Toward a Working Model of Constructivist Teaching: A Reaction to Simon

Toward a Working Model of Constructivist Teaching: A Reaction to SimonAuthor(s): Leslie P. Steffe and Beatriz S. D'AmbrosioSource: Journal for Research in Mathematics Education, Vol. 26, No. 2 (Mar., 1995), pp. 146-159Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749206 .

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Journal for Research in Mathematics Education 1995, Vol. 26, No. 2, 146-159

TOWARD A WORKING MODEL OF CONSTRUC- TIVIST TEACHING: A REACTION TO SIMON

LESLIE P. STEFFE, University of Georgia BEATRIZ S. D'AMBROSIO, Indiana University

We start this discussion of the article by Martin A. Simon with two major conjectures that we will try to substantiate. The first is that there is a kind of teaching that can legitimately be called "constructivist teaching." The second is that Simon's model of teaching his prospective elementary school teachers, if modified, would fit our understanding of constructivist teaching, even though his basic premise is that constructivism does not tell us how to teach mathematics (Simon, 1995).

The issue concerning whether constructivism tells us how to teach mathematics resides in how constructivism is understood. One way to understand constructivism is in terms of basic tenets like "knowledge is not passively received but is actively built up by the cognizing subject," and "the function of cognition is adaptive and serves in the organization of the experiential world rather than in the discovery of ontological reality" (von Glasersfeld, 1989, p. 162). These basic tenets are orienting, but as indicated by Simon, they do not stipulate a particular model of teaching mathematics. Neither do they tell us how to do family therapy or how to provide psy- chiatric counseling. People who engage in these types of human activities, however, can use the basic tenets of constructivism in building models of the realities of those with whom they interact.

If a teacher formulates a model of children's mathematical knowledge, including its construction, we claim that the model is an important part of the teacher's understanding of constructivism. Similarly, if the teacher formulates a model of how she makes sense of children's mathematical knowledge, including its construction, this would be a constructivist model of teaching. It, too, would be an important part of the teacher' s meaning for constructivism. Regarding the teacher as a learner in the activity of teaching is essential in our understanding of constructivist teaching, and it is a basic aspect of Simon's model of teaching as well.

THE ROLE OF MODELS IN A RESEARCH PROGRAM

To more deeply appreciate our claim that there is a kind of teaching that we call "constructivist teaching," it is helpful to consider how Lakatos (1970) separated the hard core of a research program from models in its protective belt. Models of mathematics teaching lie in what he called the protective belt of the hard core.

All scientific research programmes may be characterized by their 'hard core'.... We must use our ingenuity to articulate or even invent 'auxiliary hypotheses', which form a protective belt around this core.... It is this protective belt of auxiliary hypotheses which



Leslie P. Steffe and Beatriz D 'Ambrosio 147

has to bear the brunt of tests and get adjusted and re-adjusted, or even completely replaced, to defend the thus-hardened core. (Lakatos, 1970, p. 133)'

Lakatos's concepts of the hard core and protective belt of a research program are useful to us in making a distinction between the basic tenets of constructivism and the educational models that we construct using these basic tenets. In our constructive activities, however, we do not deductively apply constructivism to mathematics teaching in a way similar to Thorndike's (1924) application of his principles of learning to mathematics education. Rather, the basic tenets enter into human activity through the principle of self-reflexivity, which means that we apply the basic tenets first and foremost to ourselves in our activities (Steier, 1995).

Constructivist research concerns understanding living systems as they involve the actions and operations of the researcher. So, doing research on mathematics teaching in a constructivist research program entails the researcher establishing living models of mathematics teaching using the basic tenets of constructivism. These models are not simply examples of constructivism in action. Rather, they are an important source of the meaning of constructivism for the researcher and can deepen and transform the basic tenets.2 As indicated by Lakatos, the living models serve in bearing the brunt of "tests" and get adjusted and readjusted, or even completely replaced, to defend the thus-hardened core of constructivism. In this sense, we see Simon's research addressing issues regarding constructivist teaching. His and other research studies that focus on the analysis of constructivist teaching episodes lie in the protective belt of constructivism, where the "protective belt is constantly modified, increased, com- plicated, while the hard core remains intact" (Lakatos, 1978, p. 179).3

Two Working Models4 of Teaching in a Constructivist Research Program

It should be apparent that we expect different mathematics educators working in a constructivist epistemology to generate different specimens of teaching. Rather than a weakness, we consider this possible diversity to be an essential source of novelty in mathematics education and thus of progress in the field. We believe that there should be even more discussion and debate among constructivist researchers working on models of teaching than there is among researchers working in different research programs. This realization may come as a shock for some, because constructivist

'See Piattelli-Palmarini (1980) for further discussion of the hard core of a research program. 2We contrast living models with abstracted explanations of aspects of the living models arrived at through

conceptual analysis. It is these explanations that can deepen and transform the basic tenets of constructivism. Examples can be found in accounts of children's construction of counting schemes in Steffe and Cobb (1988).

