Maryl Gearhart

Social processes in early number development

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Teachers’ shifting assessment practices in the context of educational reform in mathematics

Abstract This paper presents a study of primary and secondary mathematics teachers’ changing assessment practices in the context of policy, stakeholder, and personal presses for change. Using survey and interviews, we collected teachers’ reports of their uses of three forms of assessment, one linked to traditional practice (exercises), and two linked to reforms in mathematics education (open-ended problems and rubrics). Findings revealed several trajectories of change in the interplay between assessment forms and the functions that they serve. Teachers may implement new assessment forms in ways that serve ‘old’ functions; teachers may re-purpose ‘old’ assessment forms in ways that reveal students’ mathematical thinking. Our developmental framework provides a way to understand the dynamics of teacher development in relation to ongoing educational reforms.

Assessment Portfolios as Opportunities for Teacher Learning

This article is an analysis of the role of assessment portfolios in teacher learning. Over 18 months, 23 science teachers developed, implemented, and evaluated assessments to track student learning, supported by portfolio tasks and resources, grade-level colleagues, and team facilitators. Evidence of teacher learning included (a) portfolios of a sample of 10 teachers and (b) the cohorts self-reports in surveys and focus groups. Teachers gained understanding of assessment planning, tasks and scoring guides, methods of analyzing patterns and trends, and use of evidence to guide instruction. Teachers made uneven progress with technical aspects of assessment and curriculum-specific assessment. Research is needed on ways to integrate the benefits of a generic portfolio with strategies to strengthen specific areas of assessment expertise.

Coordinating Numeric and Linear Units: Elementary Students’ Strategies for Locating Whole Numbers on the Number Line

Two investigations of fifth graders’ strategies for locating whole numbers on number lines revealed patterns in students’ coordination of numeric and linear units. In Study 1, we investigated the effects of context on students’ placements of three numbers on an open number line. For one group (n = 24), the line was presented in a thematic context as a “race course,” and, for a second group (n = 24), the line was presented as a conventional number line. Most students in both groups placed consecutive whole numbers at appropriate linear distances, but the thematic context group was more likely to place nonconsecutive whole numbers at appropriate linear distances. In Study 2 (n = 24), students placed numbers on lines marked with two numbers. Most students placed a third number appropriately when the marked numbers were consecutive whole numbers, but not when the labeled numbers were nonconsecutive whole numbers. The findings reveal fifth graders’ conceptual difficulties in coordinating numeric and linear units on the number line and a thematic context that can support this coordination.

Mathematics Knowledge for Teaching: Questions about Constructs

I propose that a single construct underlies all the measures described in this series of studies, and that construct is deep knowledge of mathematics, not a mathematics different from that known by mathematicians or mathematically competent adults, and not a superficial understanding of algorithms that is often the outcome of students’ learning in the current system of mathematics education. Knowledge of students and pedagogical techniques may also be important, but the study described in these articles does not address the importance of pedagogical techniques. Knowledge of students and pedagogy must be built on a foundation of deep understanding of mathematics which is not algorithmic prowess, but an ability to communicate mathematically, represent mathematics in different forms, and engage in problem-solving and reasoning mathematically.