CEPA eprint 2096 (LPS-1983a)
The constructivist researcher as teacher and model builder
Cobb P. & Steffe L. P. (1983) The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education 14(2): 83–94. Available at http://cepa.info/2096
Table of Contents
Why we, as researchers, act as teachers
The Constructivist View of Teaching
Teaching Episodes and Clinical Interviews
Macroschemes and Microschemes
Constructivist and Nonconstructivist Microschemes
Model building in a teaching experiment
Development of Jason’s Counting Schemes
Model building – The quest for generality and specificity
The educational significance of models
The constructivist teaching experiment is used in formulating explanations of children’s mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time – anywhere from 6 weeks to 2 years. The explanations we formulate consist of models – constellations of theoretical constructs – that represent our understanding of children’s mathematical realities. However, the models must be distinguished from what might go on in children’s heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children’s construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher’s understanding of the children.
Our methodology for exploring the limits and subtleties of children’s construction of mathematical concepts and operations is the primary object of attention in this paper. We argue that, in such an exploration, there is no substitute for experiencing the intimate interaction involved in teaching children. We then discuss the constructivist view of teaching and stress the importance of modeling children’s mathematical realities. Next, the similarities and differences between constructivist and nonconstructivist teaching experiments are highlighted. In the remainder of the paper, we focus on models and model building.
Why we, as researchers, act as teachers
We believe that the activity of exploring children’s construction of mathematical knowledge must involve teaching. Theoretical analysis by the researcher does play an important role in understanding the significance of children’s mathematical behavior. But knowledge gained through theoretical analysis can at best intersect only part of the knowledge gained through experiencing the dynamics of a child doing mathematics. These experiences give the researcher the opportunity to test and, if necessary, revise his or her understanding of the child. The continual tension created by inexplicable or seemingly contradictory observations leads ultimately to a knowledge of the child that supersedes the initial theoretical analysis. The insufficiency of relying solely on a theoretical analysis serves as one reason for our belief that researchers must act as teachers. Some of our most humbling experiences have occurred when knowledge gained through theoretical analysis has failed to be of value in understanding children’s mathematical realities. On the other hand, totally unexpected solutions by children have constituted some of our most exhilarating experiences.
A second reason we believe researchers must act as teachers is that the experiences children gain through interactions with adults greatly influence their construction of mathematical knowledge. The technique of the clinical interview is ideally suited to the psychological objective of investigating a sequence of steps children take when constructing a mathematical concept. In an interview, mathematical knowledge can be traced back to less abstract concepts and operations. Further, using the clinical interview, a researcher can specify structural patterns children may abstract from the experience gained through interaction with their milieu. However, the researcher conducting a clinical interview does not intend to focus on those critical moments when cognitive restructuring takes place. Consequently, the resulting accounts of abstraction are devoid of the experiential content that a teacher needs when planning specific interventions. More importantly, some children might take a different sequence of steps and construct different concepts in particular instructional settings.
A third reason for acting as teachers stems from the importance we attribute to the context within which the child constructs mathematical knowledge. The crucial role played by context can best be illustrated by example. Erlwanger (1973) found that Benny, a sixth grader, had developed a coherent rationale that accounted for his experiences in using Individually Prescribed Instruction (IPI) materials. Erlwanger concluded that, for Benny, the activity of learning and using mathematics involved formulating unrelated rules that yielded the correct answers to particular sets of problems. Benny frequently searched for patterns in the numerals of related problems. Erlwanger explains that Benny
thought these rules were invented “by a man or someone who was very smart.” This was an enormous task because, “It must have took this guy a long time … about 50 years … because to get the rules he had to work all of the problems out like that.” (p. 17)
Benny had arrived at his conception of mathematics by reflecting on, and attempting to make sense of, his past experiences of doing mathematics. This conception served as an encompassing framework within which he formulated anticipations of what sorts of events might happen and how he should respond when he worked with IPI materials (cf. Skemp, 1979, for a discussion of meta-learning).
