Creating a Mathematics Classroom Environment
that Fosters the Development of Mathematical Argumentation
Purdue University Calumet
Paper prepared for Working Group 1: Mathematics Education in Pre and Primary School, of the Ninth International Congress of Mathematical Education, July 31-August 6, 2000, Tokyo/Makuhari, Japan.
Taking the symbolic interactionist perspective that meaning is constituted as individuals interact with one another, it is essential to consider the nature of the interactions that occur in the mathematics classroom. Explicit attention to classroom social and sociomathematical norms and to classroom discourse can result in advancing childrens development of mathematical argumentation. As children learn to explain and justify their thinking to others, they develop intellectual autonomy, and in the process, mathematical power. This paper deals with issues related to classroom norms, the constitution of meaning and argumentation. Examples from a number of primary school classrooms are used to clarify and illustrate the main issues.
that Fosters the Development of Mathematical Argumentation
My purpose in this paper is to discuss ways to make sense of the mathematics classroom environment through focusing on social interaction as an interpretive framework. I will demonstrate how the sociological constructs of classroom social and sociomathematical norms can be used to characterize the mathematics classroom environment and explain the interrelationship between the nature of the classroom environment and students mathematical learning. I first discuss the basic tenets of social interactionism. Then I explain what I mean by social and sociomathematical norms, discuss how they are constituted in the classroom, and relate these constructs to students mathematical learning. In the process, I discuss the interactive constitution of instructional activities. Finally, I attempt to explain how social and sociomathematical norms characteristic of inquiry mathematics instruction contribute to students development of mathematical argumentation, intellectual autonomy and mathematical power. Throughout, examples from primary school classrooms are used for purposes of illustration and clarification.
The theoretical framework underlying my understanding of social interaction is that of symbolic interactionism. The theory of symbolic interactionism, which has its roots in the work of George Herbert Mead, John Dewey and others, has been developed extensively by Herbert Blumer (1969). Of the various approaches to social interaction, in our work, we have taken symbolic interactionism as a theoretical lens for two reasons. First, it is compatible with psychological constructivism, which forms the theoretical basis for our investigation of individual learning (Cobb & Bauersfeld, 1995). Second, as Voigt (1996) points out, the symbolic interactionist approach is particularly useful when studying students learning in inquiry mathematics classrooms because it emphasizes both the individuals sense-making processes and the social processes. Further, neither the individual nor the social is taken as primary. Thus, we do not attempt to deduce an individuals learning from social processes or vice versa. Instead, individuals are seen to develop personal understandings as they participate in the ongoing negotiation of classroom norms (Yackel & Cobb, 1996).
One of the defining principles of social interactionism is the centrality given to the process of interpretation in interaction. To put it another way, the position taken by social interactionism is that in interacting with one another, individuals have to take account of (interpret) what the other is doing or about to do. Each persons actions are formed, in part, as she changes, abandons, retains, or revises her plans based on the actions of others. It is in this sense that social interaction is seen as a process that forms human conduct instead of simply providing a setting in which human conduct takes place. As Blumer (1969) stated, "One has to fit ones own line of activity in some manner to the actions of others. The actions of others have to be taken into account and cannot be regarded as merely an arena for the expression of what one is disposed to do or sets out to do" (p. 8). Blumer goes on to clarify that the term symbolic interactionism refers to the fact that the interaction of interest involves interpretation of action. If no interpretation is involved, the interaction is referred to as non-symbolic, such as, when an individual responds in an apparent reflex response. Certainly there are many times when individuals engage in non-symbolic interaction. However, attempts to understand the meanings of one anothers actions, that is, to communicate (Rommetveit, 1985), involve symbolic interaction.
In addition to interpreting actions of others, individuals engaged in symbolic interaction attempt to indicate to others, through their actions, what their own intentions are. Thus, actions have meanings both for the person making them and the person(s) to whom the action is directed. In this sense there is a joint action that arises by the articulation of the participating actors activity. Blumer (1969) emphasized the collective nature of such joint action as follows:
A joint action, while made up of diverse component acts that enter into its formation, is different from any one of them and from their mere aggregation. The joint action has a distinctive character in its own right, a character that lies in the articulation or linkage as apart from what may be articulated or linked. Thus, the joint action may be identified as such and may be spoken of and handled without having to break it down into the separate acts that comprise it. (p. 17)
Blumer further pointed out that it is important to recognize that "the joint action of the collective is an interlinkage of the separate acts of the participants" (p. 17). As such, it has to undergo a process of formation and, even though it may be well established as a form of social action, each instance of it has to be formed once again. Consequently, the meanings and interpretations that underlie joint action are continually subject to challenge. This fact makes it possible to explain how it is that both individuals actions and the joint (collective) action of a group can change over time. Furthermore, this view of joint action supports the position that social rules, norms and values are upheld by a process of social interaction and not the other way around. Nevertheless, as we will demonstrate in the case of classroom social norms, the social norms both enable and constrain social interaction.
When there is no apparent confusion or misunderstanding in joint action, communication is effective. Otherwise it is ineffective and interaction is impeded. Following the symbolic interactionist perspective, any (human) group life is a formative process. The joint activity of the group and the activity of the individuals that make up the group are continually being formed in and through the process of interpretation and articulation. Accordingly, analysis of group activity necessarily involves attempting to make sense of the intentions and interpretations of the participants in interaction in an ongoing and dynamic manner.
