Practicing Representation: Learning with and about Representational Forms

Abstract

Every student's educational activities should include the rich variety of experience and learning made possible through participation in multiple practices of representation, Messrs. Greeno and Hall suggest. Our title is deliberately ambiguous. In one interpretation, practicing refers to a kind of exercise that is required of students, especially in mathematics and science. They have to practice using the standard forms of representation, such as arithmetic expressions, tables, graphs, and equations. In another interpretation, practicing must always be part of a social practice. Viewed in this way, learning to construct and interpret representations involves learning to participate in the complex practices of communication and reasoning in which the representations are used. This learning involves much more than simply learning to read and write symbols in arrangements corresponding to the accepted forms. A theoretical perspective - sometimes called the situative perspective - that is currently developing in research emphasizes those practices in which students participate as they learn. Teachers know the importance of student participation in the activities of learning, and they commit much effort to designing activities that will engage the participation of students. What is often overlooked is that, in their participation, students learn the prevailing practices of learning and knowing that occur in their school setting. For example, if students' learning activities include formulating questions and proposing and explaining alternative solutions, they can learn how to participate in those activities of inquiry. However, if they learn to give only the answers and explanations that are specified by teachers and textbooks, they are likely to learn the practices of memorizing. To be still more specific, if students need to construct tables and graphs to complete a project report in mathematics or science, they can learn how to consider whether and how these forms are effective in communicating the information that they think is important. Or if students draw graphs corresponding to equations for, say, linear versus exponential growth in order to explore the effects of changes in their parameters, they can learn how to use graphs to consider differences in meaning between the concepts of additive and multiplicative change. But if students simply complete assignments of constructing representations in forms that are already specified, they do not have opportunities to learn how to weigh the advantages and disadvantages of different forms of representation or how to use those representations as tools with which to build their conceptual understanding. We discuss here the process of learning to construct and interpret representations from the perspective of social practice. We and others have been conducting research on practices of representation in schools and in other work settings, and we believe that this research has implications for curriculum and instruction in school mathematics and science. Our discussion makes three main points. * Forms of representation need not be taught as though they are ends in themselves. Instead, they can be considered as useful tools for constructing understanding and for communicating information and understanding. * When representations are used as tools for understanding and communicating, they are constructed and adapted for the purposes at hand. Often a nonstandard representation serves these purposes better than a standard form, and students should learn how to generate representations flexibly for their use. * For something to function as a representation, people must interpret it to give it meaning. Of course, standard forms of representation are valuable; a sizable community shares their conventions of interpretation, and it is important for students to learn how to use these standard forms of representation. …