Conjectures and Refutations in Grade 5 Mathematics
David A. Cramer
This article makes a contribution toward clarifying what mathematical reasoning is and what it looks like in school contexts. It describes one pattern of reasoning observed in the mathematical activity of students in a Grade 5 class and discusses ways in which this pattern is or is not mathematical in order to clarify the features of a pattern of reasoning that are important for making such a judgment. The pattern involves conjecturing a general rule, testing that rule, and then either using it for further exploration, rejecting it, or modifying it. Each element of reasoning in this pattern is described in terms of the ways of reasoning used and the degree of formulation of the reasoning. A distinction is made between mathematical reasoning and scientific reasoning in mathematics, on the basis of the criteria used to accept or reject reasoning in each domain.
Young Children's Representations of Groups of Objects: The Relationship Between Abstraction and Representation
Yasuhiko Kato, Constance Kamii, Kyoko Ozaki, Mariko Nagahiro
Sixty Japanese children between the ages of 3 years 4 months and 7 years 5 months were individually interviewed to investigate the relationship between their levels of abstraction (as assessed by a task involving conservation of number) and their levels of representation (as assessed by a task asking for their graphic representation of small groups of objects). The investigation concluded that abstraction and representation are closely related and that children can represent at or below their level of abstraction but not above this level. The educational implication is that educators need to focus more on the mental relationships children make (i.e., their abstraction) because the meaning children can give to conventional symbols depends on their level of abstraction.
Untangling Teachers' Diverse Aspirations for Student Learning: A Crossdisciplinary Strategy for Relating Psychological Theory to Pedagogical Practice
The Learning Principle propounded in Principles and Standards for School Mathematics (NCTM, 2000) rehearses the familiar distinction between facts/procedures and understanding as a central guiding principle of teaching reform. This rhetorical stance has polarized mathematics educators in the "math wars," (Becker & Jacob, 1998), while creating the discursive space for mathematics teaching reform to be reified into a unitary "reform vision" (Lindquist, Ferrini-Mundy, & Kilpatrick, 1997--a vision that teachers can all too easily come to see themselves as implementing rather than authoring. Crossdisciplinarity is a strategy for highlighting the discrete notions of learning that psychology thus far has succeeded in coherently articulating. This strategy positions teachers to consult their own values, interests, and strengths in defining their own teaching priorities, at the same time marshaling accessible, theory-based guidance toward realization of its diverse possibilities.
A Framework for Analyzing Teaching
|A review of <i>Teaching Problems and the Problems of Teaching</i> by Magdalene Lampert(2001).