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• Author or Editor: Amy Roth McDuffie
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## Flying through Graphs: An Introduction to Graph Theory

The mathematics education community has called for changes in the high school curriculum to increase the emphasis on meaningful problem solving and on topics in discrete mathematics (National Council of Teachers of Mathematics 1989, 1991, 2000). This recommendation resulted from changes in knowledge and revisions in problem-solving needs because of advances in such fields as information processing and computer technology. Including graph theory in the curriculum is one way to meet these goals. Graphs present an opportunity to model and analyze such problem situations as networks and circuits. This activity incorporates basic terminology, concepts, and solution methods of graph theory in the context of solving problems related to air travel.

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## Break the Area Boundaries: Reflect and Discuss

Understanding the concept of area is a challenge for children. Beyond following procedures with formulas, studying area has been a source of confusion for students at all mathematical levels (Nitabach and Lehrer 1996; Thompson and Preston 2004). In fact, National Assessment of Educational Progress (NAEP) data from fourth and eighth graders suggest that many students have incomplete or superficial understanding of area (Martin and Strutchens 2000). Although students continue to struggle, the 2003 NAEP data shows that performance on area items is significantly improving, which may be due to an increased curricular focus on area concepts and the use of manipulatives and visualization (Blume, Galindo, and Walcott 2007).

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## Supporting All Learners in Productive Struggle

To support a growth mindset in students, consider components involving cognitive, social, and emotional aspects so that students can work within their zone of productive stuggle.

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## Tailoring Tasks to Meet Students' Needs

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

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## Lesson Planning with the Common Core

Research, theory, and the Common Core State Standards apply to lesson planning'a process similar to planning a road trip.

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## Supporting Teacher Learning: Teachers Learning about Data Display: Connecting Mathematics and Science Inquiry

Collecting, analyzing, and displaying data provide rich opportunities to connect mathematics and science concepts. However, mathematics and science teacher educators rarely work together to design tasks that connect mathematics and science. In this article, we describe collaboration between a mathematics teacher educator and a science teacher educator that included the design of an inquiry-based project for preservice elementary teachers to draw on the natural connections of these disciplines. We also discuss preservice teacher learning outcomes from the project and present recommendations for teacher educators.

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## Investigations: Connecting Multiplication to Contexts and Language

This department features children's explorations in mathematics and presents teachers with open-ended investigations to enhance mathematics instruction. The tasks invoke problem solving and reasoning, require communication skills, connect various mathematical concepts and principles, and have been classroom tested.

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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).

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## Promoting Mathematical Discourse through Children's Literature

Discourse in mathematics instruction has received considerable attention since the Standards were first published (NCTM 1989, 1991, 2000); however, prompting mathematical discussions and creating an environment that fosters discourse are challenging tasks for teachers (Corwin 1996). Moreover, students who are not used to talking about mathematics may be uncomfortable with or reluctant to participate in discussions. Discourse in mathematics involves expressing and justifying mathematical thinking and ideas. The primary purposes of facilitating discourse are to help students become aware of others' perspectives and strategies, and to clarify and expand students' own thinking and approaches (NCTM 2000).

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## Connecting Children's Mathematical Thinking with Family and Community Knowledge in Mathematics Instruction

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.