3We do not interpret "remains intact" to mean "fixed and unchanging." The basic tenets of constructivism can be modified in their use as can any other conceptual model.

4A working model of mathematics teaching refers to a description of a teacher's general ways and means of operating when teaching. It is what an observer (which may be the teacher observing him- self) might say about living models of mathematics teaching. As descriptions, working models are not scientific models.



148 Constructivist Teaching

researchers typically have engaged in debate with nonconstructivist researchers rather than with each other (e.g., Confrey, 1986, and Brophy, 1986). As constructivism has become established as an epistemology in mathematics education, this situation is rapidly changing (e.g., Steffe & Gale, 1995). A goal now is to formulate models of teaching and to resolve possible inconsistencies among these models (Lakatos, 1970).

In another paper (D'Ambrosio & Steffe, in press), we called teachers who study the mathematical constructions of students and who interact with students in a learning space whose design is based, at least in part, on a working knowledge of students' mathematics "constructivist teachers," and the activity in which they engage "constructivist teaching." We regard Simon's comment, that "the principal currencies of the mathematics teacher (if lecturing is rejected as an effective means of promoting concept development) are the posing of problems or tasks and the encouragement of reflection" (p. 141), as being his working model of teaching in a constructivist framework. These two working models (Simon's and ours) of mathematics teaching stress different aspects, and it is here that we wish to focus.

STUDENTS' MATHEMATICAL KNOWLEDGE

Students' Knowledge as Perturbations5 for the Teacher

In both working models, it is constraints such as those Simon experienced as he inter- acted with his students that force teachers to make a distinction between their knowledge and the knowledge of their students. A point of contact with his students' knowledge of area occurred when Simon posed the question concerning measuring the two tables in the hall using the method of turning the rectangle to measure "down the other way" (p. 129). He reports that at least some students remained unshaken in their resolve that the 32 and 22 (the measurements of the two tables) did not count for anything that was meaningful and that this method generated overlapping rectangles.

If the students could solve any area task that Simon posed, there would be no observ- able basis for Simon to make a distinction between his knowledge of area and his students' knowledge of area. Simon asks,

My attempts at creating disequilibrium with my current students, a key part of my the- ory and practice, had been ineffectual. How could I understand the thinking of these students, and how could I work with them so that they might develop more powerful understandings? (pp. 129-130)

Attempts to create disequilibrium of the nature Simon envisioned can be very difficult. Vinner (1990) expresses this difficulty in the following passage:

If at first it appears that inconsistencies can be very helpful in the learning of mathematics, ... this is not necessarily the case. It is true that a student will try to

"5We contrast the concept of perturbation with cognitive conflict. "Perturbation" refers to any distur- bance in components of an interacting system created through the functioning of the system. They include, but are not exhausted by, cognitive conflicts that may be experienced by the interacting system.




accommodate a recognized contradiction. But this will happen only if the student is convinced there is a contradiction. Secondly, even if the student recognizes a contradiction and tries to accommodate, there is no guarantee that the accommodation will be in the desired direction. (pp. 91-92)

Creating provocations that might lead students to make accommodations in their knowledge involves a deep appreciation by teachers of their students' mathematical knowledge. It entails an ability to create situations of learning through which the teacher might bring forth the students' mathematical knowledge as well as an ability to engage in the interactive communication that leads students to modify their knowledge. Simon provides a plausible explanation of his students' knowledge of area in terms of iterable composite units (Simon, p. 130), and he attempts to pose tasks that would engender accommodations in this concept of area. In this, his teaching acts are compatible with our understanding of constructivist teaching. In our model, however, we focus on activating schemes that we believe will serve the students in assimilating the situations of learning they encounter.

