Confrey J. (1990) What constructivism implies for teaching. In: Davis R. B., Maher C. A. & Noddings N. (eds.) Constructivist views on the teaching and learning of mathematics. National Council of Teachers of Mathematics, Reston VA: 107–124. Available at http://cepa.info/3879

Table of Contents

An Analysis of Direct Instruction

Constructivism

Some Implications for Mathematics Instruction

An Alternative Set of Assumptions

The Context

The Method

The Results

1. Promotion of Autonomy and Commitment

2. Development of Students’ Reflective Processes

3. Construction of a Case History

4. Identification and Negotiation of a Tentative Solution Path

5. Retracing and Reviewing the Solution Path

6. Adherence to the Intent of the Materials

Conclusions

References

In this chapter, a critique of direct instruction is followed by a theoretical discussion of constructivism, and by a consideration of what constructivism means to a classroom teacher. A model of instruction is proposed with six components: the promotion of student autonomy, the development of reflective processes, the construction of case histories, the identification and negotiation of tentative solution paths, the retracing and group discussion of the paths, and the adherence to the intent of the materials. Examples of each component are provided.

An Analysis of Direct Instruction

The form of instruction in mathematics that has been most thoroughly examined has been “direct instruction” (Good & Grouws, 1978; Peterson, Swing, Stark & Waas, 1984; Rosenshine, 1976). With this form of instruction, one finds a relatively familiar sequence of events: an introductory review, a development portion, a controlled transition to seatwork and a period of individual seatwork. I suggest that three key assumptions about mathematics instruction underlie direct instruction and are subject to challenge from a constructivist perspective:

Relatively short products are expected from students, rather than process-oriented answers to questions; homework assignments and test items are accepted as providing adequate assessment of the success of instruction.Teachers, for the most part, can simply execute their plans and routines, checking frequently to see if the students’ responses are within desirable bounds, and only revising instruction when those bounds are exceeded (Peterson & Clark, 1978; Snow, 1972).The responsibility for determining if an adequate level of understanding has been reached lies primarily with the teacher.

There has recently appeared an increasing amounts of evidence that direct instruction may not provide an adequate base for students’ development and for student use of higher cognitive skills. Doyle, Sanford and Emmer (1983) examined students’ views on the “academic work” in traditional classrooms and found that, as students convince teachers to be more direct and to lower the ambiguity and risk in classroom tasks, the teachers may inadvertently mediate against the development of higher cognitive skills. Other challenges to direct instruction come from the research on misconceptions (Confrey, 1987) wherein researchers have documented severe student misconceptions across topics and achievement levels. These misconceptions appear to be resistant to traditional forms of instruction (Clement, 1982; Erlwanger, 1975; Vinner, 1983). These studies point to a need to develop alternative forms of instruction.

Efforts to develop forms of instruction that overcome misconceptions have focused on the need to have students make their conceptual models explicit (Baird & White, 1984; Lochhead, 1983; Novak & Gowin, 1984; Nussbaum, 1982). Instructional models to encourage problem solving in the classroom have emphasized the need to help teachers take risks and to develop flexibility in the subject matter (Stephens & Romberg, 1985). All of this research shares a commitment to the importance of an active view of the learner. The philosophical approach that argues most vigorously for an active view of the learner is constructivism.

Constructivism

A theory that seems to be a powerful source for an alternative to direct instruction is that of constructivism (Confrey, 1983, 1985; Kelly, 1955; Glasersfeld, 1974, 1983, 1990). Put into simple terms, constructivism can be described as essentially a theory about the limits of human knowledge, a belief that all knowledge is necessarily a product of our own cognitive acts. We can have no direct or unmediated knowledge of any external or objective reality. We construct our understanding through our experiences, and the character of our experience is influenced profoundly by our cognitive lenses. To a constructivist, this circularity is both acceptable and unavoidable. One’s picture of the world is not, however, static; our conceptions can and do change. The essential fact that we are engaged in living implies that things change. By coordinating a variety of constructions from sensory inputs to meditative reflections, we adapt and adjust to these changes and we initiate others.

