This paper proposes a framework for identifying the mathematical knowledge teachers activate when using curriculum resources. We use the term knowledge of curriculum embedded mathematics (KCEM) to refer to the mathematics knowledge activated by teachers when reading and interpreting mathematical tasks, instructional designs, and representations in mathematics curriculum materials. The KCEM framework is situated within existing research on content-specific teacher knowledge. Through analysis of elementary mathematics teacher's guides from the USA, we identified elements of curriculum resources teachers interact with when using them to plan instruction. These findings were complemented by interviews of teachers using curriculum guides to plan lessons, uncovering how they interacted with different elements of their guides. From this analysis, we propose four overlapping dimensions of KCEM: foundational mathematical ideas, representations and connections among these ideas, problem complexity, and mathematical learning pathways. Using an excerpt from one elementary curriculum guide and one of the teacher interviews on using this guide to plan a lesson, we illustrate how these dimensions might be activated as teachers read and use their guides.
In this article, we describe how using prediction during instruction can create learning opportunities to enhance the understanding and doing of mathematics. In doing so, we characterize the nature of the predictions students made and the levels of sophistication in students' reasoning within a middle school algebra context. In this study, when linear and exponential functions were taught, prediction questions were posed at the launch of the lessons to reflect the mathematical ideas of each lesson. Students responded in writing along with supportive reasoning individually and then discussed their predictions and rationale. A total of 395 prediction responses were coded using a dual system: sophistication of reasoning, and the mechanism students appeared to utilize to formulate their prediction response. The results indicate that using prediction provoked students to connect among mathematical ideas that they learned. It was apparent that students also visualized mathematical ideas in the problem or the possible results of the problem. These results suggest that using prediction in fact provides learning opportunities for students to engage in mathematical sense making and reasoning, which promotes students' understanding of the mathematics that they learn.
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