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Laurie Hart Reyes and George M. A. Stanic

Edited by Glenda Lappan

A number of concerns have been expressed about the mathematics performance of all students in our schools (see, e.g., McKnight et at. [1987]), but the performance of certain groups is particularly trouble-some. For example, on standardized tests of mathematics achievement, Hispanic students and black students consistently score below their white counterparts during the primary and middle school years. In addition, although female students perform at least as well as male students during the primary school years, some evidence suggests that gender differences in favor of boys begin to appear during the middle school years, particularly on problem-solving and applications task. Are female students black, and Hispanics naturally less able in mathematics? Or do families, schools, and the create of society offer experiences that create these differences? Individual differences in mathematics performance are normal, inevitable. and obviously related to natural ability; but no reason exists to believe that female students, blacks, and Hispanics, as groups, are by nature less able in mathematics. The problem is one of equity.

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Rodney Y. Stutzman and Kimberly H. Race

Teachers spend considerable time and energy working to clarify and communicate their expectations to students. Those expectations are typically communicated when teachers present assignments. Reinforcing the assignments with an assessment process that narrows the feedback focus can benefit teachers and enrich students. For individual assignments, students really need to know the answers to the following questions: Is my work at an acceptable level? Am I on track to earn the grade to which I aspire in this course? They do not need feedback carved into razor-thin slices to make those determinations. We looked for an alternative that allows us to effectively evaluate responses to rich problems within a standards-based curriculum. We hoped to be able to assess written communication, as well as mathematical computation. Promoting student responsibility with a student-centered system was also important. Ultimately, a revised assessment system, which we have come to call EMRF, emerged.

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Eric “Rico” Gutstein

This article provides an example of, and lessons from, teaching and learning critical mathematics in a Chicago public neighborhood high school with a social justice focus. It is based on a qualitative study of my untracked, 12th–grade mathematics class, a full–year enactment of mathematics for social and racial justice. Students were Black and Latin@ from a low–income, working–class community with a tradition of resistance. Any neighborhood student could enroll without selection criteria. The class goal was for students to cocreate a classroom in which they would learn and use collegepreparatory, conceptually based mathematics to study and understand social reality to prepare themselves to change it. Through analyzing my practice, I address possibilities and challenges of curriculum development and teaching, examine student learning, and pose questions and directions for further research and practice.

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Elise Lockwood and Branwen Purdy

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, the authors used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and they sought to answer the following research questions: How do students come to understand and make sense of the MP? Specifically, while a pair of students reinvented a statement of the MP, how did they attend to and reason about key mathematical features of the MP? The students participated in a paired 8-session teaching experiment during which they progressed from a nascent to a sophisticated statement of the MP. Two key mathematical features emerged for the students through this process, including independence and distinct composite outcomes, and we discuss ways in which these ideas informed the students' reinvention of the statement. In addition, we present potential implications and directions for future research.

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Linda B. Griffin

Strategic instructional choices can simultaneously address common decimal misconceptions and help students race toward decimal understanding.

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Diana V. Lambdin, R. Kathleen Lynch and Heidi McDaniel

When asked, “How long do you think it would take you to ride a fifty-mile bicycle race?” a class of Indiana sixth graders offered answers ranging from two to twelve hours. Some admitted that their answers were blatant guesses, whereas others offered interesting and thoughtful rationales for their estimates: a car can drive fifty miles in about one hour, and a cyclist might take twice as long for the same distance; because bicycling on hills is tiring, race times will depend on how hilly the course is; and an individual race takes longer than a relay race because team members can take turns bicycling and resting.

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Joanne Harris

MEET MELANIE PATERSON, race car driver. Melanie currently races in the Canadian Formula Ford Championship series. This form of racing is an entry-level professional open-wheel series. The cars have 1600cc/1.6L engines, race with no aerodynamic aids, and use street radial tires. From Formula Ford 1600, a racer can proceed into Formula 2000, Formula Atlantic, Indy Lights, and the CART (Championship Auto Racing Teams) racing series. With each progression in race series, more powerful engines and higher technology come into play. Melanie's goal is to eventually race in the CART series as well.

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Kate Kline

Childhood isn't a race; it's a journey. We need to think about that and give children adequate time to develop as thinkers, as knowers, as mathematicians.

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Laura E. Christman

Edited by Howard Eves

The history of mathematics has many valuable suggestions for mathematics teachers as well as for students. This follows naturally since we are endeavoring to teach mathematics to young people whose development largely repeats the cultural evolution of the race. So we ask, “How did the race acquire mastery of this or that concept which we would pass on to our pupils?”

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Jill Duea Bergman

Tops and Bottoms by Janet Stevens (1995) presents many opportunities for students to think mathematically within an entertaining, fictional context (fig. 1). After losing a bet in his race with Tortoise.