A consequence of a constructivist’s denial of direct and assured access to “the way things really are” is that authority resides in the persuasiveness of one’s argument and in how well one marshals evidence in support of a position. Constructivists recognize that these forms of argument also exist within the culture of discourse. Although constructivism is often equated with skepticism, the skill the constructivist must truly develop is flexibility. While it is accurate to say that the constructivist rejects any claim which entails the correspondence of an idea with an objective reality, the most basic skill a constructivist educator must learn is to approach a foreign or unexpected response with a genuine interest in learning its character, its origins, its story and its implications. Decentering, the ability to see a situation as perceived by another human being, is attempted with the assumption that the constructions of others, especially those held most firmly, have integrity and sensibility within another’s framework. The implications for work with students are stunning.

Piaget provided an essential key to a constructivist perspective on teaching in his work, wherein he demonstrated that a child may see a mathematical or scientific idea in quite a different way than it is viewed by an adult who is expert or experienced in working with the idea. These differences are not simply reducible to missing pieces or absent techniques or methods; children’s ideas also possess a different form of argument, are built from different materials, and are based on different experiences. Their ideas can be qualitatively different, which can sometimes mean that they make sense only within the limited framework experienced by the child and can sometimes mean they are genuinely alternative. To the child, they may be wonderfully viable and pleasing. They will not be displaced by any simple provision of the “correct method,” for, by their existence for the child, they must have served some purpose. Before children will change such beliefs, they must be persuaded that the ideas are no longer effective or that another alternative is preferable.

When one applies constructivism to the issue of teaching, one must reject the assumption that one can simply pass on information to a set of learners and expect that understanding will result. Communication is a far more complex process than this. When teaching concepts, as a form of communication, the teacher must form an adequate model of the students’ ways of viewing an idea and s/he then must assist the student in restructuring those views to be more adequate from the students’ and from the teacher’s perspective.

Constructivism not only emphasizes the essential role of the constructive process, it also allows one to emphasize that we are at least partially able to be aware of those constructions and then to modify them through our conscious reflection on that constructive process. As Toulmin (1972) wrote “(Wo)Man knows and (s)he is conscious that (s)he knows” (p. 7). (Parentheses added.) Thus, not only can we assert that a constructive process is involved in all acts of perception and cognition, but also that we can gain a measure of access to that constructive process through reflection.

In mathematics the reflective process, wherein a construct becomes the object of scrutiny itself, is essential. This is not because, as so many people claim, mathematics is removed from everyday experience. It is because mathematics is not built from sensory data but from human activity (mathematics is a language of human action): counting, folding, ordering, comparing, etc. As a result, to create such a language we must reflect on that activity, learning to carry it out in our imaginations and to name and represent it in symbols and images. Reflection, as the “objectification” of a construct, functions as the bootstrap by which the mathematician pulls her/himself up in order to stabilize the current construction and to obtain the position from which the next construct can be created. Mathematicians act as if a mathematical idea possesses an external, independent existence; however the constructivist interprets this to mean that the mathematician and his/her community have chosen, for the time being, not to call the construct into question, but to use it as if it were real, while assessing its worthiness over time.

Frequently, constructivism is criticized for being overly relativistic. The argument is that, if everyone is a captive of their own constructs, and if no appeal to an external reality can be made to assess the quality of those constructs, then everyone’s constructs must be equally valid. Two replies to this argument are offered. First, the constructive process is subject to social influences. We do not think in isolation; our choice of problems, the language in which we cast the problem, our method of examining a problem, our choice of resources to solve the problem, and our acceptance of a level of rigor for a solution are all both social and individual processes. Thus, a constructivist assumes there are shaping influences on his/her constructions. The criteria for assessing the strength of an individual’s construct are discussed later in the section on powerful constructions.

Secondly, a person can never know what another person’s constructs are with any certainty. Communications between people function in two ways: people try to assess the congruency between their constructs through their use of language, choice of references, and selection of examples; concurrently, they try to assess the strength of the other person’s constructs as an independent system by considering the apparent level of internal consistency of those constructs. For example, in mathematics education a teacher needs to construct a model of a student’s understanding given what the student knows, while gauging how like the teacher’s own constructs the student’s constructs are. Thus, a teacher must always give consideration to the possibility that a student’s constructs, no matter how different they appear from the teacher’s own constructs, may possess a reasonable level of internal validity for that student and therefore must adapt the instruction suitably.

