Interpreting Responses to Problems with Several Levels and Types of Correct Answers
Susan J. Lamon and Richard Lesh
The production and interpretation of model-eliciting activities (see Chapter 2) is a complex endeavor. Problem formulation and response interpretation within complex mathematical domains require a strong supporting framework. That framework should provide a blueprint for creating problems, a basis by which to interpret students' reasoning as they interact with the problems, and a guide to instructional decision making. In this chapter, we will use examples from the domain of ratio and proportion to explore some of the issues surrounding problem formulation and the interpretation of student thinking. A framework for problem formulation and scoring will be proposed, and levels of responses to a variety of problems will be discussed in light of that framework. We begin by discussing some issues that are arising as the mathematics education community considers alternatives to current assessment practices.
The campaign for a problem-solving curriculum during the 1980s taught us that early responses to pressures for reform sometimes create the illusion of novelty but stop short of producing substantial changes. Immature ideas and meager guidelines allow nearly any current practice to be rationalized to a good fit or to sanction cosmetic changes that fail to reform.