Confrey (1982) also found it necessary to consider the context within which students do mathematics when she investigated ninth graders’ mathematical abilities. She concluded that “in order to understand a student’s mathematical performances and to judge their abilities, one must consider the influence which the context of classroom instruction has on those performances” (p. 27). By acting as teachers, and by forming close personal relationships with children, we help them reconstruct the contexts within which they learn mathematics. In particular, we help them differentiate between the contexts of doing mathematics in class and doing mathematics with us. This is essential given our objective of exploring the limits and subtleties of children’s creative possibilities in mathematics.
In summary, we believe that the failure to observe children’s constructive processes firsthand denies a researcher the experiential base so crucial in formulating an explanation of those processes. Researchers who do not engage in intensive and extensive teaching of children run the risk that their models will be distorted to reflect their own mathematical knowledge.
The Constructivist View of Teaching
The actions of all teachers are guided, at least implicitly, by their understanding of their students’ mathematical realities as well as by their own mathematical knowledge. The teachers’ mathematical knowledge plays a crucial role in their decisions concerning what knowledge could be constructed by the students in the immediate future. Through reflecting on their interactions with students, they formulate, at least implicitly, models of their students’ mathematical knowledge. Constructivist and nonconstructivist views of teaching differ in the emphasis they place on the activity of modeling children’s realities. In the constructivist view, teachers should continually make a conscious attempt to “see” both their own and the children’s actions from the children’s points of view. This emphasis stems from an analysis of teaching as primarily the activity of communicating with students. As Schubert and Lopez Schubert (1981) point out, teaching in this sense is rooted in action:
It is in the subtly powerful interaction of some teachers … with their students. It is in the daily striving of teachers who try to understand their students’ sources of meaning, their out-of-school curricula, their personal “theories” or sense-making constructs. It exists in attempts made by teachers to determine how their experience and knowledge can bolster their students’ quest for meaning. (p. 243)
In the course of an interaction, both the teacher and the children attempt to make sense of each others’ verbal and nonverbal activity. The children, for example, interpret and give meaning to the teacher’s actions in terms of their current conceptual structures. In some cases, these interpretations are influenced by the children’s intuitions about the teacher’s motives and intentions. In any event, the teacher acts with an intended meaning, and the children interpret the actions within their mathematical realities, creating actual meanings (MacKay, 1969; von Glasersfeld, 1978).
Obviously, to communicate successfully with children, there must be some fit between the intended and the actual meanings. The likelihood that a teaching communication will be successful is increased whenever the teacher’s actions are guided by explicit models of the children’s mathematical realities. From this perspective, the activity of teaching involves a dialectic between modeling and practice. The teacher’s actions are formulated within the framework of his or her current models. The plausibility of these models is in question when the teacher attempts to make sense of observations of the children’s behavior in subsequent encounters.
Teaching Episodes and Clinical Interviews
In our work, we use teaching episodes as well as occasional clinical interviews as an observational technique. The interviews are used when we want to update our models of the children’s current mathematical knowledge, usually after a vacation or when the child has been absent for some time. However, the main emphasis is on the teaching episodes, as these give us a better opportunity to investigate children’s mathematical constructions. Our primary objective is to give the children opportunities to abstract patterns or regularities from their own sensory-motor and conceptual activities. Guided by our current model, we hypothesize certain patterns or regularities that it is possible for a child to abstract. Activities are then initiated in the hope that the child will reflect on and abstract those patterns or regularities from his or her experiences. For the constructivist teacher, the key is to help children hold their own mathematical activity at a distance and take it as its own object. This is the crucial aspect of reflection (von Glasersfeld, 1982).
The teaching episodes (and the occasional clinical interviews) are routinely videotaped. These tapes serve as a record of the episodes and permit a longitudinal analysis of a child’s mathematical development. Members of the research team can discuss interpretations of the child’s behavior when viewing the tape. Although the researcher responsible for making the video-recording of the episode is free to intervene during or after an episode, the teacher has the right to ignore interventions made during an episode. The research team try to help the teacher in two ways. First, they help the teacher explicate his or her intentions and interpretations by asking appropriate questions. Second, they suggest alternative interpretations and propose activities that the teacher might wish to initiate. During these discussions, attention is given to the child’s conception of the activity of doing mathematics as well as to his or her mathematical knowledge.
