Teaching the Conceptual Structure of Mathematics

Educational Psychologist (Impact Factor: 3.29). 07/2012; 47(3):189-203. DOI: 10.1080/00461520.2012.667065


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Available from: Lindsey Engle Richland
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    • "Providing conceptual information behind a certain mathematical procedure will help students acquire flexible knowledge which is not bound to the mathematical situation in which the algorithm was learned. Students will therefore be more likely to be able to successfully transfer the acquired procedures to other mathematical problems (Richland et al. 2012). "
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    ABSTRACT: In two studies, we investigated the impact of instructors’ different knowledge bases on the quality of their instructional explanations. In Study 1, we asked 20 mathematics teachers (with high pedagogical content knowledge, but lower content knowledge) and 15 mathematicians (with lower pedagogical content knowledge, but high content knowledge) to provide an explanation about an extremum problem for students. We found that the explanations by teachers and mathematicians mainly differed in their process-orientation. Whereas the teachers mainly presented the solution steps for the problem (product-orientation), the mathematicians also provided information to clarify why a certain step in the solution was required (process-orientation). In Study 2, we investigated the effectiveness of these differing explanations. Eighty students either received a process-oriented mathematician’s explanation, a product-oriented mathematics teacher’s explanation, or no explanation for learning. We found that students who learned with a process-oriented explanation outperformed students who learned with a product-oriented explanation on an application test. Students who only had the problem but no explanation for learning showed the lowest learning gains. Apparently, deep content knowledge helped instructors generate explanations with high process-orientation, a textual feature that served as a valuable scaffold for students’ understanding of mathematical procedures.
    Full-text · Article · Dec 2015 · Instructional Science
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    • "We use problem solving more generally to refer to any process of working from an initial problem state to a solution state. Practice is often incorporated into instruction in math and science classes in the form of problem-solving activities (Richland et al. 2012). Practice is a common pedagogical feature built into most disciplines for the development of expertise (Brown et al. 1989), and there is a very large literature on practice in the cognitive and educational sciences (Chi and Ohlsson 2005; Ericsson et al. 1993; Newell and Rosenbloom 1981; VanLehn 1996). "
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    ABSTRACT: Robust knowledge serves as a common instructional target in academic settings. Past research identifying characteristics of experts’ knowledge across many domains can help clarify the features of robust knowledge as well as ways of assessing it. We review the expertise literature and identify three key features of robust knowledge (deep, connected, and coherent) and four means of assessing these features (perception, memory, problem solving, and transfer). Focusing on the domains of math and science learning, we examine how four instructional techniques—practice, worked examples, analogical comparison, and self-explanation—can promote key features of robust knowledge and how those features can be assessed. We conclude by discussing the implications of this framework for theory and practice.
    Full-text · Article · Mar 2014 · Educational Psychology Review
    • "A more recent classroom case study illustrated how a high school biology teacher expansively framed his classroom by (a) making links to settings outside of school; (b) extending temporal horizons to the past, where content was learned, and to the future, where it remains relevant; (c) connecting curriculum units across time; (d) training students to make connections across topics (cf. Richland et al., 2012/this issue); and (e) positioning students as part of a larger learning community (Engle, Meyer, Clark, White, & Mendelson, 2010; Meyer, Mendelson, Engle, & Clark, 2011). These students scored well on researcher-designed transfer tests as well as on end-of-year standardized tests. "
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    ABSTRACT: When contexts are framed expansively, students are positioned as actively contributing to larger conversations that extend across time, places, and people. A set of recent studies provides empirical evidence that the expansive framing of contexts can foster transfer. In this article, we present five potentially complementary explanations for how expansive framing may promote transfer and outline a research agenda for further investigating them. Specifically, we propose that expansive framing may: (a) foster an expectation that students will continue to use what they learn later, which may affect the learning process in ways that can promote transfer; (b) create links between learning and transfer contexts so that prior learning is viewed as relevant during potential transfer contexts; (c) encourage learners to draw on their prior knowledge during learning, which may involve them transferring in additional examples and making generalizations; (d) make learners accountable for intelligently reporting on the specific content they have authored; and (e) promote authorship as a general practice in which students learn that their role is to generate their own solutions to new problems and adapt their existing knowledge in transfer contexts.
    No preview · Article · Jul 2012 · Educational Psychologist
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