A second defining principle of symbolic interactionism, in addition to the centrality of interpretation, is that meaning is seen as a social product. Blumer (1969) elaborated this point as follows:
It [symbolic interactionism] does not regard meaning as emanating from the intrinsic makeup of the thing that has meaning, nor does it see meaning as arising through a coalescence of psychological elements in the person. Instead, it sees meaning as arising in the process of interaction between people. The meaning of a thing for a person grows out of the ways in which other persons act toward the person with regard to the thing. Their actions operate to define the thing for the person. Thus, symbolic interactionism sees meanings as social products, as creations that are formed in and through the defining activities of people as they interact. (p. 4, 5)
This view of meaning has important implications for how we interpret what we typically call "environment." If we take environment to consist of ones surroundings, the nature of that environment is formed in and through the meanings of the objects in that environment. But, since meanings grow out of social interaction, ones environment is formed in and through the process of interpretation. Thus, two individuals who are in what may appear to be the same physical setting, may be in two very different environments. Therefore, it may seem odd that the title of this paper includes the phrase " creating a classroom environment" (emphasis added). As I will explain later in the paper, a critical aspect of the classrooms in which I have worked is that they are characterized by certain social and sociomathematical norms. That is, there were normative understandings regarding expectations and obligations for social interactions and for specifically mathematical interactions. As these normative understandings were constituted, students in the class developed their own individual interpretations of them. Nevertheless, what we mean by saying that these understandings were normative is that there is evidence from classroom dialogue and activity that students interpretations were compatible. The language that we have used to describe this is to say that their interpretations are "taken-as-shared" (Cobb, Wood, Yackel, & McNeal, 1992). It is in this sense, then, that we can speak of a classroom environment. We understand that while the childrens individual "environments" may not be identical, we have reason to believe that they are, for the most part, compatible.
Over the past decade and a half, my colleagues and I have worked extensively in a number of elementary school classrooms to investigate the teaching and learning of mathematics in the classroom setting. In the process, we have conducted a number of year-long classroom teaching experiments as well as several of shorter duration. Instruction in project classrooms focuses on conceptual development as opposed to procedures and skills. A typical class session consists of teacher-led discussions of problems posed in a whole class setting, collaborative small-group problem solving, and follow-up whole class discussions in which children explain and justify the problem interpretations and solutions they developed during small group work. The instructional tasks and strategies used in project classrooms were developed through the teaching experiments. Both Steffe and colleagues seminal work on childrens mathematical conceptual development (Steffe, Cobb, & von Glasersfeld, 1988; Steffe, von Glasersfeld, Richards, & Cobb, 1983) and Realistic Mathematics Education instructional design theory, developed at the Freudenthal Institute, The Netherlands (Gravemeijer, 1994), were critical to the conceptualization of these tasks and strategies.
Through analyzing these teaching experiments, we have come to understand the importance of taking account of the social aspects of learning, including social interaction (Cobb, Yackel, & Wood, 1989; Yackel, Cobb, Wood, Wheatley, & Merkel, 1990). We have developed an interpretive framework for analyzing classrooms and explicated theoretical constructs within that framework in terms of our experiential base (Cobb & Yackel, 1996). In addition to the teaching experiments, we have collaborated with a number of elementary school teachers to support them as they radically revise the way they teach mathematics based on our findings from the classroom teaching experiments. These situations have provided opportunities for us to confirm the relevance of the theory we have developed for guiding classroom instruction in typical, non-project situations (Cobb, Yackel, & Wood, 1991).
One of our goals was to develop classroom situations where children engaged in genuine mathematical discussions with each other and with the teacher. Following Richards (1991), we have referred to this as the inquiry mathematics tradition (Cobb, et al., 1992). Richards distinguished between four distinct domains of mathematical discourse: research math, inquiry math, journal math, and school math. Research math refers to the spoken mathematics of professional mathematicians and scientists and relies heavily on technical language and underlying content assumptions. Journal math is the language of publications and papers. The emphasis is on formal communication. Richards describes inquiry math as the language of mathematical literacy. It includes asking mathematical questions, solving unfamiliar mathematical problems, proposing conjectures, and listening to mathematical arguments. By contrast, he describes school math as the discourse of traditional classrooms that relies on the initiation-reply-evaluation sequence and that focuses on "habitual, unreflective, arithmetic problems" (p. 16). According to Richards, this type of discourse does not produce mathematics or mathematical discussions but produces, instead, what he calls a type of number talk that is driven by computation. As will become apparent in the discussion of social norms and sociomathematical norms, there is a clear link between the nature of these norms and inquiry mathematics. In fact, the constructs of social norms and sociomathematical norms can be used to more clearly describe what we might mean by an inquiry mathematics tradition and how such a tradition can be fostered in the classroom. I will return to this topic later in the paper.
Classroom Social Norms
Our initial intention when conducting the first classroom teaching experiment during the 1986-87 school year, in a second-grade classroom, was to use psychological constructivism (von Glasersfeld, 1984) as a primary theoretical lens for interpreting and analyzing classroom activity. Our goal was to account for the childrens development of mathematical ways of acting and knowing as they interacted with the teacher and with their peers in the classroom. Using constructivism as a framework, we intended to analyze the childrens cognitive restructuring. However, on the very first day of school, an episode occurred making it apparent that we would also need to develop ways to take account of the social aspects of the classroom. This episode has been discussed in Cobb, Yackel, Wood, and Wheatley (1988). I repeat it here for two reasons. The first is its significance in highlighting the importance of social aspects of learning. Second, it provides an opportunity to demonstrate an analysis from the symbolic interactionist perspective.
Episode 1. The students had worked in pairs to attempt to solve several nonroutine problems. Subsequently, the teacher initiated a whole class discussion for the purpose of having the students explain their problem interpretations and solutions. The following dialogue occurred as Peter came to the front of the room to explain how he and his partner solved the following problem.
How many runners are there altogether? There are six runners on each team. There are two teams in the race.
Teacher: Peter. What answer did you come up with?
Teacher: Fourteen. How did you get that answer?
