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Cynthia W. Langrall

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Excerpts from discussion threads on the online MyNCTM community

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Neet Priya Bajwa and Jennifer M. Tobias

Two types of tasks have the potential to help second graders attend to structure and use initialmultiplicative ideas in their strategies to find the total.

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Jessica Hunt and Mary Kay Stein

Learning goals differ from performance goals. We elaborate on their function and importance as the guiding force behind maintaining cognitive rigor during mathematics learning.

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Rui Kang, Sheri Johnson, Emily Lambert, and Candi Davidson

Inspired by earlier articles on the classic drawing toy that targeted higher grade levels, the authors adopt and create activities that make Spirograph mathematics accessible to middle-grade learners.

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Sam Rhodes

Teachers can use the SCAMPER framework to help students understand and appreciate rich mathematical connections in topics such as functions. The framework facilitates critical and creative thinking by allowing students to explore concepts through open mathematics.

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Jennifer Ward

Four-year-old children used mathematics to examine the location of and access to public playgrounds in their community. They collected and recorded data, using the information to advocate for a social justice issue.

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Per Nilsson

This study introduces inferentialism and, particularly, the Game of Giving and Asking for Reasons (GoGAR), as a new theoretical perspective for investigating qualities of procedural and conceptual knowledge in mathematics. The study develops a framework in which procedural knowledge and conceptual knowledge are connected to limited and rich qualities of GoGARs. General characteristics of limited GoGARs are their atomistic, implicit, and noninferential nature, as opposed to rich GoGARs, which are holistic, explicit, and inferential. The mathematical discussions of a Grade 6 class serve the case to show how the framework of procedural and conceptual GoGARs can be used to give an account of qualitative differences in procedural and conceptual knowledge in the teaching of mathematics.

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Paul A. Frisoli and Richard A. Andrusiak

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.