Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction

Abstract

In this article we propose that an understanding of students' thinking can provide coherence to teachers' pedagogical content knowledge and their knowledge of subject matter, curriculum, and pedagogy. We describe a research-based model of children's thinking that teachers can use to interpret, transform, and reframe their informal or spontaneous knowledge about students' mathematical thinking. Our major thesis is that children enter school with a great deal of informal or intuitive knowledge of mathematics that can serve as the basis for developing much of the formal mathematics of the primary school curriculum. The development of abstract symbolic procedures is characterized as progressive abstractions of students' attempts to model action and relations depicted in problems. Although we focus on one facet of teachers' pedagogical content knowledge, we argue that understanding students' thinking provides a basis for teachers to reconceptualize their own knowledge more broadly.

Beginning teachers moving toward effective elementary science teaching

“…Considering one's students and trying to anticipate their ideas, identify their actual ideas, and work with them in instruction all contribute to better understanding learners (Carpenter, Fennema, & Franke, 1996;Smith & Neale, 1989). Smith (1999) advocated that interviewing children about content is crucial for preservice teachers, noting, Reading articles that describe research on children's thinking about how plants get their food is interesting.…”

Evaluation of Automated Diagnosis and Instruction for Mathematics Problem‐solving

“…The latter approach reflects a more traditional, summative approach to assessment; or, what they "know" that happens to be incorrect, which reflects formative assessments that focus on identifying common errors and providing feedback based on those assessments. Underwood and Underwood (2007) described an approach with the idea that to build on the KSAs a student has, it is essential to first find out what they are (Carpenter, Fennema, & Franke, 1996). Trying to build on KSAs that a student does not have is misguided.…”

From Data Through Coding Toward Categorizing: Negotiating Data Analysis in Constructivist Grounded Theory

“…There is a core of research (Carpenter, Fennema, & Franke, 1996;Fagnant, 2005;Julo, 2002;Ng & Lee, 2009;Novotná, 1999;Stacey & MacGregor, 1999) that examines various ways of supporting students in problem-solving activities, including representing and modeling. Arithmetic problems with simple additive structures that can be solved using one addition or subtraction operation are not usually considered to require any mathematical modeling.…”