Peter: Because six plus six is twelve. Six runners on two teams... (Peter stops talking, puts his hands to the side of his face, and looks down at the floor.)
Teacher: Would you say that again. I didnt quite get the whole thing. You had... Say it again, please.
Peter: (Softly, still facing the front of the room with his back to the teacher.) Its six runners on each team.
Peter: (Turns to look at the teacher.) I made a mistake. Its wrong. It should be twelve.
Our interpretation both at the time of the incident in the classroom and upon later analysis was that Peter was highly embarrassed at having made a mistake. Such embarrassment is in keeping with an interpretation of the situation as one in which errors should not be made. In this case, we infer that in Peters prior school experience it was expected that you should not make a mistake when you give an answer in class. In this case, embarrassment is warranted when one makes an error. However, in this class, the teacher did not want such an expectation to be operative. Rather, in keeping with the goals that we had for the teaching experiment, she wanted students to talk about their mathematics (Steffe, 1992). Consequently, the teacher made a decision-in-action (Cobb, Yackel, et al., 1991) to use this incident to talk about her expectations.
Teacher: (Softly.) Oh, okay. Is it okay to make a mistake?
Andrew (another student): Yes.
Teacher: Is it okay to make a mistake, Peter?
Peter: (Still facing the front of the room.) Yes.
Teacher: You bet it is. As long as youre in my class it is okay to make a mistake. Because I make them all the time, and we learn a lot from our mistakes. Peter already figured out, "Oops. I did not have the right answer the first time" (Peter turns and looks at the teacher and smiles), but he kept working at it and he got it right.
Analysis of the above episode makes it clear that there is a distinct shift in the topic of discussion after Peter acknowledged that he made a mistake. The topic shifts from a discussion of the mathematical aspects of the problem to a discussion of what the teachers expectations are for the childrens ways of working. Previously, we have referred to this as the distinction between talking about math and talking about talking about math (Cobb, et al., 1989). When the discussion is about talking about talking about math, the subject of discussion is social in nature. It is about how to act and contribute, but not about the mathematical aspects of the contribution.
We infer that when the teacher asked the question, "Is it okay to make a mistake?" she did so because she interpreted Peters actions as indicating embarrassment. Accordingly, her question was a deliberate attempt to confront what she inferred to be Peters interpretation and to make it possible for him to develop an alternative. That is, her question and further comment about the potential value of making mistakes were oriented to her inference about Peters interpretation of the situation and her desire to signal to Peter that another interpretation was possible. We note that her question was first answered by Andrew, a student sitting in the class listening to the exchange between Peter and the teacher. Andrews affirmative response indicates that other students in the class, besides Peter, were also involved in interpreting the situation at hand. Thus, the interaction that was taking place involved more than Peter and the teacher. It involved the whole class.
The emphasis on interpretation in the preceding analysis demonstrates the essence of the symbolic interactionist perspective. In addition to identifying the individual interpretations and actions in the episode, we can ask about the joint action, that is, the collective action of the group. In this case, the analysis is that in the joint action we see the initiation of taken-as-shared expectations for how classroom activity and classroom discourse proceed. Earlier we noted that, because the meanings and interpretations that underlie joint action are continually subject to challenge, it is possible to explain how both individual actions and the joint action of the group change over time. We refer to this process as the renegotiation of classroom social norms (Cobb & Bauersfeld, 1995). I do not contend that through this episode alone classroom expectations were established that would be in place for the remainder of the school year. Rather, in this episode we have an example illustrating how such renegotiation is initiated. Further, it should be remembered that symbolic interactionism posits that the joint activity of the group and the activity of the individuals who make up the group are continually being formed and reformed in an ongoing dynamic manner. Thus, every interaction gives rise to new occasions to (re)constitute the expectations that were initiated in this instance.
I have used the term "classroom social norms" several times in this paper without giving a definition. "Norm" is a sociological construct and refers to understandings or interpretations that become normative or taken-as-shared by the group. A norm is not a rule that prescribes individual action. Rather, it is a collective notion. One way to describe norms, in our case classroom norms, is to describe the expectations and obligations that become "normative." In saying this, I mean to imply that these are the expectations and obligations that become constituted in the classroom. Classroom interactions can be characterized in terms of these expectations and obligations. For example, in the above episode, we described the interaction between Peter and the teacher in terms of the expectations that Peter thought the teacher might have, in terms of the expectations the teacher did have, and the initiation of taken-as-shared expectations for how to engage in mathematical activity and discussion.
By analyzing data from our initial teaching experiments, we were able to identify a number of classroom social norms that characterized the interactions in those classrooms. Described in terms of expectations and obligations, these include the expectations that students develop personally-meaningful solutions to problems, explain and justify their thinking and solutions, listen to and attempt to make sense of each others interpretations and solutions, ask questions and raise challenges in situations of misunderstanding or disagreement, and persist in solving challenging problems. In saying that these social norms characterized the classroom interactions, I mean that these ways of acting and of interpreting the actions of others became taken-as-shared.
It is important to distinguish norms from rules. Norms are not rules that are set out in advance to govern classroom activity. Rather, norms are formed or constituted in and through the actions of the participants as they interact with one another. I prefer to use the word constituted rather than the word established to emphasize that norms are not "established" once and for all. Norms are continually reconstructed in concrete situations. They do not exist apart from the interactions that give rise to them. The importance of this understanding becomes apparent when we analyze individual classrooms. The specific norms that become constituted are peculiar to each classroom. Nevertheless, to the extent that classroom social norms constrain and enable learning, it is possible for teachers to anticipate which norms they might wish to foster. In this regard, we note with Blumer (1969) that in any collective body "there is one group or individual who is empowered to assess the operating situation and map out a line of action" (p. 56). In the classroom, typically, it is the teacher who initiates and guides the constitution of norms. Consequently, after the initial classroom teaching experiments, in our subsequent work with teachers, we have encouraged them to initiate the constitution of the classroom social norms outlined above.