Activating Prior Knowledge in Situations of Learning

To be more precise, we are interested in the schemes students use in assimilating the teacher's situations. Our interest stems from our view of learning as accommodations the students make in their functioning schemes. A possible reason why the students felt that 32 and 22 did not count for anything meaningful in turned rectangle was that the situation contained an element (vertical rectangles along the left edge) that may have blocked activation of the students' iterative scheme for finding areas. If this is the case, we would say that the students had a problem- where, as Polya (1962) suggests, to have a problem means "to search consciously for some action appropriate to attain a clearly conceived but not immediately attainable aim" (p. 117). Engaging in a successful search would definitely involve constructive activity of a rather major kind. However, it is possible to identify modifications that students make in their functioning schemes that do not fit Polya's idea of having a problem (Steffe, 1991). There is a candidate in Simon's paper for this kind of learning.

Simon reports that a student named Molly gave a convincing explanation of why multiplying the number of rectangles along the length and width of the table would always work in (Problem 1, p. 124). To make such an explanation, we believe it would be necessary for her to have already constructed the operations involved in producing a unit of units of units and to have used these operations in the assimilation of Problem 1. We are convinced that she understood Problem 1 multiplicatively. She also understood Problem 2 multiplicatively, as indicated by her response to Simon's question concerning Problem 2: "Why does that work, that when we multiply the number of columns times the number of rows we get the area?"

Well, I thought again it referred back to when you're using a row to represent the units in a group, and the columns to represent the number of groups, and since multiplication is the same as repeated addition, that when you multiplied the number of units in a group by the number of groups, you would get the total number of parts in the whole. (p. 126)




As we try to imagine an event of learning that led to Molly's comment, we look to the comments she made concerning counting the corner rectangle twice:

Because it ... the comer not only represents a one, it's just one numbering of a group, or it's also numbering a part of that unit-a unit in that group-so it's not, it's two different things, just like when they were saying it's a row and a column, well, it's two different things, it's a unit and also representing a group. (Simon, p. 126)

From this, it is not too hard to imagine a perturbation that Molly may have experienced when confronted with the possibility that Bill counted the corner rectangle twice. She did not simply disregard the possibility. Rather, if she experienced a perturbation, she neutralized it by making a distinction between the comer rectangle representing a group and a unit in the group. In making this distinction, we would not say that Molly solved a problem in the sense explained by Polya. Nevertheless, we consider Molly's distinction to be an event of learning whenever it is novel. In this case, we call it a generalizing assimilation, the constructivist way of speaking of transfer of learning (Steffe & Wiegel, 1994). The assimilation is called generalizing because of the change in her concept of multiplication that enabled her to neutralize her perturbation.

We definitely agree with Simon that posing problems or tasks is a principal currency of a mathematics teacher. In this, however, we would like to replace "problems or tasks" by "situations," with the understanding that this phrase includes situations that the students regard as genuine problems. We also want it to include situations that lead to generalizing assimilation and functional accommodations (Steffe, 1991).

We do not know if a learning event such as we imagine transpired in Molly's case, and it isn't our intention to argue that it did. Rather, we use our example to argue a different point. We know that by the age of 11 years, many children have constructed the operations necessary to produce a unit of units of units (Steffe, 1992). So, it seems plausible to us that most of the preservice teacher education students that Simon taught would have constructed these operations. If so, it also seems plausible that activating the students' multiplying schemes in situations involving col- lections of discrete items before asking them to solve Problem 1 would maximize the possibility of their using these operations in assimilation, as Molly did. A possible situation would be to ask the students to find the number of items in an array, say, where the 4th row and the 7th column are visible as in Figure 1.

It is not necessarily true that Simon's students had difficulty because their understanding of multiplication was not powerful. Rather, it may be that they did not spontaneously activate the necessary multiplying schemes needed to interpret Simon's situation as Molly did.

The Hypothetical Learning Trajectory

To contrast our two models of teaching even further, we turn to Simon's hypothetical learning trajectory (HLT). We find it to be quite appealing, especially when considered in conjunction with the four themes concerning teacher decision




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making. How the HLT might differ from our working model of constructivist teaching is an interesting question and one that we do not attempt to answer definitively. We do suggest that there may be a difference in emphasis in the teacher's understanding of the mathematics of students in the two models of constructivist teaching.