Comparing mathematics to a tool is perhaps useful in seeing how mathematics is not a description of an external reality, but is, on the contrary, a human construction, invented to achieve human purposes. Consider a common tool such as a spoon, a can opener or a computer. A tool is always used to act on something else, to shape it or to move it. It has an impact on the object; its role is not neutral. One does not come to know a tool through a description of it, but only through its activity. Its structure and its function are interrelated. A powerful tool is one of broad application. If it is too specialized or too inaccessible to use with ease, it will fall into disuse. Anthropologists examining ancient cultures found tools to have been durable and to have been tied into the daily lives of the individuals in a culture in essential ways. A tool is designed to save people effort. Once one gains facility in its use, one gains time to undertake further activities. Mathematics is such a tool and its changes reflect the changes in the kinds of activities human beings are engaged in.

Thus, as a constructivist, when I teach mathematics I am not teaching students about the mathematical structures which underlie objects in the world; I am teaching them how to develop their cognition, how to see the world through a set of quantitative lenses which I believe provide a powerful way of making sense of the world, how to reflect on those lenses to create more and more powerful lenses and how to appreciate the role these lenses play in the development of their culture. I am trying to teach them to use one tool of the intellect, mathematics.

Some Implications for Mathematics Instruction

A constructivist theory of knowledge has dramatic implications for mathematics instruction. It follows from this theory that students are always constructing an understanding for their experiences. The research on students’ misconceptions, alternative conceptions, and prior knowledge provides evidence of this constructive activity. From a more knowledgeable vantage point, we can claim that these constructions of our students are weak; they both lack internal consistency and explain only a limited range of phenomena. As mathematics educators, we must thereby promote in our students the development of more powerful and effective constructions. To do this, we must define what is meant by a more powerful and effective construction and attend to how the promotion of these constructions might be achieved.

I want to suggest that the most fundamental quality of a powerful construction is that students must believe it. Ironically, in most formal knowledge, students distinguish between believing and knowing. To them there is no contradiction in saying, “I know that such and such is considered to be true, but I do not believe it.” To a constructivist, knowledge without belief is contradictory. Thus, I wish to assert, that personal autonomy is the backbone of the process of construction.

In addition to the necessary quality of commitment by the construer, a powerful construction exhibits other significant qualities. The list which follows is intended to be illustrative, not exhaustive. Powerful constructions are typically characterized by:

A structure with a measure of internal consistency;An integration across a variety of concepts;A convergence among multiple forms and contexts of representation;An ability to be reflected on and described;An historic continuity;Ties into various symbol systems;An agreement with experts;A potential to act as a tool for further constructions;A guide for future actions; and An ability to be justified and defended.

Students of mathematics often apply only one criterion to their evaluation of their own constructs, asking “is it in agreement with the experts?” (Or, in less constructivist terms, “Is it right?”) As a result, their knowledge of mathematics becomes isolated and formalized from the rest of their experience, which is constructed from their action on the world in a more spontaneous and interactive fashion. Memorization and imitation of examples produce the “right answer,” the desired outcome, in a local, well-defined problem space and thereby outpace the more difficult endeavors of constructing the idea and of coordinating its interactions with the other qualities of powerful constructions. The thesis of this paper is that students must learn to construct powerful ideas and that this constructive process requires the coordination and convergence of the ways of knowing identified in the above list of the qualities of powerful constructions.

An Alternative Set of Assumptions

Constructivism commits one to teaching students how to create more powerful constructions. Variations are expected and nurtured, and the student is given primary responsibility for assessing the quality of a construction. The goal of instruction can be stated as:

An instructor should promote and encourage the development for each individual within his/her class of a repertoire for powerful mathematical constructions for posing, constructing, exploring, solving and justifying mathematical problems and concepts and should seek to develop in students the capacity to reflect on and evaluate the quality of their constructions.