In the following paragraphs, I include an example to illustrate the social norms described above and the learning opportunities that these norms can afford. This example illustrates the interactions that occurred between two second-grade children.
Episode 2. This dialogue took place in the fifth month of the school year as Craig and Karen worked together during small group work. They had worked as partners for a number of months. Therefore, they had developed a relatively stable, though often contentious, working relationship. They were attempting to solve the following problem.
39 + 19 = _____
Craig: (Uses a hundreds board and starts counting at 40 on the hundreds board.) Forty, 41, 42, 43, ..., 57, 58, 59. (While counting, he does not visibly keep track of his counting acts.)
Karen: Thirty-nine, 49. Thats 10. (She points to the hundreds board. She continues counting on her fingers, starting at 50. She puts up one additional finger with each number word uttered and stops when nine fingers are up.) Fifty, 51, 52, 53, 54, 55, 56, 57, 58.
In attempting to resolve the disagreement between their individual answers, each child repeats his or her solution several times. My inference is that, during this time, each child is simply trying to advance his or her own agenda. Neither is attempting to interpret the others activity. However, the next remark, made by Karen, indicates a shift in that she now refers explicitly to her interpretation of Craigs activity. Further, she offers an explanation that is based on her previously reported solution activity but has modified it in an apparent attempt to accommodate Craigs thinking.
Karen: Youre not even counting [meaning "keeping track of your counting"]. Come here, Ill explain how I got my number. See, you have 39 and you plus 10 more and thats 49; 50, 51, 52, 53, ...58. (This time Karen is pointing to numerals on the hundreds board with her pencil as she utters the words. Simultaneously, she uses fingers of both hands to keep track of her counting acts. She stops when nine fingers are up.) Theres 19. So it has to be 58.
The adaptation Karen made is to point to the hundreds board as she counts up to 58, whereas previously she had used only her fingers from 49 to 58. However, like Craig, she did not use the hundreds board to keep track of how many she counted. She still used her fingers for that purpose. Thus, from the observers perspective, her adaptation did not serve a clarifying function for Craig, even though she repeated her explanation several times. Her activity of keeping track of the 19 in the double counting process remained covert since she relied on her knowledge of finger patterns to stop when she had nine fingers up. This important component of her solution procedure did not become an explicit topic of conversation. Nevertheless, as the episode continues we see evidence that Craig is attempting to interpret Karens activity and relate it to his own. Her repeated attempts to explain are followed by this remark.
Craig: Look, okay. Ten plus that, 49 (pointing on the hundreds board to 39, then 49). And then, look, 50, 51, (still pointing to the hundreds board) 52, 53, 54, 55, 56, 57, 58. Fifty-ni&endash; (hesitates).
For the first time in this episode, Craig counts 10 as a single unit item, rather than as 10 separate units of one. His hesitation at 59 seems to indicate some uncertainty, suggesting that Karens explanations might have caused him to reflect on his counting activity. Meanwhile, as Craig counted, Karen kept track of his counting using her fingers. In effect, she carried out that part of the activity that she inferred Craig was omitting. We might infer that Karen assumed that Craig would become aware of her activity and, in doing so, realize that he was not keeping track of his counting acts. (Un)fortunately Karen made an error herself in keeping track of Craigs counting. She started recording on her fingers when he said "49" rather than 50, prompting the next comment.
Karen: You went 11. You had to go only ten.
This error had the effect of bringing to the fore the double counting that Karen was engaged in. Now for the first time in the episode, Craig indicates that he understands the need to keep track of his counting acts.
Craig: Look, 50, 51, ... (counting on the hundreds board) ... 55. Okay! Fifty (Now Craig starts keeping track on his fingers), 1, 2.
Karen takes over and once more demonstrates the solution method she has been advocating.
Karen: 39, 49, 50 (starts keeping track on her fingers), 51, ..., 58.
Craig also keeps track but begins with Karen counting from 51 rather than from 50. The following exchange ensues.
Craig: That was only eight. Look Ill show you.... Forty, 41, ..., 50 (pointing to each number on the hundreds board).
Karen: Thats a ten.
Craig: Fifty-one, 52, 53, 54, 55, 56, 57, 58, 59. That is a nine.
Karen: Thats 20.
In this brief exchange we see that Craig now interprets the task as one that requires double counting. The two children exchange several more remarks before they resolve their differences.
Craig: See, look, 1, 2, 3, ..., 9 (pointing on the hundreds board to 58, 57, 56, ..., 50).
Karen: (Taking over the counting task from Craig,) 10 (pointing to 49),11, 12, ..., 20 (pointing to 39). Twenty! Twenty! Twenty! But this [number in the problem] is only 19. So&endash;58.
After a long pause, Karen repeats her original explanation once more.
Karen: Thirty-nine, 49. Thats 10.
Craig: Forty-nine [meaning the answer is 49].
Karen: No, its not. Im trying to show you how I got the number. Thirty-nine, 49 (pointing on the hundreds board). Then one (holds up one finger) 50, 51, 52, 53, ..., 58. (Karen holds up an additional finger each time she says a number word starting at 50 and stops when she has 9 fingers up.)
After further dialogue, Craig counts silently on the hundreds board, erases the 59 he had written down on the activity page and says, "Fifty-eight, because I know why. Youre counting this one up here" [referring to 40].
Throughout this episode, there is ample evidence within the interactions of the expectations to develop personally-meaningful solutions, to explain and justify solutions, to listen to and attempt to make sense of others interpretations and solutions, to ask questions and raise challenges in situations of misunderstanding or disagreement, and to persist in solving challenging problems. These expectations stand in stark contrast to traditional instruction, where the interactions clearly indicate expectations of following pre-given rules and procedures and avoiding errors.