Like Vygotsky (1987), Simon takes the current knowledge of students seriously, and it is given a central place in the design of instruction. In Vygotsky's system, the student's actual level of development is defined by what the student can do without the aid of the teacher. The zone of proximal development is what the student can do with the help of the teacher. As we read Simon's account of his thought processes concerning his HLT, we see at least a partial fit with how Vygotsky might have thought about a similar type of teaching problem. Starting from what he regarded as the current level of development of his students' concept of area, Simon carefully artic- ulated his design of a set of activities with his students' thinking. In doing so, he defined a zone of proximal development for his students.

Of course, we don't mean to imply that Simon is a neo-Vygotskian. Given his social constructivist approach, however, we do find similarities between his and a Vygotskian approach. A fundamental difference in the two approaches resides in Simon's view of the teacher as a learner: "The continually changing knowledge of the teacher (see #2) creates continual change in the teacher's hypothetical learning trajectory" (p. 141).

In this, we see Simon's model of teaching as being compatible with our model because it includes the basic tenet of constructivism that knowledge is not passively received but is actively built up by the cognizing subject, who in this case is the teacher. Still, there is a difference in emphasis that we have tried to make explicit in the fore- going discussion on activating prior knowledge.

An implicit concept of multiplication. To illustrate the difference in emphasis we turn to an example of a concept of multiplication that has been called implicit (Steffe,




1992). This concept was learned when working with an 8-year-old child, Maya. After Maya said that six blue rectangles would go on a larger red rectangle (they fit exactly) and that two orange triangles would go on one of the blue rectangles (which also fit exactly), she was asked how many of the orange triangles would go on the red rectangle (Steffe, 1992, p. 279). After looking straight ahead, she subvocally uttered number words and said "12." In explanation, she tapped the table twice with each of six fingers synchronously with uttering number words "1, 2; 3, 4; 5, 6; 7, 8; 9, 10; 11, 12."

This description of Maya's language and actions illustrates one thing that we mean by "studying the mathematical constructions of students" in our working model of constructivist teaching. In our view, a teacher regards the students' mathematical language and action to constitute a living mathematics and interacts with students in a learning space whose design is in part based on that language and action. We believe that students have a mathematical reality of their own independent of us as teachers. We do not deny students their own mathematical reality. In fact, we find that we are constrained by the students' mathematical language and action and cannot do just anything we want to do when teaching. More important, the students' mathematics as we experience it when teaching provides occasions for our teaching actions.

What the teacher makes of students' mathematical language and actions is a function of his or her own interpretive constructs. For example, we believe that Maya's language and actions should not be interpreted as an example of the cross product of two sets. Although we can see echos of the cross product, Maya's concept of mul-

tiplication was based on conceptual material different from that of our concept of the cross product and at a lower level of abstraction. The issue is to come up with an operative multiplying scheme that might underpin her mathematical activity.

Based on the way Maya counted in patterns of two, it is plausible that she established "six twos" prior to counting. So, an operation had to be constructed that would account for how she established six twos. In this, it was essential to attribute to Maya the ability to distribute a unit of two across the units of her concept of six. But there seems to be a more fundamental operation that produces the operation of distribution, because other children being taught who were of the same age as Maya could not make the distribution. The critical difference between Maya and these other children was that Maya could use her number sequence as its own material of operating. That is, she could start at, say, "23" and count seven more times, constitut-

ing each number word after "23" as a countable unit item and also as a counted unit item ("24 is 1"). This ability to use her number sequence as its own material of operating created the possibility of the novelty of inserting a unit of two into each of the unit items of the composite unit, six.6 So, in our way of thinking, Maya constructed the distribution operation as an accommodation of her number sequence.

"6Because she could use her number sequence as its own material for operating, Maya could re-present the items of the composite unit, six, and then count these units "1, 2, 3, 4, 5, 6." Although there was no indication that she actually counted in this way, it is sufficient for her to be able to do so to distribute a unit of two across the individual units of six.




Maya's distribution operation was called implicit to emphasize the belief that she was not aware of how "six twos" appeared to her. This is one reason why we believe that her distribution operation was constituted at a lower level of abstraction than the operation of cross product would be. Another reason is that the material of her distribution operation consisted of the figurative material pointed to by the unit items of her number concepts. In any event, the explanation of Maya's multiplying scheme was formulated in a language that would not be used to explain our concepts of multiplication of whole numbers.

