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• Author or Editor: Erik Tillema
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## Chinese Algebra: Using Historical Problems to Think About Current Curricula

Analyzing three problems from the Jiu Zhang Suanshu, a Chinese mathematics text from the first century BCE. Each problem examines how the Chinese generated equivalent expressions, an important concept in algebra today. Historical text is be adapted thereby algebraic thinking can be honed.

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## Cultivating an Area Model

A rich understanding of multiplying binomials can be developed by working through a series of concrete problems.

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## Math for Real: A Three-Girl Family

Mr. and Mrs. Ortenzi want to have three children. They have picked out three names for a boy and three names for a girl. Since Mrs. Ortenzi wants to have three girls, she would like to learn about the chance that this might occur. To help her determine her chance of having three girls, solve the problems below.

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## Math for Real: A Speck of Dust

Think of any dusty surface. How did the dust get there? What is in dust? How many specks of dust can be found in one swipe of a dust cloth?

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## Designing for Voice and Agency

Specific teacher moves and lesson planning can facilitate student empowerment in the middle school classroom.

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## What Is the Difference? Using Contextualized Problems

Real-world situations involving layers of subtraction will help students in their future work with algebra.

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## Teaching and Learning Fraction Addition on Number Lines

We present a case study of teaching and learning fraction addition on number lines in one 6th-grade classroom that used the Connected Mathematics Project Bits and Pieces II materials. Our main research questions were (1) What were the primary cognitive structures through which the teacher and students interpreted the lessons? and (2) Were the teacher's and her students' interpretations similar or different, and why? The data afforded particularly detailed analyses of cognitive structures used by the teacher and one student to interpret fractions and their representation on number lines. Our results demonstrate that subtle differences in methods for partitioning unit intervals did not seem important to the teacher but had significant consequences for this student's opportunities to learn. Our closing discussion addresses knowledge for teaching with drawn representations and methods for examining interactions between teachers' and students' interpretations of lessons in which they participate together.