The example also illustrates the learning opportunities that can arise for children in classrooms that are characterized by these social norms. In this case, it is clear that both Craig and Karen extended their own initial conceptual framework as they attempted to make sense of each others solution activity. Throughout the episode, Karens ability to articulate her counting activity improved. For his part, Craig became explicitly aware of the need for double counting and developed a way to accomplish this. Further, there is some indication that Craig was beginning to develop a tens structure for number. At least once he counted a unit of ten rather than ten individual unit items. In addition, he clearly had partitioned the 19 to be added into 10 and 9 more by the end of the dialogue.
As a final comment about this episode, I remind the reader that, as social products, the social norms that are evident throughout do not determine the social interactions. Instead, they are themselves continually being (re)constituted by and through the interactions. The social norms do not exist apart from the interactions that give rise to them.
The social norms discussed in the preceding section are general classroom social norms that could apply to any subject matter area. They are not unique to mathematics. For example, we might expect children to explain their thinking and challenge each others interpretations in science or history classes as well as in mathematics. To distinguish those normative aspects of mathematics discussions that are specific to students mathematical activity from general social norms, we use the term "sociomathematical norms." Normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient, and mathematically elegant are examples of sociomathematical norms. Similarly, what counts as an acceptable mathematical explanation and justification is a sociomathematical norm. The distinction between social norms and sociomathematical norms is a subtle one. For example, the understanding that students are expected to explain their solutions and ways of thinking is a social norm, whereas the understanding of what counts as an acceptable mathematical explanation is a sociomathematical norm.
It is legitimate to ask how notions such as mathematical difference, sophistication, efficiency, elegance and explanation come to have meaning for students. To answer this question, we return to the symbolic interactionist position on meaning. This position is that meaning arises through interactions. Meaning is a social product, a creation that is formed as people interact. Accordingly, the meaning of mathematical difference, for example, is not something that can be outlined in advance for students to "apply." Instead, the meaning of mathematical difference must be formed in and through the interactions of the participants (the teacher and the students) in the classroom.
We can make a distinction between those classrooms where teachers are giving explicit attention for the first time to the sociomathematical norms that are constituted and those where teachers have a history of initiating the negotiation of certain sociomathematical norms. In the first case, the teacher has no precedence for knowing what types of contributions the students might offer so she may have to make on-the-spot decisions with little or no background information. On the other hand, teachers in the second case can use their experience to anticipate students contributions and the types of interactions that might result. Because of this, it is advantageous in teacher development to provide teachers with experiences so they can think about potential student offerings and anticipate ways of responding to them. In this regard, the Japanese style lesson plan (Stigler, Fernandez, & Yoshida, 1996) is extremely helpful to teachers in planning classroom instruction in which they intend to give explicit attention to classroom norms.
To illustrate sociomathematical norms and how they are constituted in the classroom, I use the example of what counts as different mathematically. As part of the process of guiding the development of a classroom environment where students developed personally-meaningful solutions and explained and justified their thinking, project classroom teachers regularly asked students if anyone had solved a task in a different way and then questioned contributions that they did not consider to be mathematically different. It was while we were analyzing the teachers and students interactions in these situations that the importance of sociomathematical norms first became apparent.
The following episode, from a second-grade classroom, took place several months after the beginning of the school year. For emphasis, all of the explicit references to mathematical difference are shown in italics.
Episode 3. The number sentence 16 + 14 + 8 = _____ was written on the chalkboard. The childrens task was to solve the problem mentally, without access to paper and pencil. They were instructed to figure out a solution "without counting."
Lemont: I added the two ones out of the 16 and [the 14] ... would be 20 ... plus 6 plus 4 would equal another 10, and that was 30 plus 8 left would be 38.
Teacher: All right. Did anyone add a little different? Yes?
Ella: I said 16 plus 14 would be 30 ... and add 8 more would be 38.
Teacher: Okay! Jose. Different?
Jose: I took two tens from the 14 and the 16 and that would be 20 ... and then I added the 6 and the 4 that would be 30 ... then I added the 8, that would be 38.
Teacher: Okay. Its almost similar to&endash; (Then, addressing another student,) Yes? Different? All right.
This response to Jose suggests that the teacher is in the process of working out the meaning of mathematical difference for himself. At the same time, it gives the students some insight into the teachers interpretation of difference. Joses solution was questionable in this regard. Nevertheless, because the teacher does not elaborate, the students are left to develop their own interpretations.
Rodney: I took one off the 6 and put on the 14 and I had ... 15 [and] 15 [would be] 30, and I had 8 would be 38.
Teacher: Yeah! Thirty-eight. Yes. Different?
Tonya: I added the 8 and the 4, that was 12 ... So I said 12 plus 10, that would equal 22 ... plus the other 10, that would be 30&endash;and then I had 38.
Teacher: Okay! Dennis. Different, Dennis?
Six additional responses were given, two by children who had already responded, before the teacher went on to another number sentence. In each case, the teacher continued to ask for something different.
Through participating in interactions of this type, the teacher and children interactively constituted meaning for "mathematical difference." For their part, the children learned that the teacher legitimized solutions that consisted of partitioning the summands and recombining the results in various ways but offered sanctions against those that were restatements of previously-given solutions. On the other hand, the teacher learned which types of solutions the students were able to develop and what they offered as potentially different. Thus, we can say that in the process of the interaction both the children and the teacher were indicating their interpretations and were adjusting their interpretations to those of the other participants. It is also apparent from this example that in another situation the meaning of difference would need to be (re)constituted anew. To the extent that there are similarities between this situation and others, the participants in the interaction have some basis for anticipating others interpretations and making interpretations of their own that are compatible. In this way, taken-as-shared understandings can develop over time.