Our emphasis on formulating and revising explicit models of children’s mathematical realities in the context of acting as teachers is in harmony with Vygotsky’s research as modeling rather than empirically studying mathematical processes (El’konin, 1967, p. 36). Unfortunately, this emphasis has been submerged in the literature dealing with the methodology of the teaching experiment (Kantowski, 1978; Kieran, 1982; Menchinskaya, 1969a). In the following sections, we first consider the characteristics shared by all teaching experiments and then differentiate between the constructivist teaching experiment and other variants of the method.
One general characteristic shared by constructivist and nonconstructivist teaching experiments is the “long-term” interaction between the experimenters and a group of children. A second is that the processes of a dynamic passage from one state of knowledge to another are studied. What students do is of concern, but of greater concern is how they do it. A third characteristic is that the data are generally qualitative rather than quantitative. The qualitative data emanate from two possible sources. The first source is teaching episodes with the children. For example, Davydov (1975) reports anecdotal data obtained from his observations of teaching in classes for which he had designed learning material. The data took the form of verbatim exchanges between the teacher and her students as well as descriptions of the instructional contexts and the students’ responses in those contexts. The second source is clinical interviews conducted at selected points in the teaching experiment.
Macroschemes and Microschemes
Menchinskaya (1969a) identifies two types of teaching experiments reported in the Soviet literature. She calls the first type a macroscheme: “Changes are studied in a pupil’s school activity and development as he makes the transition from one age level to another, from one level of instruction to another” (p. 5). This type is exemplified by Davydov’s (1975) teaching experiment. He constructed teaching material that reflected his view of quantity, a view derived from his previous work with children. Children were expected to compare objects on various attributes, reverse the sense of an equality, reverse the sides of an equality, reason transitively, add a quantity to equalize inequalities, and so on. Because Davydov was interested in processes expressible in terms of an ontological notion of quantity, his experiment involved teaching children to behave in certain ways so that they would experience these processes. Consequently, he used intact classes for his teaching experiment.
The second type of teaching experiment identified by Menchinskaya is the microscheme, where “in a single pupil the transition is observed from ignorance to knowledge, from a less perfect mode of school work to a more perfect one” (p.6). Kantowski (1977) conducted a teaching experiment of this type in the United States. She went outside the realm of mathematics for processes. Goal-oriented heuristics, analysis, synthesis, persistence, and looking-back strategies were among the processes of interest in her study. These processes, being “thought” oriented rather than “content” oriented, led her to investi‑gate individual students as they attempted to acquire the processes in the context of her instruction in geometry.
These examples illustrate the considerable variation in how the passage from one state of knowledge to another is investigated. In general, a macro- scheme such as Davydov’s has a curriculum orientation, and a microscheme such as Kantowski’s has a psychological orientation. Given our focus on children’s constructive activity, it is clear that constructivist teaching experiments are microschemes.
Constructivist and Nonconstructivist Microschemes
Kantowski’s outstanding investigation exemplifies a characteristic shared by nonconstructivist teaching experiments. The processes studied are determined a priori to be the ones of interest. Alternative processes are of secondary importance. This characteristic reflects beliefs about how learning and teaching are related. Menchinskaya (1969b) states the Soviet position succinctly when she says that “neither scientific nor everyday concepts spring forth spontaneously; both are formed under the influence of adult teaching,” (p. 79). One can see, then, that she believes that children form scientific concepts as a result of receiving instruction in specific school subjects and that the processes of mastery can be studied only in the context of these subjects.
We, too, believe that adults can help children as they attempt to learn mathematics. However, it is not the adult’s interventions per se that influence children’s constructions, but the children’s experiences of these interventions as interpreted in terms of their own conceptual structures. In other words, the adult cannot cause the child to have experience qua experience. Further, as the construction of knowledge is based on experience, the adult cannot cause the child to construct knowledge. In a very real sense, children determine not only how but also what mathematics they construct. Consequently, we do not attempt to study children’s construction of certain preselected processes in instructional contexts. Instead, we attempt to understand the constructions children make while interacting with us.
