Constructivist Views on the Teaching and Learning of Mathematics || Chapter 8: What Constructivism Implies for Teaching

  • Published on

  • View

  • Download

Embed Size (px)


<ul><li><p>Chapter 8: What Constructivism Implies for TeachingAuthor(s): Jere ConfreySource: Journal for Research in Mathematics Education. Monograph, Vol. 4, ConstructivistViews on the Teaching and Learning of Mathematics (1990), pp. 107-122+195-210Published by: National Council of Teachers of MathematicsStable URL: .Accessed: 03/10/2013 01:54</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .</p><p> .</p><p>JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact</p><p> .</p><p>National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to Journal for Research in Mathematics Education. Monograph.</p><p> </p><p>This content downloaded from on Thu, 3 Oct 2013 01:54:02 AMAll use subject to JSTOR Terms and Conditions</p></li><li><p>Chapter 8: What Constructivism Implies for Teaching </p><p>Jere Confrey Cornell University </p><p>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. </p><p>An Analysis of Direct Instruction The form of instruction in mathematics that has been most thoroughly examined has been "direct instruction" (Good &amp;Grouws, 1978; Peterson, Swing, Stark and 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: </p><p>1. 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. </p><p>2. 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 &amp; Clark, 1978; Snow, 1972). </p><p>3. The responsibility for determining if an adequate level of understanding has been reached lies primarily with the teacher. </p><p>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 </p><p>107 </p><p>This content downloaded from on Thu, 3 Oct 2013 01:54:02 AMAll use subject to JSTOR Terms and Conditions</p></li><li><p>108 </p><p>to traditional forms of instruction (Clement, 1982; Erlwanger, 1975; Vinner, 1983). These studies point to a need to develop alternative forms of instruction. </p><p>Efforts to develop forms of instruction that overcome misconceptions have focused on the need to have students make their conceptual models explicit (Baird &amp; White, 1984; Lochhead, 1983a; Novak &amp; 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 &amp; 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. </p><p>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; von Glasersfeld, 1974, 1983, this monograph). 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. </p><p>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. </p><p>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 </p><p>This content downloaded from on Thu, 3 Oct 2013 01:54:02 AMAll use subject to JSTOR Terms and Conditions</p></li><li><p>109 </p><p>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. </p><p>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. </p><p>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. </p><p>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. </p><p>This content downloaded from on Thu, 3 Oct 2013 01:54:02 AMAll use subject to JSTOR Terms and Conditions</p></li><li><p>110 </p><p>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. </p><p>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. </p><p>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. </p><p>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 </p><p>This content downloaded from on Thu, 3 Oct 2013 01:54:02 AMAll use subject to JSTOR Terms and Conditions</p></li><li><p>111 </p><p>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. </p><p>Some Implications for Mathematics Instruction A constuctivist 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. </p><p>I want to suggest that the most fundamental quality of a powerful construction is that students must believe it. Ironically, in mos...</p></li></ul>


View more >