From this same classroom, there is evidence that such taken-as-shared understandings do develop. Some weeks later, in a discussion involving another number sentence task, in which no counting was to be used to arrive at a solution, the teacher first called on Dennis to explain how he solved the task. Subsequently the teacher called on Alex, who responded with, "I did the last one like Dennis." A few moments later, after Ella offered her solution, Dennis protested to the teacher, "Mr. K., thats the same thing I said." Both Alexs response and Dennis protest indicate that a taken-as-shared meaning for mathematical difference had begun to be constituted. Further, their comments served to perpetuate the expectations that had been initiated with respect to mathematical difference.
The importance of sociomathematical norms, such as what counts as mathematical difference, for student learning can be seen in considering the cognitive activity that was necessary before Alex and Dennis could make their comments. Before Alex could say what he did, he needed to understand Dennis explanation, compare it to the solution he himself had developed and judge the extent of their similarities and differences. Dennis remark required similar activity in relation to Ellas explanation. Thus, the instructional activity, as it was interactively constituted in the classroom, had the potential for cognitive activity that extended beyond solving the task itself. For at least some of the children in the class, the solution process itself became an object of reflection. Further, the emphasis on different solutions had the effect of encouraging the children to partition numbers in many different ways, since reporting various ways of combining the tens and ones was regularly accepted as different by the teacher. As a result, the children in the class developed considerable flexibility in mental computation of two-digit addition and subtraction tasks.
The interaction that took place between the children and the teacher also served to sustain the social norms of developing personally-meaningful solutions and listening to and attempting to make sense of each others explanations. In this way, this example also illustrates the interrelationship between social and sociomathematical norms. Finally, we note that Dennis protest demonstrates that, while the teacher may initiate the negotiation of norms, the children in the class join in the process of making them explicit topics of conversation.
One of the sociomathematical norms that has been most useful in clarifying the relationship between the social features of the classroom and the quality of students mathematical learning, from elementary school through university-level mathematics, is the norm of what constitutes acceptable mathematical explanations and justifications (Yackel & Cobb, 1996; Yackel & Rasmussen, 1999). Here I take explanation and justification to be social constructs that serve communicative functions. Students and the teacher give mathematical explanations to clarify aspects of their mathematical thinking that they think are not readily apparent to others. They give mathematical justifications in response to a challenge that there is an apparent violation of normative mathematical activity (Cobb, et al., 1992). For example, if a child said that he solved 16 + 8 + 14 by taking one from the 16 and adding it to the 14 to get 15 and 15, adding the 15 and 15 to get 30, and adding the remaining 8 to get 38, we would infer that he was explaining his solution to others in the class. However, a challenge to the solution that says that you first have to add the 16 and 8 and then add 14 to that sum is a request for a justification of the solution.
Elsewhere, I have documented second-grade childrens evolving understanding of mathematical explanation and justification (Yackel, 1992). The analysis showed that initially children had to learn that their explanations and justifications should be based on mathematical thinking and activity. For example, early in the school year one child attempted to convince his partner that his answer was correct by arguing that he had the better pencil. Another child changed her answer in a whole class discussion when the teacher asked the class if they agreed. She later explicitly said that she interpreted the teachers question as an indication that she was wrong. Incidents such as these provide opportunities for the teacher to initiate the negotiation of what constitutes an acceptable explanation and an acceptable justification. In the classroom that was the subject of the above analysis, by the second month of the school year the expectation that explanations should be about actions on (what had become) mathematical objects (for the students) was beginning to be taken-as-shared. The following example serves as an illustration.
Episode 4. The episode is taken from a whole class discussion in which children discussed their solutions to problems they had attempted to solve in prior small group work. The problem under discussion is the following.
Roberto had 12 pennies. After his grandmother gave him some more, he had 25 pennies. How many pennies did Robertos grandmother give him?
As Travonda explains her response, she asks the teacher to notate her solution method using the standard vertical format for addition. At Travondas direction, the teacher writes the following on the overhead.
Thus far, Travondas explanation has involved only specifying the details of the conventional vertical format. She continues.
Travonda: I said, one plus one is two, and 3 plus 2 is 5.
Teacher: All right. She said &endash;
Rick: (Interrupting.) I know what she was talking about.
Teacher: Three plus 2 is 5, and one plus one is two.
Travondas explanation is only procedural in nature. It is about combining digits and makes no reference to the quantities involved in the problem. Further, in reiterating her solution, the teacher modifies it by proceeding from right to left. In doing so, he makes it conform even more closely to the standard algorithm. Neither one refers to the answer as 25. Several children simultaneously challenge the explanation.
Jameel: (Jumping from his seat and pointing to the projection screen.) Mr. K. Thats 20. Thats 20.
With this remark, Jameel is apparently challenging the teachers remark, "[O]ne plus one is two." Rick, and others, on the other hand, challenge the entire solution.
Rick: (Simultaneously.) Un-uh [no]. Thats 25.
Several students at once: Thats 25. Thats 25. Hes talking about that.
Jameel: Ten. Ten. Thats taking a 10 right here. (Walking up to the overhead screen and pointing to the numbers as he talks.) This 10 and 10 (pointing to the ones in the tens column). Thats 20 (pointing to the 2 in the tens column).
Jameel: And this is 5 (pointing to the 5 in the ones column) more and its 25.
Teacher: Thats right. Its 25.