This goal suggests acceptance of three assumptions:

Teachers must build models of student’s understanding of mathematics. To do this, teachers need to create as many and as varied ways of gathering evidence for judging the strength of a student’s constructions as possible. The result will be that a teacher creates a “case study” of each student.Instruction is inherently interactive; through their interactions with students regarding their knowledge of subject matter, teachers construct a tentative path upon which students may move to construct a mathematical idea more consonant with accepted mathematical knowledge. Teachers, however, must already be prepared for the likelihood that the students’ constructions will not coincide with their own, and encourage the students’ expression of their beliefs so that teachers come to understand student beliefs. Teachers then must be prepared to revise their own beliefs or to negotiate with the student to find a mutually acceptable alternative (which may or may not endorse the conventions of mathematical practice). If the student advocates a solution that is clearly lacking adequate argument, teachers will need to signal firmly that, in their judgment, the student’s position lacks legitimacy.Ultimately, the student must decide on the adequacy of his/her construction.

Using these alternative assumptions, an examination of a teacher with constructivist beliefs is undertaken in the second part of the article. Specific examples of the methods used by this outstanding teacher are provided, with a more detailed discussion of each of the assumptions.

The Context

The study took place at the SummerMath program, an experimental summer program for young women in high school offered by Mount Holyoke College. The program, described in earlier papers (Confrey, 1983), has explicit constructivist underpinnings. Most of the young women who attended were academically capable but had experienced difficulties with mathematics. Since no scholarships were available at the time, the students were, for the most part, upper middle class or wealthier. Follow-up evaluations of the program indicate that students’ scores on SAT (math) showed considerable improvement and that alumnae report they are more persistent, more confident, and ask more questions.

The intent of this study was to construct a model of the practices of a teacher committed to constructivist beliefs. The instructor was selected for his excellence in teaching, as evaluated by the students in the program for two summers and confirmed again during the third year, the year of the study. He consistently received strong evaluations both at the end of the program and in follow-up evaluations four months later.

The study was conducted during the second of six weeks of the program. A class entitled Fundamental Mathematics Concepts, Level I, was selected for investigation. The topic for the week was the representation of fractions; the materials were designed to reveal misconceptions and to promote the coordination of arithmetical manipulations of fractions with actions on pictures.

The eleven young women in the class ranged from ninth to eleventh graders. In order to be in this class, the student had to have scored less than 45% on a multiple choice placement test made up of twenty-five items from the high school curriculum.

Frequently the organization and pace of the class differed significantly from typical classrooms. The students in the class worked in pairs on the curriculum materials provided each day. Using the paired problem solving method of Whimbey and Lochhead (1980), the students took turns solving the problems. One student was supposed to talk through the problem while the other asked questions about the method. Often, in spite of one week of focused training on the method, the students would solve the problems together.

In this study, the focus is on the teacher-student interactions. The data are taken from videotapes of these interactions. Consistently on the videotapes one sees that the students do most of the talking and writing. The pace at which the content is covered is dramatically slower than in traditional classes; it is not unusual for the students to solve only two or three problems in a class.

The Method

The model for the research is described by Donald Schön (1983) in his book, The Reflective Practitioner. Schön argues that the professional engages in an art of practice which is not easily or accurately characterized by a technical analysis. He searches for an “epistemology of practice implicit in the artistic, intuitive processes which some practitioners do bring to the situations of uncertainty, instability, uniqueness and value conflict” (p. 49).

Schön developed the idea of “reflection in action.” He suggested that much of the practitioner’s knowledge is tacit; it operates on his/her actions, decisions and judgments but cannot be stated under the usual circumstances. However, when stimulated to reflect on those actions through surprise, puzzlement, or perhaps intention, the practitioner may ask questions such as: “What features do I notice when I recognize this thing? What are the criteria by which I make this judgement? What procedures am I enacting when I perform this skill? How am I framing the problem?” In answering these questions tentatively, the professional, according to Schön, “also reflects on the understandings which have been implicit in his action, understandings which he surfaces, criticizes, restructures and embodies in further actions” (p. 50).

In concert with Schön’s perspective, a researcher (who also taught in the program) and the teacher together examined the teacher’s practice. No attempt was made to identify predetermined categories for discussion in the interviews. The intent was to try to develop, through the use of videotapes, a model of this particular teacher’s instruction which was acceptable to both the teacher and the researcher.

