Literature Index

This article focuses on a form of instructional design that is deemed fitting for reform<br>mathematics education. Reform mathematics education requires instruction that<br>helps students in developing their current ways of reasoning into more sophisticated<br>ways of mathematical reasoning. This implies that there has to be ample room for<br>teachers to adjust their instruction to the students' thinking. But, the point of departure<br>is that if justice is to be done to the input of the students and their ideas built on, a<br>well-founded plan is needed. Design research on an instructional sequence on addition<br>and subtraction up to 100 is taken as an instance to elucidate how the theory for<br>realistic mathematics education (RME) can be used to develop a local instruction theory<br>that can function as such a plan. Instead of offering an instructional sequence that<br>"works," the objective of design research is to offer teachers an empirically grounded<br>theory on how a certain set of instructional activities can work. The example of addition<br>and subtraction up to 100 is used to clarify howa local instruction theory informs<br>teachers about learning goals, instructional activities, student thinking and learning,<br>and the role of tools and imagery.

Arguments are presented to support increased emphasis on logical aspects of formal methods of analysis, depending on probability in the sense of R. A. Fisher. Formulating probabilistic models that convey uncertain knowledge of objective phenomena and using such models for inductive reasoning are central activities of individuals that introduce limited but necessary subjectivity into science. Statistical models are classified into overlapping types called here empirical, stochastic and predictive, all drawing on a common mathematical theory of probability, and all facilitating statements with logical and epistemic content. Contexts in which these ideas are intended to apply are discussed via three major examples.

This study follows an earlier study of school students' abilities to draw inferences when comparing two data sets presented in graphical form (Watson and Moritz, 1999). Forty-two students who were originally interviewed in grades 3 to 9, were subsequently interviewed either three or four years later. The results for individual student development add to the credibility of the cross-age observations, as well as support the hierarchical framework suggested by the original study. Changes in levels of performance and strategies for drawing conclusions are documented. A further step from the original study is the consideration of how students used the variation displayed in the graphical presentation of the data sets as a basis for decision-making. Implications for teaching and for further research are discussed.

Students of all ages seem fascinated by the lottery, making it a ready tool for illustrating basic probabilistic concepts. The author has developed a program called "Lotto Luck" for IBM PC compatibles which has been used on over 100 classrooms from grades 6 through 12 and with dozens of college classes and civic groups to demonstrate what happens to the "earnings" of the frequent lottery player over a period of time. We discuss how to use the program and provide information for obtaining the complied code by ftp.