Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning
Marjorie Henningsen, Mary Kay Stein
In order to develop students' capacities to "do mathematics," classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.
Inconsistency Between a Beginning Elementary School Teacher's Mathematics Beliefs and Teaching Practice
Anne M. Raymond
This study investigates relationships between a beginning elementary school teacher's beliefs and mathematics teaching practices. A proposed model of relationships between beliefs and practice provided a conceptual framework for the examination of factors that influence beliefs, practice, and the level of inconsistency between them. Data were gathered over 10 months through audiotaped interviews, observations, document analysis, and a beliefs survey. Analyses included the categorization and comparison of beliefs and practice and the development of a revised model of relationships between beliefs and practice. Findings indicate that this teacher's beliefs and practice were not wholly consistent. Rather, her practice was more closely related to her beliefs about mathematics content than to her beliefs about mathematics pedagogy. Her beliefs about mathematics content were highly influenced by her own experiences as a student and her beliefs about mathematics pedagogy were primarily influenced by her own teaching practice. However, the extent to which her teacher preparation program influenced either her beliefs or practice was limited.
Teaching Realistic Mathematics Modeling in the Elementary School: A Teaching Experiment With Fifth Graders
Lieven Verschaffel, Erik De Corte
Recent research has convincingly documented elementary school children's tendency to neglect real-world knowledge and realistic considerations during mathematical modeling of word problems in school arithmetic. The present article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution. The learning and transfer effects of an experimental class of 10- and 11-year-old pupils--compared to the results in two control classes--provide support for the hypothesis that it is possible to develop in elementary school pupils a disposition toward (more) realistic mathematical modeling.