What kind of statistical models seem particularly promising to describe children's mathematical thinking? How can measurement procedures capture distinctions between critical states of understanding?
In chapter 13, Lesh and Lamon use the theoretical perspectives described in earlier chapters to focus on specific problems about ratio and proportion. Questions that are addressed include the following: What are some of common misconceptions about the formulation of "good" problems and "good" responses? What are some specific cognitive objectives of instruction about ratios and proportions? How can reliable scores be produced for problems with multiple solution paths which have different levels of difficulty?
In chapter 14, Lesh, Lamon, Gong, and Post describe one way to represent complex profiles of student abilities and achievement. They use computer-generated "learning progress maps" which succeed in being simple (from the point of view of educational decision makers) because they are graphics-based, interactive, and inquiry oriented, with details that are displayed only when they are requested by individual decision makers. That is, the reports are versatile enough to aggregate and display information in alternative ways to address a variety of decision-making issues.
In chapter 15, Lesh, Lamon, Lester, and Behr describe some of the most important assumptions underlying traditional types of standardized testing compared with the types of alternative assessments emphasized in this book. Chapter 15 also describes several priorities for future research (with special attention being given to issues related to equity, technology, and teacher education), and it concludes with specific examples taken from three current closely related projects that were explicitly designed to find practical ways to implement recommendations made in other chapters of this book.
In chapter 16, Schwartz summarizes conclusions reached by a series of recent projects focusing on "The Prices of Secrecy. The Social, Intellectual, and Psychological Costs of Current Assessment Practice" ( Schwartz and Viator, 1990).
It is our hope that this book will provide both general principles and specific examples to help support curriculum reform efforts that