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M. Lynn Breyfogle

This department publishes brief news articles, announcements, and guest editorials on current mathematics education issues that stimulate the interest of TCM readers and cause them to think about an issue or consider a specific viewpoint about some aspect of mathematics education. This month, the chair of the TCM Editorial Panel welcomes readers to a new academic year; and the Coaches' Corner suggests ways for math specialists to intrinsically motivate teachers.

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Renee Parker and M. Lynn Breyfogle

This student-friendly rubric helped improve third graders' competencies when explaining solution strategies in writing.

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M. Lynn Breyfogle and Barbara Spotts

What are you doing to grow professionally and improve instruction when no one is looking?

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M. Lynn Breyfogle and Courtney M. Lynch

To analyze students' geometric thinking, use both formative and summative assessments and move students along the van Hiele model of thought.

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M. Lynn Breyfogle and Judith Quander

Share news about happenings in the field of elementary school mathematics education, views on matters pertaining to teaching and learning mathematics in the early childhood or elementary school years, and reactions to previously published opinion pieces or articles. Find detailed department submission guidelines at

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Beth A. Herbel-Eisenmann and M. Lynn Breyfogle

Teachers pose a variety of questions to their students every day. As teachers, we recognize that some questions promote deeper mathematical thinking than others (for more information about levels of questions, see Martens 1999, Rowan and Robles 1998, and Vacc 1993). For example, when asking, “Is there another way to represent or explain what you are saying?” students are given the chance to justify their thinking in multiple ways. The question “What did you do next?” focuses only on the procedures that students followed to obtain an answer. Thinking about the questions we ask is important, but equally important is thinking about the patterns of questions that are asked.

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Amy Roth McDuffie, Kay A. Wohlhuter, and M. Lynn Breyfogle

Thread small changes seamlessly into high-level reasoning tasks to reach all students.

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Brittany L. Hoffman, M. Lynn Breyfogle, and Jason A. Dressler

To bolster students' ability to prove as well as develop mathematical argumentation skills, create an environment in which students must regularly explain and justify their thinking.

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M. Lynn Breyfogle and Lauren E. Williams

Teachers often need to alter mathematical tasks that they find in their district-adopted set of curriculum materials or develop new ones if none is present on a particular topic. However, how to best go about this work is not always clear. How do you make effective decisions about alterations? What should you keep in mind as you consider developing tasks to help your students with a particular idea or misconception? These and other questions were central in our minds as we developed a task to help students learn about elapsed time.

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Kay A. Wohlhuter, M. Lynn Breyfogle, and Amy Roth McDuffie

What mathematics must teachers understand in order to teach elementary school mathematics? Historically, the answer to that question was that they needed to know only the mathematics concepts and procedures they taught. Research on the teaching and learning of mathematics challenges that myth and indicates that the role and substance of mathematics knowledge needed for teaching has expanded. In addition to a profound understanding of fundamental mathematics content (Ma 1999), teachers need deep knowledge about how students learn particular math concepts and processes and, correspondingly, which teaching approaches and strategies are most effective in meeting students' learning needs. Ball, Hill, and Bass (2005) refer to this kind of knowledge as mathematics knowledge for teaching. For example, to determine whether a child's unique approach for adding fractions can be generalized or how an area model might be used to develop the partial-product algorithm for multiplying two-digit numbers, teachers need a form of mathematical knowledge different from other professionals (e.g., engineers). Aligned with this research, the Teaching Principle identifies the necessity for teachers to develop deep knowledge and understanding of mathematics to meet the needs of their students today and in the future (NCTM 2000).