he had understood correctly.

At other times, however, Sam seemed to be seeking hints for how to proceed. In problem 2A-2 (Figure 1 below), he asked a number of questions:

S: You want me to draw this?

G: I want you to read what this problem is and see what you can do with this problem.

S: Now, I'm going to look over this again.

G: Okay. [he reads the problem]

S: This is like that x + 1 stuff equal y.

G: You got it.

S: So that's what the function rule is?

G: The function rule is like what you were doing in class when you were trying to find the function rule.

S: [pause] Okay, now I remember. It'd be like x [pause], two x equal [pause] y. [writes 2x = y] I don't know. I don't know if I'm right. I can't remember.

For the table below, what is the function rule? What is the output for input 15? input 30?

input output

-------------------------------

1 | 10

2 | 5

3 | 0

5 | -10

10 | -35

Figure 1. Problem 2A-2

Sam had seemed to connect this problem to graphs in a previous problem in the same interview, so he asked if I wanted a graph, a clarifying question. After I told him to just read the problem, he checked his understanding that a function rule was an equation written y equals an expression involving x. When I agreed with him, he asked again about what the function rule was. At the time I sensed he was fishing for a direction to proceed, and so simply mentioned we had done function rules in class and gave completely redundant information. My response no longer added anything useful to the problem solution. Sam then made a guess, but his announcement that he did not really know if his guess was right indicated he was stuck and hinting for additional help. This strategy was wise since when students were stuck, I usually provided some question or bit of information that would help them get going again. We continued the problem with my giving him help by asking if his rule fit the table. By that time Sam had figured out to look for a pattern of change in the data and immediately computed the pattern of change. Our question and answer session had given him enough information, or possibly just time to think, about the problem to get him started.

In the Profit Problem in interview 3, a long complex, situated problem involving comparison of two polynomial functions, Sam also engaged in a long sequence of questions and comments before he chose any plan of action. In short, Sam's use of a legitimate school routine, asking questions to clarify the problem statement, served both to advance the interview in a non-threatening way and to gather information to make progress toward producing what was expected by the researcher.

Researcher's Comments and Questions Triggering Responses

Sam seemed attuned to my questions as well, sometimes taking them directly as hints. At other times they seemed to trigger a change in the direction of his problem solving. As we continued problem 2A-2 above, one of my questions helped him again. He had just described the pattern of change in the table and conjectured a rule, y = -5x, which matched the change pattern in the dependent variable, but not the table as a whole. Sam quickly scratched it out. I told him to just put an x through his work so I had a record of his thinking. Then because I wanted him to verbalize his thinking, I asked him why he scratched out the -5x. He explained that when he had substituted 1 for x, the result did not match the table entry. Unbeknownst to me, this question caused him to reconsider his choice and quickly add the necessary 15 to verbally give the correct rule. As he continued, he made a number of recording errors, but finally wrote the rule correctly as y = -5x + 15 and checked its accuracy with x = 2 and x = 3. When I asked Sam if he was sure about it, he checked the rest of the table entries in his rule. He took my question, which was to find out how certain he was of his answer, to indicate the possibility that the rule could be wrong. At the end of the problem he announced that he did the problem with "that hint" from me. When I asked what hint, he said, "When you said why did you scratch out that five, negative five." This question became a hint not to abandon y = -5x. Sam had started this problem with several questions that gave him additional information. When he was stuck, he got a suggestion about how to proceed. He was explicit about how my question caused him to reconsider his first choice. Finally, when I asked if he were sure about a response, it triggered additional checking for a correct solution with the table values. Until I had asked, Sam had omitted checking the remaining table entries in the rule, a step that was necessary unless he gave some other argument to confirm the match between the equation and all table entries.

Sometimes my questions about whether he was sure about a response triggered checks that were not necessary. In one problem to choose either Alice's, Bea's, or Carolina's graph to match a table, Sam had used the units from a sheet of graph paper to plot the points in the given table on the axes for Alice's graph. He then chose Bea's graph as correct, and when I asked why, he explained his choice well:

S: Because the form right there follows the pattern that she got. (shape of the graph)

G: The pattern of the dots that you drew [T: um-huh] looks more like Bea's pattern?

