Kristie J. Newton

An Extensive Analysis of Preservice Elementary Teachers’ Knowledge of Fractions

The study of preservice elementary teachers’ knowledge of fractions is important because fractions are notoriously difficult to learn and teach. Unfortunately, studies of preservice teachers’ fraction knowledge are limited and have focused primarily on division. The present study included all four operations to provide a more comprehensive understanding of this knowledge. Because knowledge is complex, it was examined in five ways: computational skill, basic concepts, word problems, flexibility, and transfer. To further capture the complexity of knowledge, solution methods were examined for patterns that might reveal understandings and misconceptions. Data were gathered before and after a course designed to deepen preservice teachers’ knowledge. Quantitative and qualitative shifts occurred during the semester, but flexibility and transfer were low. Implications for teacher education are discussed.

Understanding the development of flexibility in struggling algebra students

The ability to flexibly solve problems is considered an important outcome for school mathematics and is the focus of this paper. The paper describes the impact of a three-week summer course for students who struggle with algebra. During the course, students regularly compared and contrasted worked examples of algebra problems in order to promote flexible use of solution strategies. Assessments were designed to capture both knowledge and use of multiple strategies. The students were interviewed in order to understand their rationales for choosing particular strategies as well as their attitudes toward instruction that emphasized multiple strategies. Findings suggest that students gained both knowledge of and appreciation for multiple strategies, but they did not always use alternate strategies. Familiarity, understandability, efficiency, and form of the problem were all considerations for strategy choice. Practical and theoretical implications are discussed.

Early Mathematics Learning in Perspective: Eras and Forces of Change

The character of early mathematics education has changed over the last century not only in terms of the pedagogical approaches to teaching young children, but also in relation to the content and goals of that instruction. In this chapter, we explore psychological, sociocultural, and neurophysiological developments that may have helped to shape these pedagogical trends in early mathematics education. For example, early childhood educators have traditionally advocated for learning through play, but views of its nature and purpose have varied markedly among educational researchers and practitioners. In part, these variations exist as a function of the changing beliefs about children’s cognitive capabilities and about the role of early educational experiences in enhancing the capabilities of young minds. They also mirror shifting psychological orientations toward learning and philosophical perspectives toward knowledge and knowing. An examination of the writings of such influential theorists as Dewey, Thorndike, Piaget, Vygotsky, Flavell, and Rogoff serve to illuminate these shifting orientations toward young minds, mathematics, and the learning of mathematics. Drawing on these writings, we chart the course of early mathematics education in relation to these theoretical underpinnings, consider emerging trends, and address the implications for early mathematics research and practice.