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This study investigated teachers’ use of knowledge from research on children’s mathematical thinking and how their students’ achievement is influenced as a result. Twenty first grade teachers, assigned randomly to an experimental treatment, participated in a month-long workshop in which they studied a research-based analysis of children’s development of problem-solving skills in addition and subtraction. Other first grade teachers (n = 20) were assigned randomly to a control group. Although instructional practices were not prescribed, experimental teachers taught problem solving significantly more and number facts significantly less than did control teachers. Experimental teachers encouraged students to use a variety of problem-solving strategies, and they listened to processes their students used significantly more than did control teachers. Experimental teachers knew more about individual students’ problem-solving processes, and they believed that instruction should build on students’ existing knowledge more than did control teachers. Students in experimental classes exceeded students in control classes in number fact knowledge, problem solving, reported understanding, and reported confidence in their problem-solving abilities.

The Implicit Association Test (IAT) is widely used in psychology. Unfortunately, the IAT cannot be run within online surveys, requiring researchers who conduct online surveys to rely on third-party tools. We introduce a novel method for constructing IATs using online survey software (Qualtrics); we then empirically assess its validity. Study 1 (student n = 239) found good psychometric properties, expected IAT effects, and expected correlations with explicit measures for survey-software IATs. Study 2 (MTurk n = 818) found predicted IAT effects across four survey-software IATs (d's = 0.82 [Black-White IAT] to 2.13 [insect-flower IAT]). Study 3 (MTurk n = 270) compared survey-software IATs and IATs run via Inquisit, yielding nearly identical results and intercorrelations expected for identical IATs. Survey-software IATs appear reliable and valid, offer numerous advantages, and make IATs accessible for researchers who use survey software to conduct online research. We present all materials, links to tutorials, and an open-source tool that rapidly automates survey-software IAT construction and analysis.

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This study documents how teachers who participated in a professional development program on understanding the development of students’ mathematical thinking continued to implement the principles of the program 4 years after it ended. Twenty-two teachers participated in follow-up interviews and classroom observations. All 22 teachers maintained some use of children’s thinking and 10 teachers continued learning in noticeable ways. The 10 teachers engaged in generative growth (a) viewed children’s thinking as central, (b)possessed detailed knowledge about children’s thinking, (c) discussed frameworks for characterizing the development of children’s mathematical thinking, (d) perceived themselves as creating and elaborating their own knowledge about children’s thinking, and (e) sought colleagues who also possessed knowledge about children’s thinking for support. The follow-up revealed insights about generative growth, sustainability of changed practice and professional development.

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.

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