To this end, for five days the class was videotaped for its hour and a half duration. Two students in the class were selected and paid to be interviewed each day, and their work was collected and copied each day. They were also asked to keep journals over the summer session. Each afternoon of the five days, the instructor and the researcher discussed the day’s instruction. Each day the instructor would describe what he felt were significant issues, would specifically describe his interactions with the two students who were to be interviewed, and then would view portions of the tape, answering the researcher’s questions and commenting on portions that he recalled as significant. The focus was on explaining how he viewed his role in the class. The interview lasted from sixty to ninety minutes daily.

For the last four days of the study, after the interview with the instructor, the researcher would interview separately the same two students who worked together every day in class. This interview would begin with a clinical interview on one of the problems discussed on the tape. Then the taped interactions between the instructor and these students would be viewed and discussed with an emphasis on what the student thought the instructor said or meant. The interviews lasted from forty-five minutes to an hour.

The Results

The results of the study will be presented in the form of a model of the teacher’s instruction. The following six components of the model will be described and illustrated:

Promotion of autonomy and commitment in the students;Development of students’ reflective processes;Construction of case histories;Identification and negotiation of tentative solution paths with the student;Retracing of those solution paths; andAdherence to the intent of the materials.

For each component, specific techniques used by the teacher will be discussed and illustrated with examples drawn from the videotapes and the interviews.

1. Promotion of Autonomy and Commitment

Earlier I stated that personal autonomy is the backbone of the process of construction. Baird and White (1984) argued that a significant improvement in student learning depends on “a fundamental shift from teacher to student in responsibility for, and control over, learning” (p. 2). In this teacher’s interactions with students, he consistently demanded that the students make a commitment to their answers. He used four techniques to accomplish this goal: he questioned students’ answers whether they were right or wrong; he insisted that students engage in a problem at least to the extent of explaining what they had tried; he would remain with a group long enough to get them started in a potentially productive direction; and he emphasized the importance of having a student evaluate his/her own success. The following quotes from the interviews with the teacher illustrate these techniques:

But, she’s given me nothing to work from other than saying, “You only give us stuff that is too hard for us to do, and we are stupid.” I can’t deal with that until she starts putting forth effort, and that gets a starting point to discuss what she understands and what she doesn’t understand. Why? She’s got to get over that herself.

If I stand there, they are going to continually look up at me to see if every line they draw at this point is right. I’m leaving them there; let them see what they can do with what’s left [after I walk away].

The students have a success, and one way to treat that is, we can tell them they’ve just had a success. But they’ve got to sit there and go, “I’ve just done this all by my little old self.” I think that’s the point that’s very important. It’s their reflection on what “I’ve done,” what they’ve done, and their admission they’ve done something; they’ve beaten the problem.

The need to increase the level of student autonomy in relation to mathematics is continually addressed in this teacher’s instructional methods. He believed that a measure of autonomy is a prerequisite to developing the self-awareness one needs to be a successful problem solver. Once a student began to take responsibility for her thinking, the teacher felt that he could move to develop her powers of reflection.

2. Development of Students’ Reflective Processes

I posited that reflection is the bootstrap for the construction of mathematical ideas. In order to promote a student’s awareness of her problem solving, the teacher asked three categories of questions, which correspond roughly to the three stages of construction posited in Confrey (1985): the problematic, action and reflection. In that work, I indicated that, for students to modify and adapt their constructions, they must: (1) encounter a situation that they experience as personally problematic, as a roadblock to where they wish to be; (2) act to resolve the problematic, often using multiple forms of representation and (3) assess the success of their action in resolving the problematic or determine what problematic remains. In this teacher’s instruction, there was evidence that he used three levels of questioning to increase his students’ awareness of their own strategies and methods. Each level is discussed and illustrated.

