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Esther M. H. Billings

Explore how this teacher incorporates the Standards for Mathematical Practice (SMP) into her instruction to help young learners use representations, look for patterns, notice underlying mathematical structure, and begin to generalize relationships.

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Esther M. H. Billings

Several problems promoting proportion sense, reasoning about quantities, and the various relationships among quantities in proportions.

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Esther M. H. Billings and Melanie Schultz McClure

How the context of mailing a package was used with seventh-grade students to explore and connect the different representations of step and linear functions. Students will learn geometry concepts and algebra concepts through this hands-on, real world application.

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Esther M. H. Billings and Charlene E. Beckmann

Most students love a good story! When we use children's literature as a context to explore functional relationships, our students are more motivated to learn. In this article, we share how we use children's literature in courses for prospective middle-grades teachers to explore and deepen these teachers' understanding of functions. We expect prospective teachers to teach functions to their future middle-grades students with understanding; therefore, we use literature to help them develop a personal understanding of functions that is connected and comprehensive.

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Esther M. H. Billings and Tracy Lakatos

Students' informal algebraic encounters in grades K–8 lay the foundation for more formalized study of algebra in the future (NCTM 2000; Yackel 1997). The study of functions and relationships is essential in algebra. Functions and relations involve variables and describe how the change in one variable causes a change in a second variable. Words, tables, graphs, and formulas can be used to describe these relationships, which can also be found in many aspects of real life. Consider the following two examples: (1) the darkness of a piece of toast is a function of the setting on the toaster (Ritchhart 1997) and (2) the length of time required to ride to school on the bus can also be a function of the distance between the bus stop and the school. These relationships were described in words, but they can also be described by using a graph, a table, and depending on the exact nature of the relationship, even a formula. Many children do not realize that their everyday lives are filled with such functions and relations (Ritchhart 1997), and they have limited opportunities to interpret and analyze change graphically. Because students experience difficulty in interpreting graphs (Janvier 1981), they must be given opportunities to both interpret and create them.

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David C. Coffey and Esther M. H. Billings

Why is it so difficult for student teachers to consistently understand, apply, and retain important content from professional literature? They must be taught strategies to read in this specific genre.

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Esther M.H. Billings, Tarah L. Tiedt, and Lindsey H. Slater

Dorothy, a second grader at an at-risk school where 87 percent of the students receive free or reduced lunches, watched intently as Mrs. T laid out pattern blocks to create a sequence of figures shaped like people (see fig. 1). When Mrs. T asked Dorothy, “What will the next one look like?” Dorothy replied, “Well, if you wanted to go from [the first person] all the way to four … you would put a head, then the body, hands, then the feet. And then you would put a hat, and another hat, and another one, and another one …. I think that's heavy for a person.” Dorothy thought the hats looked a bit top-heavy; nevertheless, she extended the pictorial growth pattern by correctly building the fourth person (see fig. 1).

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Esther M. H. Billings, David C. Coffey, John Golden, and Pamela J. Wells

A professional development workshop supports teachers' understanding of the Standards for Mathematical Practice and helps them transfer this knowledge to the middle school classroom.