Model building in a teaching experiment
We contend that children’s mathematical knowledge can be modeled in terms of coordinated schemes of actions and operations (von Glasersfeld, 1980). Our goal is to specify these schemes and to intervene in an attempt to help the children as they build more sophisticated and powerful schemes.
We offer a brief glimpse of the steps that Jason, a six-year-old child in 1980, took when constructing his counting scheme. We also mention some of our concomitant intentions when teaching Jason. However, this brief illustration focuses on Jason’s counting behavior and our interpretation of it rather than on our teaching episodes (cf. Steffe & Cobb, 1982, for an elaboration of a teaching episode). We make this choice to emphasize that our models are of children’s understanding. It will be seen that we conduct an experiential analysis of Jason’s progress within the framework of a theoretical model of children’s counting types.
Development of Jason’s Counting Schemes
We distinguish between the activities of counting and of reciting a sequence of number words. When we say a child counts, we mean the child coordinates the production of a sequence of number words with the production of a sequence of unit items (items that are equivalent for the child in some way). This activity of counting follows the establishment of a collection of countable items. The result of the counting activity is a collection of counted items.
Our investigations indicate that the quality of the items that children create while counting undergoes a developmental change (Steffe, von Glasersfeld, Richards, & Cobb, 1983). In October 1980, Jason was what we call a counter of motor unit items. His most sophisticated counting activity involved counting his movements as substitutes for visual items screened from view. For example, in a clinical interview, we asked Jason to find out how many marbles there were in all when four were hidden beneath the interviewer’s hand and seven were visible. Jason counted the visible marbles, pointing to each in turn. He continued his count of the entire collection by pointing rhythmically over the interviewer’s hand while synchronously uttering the number words “8-9-10-11.” Jason was successful because his counting acts completed a temporal pattern. He also counted a partially screened collection of seven squares, where three were visible, by first counting the three visible ones and then continuing over the screen, stopping when his counting acts completed a square pattern. When five squares were screened, however, he did not know when to stop counting.
Our immediate intention was to help Jason develop spatial and temporal patterns in counting activity involving more than four items. He might then continue until the pointing acts in the continuation completed, say, a domino five pattern. This decision was based on our belief that the coordination of spatial and temporal patterns with the counting scheme can play a crucial role in the process of constructing more sophisticated types of unit items. Our technique was to hide a collection of felt squares beneath a cloth, lift the cloth briefly, and then ask Jason how many squares he thought were hidden. When Jason did not recognize the pattern, the teacher would ask Jason to count “what you saw.” Jason counted by pointing over the cloth where his points of contact completed a spatial pattern analogous to what he had seen. The squares were arranged both randomly and in regular patterns, such as in linear and square four patterns (…. and : : ). Within three sessions (in December 1980) Jason had constructed patterns for the number words up to “seven.” Further, these patterns eventually were coordinated with his counting scheme. For example, on 8 April 1981, to find out how many checkers were under two cloths after being told that nine were under one and five under the other, Jason pointed over the first cloth while synchronously uttering “1-2-…-9” and then continued by pointing over the second cloth while uttering “10-11-12-13-14.” He stopped when his points over the second cloth completed a domino five pattern.
We hypothesized that Jason always counted starting from 1 because what was significant for him while counting was the perceptual and motor activity. We thought that Jason would not be able to find out how many items there were in all by counting-on over the second cloth until the verbal aspect of counting became significant for him (Steffe et al., 1983). Starting at 1 was not merely a habit. From his perspective, he had to count all the items – he had no alternative. His failure to count-on was ultimately related to the types of unit items he could create while counting.
The difficulty Jason had in curtailing his counting activity over the first cloth was exemplified in the 8 April teaching episode. Our goal was to help Jason take his utterances as substitutable items (i.e., to count verbal unit items) and thus curtail his motor activity in counting. We presented a sequence of ten tasks in which Jason was asked to count all the items hidden under two screens. Jason folded his hands while counting to perform the last two tasks, indicating that he was counting verbal unit items. He finally showed some awareness of the superfluousness of counting over the first cloth when performing the last task. After uttering “1-2-…-12,” he said “wait,” pointed to the cloth, and said “12” before continuing. Immediately afterwards, the interviewer presented three addition sentences (9 + 3 = \_, 17 + 3 = \_, and 25 + 3 = \_). Jason counted to solve them starting with 1, gesturing in the air in synchrony with uttering number words. In the teaching episode, Jason made progress toward becoming a counter of verbal items (i.e., he could substitute vocal utterances as well as movements for visual items) in the context of counting the hidden items. But his solutions to the number sentences indicate that his progress in this episode was limited to particular local contexts.