Level One: The Interpretation of the Problem

These questions involved the request to reread or restate the question. The teacher would ask the students questions such as, “What are we doing?” “What is the problem?” or “What does this problem say?” In asking these, the teacher would often focus the student’s attention on the language the student was using. These questions appear deceptively simple to an observer, for the students, it was often difficult to repeat the problem or describe it in any fashion. The students appeared so unaccustomed to speaking mathematically that the questions on this level served a subtle and essential role. Furthermore, it was apparent that the students’ responses to this level of questioning had a

significant impact on their success in solving the problem. Often what sounded to the listener to be multiple re-readings of the problem had the effect of curtailing the amount of time the student needed to undertake a solution to the problem, possibly indicating that what was verbalized as repetition represented significant cognitive processing on the part of the student.

Level Two: Cognitive Strategies

The teacher would ask a student to describe what she was doing. When working with students at this level, he would use the level of precision of the student’s statements as a standard, requesting slightly more. He would not allow the students to introduce mathematical terminology or formulae without explaining them to him. In one interview he commented, “I think one of the things that happens as students learn to relate to teachers is that they come close, and teachers fill in the blanks.” A typical teacher/student interaction, in which they discuss pictorial comparisons of 13/5 and 21/10, illustrates Level Two questioning. The student is currently attempting to draw a picture to compare 2/3 and 5/7 and contrasting it to the solution strategy she used to solve 13/5 and 21/10:

Teacher: How did we do the fifths and the tenths?Student: But those were in proportion.Teacher: What do you mean, “in proportion?”Student: Not proportion; they were equal. At least, urn, I mean they weren’t equal, but – I know what I mean.Teacher: I know what you mean too, but now you’ll have to tell me.Student: I mean, I can’t think of a word. I mean five is half of ten; therefore we divided the fives in half. It would be just like adding five more. I don’t know how to explain it. I mean you have like five parts and you divide it in half, and it was like double. But like if you took three things and divided it in half, you’d have six things and not seven. Therefore, I mean, you’d have a different problem.

When the teacher requires the student to explain her meaning of proportion, she reveals her tendency to think of an increased denominator as an additive operation, and then she revises her approach to a multiplicative one, doubling. If she had continued to think of the change from fifths to tenths as additive, the change from thirds to sevenths would have posed no difficulty, but her method would most likely have failed. After this exchange, the student pair and the teacher work on a method to divide the rectangle vertically in thirds and horizontally in sevenths to make units of twenty-firsts and compare their relative size.

Level Three: Justification of Strategies

Once a student was able to tell the teacher what the problem was about and how she was going about solving it, the teacher began to ask the student to defend her answer. Again, the level of rigor demanded depended on the knowledge the teacher had of the student; he demanded an explanation which adequately fit the student’s interpretation of the problem and the methods and strategies she had constructed.

Examples of the questions he used included: “Why? What does that tell you? What can you tell me? Why not?” and “What do you mean it doesn’t work?”

The development of the students’ reflective processes was a primary goal of this teacher’s instruction. Generally he used three questioning strategies to develop these processes. Much of his time was spent asking students to discuss their interpretation of the problem and to describe precisely their methods of solution. Once they had carefully described their interpretation and methods, he asked students to defend their answers. Two characteristics of his interactions were prevalent: a focus on the language used by the students, and an acceptance and exploration of the students’ visions of the problems. He worked primarily within his understanding of their framework, and in order to do this effectively, he developed a model to allow him to understand a particular student’s case.

3. Construction of a Case History

Not only did questioning of the students promote their reflection, it allowed the teacher to gather substantial knowledge and insight into their understanding of mathematics. The teacher often spoke of a student’s general tendencies in problem solving, and used this information to design appropriate solution strategies. Because this knowledge involved cognitive, affective and personal dimensions, we developed the habit of referring to this as the teacher’s construction of a case history. As pointed out by Cobb and Steffe (1983), researchers and teachers alike build models of their students’ mathematical knowledge, attempting “to ‘see’ both their own and the children’s actions from the children’s point of view” (p. 85). Models which can be used effectively to interpret the students’ performances over time can be thought of as case histories (e.g., Confrey, 1983; Erlwanger, 1973).