This explanation of Maya's multiplying scheme was not available when Maya was being taught multiplication. All that was available was the observation of Maya's method of multiplying. One of the basic reasons we do teaching experiments is to formulate models of the students' mathematics. During a teaching experiment, we usually proceed on the basis of our mathematical knowledge and observations of the students' ways and means of operating. Everything we do is experimental and is done with the intention of constructing an explanatory model of the constructive activity of the children and our own ways and means of operating as teachers.

If Simon's HLT is any indication, he seems to share in our goals. However, we believe that Simon's emphasis on the social processes involved in teaching mathematics makes it quite difficult to focus on the mathematics of his students. The distinction we are trying to make is clarified by Maturana's (1978) idea of nonintersecting phenomenal domains of interaction.

A scientist must distinguish two phenomenal domains when observing a composite unity (a) the phenomenal domain proper to the components of the unity, which is the domain in which all the interactions of the components take place; and (b) the phenomenal domain proper to the unity, which is the domain specified by the interactions of the composite unity as a simple unity. (p. 37)

If we regard the composite unity under consideration to be the mathematical concepts and schemes of individual students, then the first phenomenal domain would be a domain of psychological phenomena and the second a domain of sociological phenomena.

When engaged in the activity of teaching, a teacher must unavoidably focus on sociological phenomena and is necessarily restricted to making descriptions of the language and actions of students. Explanations (components of the composite uni- ties in Maturana's system) arise through retrospective analyses of the mathematical interactions that transpire in the classroom discourse. Given the complexity and ambiguity of classroom discourse, making explanations of the sort that we offered for Maya's language and actions often remains beyond the reach of the teacher/researcher. So, it is to Simon's credit (cf. Simon & Blume, 1994) that he has attempted to explain his students' mathematical knowledge.

We find Simon's explanations, however, to be couched in terms of the mathematical concepts and operations of the teacher. We do not consider this to be wrong, because it is exactly what we do when we engage in teaching experiments. But we wonder if Simon intends to go further and explain the mathematics of his students and mathematical learning in terms of the students' schemes of action and operation.




Doing this would involve conducting a teaching experiment of a different kind; one in which the focus is on the interactions that might occur in the first of Maturana' s two phenomenal domains of interaction.

Zone of potential construction. In our working model, a hypothetical learning trajectory developed by a teacher in planning for teaching would include several aspects of the learners. First, the teacher's descriptions (i.e., a working model of the mathematics of the students) of the students' schemes of action and operation would be at the forefront. In this, the mathematical schemes of the students that might be relevant in their assimilation of the planned situations of learning would be specified in the form of situations that might activate those schemes. Second, the actions of students elicited by the situations posed should be in the foreground of the HLT, including possible modifications of those actions. Student actions shape the inferences drawn about student knowledge, and those inferences should shape the teacher's plan of action.

The diagram in Figure 2 would fit somewhere inside Simon's HLT, and we feel it represents a dimension of teacher knowledge that is essentially implicit in Simon's HLT.

Mathematics of students

Zone of potential construction Situations

Student actions and modifications

of actions

Figure 2. A piece of the hypothetical learning trajectory.

In our model we use the phrase "zone of potential construction" to refer to a teacher's working hypotheses of what the student can learn, given her model of the student's mathematics. The zone of potential construction is determined by the teacher as she interprets the schemes and operations available to the student and anticipates the student's actions when solving different tasks in the context of interactive mathematical communication. The anticipation is based on the teacher's knowledge of other students' ways of operating, on the teacher's knowledge of the particular mathematics of that student, and on the results of the teacher's interactions with the student. The




zone of potential construction is based primarily on the student's mathematical knowledge rather than on the mathematical knowledge of the teacher that, for the moment at least, she would not attribute to the student.

We agree with Simon that the role of anticipation in the teacher's understanding of the mathematics of the learner is an important piece of the HLT. We point out that this anticipation, when contrasted with the students' unexpected actions, can

generate perturbations for the teacher. The nature of the strategies taken by the teacher towards neutralizing these perturbations characterizes constructivist teaching. How does the teacher try to activate schemes she conjectures that the students have? How does the teacher interpret students' actions? How does the teacher modify a task that fails to activate certain schemes? These are but a few questions that point to an important aspect of constructivist teaching-the ability of the teachers to test their hypotheses about the nature of the mathematical knowledge of their students and the ability of the teachers to neutralize the perturbations that emerge in testing their hypotheses. Perturbations are neutralized as teachers refine and modify their

hypotheses while intrepreting the mathematics of their students.

