Bruce, Catherine D

Spatial reasoning and spatial structures are relatively new areas of research in mathematics education. In this study of children exploring spatial conceptions of grid structures, twenty-one children (ages 4-9) were given a series of tasks involving square grids during virtual interviews. As a result of an ideal-type analysis of the qualitative data, a typology of conceptions of grids emerged showing five distinct categories sequenced from very early conceptions of square grids (as a series of isolated cells) to more coordinated structuring (as related and intersecting rows and columns). The five categories - Single Cell Structuring, Partial Unit Building, Whole Figure and Parts-of-Figure Noticing, Composite Unit Structuring and Coordinated Structuring - are described through illustrative examples. Students' gestures, language and diagrams were considered together when constructing the types. Interestingly, the spatial structure of grids was not readily apparent to many students and in fact was found to be complex for students to conceptualize. With minimal research on grids as a spatial structure in the mathematics education research field, there is strong potential for further investigation in this area.

Author Keywords: Grids, Mathematics, Spatial reasoning, Young Children

This study investigates the potential for a responsive model of professional development in mathematics education which acknowledges how teachers perceive the value of professional learning, and examines how those perceptions are connected to teacher efficacy. Three fields of educational research ground this study: (i) professional development strategies in mathematics education, (ii) teacher efficacy, and (iii) self-determination theory and andragogy. Data collection and analysis involved four detailed case studies and a cross-case analysis of similarities and distinctions among the cases, in an instrumental-multiple-case study design. Results suggest: (1) some characteristics of professional development were consistently designated as high or low value, independent of efficacy ratings, (2) other professional learning experiences were valued relative to the participants' sense of efficacy at different times in their careers, and (3) characteristics of professional development designated as high value during periods of low efficacy were fundamentally teacher-centric, but during periods of high efficacy, they were fundamentally student-centric.

Author Keywords: efficacy, mathematics education, mathematics teachers, professional development, professional learning, teacher efficacy