Although no elaborate case history can be provided here, a brief summary might be helpful. One of the two students who participated in the interview, Joyce, consistently rejected the use of pictures, relying heavily on borrowed algorithms and half-remembered rules. On the first day of class, the teacher commented that he was having some difficulty getting this pair of students to draw the pictures. On the second day, the student had difficulty solving a ratio problem. She abruptly switched and tried to use a percent, still unable to explain what the problem was about. The teacher commented that she was overworking the problem. In the interview, he explained his comment, saying:

What I am trying to get her to do is to see that she can think her way through the problem … without heading off into Percent Land or whatever. That she can think her way through it and will have a fair idea of what’s going on without working the problem. The reason I say that is, in working with her the last few days, her tendency is to use those poorly taught algorithms from arithmetic whenever possible. You drop one in, and something comes out, and you say, “Oh, something came out; I’m happy,” and you have no idea what’s going on.

In the interview with Joyce, she expressed her ambivalence with the drawing of pictures and described the differences in this type of instruction:

I don’t know exactly how it would work if you would have to, like in algebra; you have different types of formulas and things like that. When you come across a problem like that it seems to me you would need a formula, and its not something that you would just figure out on the spur of the moment, without going through a really long, extensive proof or something like that.

On the last day of the study, the teacher discussed his evaluation of Joyce’s situation:

Teacher: I’m really stuck. I don’t know how to get her off that, and I don’t really think that I can. I really think that she has got to make the decision that the way she has been learning math in terms of algorithms is not helping her. Her reliance on calculation is really not helping her. Until she decides and makes that recognition, the best we can do is just try daily to break her of it. [I can] Say, look at it this way, and she will probably sit there and draw the pictures [for me]. [And] She will sometimes do it [for herself].Researcher: Were you discouraged?Teacher: No it’s going to take her a while. I mean, what are we doing? We’re taking two weeks of experience in drawing pictures and trying to downset against nine years of “It’s important to do the calculations quickly in your head.” She may or may not come around. I will keep hoping that, eventually, when she sees more and more examples of her algorithms falling down because she looks at the way a result comes out and she doesn’t like they way the result looks, so she questions the algorithms.

By interacting with the students primarily in one-on-one (or -two) settings, the teacher was able to form a powerful model of the students’ characteristic approaches to solving problems. In order to do so, he created multiple sources of evidence from which to build his models, using his interactions with them, their performance on key items from the curriculum, and his observations of their interactions with the other students.

At this point of the paper, our model of this teacher’s instruction appears relatively static; he has created a powerful model of the student’s mathematical ideas, he has insisted on her autonomy, and he has focused her thinking on her own thinking, increasing her powers of reflection. We might ask, “when does he begin to teach?” In asking this, we find ourselves back in a traditional interpretation of teaching as “telling.” In fact, as one examines the videotapes of this instruction, it appears that he is indeed teaching already, for the students are learning to solve these problems successfully, often as a result of their interactions with him.

Perhaps this “progress” on the part of the students was attributable to the particular context. It could be claimed that, since the topic was the representation of fractions, the teacher is remediating, not teaching new material. I think the question of whether this instructional model is useful in teaching new material must remain an open question in need of further examination. In many ways, I would argue that, for these students, this was new material. Furthermore, I suggest that, if it does indeed prove true that the model seems to be most appropriate for “remediation,” its value is still not easily dismissed. Our elementary and secondary curriculum has massive amounts of repetition built into it, simply because the students never learn or retain what they have been taught.

This teacher’s instruction was concerned with challenging and changing his students’ current conceptions. The teacher would try to aid the student in building a more powerful construction, from the student’s point of view. Because the teacher’s own constructions formed the framework for examining student constructions, their influence on the direction of the interaction was unavoidable. However, as evidenced by the substantial variation witnessed in the solution paths, the teacher’s conceptions were either very flexible, possessing multiple perspectives, or were in fact altered over the course of the interactions. Because of the students influence on the method of solution, I have labeled this component of instruction as the identification and negotiation of a tentative solution path.

4. Identification and Negotiation of a Tentative Solution Path

At times, the teacher was unable to promote a successful resolution of a problem using the techniques cited above. However, from the information gained from this questioning and from his broader knowledge of the student’s case, he would have developed a model of the student’s understanding of the problem. Assuming that he judged that the student was investing enough initial effort, he would intervene more directly.

