Glendon W. Blume

Justification in the mathematics classroom: A study of prospective elementary teachers

Abstract This study focuses on mathematical justification in the context of a mathematics course for prospective elementary teachers. The course was run as a whole-class constructivist teaching experiment. The analysis of the data provides a detailed look at how classroom norms for mathematical justification were established given the prospective teachers traditional expectations of school mathematics and the teachers reform-oriented notions. Three episodes from the class transcripts are analyzed in this article. They reveal the interactive process of negotiating new classroom norms for doing mathematics, the range of student responses to situations for justifications, and the complex interrelationship between the investigation of mathematical validity and development of mathematical understanding in the classroom community.

Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers

Abstract The authors employed a whole-class teaching experiment in a mathematics class for prospective elementary teachers to study the development of the ability to identify a ratio as the appropriate measure of a given attribute (“ratio-as-measure”). Prior studies as well as pretest data for this study demonstrated that students often invoke additive comparisons when multiplicative comparisons are appropriate. Analysis of the difficulties which students encountered in considering a ratio as an appropriate measure of an attribute of a physical situation led to the hypothesis that an understanding of ratio-as-measure requires, in addition to an understanding of aspects of multiplicative relationships, an understanding of mathematical modeling.

Factors that influence teachers learning to do interviews to understand students' mathematical understandings

Understanding students understanding of mathematical ideas can inform mathematics teaching, and task-based interviews are one way in which teachers can learn more about their students understandings. The CIME project was designed to empower mathematics teachers to use interviews to understand their students mathematical understandings as well as to prepare teachers to use technology-intensive curricula. This study examined the influences on three high school mathematics teachers as they learned to use task-based interviews to understand students mathematical understandings. The areas of teacher knowledge and conceptions that influenced the teachers we studied were: teachers mathematical understandings and knowledge of technology and the perceived importance of curriculum topics; teachers views of knowing mathematics; teachers perceptions of students characteristics and needs; and teachers perceptions of interviewing and the role of questioning in interviews.

A Teacher’s Conception of Definition and Use of Examples When Doing and Teaching Mathematics

To contribute to an understanding of the nature of teachers’ mathematical knowledge and its role in teaching, the case study reported in this article investigated a teacher’s conception of a metamathematical concept, definition, and her use of examples in doing and teaching mathematics. Using an enactivist perspective on mathematical knowledge, the authors give an account of the case of Lily, a prospective, then beginning, teacher who conceived of mathematical definition as an object with particular form and function and engaged in purposeful, specialized use of examples when doing and teaching mathematics. Lily’s case illustrates how a teacher’s interpretation of examples (as exemplifications or single instances) and conception of the form and function of definitions can influence her doing and teaching mathematics. An implication is that teacher preparation should foster teachers’ abilities to use examples purposefully to provide students with rich opportunities to engage in mathematical processes such as defining.