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Fairness of Dice: A Longitudinal Study of Students' Beliefs and Strategies for Making Judgments

Jane M. Watson and Jonathan B. Mortiz

One hundred eight students in Grades 3, 5, 6, 7, and 9 were asked about their beliefs concerning fairness of dice before being presented with a few dice (at least one of which was “loaded') and asked to determine whether each die was fair. Four levels of beliefs about fairness and four levels of strategies for determining fairness were identified. Although there were structural similarities in the levels of response, the association between beliefs and strategies was not strong. Three or four years later, we interviewed 44 of these students again using the same protocol. Changes and consistencies in levels of response were noted for beliefs and strategies. The association of beliefs and strategies was similar after three or four years. We discuss future research and educational implications in terms of assumptions that are often made about students' understanding of fairness of dice, both prior to and after experimentation.

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Experimental Programs: An interim report on the national longitudinal study of mathematical abilities*

Leonard S. Cahen

Edited by Eugene D. Nichols

In September of 1962 approximately 110,000 students took their first set of inventories as participants in the National Longitudinal Study of Mathematical Abilites (NLSMA).

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The Development of Mathematics Achievement Tests for the National Longitudinal Study of Mathematical Abilities

Thomas A. Romberg and James W. Wilson

The purpose of this paper is to demonstrate by a few examples how the mathematics achievement tests used in the National Longitudinal Study of Mathematical Abilities (NLSMA) were developed.

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The Results of a Longitudinal Study in Mathematics in New Hampshire

Gilbert R. Austin and Fernand Prevost

A series of longitudinal studies of mathematical computational ability at grades eight and ten in the state of New Hampshire produced the following results. In the fall of 1963. using the Metropolitan Achievement Test and the Arithmetic Computational subtest of that battery, the median raw score for eighth graders in New Hampshire equalled a grade equivalent of 8.8.

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Sex and Ethnic Group Differences in Mathematics Achievement: Results from the National Longitudinal Study

Elsie G. J. Moore and A. Wade Smith

The mathematics test achievement of 11 914 young men and women, aged 15 to 22, who participated in the National Longitudinal Study of Youth Labor Force Behavior in 1980 was analyzed to assess sex, education, and ethnic group effects. The mathematics knowledge and arithmetic reasoning subtests of the Armed Services Vocational Aptitude Battery were the measures of achievement. A multivariate analysis of variance revealed significant main effects for sex, education, and ethnic group membership. Generally, the mathematics achievement of all groups increased the longer they were in school. White males showed the greatest benefit of continued education, and blacks as a group showed the smallest. However, significant two-way interaction effects confounded these general trends.

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A Longitudinal Study of Invention and Understanding in Children's Multidigit Addition and Subtraction

Thomas P. Carpenter, Megan L. Franke, Victoria R. Jacobs, Elizabeth Fennema, and Susan B. Empson

This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1—3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.

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A Longitudinal Study of Students Completing Four Years of UCSMP Mathematics

Daniel B. Hirschhorn

This article reports on the achievement and attitude aspects of a study comparing students who had the first 4 years of the University of Chicago School Mathematics Project (UCSMP) secondary curriculum to two distinct groups of comparable students, an age cohort and a mathematics course level cohort. A case study design with schools in three different sites (one urban, two suburban) was used. Three instruments were given: a Mathematics Level I Achievement Test from the Educational Testing Service, a 30-item Applications Test, and a 25-item Student Opinion Survey. At two of the sites, the UCSMP students outperformed both the age and course level cohorts by substantial amounts on both the achievement and application tests. At the third site, both comparison cohorts outperformed the UCSMP students on the achievement test, but results on the application test were mixed. At all three sites, there was little difference in the attitude items except on items concerning calculator use.

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A Longitudinal Study of Learning to Use Children's Thinking in Mathematics Instruction

Elizabeth Fennema, Thomas P. Carpenter, Megan L. Franke, Linda Levi, Victoria R. Jacobs, and Susan B. Empson

This study examined changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period in which the teachers participated in a CGI (Cognitively Guided Instruction) teacher development program that focused on helping the teachers understand the development of children's mathematical thinking by interacting with a specific research-based model. Over the 4 years, there were fundamental changes in the beliefs and instruction of 18 teachers such that the teachers' role evolved from demonstrating procedures to helping children build on their mathematical thinking by engaging them in a variety of problem-solving situations and encouraging them to talk about their mathematical thinking. Changes in the instruction of individual teachers were directly related to changes in their students' achievement. For every teacher, class achievement in concepts and problem solving was higher at the end of the study than at the beginning. In spite of the shift in emphasis from skills to concepts and problem solving, there was no overall change in computational performance. The findings suggest that developing an understanding of children's mathematical thinking can be a productive basis for helping teachers to make the fundamental changes called for in current reform recommendations.

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Psychological Imprisonment or Intellectual Freedom? A Longitudinal Study of Contrasting School Mathematics Approaches and Their Impacton Adults' Lives

Jo Boaler and Sarah Kate Selling

In a previous study of 2 schools in England that taught mathematics very differently, the first author found that a project-based mathematics approach resulted in higher achievement, greater understanding, and more appreciation of mathematics than a traditional approach. In this follow-up study, the first author contacted and interviewed a group of adults 8 years after they had left the 2 schools to investigate their knowledge use in life. This showed that the young adults who had experienced the 2 mathematics teaching approaches developed profoundly different relationships with mathematics knowledge that contributed towards the shaping of different identities as learners and users of mathematics (Boaler & Greeno, 2000). The adults from the project-based school had also moved into significantly more professional jobs, despite living in one of the lowest income areas of the country. In this article, we consider the different opportunities that the 2 school approaches offered for longterm relationships with mathematics and different forms of mathematical expertise that are differentially useful in the 21st century (Hatano & Oura, 2003).

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The Effects of Changes in Mathematical Knowledge on Teaching: A Longitudinal Study of Teachers' Knowledge and Instruction

Yasemin Copur-Gencturk

In this study, I examined the relationship between teachers' mathematical knowledge and instruction. Twenty-one K—8 teachers who were enrolled in a master's program were followed for 3 years to study how their mathematical knowledge and teaching changed over time. The results of multilevel growth models indicated that gains in teachers' mathematical knowledge predicted changes in the quality of their lesson design, their mathematical agenda, and the classroom climate. Analyses of interviews and classroom observation data conducted with a subgroup of teachers revealed that in addition to the gains teachers made in their mathematical knowledge, their exit level of knowledge played a significant role in the quality of the changes in their practices.