From the tapes and the interviews, there was evidence that he would analyze the difficulties he anticipated the students might have with the problem and then, with his knowledge of the student’s case, he would develop a tentative solution path and negotiate it with the student. Since he would typically select certain conceptually difficult problems and review them with all students, the researcher could analyze the differences in his approach. The variety of methods used from group to group substantiated the need to conceptualize a more interactive and negotiated view of his classroom. As the teacher commented, “You can’t walk into class saying, ‘I know I’m going to do it this way. “

Thus, this teacher would gather evidence on which he could build a model of how the student was thinking about the problems. From this model, he would construct a tentative solution path. At a more global level, he would be building an understanding of the student’s “case.” As he worked towards a solution with the student, he would test the adequacy of his models of the student’s theories and of his case history. The results of these tests could lead to a revision of his tentative solution path. Over the course of the week, there were changes in how students responded. As they became more autonomous and confident, they influenced to a greater extent the direction and outcome of the interaction.

One set of strategies was designed specifically to encourage the student to form a more powerful construction as described earlier. These included relating the ideas to other concepts, challenging students’ use of symbols and the English language to communicate their ideas, pushing towards consistency within a related set of concepts, limiting the scope of the idea to allow for a local resolution of the ideas, asking for representations, or exploring the idea from multiple perspectives. He often introduced a measure of cognitive conflict in order to promote the construction of more powerful ideas.

Some of the variety of questions he asked can be illustrated by a listing in which the reader is advised to attend to the form rather than the particular substance: “How does that relate to what you were seeing up here? Is there anything you did in the last one that can help you with this one? What if I colored in seven boxes? This doesn’t look like this. Can you do something similar?

Which am Ito believe, your picture or your diagram?” These questions invite the student to attend to a previous issue, to resolve conflicts and to find analogies between different episodes.

At other times, the instructor would provide the student with more directed guidance in solving the problem. Through the use of product-oriented questions, the teacher would move a student towards a solution. These occasions seemed to arise: 1) when the student’s tolerance for frustration was low; 2) when the student needed to experience success or progress, and/or 3) when the class as a whole needed closure on a topic.

Such decisions appeared to produce unreliable outcomes. At least once when the teacher was more directive with the two interviewees, the interviews revealed that the students’ understanding of the material was weak and sketchy. There did, however, seem to be some evidence that such a decision to provide directed guidance even with deleterious cognitive consequences may be important affectively in order to lower a student’s frustration level and to encourage her to be willing to engage in the problem-solving process at a later time.

5. Retracing and Reviewing the Solution Path

When the problem was solved, the instructor would revisit the problem with the student. This strategy of “re-viewing” the problem was useful for providing: 1) opportunities for reflection; 2) an overview of the problem; 3) occasions for the teacher to advocate for his view of mathematics teaching and learning; and 4) the student with a sense of accomplishment. Examples of the questions he used in reviewing solution paths included: “Isn’t that what your picture says? How do you decide? How do you know when to reorganize?”

6. Adherence to the Intent of the Materials

Too many people assume that constructivist teaching implies a laissez-faire attitude on the part of the teacher. Teachers learning to teach within this framework, fearing that they will “tell too much,” often remain silent, while the students flounder in frustration. They mistakenly believe that a constructivist teacher lacks a specific agenda for what is to be learned in the classroom. Such a characterization did not apply to this teacher. He was committed to a particular view of mathematics learning and found many opportunities to share this with the students.

During the week, the teacher was determined to have the students come to see that one can make sense of fractions using pictures, and that the algorithms for rational numbers can be seen as actions on those pictures. If a student chose to approach an understanding of the problem which did not use pictures, the instructor would allow her to complete the investigation, but then he would relate that solution to the representation of fractions. He strove to do this without undermining the student’s initiative

Conclusions

In this paper, the assumptions behind direct instruction were examined and questioned from a constructivist perspective. After discussing constructivism, alternative assumptions for instruction were offered. Then, using videotapes of four days of instruction and methods of clinical interview and stimulated recall, I examined the practice of one teacher in the constructivist tradition who seemed exemplary from a variety of measures. A model of this particular teacher’s instruction was presented and discussed. The purpose of the work was to suggest that alternative forms of instruction can exist in mathematics that differ in their basic assumptions from the tradition of “direct instruction.”

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