Greeno, James G. and Rogers P. Hall

Phi Delta Kappan, Jan97, Vol. 78 Issue 5, pp 361-366, 1997.

Abstract

"OUR TITLE is deliberately ambiguous. In one interpretation, "practicing representation" 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 representation" 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."

Annotatie

Enkele citaten:

"... 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."

"...technical representations are often taught as though they were ends in themselves."

Learning to communicate:

• "An important educational goal is for students to learn to use multiple forms of representation in communicating with one another."
• "When students learn to use representations as tools, they are preparing for the kinds of activities that are common among mathematicians, scientists, engineers, and others who use mathematics in their professional work."

Representaties functioneren dan als:

""boundary objects" -- that is, representations that can be interpreted by people from different communities in ways that allow them to share information."

1. Representations to aid understanding:
2. "Representations are constructed for specific purposes during attempts to solve problems and communicate with others about these attempts. In addition, the meanings of representations can shift as problem-solving purposes and difficulties change. Under these circumstances, representations often match the processes of solving the problem, providing a kind of model of the students' thinking as they work. This contrasts with common methods of teaching and assessment, in which students are instructed to represent problems with standard forms that depend on a classifica-tion of problem types, rather than on the processes of solution."
3. "Students often construct representations in forms that help them see patterns and perform calculations, taking advantage of the fact that different forms provide different supports for inference and calculation. Thus solving a problem involves an interactive process in which students construct representations based on partial understanding and then can use the representations to improve their understanding, which leads to a more refined representation, and so on."
4. "Students often use multiple forms of representation in working on a problem, some of which are invented by the students and differ from forms that are explicitly taught in the curriculum. These hybrid combinations often show that students have significant partial understanding of difficult mathematical concepts. The invention of novel forms by students shows that they can use representational material constructively as they build their understanding."
5. Representations and conventions of interpretation
6. "... for a notation to function as a representation, someone has to interpret it and thereby give it meaning. (In Peirce's view, with which we agree, there are three things involved whenever there is a representation: something that is represented, the referent; the referring expression that represents the referent; and the interpretation that links the referring expression to the referent.)"
7. "School practice is now moving toward better recognition that interpretation is an essential part of representations in mathematics and science. In these practices, students not only learn to follow standard conventions of interpretation, but they also can come to understand how representations work. Understanding representations includes knowing that there can be different interpretations of the same notation. It is important to know the conventional interpretations of standard forms, but it is often productive to construct nonstandard representations with special interpretations in working on problems and communicating about ideas."

 

1. Conclusions:
2. "We believe that educational purposes are better served if students are involved in activities in which they learn to construct versions of representations flexibly and to participate in discussions in which conventions of interpretation are developed. Such an approach enables them to understand and appreciate that mathematical and scientific representations, like those in other domains, are adapted for particular uses."
3. "Across domains, these practices share important features of use, adaptability, and interpretation that should be included in classroom learning."