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• Author or Editor: Chris Rasmussen
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## Pedagogical Content Tools: Integrating Student Reasoning and Mathematics in Instruction

Teaching in a manner consistent with reform recommendations is a challenging and often overwhelming task. Part of this challenge involves using students' thinking and understanding as a basis for the development of mathematical ideas (cf. NCTM, 2000). The purpose of this article is to address this challenge by developing the notion of pedagogical content tool. A pedagogical content tool is a device such as a graph, diagram, equation, or verbal statement that a teacher intentionally uses to connect to student thinking while moving the mathematical agenda forward. We tender two examples of pedagogical content tools: Transformational record and generative alternative. These two pedagogical content tools are put forth as instructional counterparts to the Realistic Mathematics Education (RME) design heuristics of emergent models and guided reinvention, respectively. We illustrate the pedagogical content tools of transformational record and generative alternative by drawing on examples from two classroom teaching experiments in undergraduate differential equations.

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## A Method for Using Adjacency Matrices to Analyze the Connections Students Make Within and Between Concepts: The Case of Linear Algebra

The central goals of most introductory linear algebra courses are to develop students' proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students' interpretation of and connections between concepts. Three cases provide examples that illustrate the usefulness of this approach for comparing differences in the structure of the connections, as exhibited in what we refer to as dense, sparse, and hub adjacency matrices. We also make use of mathematical constructs from digraph theory, such as walks and being strongly connected, to indicate possible chains of connections and flexibility in making connections within and between concepts. We posit that this method is useful for characterizing student connections in other content areas and grade levels.

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## Effects of Standards-Based Mathematics Education: A Study of the Core-Plus Mathematics Project Algebra and Functions Strand

To test the vision of Standards–based mathematics education, we conducted a comparative study of the effects of the Core-Plus Mathematics Project (CPMP) curriculum and more conventional curricula on growth of student understanding, skill, and problem-solving ability in algebra. Results indicate that the CPMP curriculum is more effective than conventional curricula in developing student ability to solve algebraic problems when those problems are presented in realistic contexts and when students are allowed to use graphing calculators. Conventional curricula are more effective than the CPMP curriculum in developing student skills in manipulation of symbolic expressions in algebra when those expressions are presented free of application context and when students are not allowed to use graphing calculators.

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## Brief Report: Characteristics of Precalculus Through Calculus 2 Programs: Insights From a National Census Survey

We present findings from a recently completed census survey of all mathematics departments in the United States that offer a graduate degree in mathematics. The census survey is part of a larger project investigating institutional features that influence student success in the introductory mathematics courses that are required of most STEM majors in the United States. We report the viewpoints of departments about characteristics shown to support students' success as well as the extent to which these characteristics are being implemented in programs across the country. We conclude with a discussion of areas where we see the potential for growth and further improvement.

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## New Assessments for New Standards: The Potential Transformation of Mathematics Education and Its Research Implications

### NCTM Research Committee

During the past 2 decades, significant changes in mathematics curriculum standards and policies have brought greater attention to assessment instruments, practices, purposes, and results. In moving toward stronger accountability, the No Child Left Behind Act (NCLB) of 2001 (NCLB, 2002) mandates that school districts receiving funding under NCLB formulate and disseminate annual local report cards that include information on how students and each school in the district performed on state assessments. This mandate has not only facilitated a growth in state testing (Wilson, 2007) but also influenced the teaching of mathematics (Seeley, 2006). More recently, the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) crafted and launched the Common Core State Standards for Mathematics (NGA Center & CCSSO, 2010), which have been formally adopted by the vast majority of U.S. states and territories. The Common Core State Standards for Mathematics (CCSSM) specifies standards for mathematical content by grade in K–8 and by conceptual categories at the secondary level and identifies key Standards for Mathematical Practice that should be present in K–12 instruction. The CCSSM represents an unprecedented initiative to raise academic standards in school mathematics that will inevitably influence the development of curriculum materials, teaching, and assessment practices.

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## Trends and Issues in High School Mathematics: Research Insights and Needs

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## Reporting Research for Practitioners: Proposed Guidelines

The National Council of Teachers of Mathematics (NCTM) espouses priorities to foster stronger linkages between mathematics education research and teaching practice. Of the five foundational priorities, one is directly focused on research, indicating NCTM's commitment to “ensure that sound research is integrated into all activities of the Council” (NCTM, n.d.). Another priority specifically references the relationship between research and mathematics teaching; the priority on curriculum, instruction, and assessment states that NCTM pledges to “Provide guidance and resources for developing and implementing mathematics curriculum, instruction, and assessment that are coherent, focused, well-articulated, and consistent with research in the field [emphasis added], and focused on increasing student learning” (NCTM, n.d.).