Mathematical comprehension facilitated by situation models: Learning opportunities for inverse relations in elementary school

Abstract

The Common Core State Standards call for more rigorous, focused, and coherent curriculum and instruction, has resulted in students being faced with more cognitively high-demanding tasks which involve forming connections within and between fundamental mathematical concepts. Because mathematical comprehension generally relates back to one’s ability to form connections to prior knowledge, this study sought to examine the extent to which current learning environments expose students to connection-making opportunities that may help facilitate mathematical understanding of elementary multiplicative inverses. As part of an embedded mixed-methods design, I analyzed curriculum materials, classroom instruction, and student assessments from four elementary mathematics teachers’ classrooms. A situation model perspective of comprehension was used for analysis. The aim of this study was thus to determine how instructional tasks, representations, and deep questions are used for connection-making, which is the foundation of a situation model that can be used for inference-making. Results suggest that student comprehension depends more on connection-making opportunities afforded by classroom teachers, rather than on learning opportunities found solely within a curriculum. This included instruction that focused on deeply unpacking side-by-side comparison type examples, situated examples in personal concrete contexts, used semi-concrete representations to illustrate structural relationships, promoted efficiency through the sequence of presented representations, and posed deep questions which supported students’ sense-making and emphasized the interconnectedness of mathematics. By analyzing these key aspects, this study contributes to research on mathematical understanding and provides a foundation for helping students facilitate transfer of prior knowledge into novel mathematical situation.