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Angeliki Kolovou, Marja van den Heuvel-Panhuizen and Olaf Köller

This study investigated whether an intervention including an online game contributed to 236 Grade 6 students' performance in early algebra, that is, solving problems with covarying quantities. An exploratory quasi-experimental study was conducted with a pretest-posttest-control-group design. Students in the experimental group were asked to solve at home a number of problems by playing an online game. Although boys outperformed girls in early algebra performance on the pretest as well as on the posttest, boys and girls profited equally from the intervention. Implications of these results for educational practice are discussed.

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Mary E. Brenner, Sally Herman, Hsiu-Zu Ho and Jules M. Zimmer

Flexible use of multiple representations has been described as a key component of competent mathematical thinking and problem solving. In this study, 6th-grade American students are compared to 3 samples of Asian (Chinese, Japanese, and Taiwanese) 6th graders to determine if the well-documented mathematical achievement of students from these Asian nations may be due in part to a greater understanding of mathematical representations. The results show that, among all groups, Chinese students generally scored highest on the representation tasks and, except on items about the visual representations of fractions, all Asian samples scored significantly higher than the American sample. The results are discussed in terms of possible instructional antecedents and textbook differences.

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Yepling Li

To illuminate the cross-national similarities and differences in expectations related to students' mathematics experiences between the United States and China, I compared all relevant problems that followed the content presentation of addition and subtraction of integers in several American and Chinese mathematics textbooks. A 3-dimensional framework (for mathematical features, contextual features, and performance requirements) was developed in this study to analyze these textbook problems. The results show that the percentage differences in problems' dimensions, mathematical and contextual features, were smaller than the difference in problems' performance requirements. Specifically, the differences found in problems' performance requirements indicate that the American textbooks included more variety in problem requirements than the Chinese textbooks. The results are relevant to documented cross-national differences in American and Chinese students' mathematical performances.

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Julie E. Riordan and Pendred E. Noyce

Since the passage of the Education Reform Act in 1993, Massachusetts has developed curriculum frameworks and a new statewide testing system. As school districts align curriculum and teaching practices with the frameworks, standards-based mathematics programs are beginning to replace more traditional curricula. This paper presents a quasi-experimental study using matched comparison groups to investigate the impact of one elementary and one middle school standards-based mathematics program in Massachusetts on student achievement. The study compares statewide standardized test scores of fourth-grade students using Everyday Mathematics and eighth-grade students using Connected Mathematics to test scores of demographically similar students using a mix of traditional curricula. Results indicate that students in schools using either of these standards-based programs as their primary mathematics curriculum performed significantly better on the 1999 statewide mathematics test than did students in traditional programs attending matched comparison schools. With minor exceptions, differences in favor of the standards-based programs remained consistent across mathematical strands, question types, and student sub-populations.

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Michelle Stephan and Didem Akyuz

This article presents the results of a 7th-grade classroom teaching experiment that supported students' understanding of integer addition and subtraction. The experiment was conducted to test and revise a hypothetical learning trajectory so as to propose a potential instructional theory for integer addition and subtraction. The instructional sequence, which was based on a financial context, was designed using the Realistic Mathematics Education theory. Additionally, an empty, vertical number line (VNL) is posited as a potentially viable model to support students' organizing their addition and subtraction strategies. Particular emphasis is placed on the mathematical practices that were established in this setting. These practices indicate that students can successfully draw on their experiences with assets, debts, and net worths to create meaning for integer addition and subtraction.

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Debra I. Johanning

This article describes 1 prevalent practice that a group of 6th- and 7th-grade students engaged in when they used fractions in the context of area and perimeter, decimal operations, similarity, and ratios and proportions. The study's results revealed that students did not simply take the concepts and skills learned in formal fractions units and use them in these other mathematical content areas. Their understanding of how to use fractions was tied to their understanding of situations in which they could be used. Students had to take into account both mathematical and situational contexts when making choices about how to use fractions. This led students to raise questions regarding what was appropriate when using fractions in these new contexts and how fractions and the new contexts were related.

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Curtis L. Pyke

This article reports on the results of a study that investigated the strategic representation skills of eighth-grade students while they were engaged in a set of tasks that involved applying geometric knowledge and using algebraic equations. The strategies studied were derived from Dual Coding Theory (DCT) (Paivio, 1971, 1990), and they were elicited with task-specific prompts embedded in an assessment developed for the study. The purpose of the study was to test a model that highlights strategic representation as a mediator of the effects of reading ability, spatial ability, and task presentation on problem solving. The proposed model was tested using the linear structural equations modeling approach to causal analysis and the data did not reject the model. The results showed that students' use of symbols, words, and diagrams to communicate about their ideas each contribute in different ways to solving tasks and reflect different kinds of cognitive processes invested in problem solving.

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Stephen J. Pape

Many children read mathematics word problems and directly translate them to arithmetic operations. More sophisticated problem solvers transform word problems into object-based or mental models. Subsequent solutions are often qualitatively different because these models differentially support cognitive processing. Based on a conception of problem solving that integrates mathematical problem-solving and reading comprehension theories and using constant comparative methodology (Strauss & Corbin, 1994), 98 sixth- and seventh-grade students' problem-solving behaviors were described and classified into five categories. Nearly 90% of problem solvers used one behavior on a majority of problems. Use of context such as units and relationships, recording information given in the problem, and provision of explanations and justifications were associated with higher reading and mathematics achievement tests, greater success rates, fewer errors, and the ability to preserve the structure of problems during recall. These results were supported by item-level analyses.

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Robert E. Reys and Der-Ching Yang

This research provides information on the number sense of Taiwanese students in Grades 6 and 8. Data were collected with separate tests on written computation and number sense. Seventeen students were interviewed to learn more about their knowledge of number sense. Taiwanese students' overall performance on number sense was lower than their performance on written computation. Student performance on questions requiring written computation was significantly better than on similar questions relying on number sense. There was little evidence that identifiable components of number sense, such as use of benchmarks, were naturally used by Taiwanese students in their decision making. This research supports the need to look beyond correct answers when computational test results are reported.

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Kristen N. Bieda

Discussions about school mathematics often address the importance of reasoning and proving for building students' understanding of mathematics. However, there is little research examining how teachers enact tasks designed to engage students in justifying and proving in the classroom. This article presents results of a study investigating the processes and outcomes of implementing proof-related tasks in the classroom. Data collection consisted of observations of 7 middle school classrooms during implementation of proof-related tasks—tasks providing opportunities for students to produce generalizations, conjectures, or proofs—in the Connected Mathematics Project (CMP) curriculum by teachers experienced in using the materials. The findings suggest that students' experiences with such tasks are insufficient for developing an understanding of what constitutes valid mathematical justification.