Simcalc: Accelerating Students’ Engagement With The Mathematics Of Change

Introduction

A central phenomenon of the twenty-first century will be change: economic, social, and technological change. Indeed, engaging students in analysis of change and variation is a central element of nearly every chapter in this book. Today, however, the mathematics of change and variation (MCV), despite its importance in understanding and controlling this ubiquitous phenomenon, is packed away in a course, Calculus, that sits at the end of a long series of prerequisites that filter out 90% of the population. This is especially true for students from economically poorer neighborhoods and families. And even the 10% who do have nominal access to MCV in calculus courses develop mostly symbol manipulation skill but little understanding (Tucker, 1990). The traditional curriculum thus excludes most children from the concepts of rate of change, accumulation, approximation, continuity, and limit (among others). These are the very concepts children most need not only to participate in the physical, social, and life sciences of the twenty-first century, but also to make informed decisions in their personal and political lives. Even though MCV concepts were at the heart of mathematics and science historically (Bochner, 1966), in education the opposite is more nearly true. Conventional curricula neglect, delay, or deny students’ access to MCV.

The mission of our SimCalc project is to give ordinary children the opportunities, experiences, and resources they need to develop extraordinary understanding and skill with MCV. Using a combination of advanced technology and carefully reformulated curricula, we aim to democratize access to the mathematics of change. This chapter discusses the research findings and design principles guiding our approach, with specific attention to our first software product, “MathWorlds.” MathWorlds provides dynamic, direct manipulation graphs, piecewise definable functions, and animated cartoon worlds to engage elementary, middle, and high school students in qualitative and quantitative reasoning about the relationships among position, velocity, and acceleration in complex contexts. Formative evaluation experiments with diverse inner city students (the large majority of whom were in the lowest quartile of both academic achievement and socio-economic status) show that MathWorlds, coupled with an appropriate curriculum and teaching practice, can enable students to construct viable MCV concepts.