Using models of operations and equations

Summary From a general and systematic point of view, this article has attempted to answer the questions, How do models and equations relate?, How does one teach this relationship? In the process, a systematic procedure for analyzing this task of teaching operations with the use of models and developing teaching strategies has been demonstrated. In applying the several concepts of this article to the preparation of a lesson, the teacher can (a) choose models and a family of equations, (b) choose an analysis of the equations into constituent parts involving family or equation levels, (c) choose an analysis of models into their critical attributes, and (d) plan a strategy utilizing direct correlates or oblique correlates for connecting the equations and models at a general or instance level. On the basis of such planning, the teacher will develop with a class a knowledge of the parts of the equation, the critical parts of the model, and correlates for connecting the two.Without utilizing these concepts, teachers have been found to attempt the introduction of the connection between an equation and a new model simply by placing the illustration (instance of a model) and equation together on the chalkboard and reading the equation. (This may seem absurd, but the author is aware of a film designed to present exemplary teaching in which this is done and has observed student teachers overlooking the necessity for carefully connecting the model and the equation.) Such practices as these in the context of the common telling-teaching process suggest that teachers' commentaries should provide extensive analyses and direct correlates 7].The four concepts (correlates, analysis of models and families of equations at the general and instance level, oblique correlates, and adequacy of analysis of models and equations) have been referred to as general concepts. They are general in the sense that they can be applied to the use of models of any operation. As was pointed out earlier, this level of generality extends to more than forty families of models of operations and inverse operations on whole numbers. It also extends to the use of models of operations in other number systems.