We interpret the actions of Jameel, Rick and the other students as challenges to the procedural character of the explanation. Their insistence on referring to the quantities involved in the problem indicate that, for them, explanations have to describe actions on what they take as mathematical objects, not procedures. Videotape data of the solution activity of Jameel and his partner shows that they struggled in their attempt to communicate with each other as they solved this problem during small group work. Jameel used a partitioning strategy, coupled with the compensation thinking strategy. He compared 12 plus 13 to 10 plus 15. In doing so, he focused specifically on the quantities involved and not on the digits that comprise the numerals that represent the quantities. Both he and his partner used unifix cubes in their attempts to explain to each other. The teacher joined the group for an extended period of time while Jameel attempted to explain his thinking. Despite his probing questions and sincere attempt to make sense of Jameels explanation, the teacher was unsuccessful in understanding Jameels (relatively) sophisticated, but inarticulate, solution. Nevertheless, during the entire discussion with the teacher, Jameel and his partner continually referred to quantities. Thus, it is not unexpected that, in the subsequent whole class discussion, Jameel challenged the adequacy of the solution method that Travonda and the teacher proposed. Their explanations did not address the "mathematical objects" that Jameel had acted on when he solved the problem.
Having participated in the small group discussion with Jameel and his partner, the teacher was well aware of the nature of their activity. Consequently, it is not surprising that he readily acknowledged the challenges that Jameel and the other children raised and that he accepted Jameels reinterpretation. In doing so, the teacher legitimized the ongoing renegotiation of the sociomathematical norm that to be acceptable, explanations had to be descriptions of actions on taken-as-shared mathematical objects that are experientially-real for the students.
In using the language "experientially-real mathematical objects," I mean that the students act as though the mathematical entities they work with are objects in their mathematical worlds. For example, in saying, "Thats taking a 10 right here," Jameel is implying that he can treat the ten as an object that he can manipulate. He can "take it" and combine it with another ten. In doing so, Jameel creates a new mathematical object, a 20. Furthermore, he "takes" the 10 from 12, indicating that 12 is an object for him that can be partitioned into 10 and 2. Jameel acts as though these are mathematical objects within his world of experience, that is, that they are experientially real for him.
Earlier I indicated that in using the notion "inquiry mathematics tradition" we were following Richards (1991) in describing classroom situations where children engage in genuine mathematical discussions with each other and the teacher. Richards intention seems to be two-fold. First, the classroom should be one where students are expected to discuss, that is, explain their thinking and ask questions of others. This aspect relates to what we have called social norms. Second, Richards intention includes the nature of the discussions. They should be "genuine" mathematical discussions. Our analysis of sociomathematical norms makes it possible to be more specific about what this might mean. We take it to mean that the sociomathematical norm, that explanations should be descriptions of actions on taken-as-shared mathematical objects that are experientially real for the students, has been negotiated in the classroom. Subsequently, when I use the phrase inquiry mathematics instruction, I mean that this sociomathematical norm for what constitutes acceptable explanations has been negotiated.
There is ample evidence from classrooms where this sociomathematical norm is taken-as-shared to support the claim that such classrooms enable the development of sophisticated forms of mathematical argumentation, of intellectual autonomy, and hence of mathematical power. The following example is included to illustrate the level of mathematical argumentation that young children are capable of, given a classroom environment that supports an inquiry tradition. The example is taken from the same second grade project classroom as the previous example and occurred in the eighth month of the school year.
Episode 5. The teacher wrote the number sentence 46 + 38 + 54 = ___ on the chalkboard as a mental computation task. Students were to solve the task without paper and pencil and were instructed to do so without counting. The first child called on to explain how she solved the problem was Donelle. She reported that her answer was 138. However, when she attempted to explain how she figured it out, it became apparent that she had to reconstruct the solution process. She did so, describing her thinking out loud for the class to hear. In the process she made an error.
Donelle: I said 40 plus 30. Thatd be 70.
Donelle: And I said 70 plus the 8, thatd be 78.
Donelle: And I said 78 plus the 50, thatd be (long pause) a hundred and &endash; sixteen.
Several students raised their hands indicating that they had questions for Donelle. In response to a specific question about the sum of 78 and 50, Donelle recalculated. But she once again arrived at the partial sum of 116. It was at this point that Travonda put forth an argument in an attempt to convince Donelle that 116 could not be correct.
Travonda: ... you said that was 16, what other&endash; when you add 6 and the 4 you wont&endash;youll get 26 and you, you wont get, um 138.
Donelle: (To the teacher.) I didnt understand her question.
Teacher: (To Travonda.) Whats the question?
Travonda: I, I said, if you said that was 116, and I said the 6, thatd be 116 and 6 plus 4, thatll equal to 10 so the 116 plus the 10, that will make it, um, a hundred, um a hundred and twenty-six, and that wont be, um 138.
While we would anticipate that some children might challenge the result of Donelles addition, the nature of Travondas argument is remarkable since it is an argument by contradiction. Even when Donelle indicated that she did not understand, Travonda did not modify her argument to explicitly tell her what the partial sum should be. Instead, she repeated her argument which relied on developing a contradiction between having the partial sum of 116 and a total of 138. My interpretation is that Travondas intention was to help Donelle understand how she could figure out that her partial sum could not be correct.
It is not possible to tell from the ensuing dialogue whether or not Travondas argument served its intended function of convincing Donelle of her error. However, there is evidence that it provided other children with the opportunity to reflect on the form of her argument. Over the next five minutes another student, Jameel, repeatedly asked Travonda to clarify her explanation.
Jameel: Wouldnt that be, ah, that would be 128? ... Im talking about what Travonda said it was 126.
Travonda: Yea, but Im talking about when she [Donelle] said 116 and I said she only had the 6 and the 4 left, so that would make it a 10 and with that 10, and she add it to the 116 shell get 126 and she wants to get, um, 138.