The limitations of being a counter of motor items can be illustrated by Jason’s almost total lack of knowledge of the basic addition facts. Although we do not want to give the impression that counters of motor unit items cannot learn the basic facts, their spontaneous methods and strategies for doing so are very limited. For example, in a teaching episode held on 23 March 1981, Jason found the sum of 3 and 4 by simultaneously extending three fingers, simultaneously extending four fingers, and then counting them all. He clearly did not know the sum of 3 and 4. To find the sum of 7 and 4, Jason sequentially put up seven fingers in synchrony with uttering “1-2-…- 7,” and continued sequentially putting up fingers while uttering “8-9-10-11.” He reused one finger he had already put up. These primitive methods inhibit children from using addition facts they already know to help them find a given sum. There were indications that Jason knew the sums 3 + 3 and 5 + 4, but he did not spontaneously use this knowledge to find the sums 3 + 4 and 7 + 4. He found each sum independently as a new task. This was a recurrent feature of his mathematical behavior.
Throughout April and most of May 1981, Jason’s counting scheme remained sensory-motor. However, he made rapid progress during this period and could soon count verbal unit items in a variety of contexts. Further, by the end of May 1981, he had a flexible, adaptive counting scheme that he could use to solve a variety of problems. He could take a sensory-motor unit itself as a unit to be counted (or create abstract units). He solved problems by intentionally finding out how many times he counted. Counting was an expression of numerical structure. For example, in an interview in November 1981, he counted 16 units beyond 14 to find the sum of 14 and 16 by sequentially putting up fingers in synchrony with uttering “16-17-…-31” (he stopped at 31 because of an executive error). He intentionally kept track of his utterances. Moreover, he counted backwards “16-15-14 - - - 13” to find how many marbles were left in a tube containing 16 after 3 had been removed, which had been completely beyond him in April. This great flexibility in his use of the counting scheme was consistent with the adaptiveness he displayed in the November interview. Initially, he could not solve a missing addend task like 11 + \_ = 16 by counting. However, with only minor suggestions from the interviewer, he solved not only that particular task, but others as well. In addition to using his counting scheme creatively in finding sums, missing addends, and differences, Jason displayed powerful numerical strategies. For example, after counting to solve 5 + \_ = 12, he knew that 9 was the answer to 5 + \_ = 14, because 9 is 2 greater than 7!
We have discussed some of Jason’s typical mathematical behavior within the framework of our model of counting types. In particular, we accounted for Jason’s use of increasingly sophisticated solution procedures in terms of changes in his counting scheme. This experiential analysis allows us to further elaborate the counting types model.
Model building – The quest for generality and specificity
Thus far, the discussion of model building has focused on interactions between children and a teacher. The reader might well have inferred that our sole objective is to account for the mathematical progress made by the small number of children who participate in a teaching experiment. However, we strive to build models that are general as well as specific. On the one hand, the model should be general enough to account for other children’s mathematical progress. On the other hand, it should be specific enough to account for a particular child’s progress in a particular instructional setting. We attempt to attain these seemingly contradictory objectives by ensuring that there is a dialectical interaction between the theoretical and empirical aspects of our work. Although one aspect may be more prominent for a time, it should never completely dominate the other. The interaction can be seen in the following brief account of a sequence of teaching experiments.
One objective of our research program was to build a viable model to account for children’s construction of numerical concepts and operations. An initial model of children’s counting (Steffe, Richards, & von Glasersfeld, 1979) was formulated on the basis of the experience of teaching six-year-old children in two yearlong teaching experiments. Although we constantly attempted to organize and make sense of these experiences, our emphasis in the initial phase was empirical. The next phase of the modeling process involved reformulating the initial model, aided by a theoretical model of the construction of units and number (von Glasersfeld, 1981). The reformulation of the initial model into a developmental model of counting types was also aided by analyses of videotapes of children solving arithmetical tasks. Our constant return to children’s behavior via the videotapes both stimulated and modified our thinking. But it was the theoretical work that predominated during this phase.
