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Key Recommendation: Middle schools should offer a common shared pathway grounded in the use of mathematical practices and processes to coherently develop deep mathematical understanding, ensuring the highest quality mathematics education for each

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Key Recommendation: Early childhood settings and elementary schools should build a strong foundation of deep mathematical understanding, emphasize reasoning and sense making, and ensure the highest-quality mathematics education for each and every

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Ruby Bostick Midkiff and Mary McCart Cramer

Educators tend to think of children's books as being used primarily in language arts, reading, and, perhaps, science and social studies and limited to specific grade levels. However, the scope should be broadened to include literature as a stepping stone to mathematical understanding. Books that primary-grade students enjoy can facilitate mathematical understanding in older students, and book on higher reading level can be read to younger students for the same reason. Children's literature presents a natural way to connect language and mathematics.

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Jean L. Behrend

The small group of second- and third-grade students in the learning disabilities resource room chanted, “The ones are first. The ones are first. The ones are first!” “Remember,” their teacher said, “when you add numbers together, you start with the numbers in the ones place. The ones are first!” The teacher hoped that this chant would reduce computational errors. When students exhibit difficulties in mathematics, teachers often try a quick fix, such as giving a rule to follow. The rule may appear to solve the immediate problem but could actually interfere with students' development of mathematical understanding.

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Ben A. Sueltz

Several years ago the writer undertook a comprehensive study of the measurement of mathematical understandings and judgments at the elementary and junior high school levels. Phases of this study have been described and results have previously been reported.1 During the past two years the investigation has been carried into the college field. No attempt has been made to evaluate in terms of typical college-level materials. Rather, the materials developed for use in the junior high school grades have been applied to college students in order to determine what types of understandings and judgments have been retained or augmented since their initial period of study. A number of incidental studies such as the relationship of results of tests of understanding and judgment to success in college mathematics and to general college success have been made but the results of these will not be presented at this time. In this report, the implication for shifts in the curricula on the secondary level will be emphasized.

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Melissa C. Gilbert and Lauren E. Musu

This article focuses on learning environments that support students' mathematical understanding and enhance their adaptive motivation. Drawing on our experiences as teachers and teacher educators, we first describe adaptive motivation. We then introduce TARGETTS, a lesson planning and analysis tool, and describe ways in which teachers can use this tool to highlight specific instructional strategies that support both adaptive motivation and mathematical understanding.

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Laura A. Sgroi, Nancy Gropper, Mary Tom Kilker, Nancy M. Rambusch and Barbara Semonite

The mathematics education experienced by a young child in school is influenced by the nature of the child and by at least three other variables: the curriculum, the learning environment, and the means of assessment.

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Wendy S. Bray, Jillian D. Johnson, Nancy Rivera, Lee-Ann Fink, Charity Bauduin and Robert C. Schoen

Teachers engage in sustained peer collaboration about formative assessment with a focus on students' mathematical thinking.

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Dan Smith

Instruction in vectors offers many opportunities to teach fundamental mathematical ideas.

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Anne E. Adams, Jerine Pegg and Melissa Case

Help students develop skills to support comprehension of concepts and terminology.