From this, and some of Jameels later comments, we infer that he may also have been thinking about an argument by contradiction but reasoning in a different way. He may have been reasoning that if the result is 138 and there is 6 and 4 left to add, then the partial sum has to be 128. Since Donelle does not have 128, there is an error. By contrast Travonda was reasoning that if the partial sum is 116 and there is 6 and 4 left to add, then the result has to be 126, which contradicts Donelles claim that the sum is 138. After some intervening minutes in which other children "helped" Donelle arrive at the correct answer, Jameel again asked for a clarification of Travondas argument. Even though the teacher attempted to dismiss his request, Travonda once again took up the challenge to clarify her argument.
Teacher: Well, thats gone now. Weve forgotten it.
Travonda: Hes (referring to Jameel) talking about when I said that it would be 126. If she wouldve added the 6 and the 4, thatll be 126 and, and the last number, um, and she said it would be 138, so she would, so she, and thatd be 126 so she wouldnt get 138.
I contend that Travondas purpose when she repeated her argument to Jameel was not the same as when she addressed Donelle. When she addressed Donelle, her purpose was to convince her that 116 is not the partial sum. When she addressed Jameel, her purpose was to convince him that her argument was valid. Her change in intentions is indicated both by the fact that she took up the challenge to respond to Jameels second request for clarification, even though the teacher attempted to dismiss it as no longer relevant, and by the fact that in the intervening minutes other children gave explanations which clarified that the partial sum Donelle was attempting to calculate should have been 128. It appears that Jameel was focusing on the nature of Travondas argument rather than on the computational results. Further, the opportunity for him to do so was directly related to the fact that, unlike Donelle, he was not personally engaged in the process of explaining his solution to the addition problem. He was able to distance himself from the process of solving the problem and focus on the argument itself.
This example illustrates that, as early as second grade, children can and do take the arguments they hear others give as objects of reflection. In this case, the interaction between Travonda and Jameel was about the nature of the argument and not about the computational result. This is powerful evidence that, given the opportunity, (at least some) children can engage in relatively sophisticated levels of mathematical reasoning, well beyond what is typically thought of as appropriate for primary school mathematics. Further, the situation that prompted the interaction in the preceding example was a relatively simple task. Most would describe it as a computation task. Yet, in this case, the discussion of the task provided a forum for reflecting on the form of mathematical argument. It is for this reason that I find it difficult to separate out issues of classroom environment from instructional tasks. When a class follows an inquiry tradition of instruction, many of the "tasks" that children engage in are tasks that they set for themselves as they attempt to reason about the dynamic interaction that occurs in small group interactions and in whole class discussions. In a real sense, by choosing what they reflect on, students individualize the instruction for themselves in ways that only they can do.
and Mathematical Power
In the preceding sections, I have given examples of the types of mathematical argumentation that characterize mathematics classrooms that follow an inquiry tradition. Both the social norm that students are expected to explain and justify their mathematical activity and the sociomathematical norm of what constitutes an acceptable mathematical explanation and justification, as explained above, contribute to creating a classroom environment where mathematical argumentation flourishes. I use the word "creating" here cautiously. In keeping with the symbolic interactionist perspective, I intend to imply that every situation is constituted anew through the interaction of the participants. However, to the extent that the students and the teacher negotiate norms that are taken-as-shared, there is some stability in the interpretations they make and attribute to others. Consequently, they have some idea of what is expected and of their obligations. In this sense, the word "create" means to establish some regularity in the classroom social norms and sociomathematical norms. And it is in this sense that we can talk of a "classroom environment" as though it exists. It may be more appropriate to speak of negotiating a taken-as-shared classroom environment that is experientially-real to the students.
Kamii (1985) follows Piaget (1948/1973) in saying that the main purpose of education is autonomy. One of the benefits of establishing the social norms implicit in the inquiry approach to mathematics instruction is that they foster childrens development of social autonomy (Cobb, et al., 1989; Nicholls, Cobb, Wood, Yackel, & Patashnick, 1990). However, it is the analysis of sociomathematical norms that clarifies the process by which teachers foster the development of intellectual autonomy. Intellectual autonomy refers to an individuals participation in classroom interactions. Individuals who are intellectually autonomous use reasoned judgment in making decisions about the way they interact mathematically and about the contributions they make. They do not rely on external rules or an external authority to know when and how to interact. In order to achieve intellectual autonomy, children need a basis for making judgments about what is acceptable mathematically, for example, with respect to mathematical difference, mathematical sophistication, mathematical efficiency, mathematical elegance, and mathematical explanation and justification. However, these are precisely the types of judgments that the teacher and students negotiate when constituting sociomathematical norms that are characteristic of an inquiry tradition. In the process, students construct specifically mathematical beliefs and values that help them form their judgments. For example, Jameels challenge that "one and one is two" signifies "ten and ten is twenty" demonstrates that children are capable of judging what is appropriate mathematically. His challenge indicates that he had developed the belief that mathematical explanations should describe actions on experientially-real mathematical objects. Examples such as this show that it is precisely because children can make personal judgments of this type on the basis of their mathematical beliefs and values that they can participate as increasingly autonomous members of an inquiry mathematics community. As they do so, they develop mathematical power.
Mathematical power has been defined as "an individuals abilities to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve nonroutine problems" (National Council of Teachers of Mathematics, 1989, p. 5). One of the widely acclaimed goals of mathematics education is to help students develop mathematical power. Throughout this paper, I have attempted to show that one way to achieve this goal is to establish an inquiry approach to instruction. This means creating classroom environments that are characterized by social norms and sociomathematical norms of the nature that I have discussed extensively in this paper. Mathematics education research of the past several decades has provided us with an understanding of how to accomplish the task of developing inquiry approaches to instruction. It remains for us to take up the challenge to do so.
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