Even though the developmental model of counting types had an experiential basis, it did not indicate how we might help children make progress. A new teaching experiment was called for to fill out and, if necessary, refine the skeleton of some possible courses of development specified by the developmental model. When we formulated the model, we were primarily concerned with the form and structure of the counting scheme at various points in its development. The teaching experiment allowed us to conduct an analysis of the steps the children took when making progress in the construction of more sophisticated unit items. A cursory experiential analysis of children’s progress is exemplified by the above discussion of Jason’s progress in counting. Analyses of teaching episodes made possible the reconstitution of what had before been couched in terms of theoretical constructs only. The human activities that might constitute “doing mathematics,” as referred to by Plunkett (1982, p. 46), were used to specify the construction of the counting scheme. The theoretical aspect of our methodology is apparent in the final phase of the modeling process in that the theoretical model served as a guiding framework. However, the primary emphasis was, once again, empirical.
In summary, we attempt to account for the observed regularities in children’s progress by developing abstract, theoretical constructs. This constitutes our quest for generality. We strive for specificity by filling out these constructs with experiential content. At any point in the modeling process, novel, unexpected observations can lead to a reformulation of the theoretical constructs. Conversely, a theoretical reformation can lead to the novel interpretation of previous observations. The interdependence of theory and observation is consistent with Lakatos’ (1970) analysis of a scientific research program. We firmly believe that this feature of our method is essential and contributes enormously to the understanding of how children might construct their mathematical realities. However, we should not forget that a model is no more than a plausible explanation of children’s constructive activities. One can never claim a correspondence between the model and children’s inaccessible mathematical realities. Although a model can be viable, it can never be verified.
The educational significance of models
Hawkins (1973a) offers the following characterization of the activity of teaching:
The teacher begins to assemble…information over a variety of children although for thirty children the task is enormous, and even the best teachers will confess to omissions. Then there is a trial-and-error of communication, further observation, a gradual and still tentative sort of portraiture involving the child’s style, strengths, weakness, skills, fears, and the like…. What [the teacher] finds himself doing is beginning to build what I would call a map of each child’s mind and of the trajectory of his life. It is fragmentary, fallible, but it is subject always to corrections. (p. 13)
Hawkins (1973b) emphasizes the commonality of research and teaching:
The really interesting problems of education are hard to study. They are long-term and too complex for the laboratory, and too diverse and nonlinear for the comparative method. They require longitudinal study of individuals…. The investigator who can do that and will do it is, after all, rather like what I have called a teacher. (p. 135)
The researcher who conducts a teaching experiment attempts to perform the same activities as Hawkins’ teacher. The single difference between the researcher and the teacher is that the researcher interacts with fewer children and has greater opportunity and more time to make sense of their behavior.
Essentially, our models are the results of attempts to explicate our understanding of children’s constructions. These models, which were developed by intensively analyzing children’s behavior, capture our knowledge of recurrent features of the activity of doing mathematics. They also embody our suggestions on how to aid children as they attempt to construct mathematical knowledge. The counting-types model, for example, constitutes the organization that we use when interpreting children’s behavior and when planning interactions with children on the basis of those interpretations.
We believe that nothing could be more useful to teachers than the type of knowledge represented by a model. However, just as children construct mathematical knowledge, so teachers construct their own understanding of children’s mathematical realities. We can no more give teachers our counting-type model than we can give children our knowledge that subtraction is the inverse of addition. The question of how to help teachers as they strive to understand children’s mathematical realities is of critical importance.
Portions of this article are based on an interdisciplinary research program supported by the National Science Foundation under grants Nos. SED78-17365 and SED80- 16562. Any opinions, findings, and conclusions or recommendations expressed in the article are those of the authors and do not necessarily reflect the views of the National Science Foundation. Our thanks to Ernst von Glasersfeld, William E. Doll, Sigrid Wagner, and two anonymous reviewers for helpful comments on earlier. drafts.
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