REFLECTION IN TEACHING MATHEMATICS

We now turn to the second of Simon's principle currencies of the mathematics teacher-the encouragement of reflection. Von Glasersfeld (1991) has explained reflection as that "capability that allows us to step out of the stream of direct experience, to re-present a chunk of it, and to look at it as though it were direct experience, while remaining aware of the fact that it is not" (p. 47).

Reflection as explained by von Glasersfeld is essential in the construction of mathematical concepts of all kinds. However, the conscious awareness of the reflecting agent enters into this explanation of reflection, which opens up the question of the role of reflection in the construction of mathematics. As we consider Maya's implicit concept of multiplication, for example, we see that the individual who has constructed a novel operation might remain unaware of the operation. This is com-

patible with the fact that "conscious conceptualized knowledge of a given situation

developmentally lags behind the knowledge of how to act in the situation" (von Glasersfeld, 1991). In this, von Glasersfeld was discussing Piaget's (1974) findings that action

by itself constitutes know-how but not knowledge in the sense of conceptualized under-

standing. "Knowing in action" is fundamental in the construction of mathematical

knowledge and constitutes a basic source of conceptualized knowledge. So it is not always appropriate to encourage reflection in the sense explained by

von Glasersfeld. Seemingly, reflection is appropriately encouraged after one or more individuals has made what Piaget called a reflective abstraction. For us, this raises the issue of the nature of mathematical activity in the classroom. For example, although social interaction is useful throughout the constructive process, we interpret many of Simon's protocols in terms of what Piaget called "reflective thought," whose function is to bring the products of reflective abstraction into conscious awareness and to thematize those products. Molly's comment that we have cited concerning the




relationship of multiplication to the rows and columns is a particularly good example of reflective thought. To us, she was aware of how she structured the multiplicative situation ("you're using a row to represent the units in a group, and the columns to represent the number of groups"), as well as the results of multiplying ("when you multiplied the number of units in a group by the number of groups you would get the total number of parts in the whole"). Both of these comments solidly indicate reflective thought (and reflection).

Molly's explanations were offered in response to questions asked by Simon, and in this, we believe that Simon did in fact encourage reflection. This is especially apparent to us in the exchanges among the students that Simon reported. We find many instances of what Wertsch and Toma (1995) refer to as the "dialogic function of text." Rather than speaking to convey information, we see instances where the students were using "the claims or explanations of another speaker as a kind of 'thinking device' for further consideration, criticism, and so forth" (p. 169). Regarding the thoughts of another for further consideration, especially as something to understand, critique, or compare and contrast with one's own thoughts, provides a perspective on the role of social interaction in the mathematics classroom that is quite valuable. Of course, there are other functions of social interaction to which Simon points in his essay that are of equal value, but using social interaction to encourage reflection is especially relevant, and we modify our working model of teaching accordingly. As before, we call a teacher who studies the mathematical constructions of students and who interacts with students in a learning space whose design is based, at least in part, on a working knowledge of students' mathematics "a constructivist teacher." In our statement, we emphasize that the design of the learning space includes three principal currencies of a mathematics teacher: the posing of situations, the encouragement of reflection, and interactive mathematical communication.

In our previous statement, the first two of our three principal currencies of a mathematics teacher were left implicit in the design of a learning space. One reason for stressing interactive mathematical communication as a principal currency of a mathematics teacher is that social interaction underlies all teaching actions. Moreover, nonverbal interactive mathematical communication should be brought into the foreground, because knowing in action enters in multifarious forms throughout mathematical activity of all kinds. Both verbal and nonverbal interactive mathematical communication offer the teacher the opportunity to enact their mathematical knowledge in appropriate and timely ways.

DISCUSSION

One model of teaching supersedes another if it serves the same purposes better, where "better" must be judged on the basis of criteria such as the relative explanatory power of the models, the relative power of resolution the models allow, or the relative usefulness of the models. The superseding model solves not only the same problems as the preceding model, but also problems the preceding one did not solve. Finally, the superseding model is a reorganization of the previous model in



Leslie P. Steffe and Beatriz D'Ambrosio 157

that it contains the elements of the preceding model but in a new organization with other elements. In other words, it is the product of an accommodation.