S: Yeah. So I'm going to say her chart is more accurate than theirs. Now, I'm gonna recheck my stuff to see if I'm right.

Here when I simply echoed his response, substituting "Bea" for "she," he repeated the point plotting with the same units, this time on Bea's axis. Even though the units he used were different from those actually on the graph, he had judged the match correctly by the shape of the graph, so there was no need to repeat this plotting. My question had seemed to trigger the check. Sam's being tuned into my responses was a frequent occurrence, and as an interviewer, I had to work very hard to refrain from giving clues to Sam inadvertently.

I assume Sam had learned a strategy of paying careful attention to teachers' responses. It is likely that asking a large number of questions or getting a problem rephrased could result in subtle differences in a problem statement that made it clearer. Answers could also clarify "what the teacher wanted." In addition, it is likely students are familiar with a style of questioning in mathematics classes where, when students give wrong or incomplete results, their teachers echo students' responses or ask if they are sure of an answer specifically to indicate the students' answers are wrong without saying so directly. Thus, they give students an opportunity to think harder about the problem by extending the time. I had the strong sense in working with Sam that sometimes his questions were "buying time" to think, as well as to get hints, while he thought about how to proceed.

Discussion and Conclusion

Each of these interaction patterns can be associated with similar routines in school where the teacher attempts to advance the discussion of mathematics, diagnose student's difficulties, and attend to students' self-esteem so they can continue to participate. Even though I was acting in the role of researcher and asked questions for different purposes than in a classical classroom, I am also a mathematics teacher, am familiar with the routines discussed., and was strongly motivated to keep the interactions moving.

Assume a teacher in a traditional classroom is motivated to help students understand what she takes as shared by the mathematics community and at the same time likes to feel effective in her work. If a student is motivated to cooperate, as Sam seemed to be, he attempts to do what is expected of him, that is, solve mathematics problems as defined by giving correct answers. Yet he expects not to lose face while doing so. Each of these stances require that any interchange between teacher and student not reach an impasse; otherwise, one or the other, or both, is viewed as deficient in some way. The teacher is viewed as unsuccessful or ineffective at teaching because she has failed to convey the appropriate information to the student or her ability to ask appropriate questions is suspect. The student is seen as not knowing the appropriate information or means of stating it and, thus, is judged unable to do the mathematics or possibly simply dumb. Thus, both teacher and student are vested in making the interaction routines work. Given a classic style of instruction where the student is to arrive at a correct answer, it is natural that to help a student save face after several impasses, the teacher gives more and more focused tasks. The student then must divert the task from a focus on solving mathematics problems toward successfully completing the interchange--with or without progress in the mathematical task.

Most dedicated and perceptive teachers are well aware when a student is not viewing mathematics as they do and would welcome assistance in being successful in their classrooms. In a traditional classroom, however, teachers may not even be aware that although these kinds of interchanges seem to be open to student's views, they are actually a form of direct instruction where students attempt to fill in the blanks with the "correct" response. Further, they are likely not to be aware that students are involved in a task with an entirely different purpose than the problem solving intended by the teacher; rather, students see their task as completing the interaction without becoming stuck.

The consequences of orienting teaching solely by the mathematics and not recognizing that how the ways in which we think of mathematics and instruction impacts the social interaction in the classroom are dire. Teachers might question their own dedication or ability and become frustrated with all sorts of ugly results. Students often simply assume they cannot do mathematics, sometimes with devastating results to their self-esteem and, and certainly by being cut off from many rewarding life paths because they do not have the required mathematics.

References

Bauersfeld, H. (1995). "Language Games" in the mathematics classroom: Their function and their effects. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 271-291). Hillsdale, NJ: Lawrence Erlbaum Associates.

Bauersfeld, H. (1988). Interaction, construction, and knowledge--Alternative perspectives for mathematics education. In D. A. Grouws & T. J. Cooney (Eds.), Perspectives on research on effective mathematics teaching: Research agenda for mathematics education (Vol.1, pp.27-46). Reston, VA: NCTM and Lawrence Erlbaum Associates.