We view our modified working model of constructivist teaching as superseding our previous working model. How Simon might regard it relative to his working model remains an open question. In our case, the emphasis on reflection in the context of social interaction constitutes a change because it introduces new problems in the mathematics of students, especially at the elementary school level. Before the mathematics of students can be regarded as constituting conventional knowl-

edge among students, the students would need to be explicitly aware of at least some of their mathematical concepts and operations and have a sense of the logical neces- sity of these concepts and operations. This is a tall order, but it is needed if there is to be a mathematical culture of the classroom produced through interactive mathematical communication, as advocated by Bauersfeld (1995). This is a fascinating question, because it implies that the students would have a sense of autonomy and power of ownership of the mathematical concepts of their community and would engage in independent mathematical activity in the sense of Bauersfeld' s view of mathematizing as the interactive constitution of a social practice. The role of the mathematics teacher is at issue here, but we feel that the posing of situations and the encouragement of reflection would remain principal currencies of the teacher. However, the purposes and the means of posing situations and encouraging reflection would be modified as we understand it in Simon's model.

We advocate that situations be posed by teachers to bring forth, sustain and encourage, and modify the mathematics of students. As such, our situations include what we regard as potential problems for the students we teach, but they are not restricted to that category. Given our emphasis on the independent but interactive mathematical activity of our students, we pose situations that help students sustain their mathematical activity and develop confidence in their ability to do mathematics without the help of a teacher or more able peers. We are not advocating long lists of mindless exer- cises. Rather, we are advocating situations that involve assimilating generalization as we have exemplified it above. The situations should be interesting and challenging for the students, but not so far beyond their current concepts and schemes as to require a major reorganization for solution. In fact, this category of situations would be in the students' zone of actual construction-their comfort zone. A goal in this is for the students to produce their own situations that reflect their zones of actual construction. The primary purpose of the students' production of situations is for the students' to make their mathematical knowledge explicit. The particular situations produced serve as symbols of the students' mathematical knowledge if the situations are instances of categories of situations the students could produce. A second purpose is for the students to come to understand mathematics as something that belongs to them rather than something that belongs to the teacher or that appears only in workbooks.

Including student-generated situations and students' independent but interactive mathematical activity in the design of a learning space does not preclude the teacher's responsibility to pose situations. We understand the student-generated situations




as co-emerging in interactive communication with the teacher. The situations serve as instruments of communication and as means for the teacher to support and sustain the students' mathematical activity. However, learning how to act in such a way that the students' mathematical activity is the focus of attention in the classroom is a basic goal of a constructivist teacher.

We expect many functional modifications to occur incidentally or spontaneously as students engage in independent but interactive mathematical activity. Nevertheless, it is the responsibility of the teacher to intentionally pose situations of learning that serve in defining the students' zones of potential construction. These situations are of two broad types. First there are those situations the teacher chooses to be within the assimilatory power of the students, but which contain elements that would block a solution of the situation unless the students modified their mathematical activity using operations "within" the activity. An example of this kind of situation would be Simon's Problem 2 if the students' multiplying scheme for discrete situations was activated. These are situations the teacher believes the students can solve within the framework of the ongoing mathematical communication of the classroom. The second type of situations are those situations that are also chosen to be within the assimilatory power of the students, but which contain elements that would block a solution of the situation unless the students modified the mathematical activity using operations "outside" of the activity. These kinds of situations constitute genuine problems for the students, and we use them to encourage the students to reor- ganize their activity.

We have tried to say enough to substantiate our claim that there is a kind of teaching called "constructivist teaching." This way of teaching manifests the basic tenets of constructivism in living forms and constitutes meaning of the tenets in the context of teaching mathematics. The basic tenets do not and cannot tell a teacher what their meaning might be-they do not define the content of the mathematics of students, nor do they prescribe how to bring it forth, to sustain and encourage it, or to modify it. Insofar as these things constitute meanings of the basic tenets of constructivism, the issue of whether there is a form of teaching called "constructivist teaching" turns on how one chooses to understand constructivism.

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AUTHORS

LESLIE P. STEFFE, Research Professor, University of Georgia, 105 Aderhold Hall, Athens, GA 30602 BEATRIZ S. D'AMBROSIO, Associate Professor, Indiana University at Indianapolis, 902 West New

York Street, Indianapolis, IN 46202



Documents

Toward a Working Model of Constructivist Teaching: A Reaction to Simon