An understanding of fractions eludes many U.S. students, and research-based knowledge about fractions, such as the utility of linear representation, has not broadly influenced instruction. This randomized trial of lesson study supported by mathematical resources assigned 39 educator teams across the United States to locally managed lesson study supported by a fractions lesson study resource kit or to 1 of 2 control conditions. Educators (87% of whom were elementary teachers) self-managed learning over a 3-month period. HLM analyses indicated significantly greater improvement of educators' and students' fractions knowledge for teams randomly assigned to lesson study with resource kits. Results suggest that integrating researchbased resources into lesson study offers a new approach to the problem of “scale-up” by combining the strengths of teacher leadership and research-based knowledge.

### Catherine Lewis and Rebecca Perry

### Carly Ziniuk

Linear models for longevity and height relate family information.

### Christopher Columbus

The purpose of this note is to question the almost exclusive use of linear models in mathematics education research and to urge thoughtful exploration of alternate models when a linear model fits the data of a study poorly. (Here “linear model” refers to a model linear in one or more continuous independent variables, not to a model linear in the parameters to be estimated.)

### Gerhard Sonnert, Melissa D. Barnett and Philip M. Sadler

.5, C– = 71, D+ = 68, D = 64.5, D– = 61, F = 40), an approximation that did introduce some additional uncertainty. Institutional differences and idiosyncrasies in grading systems and policies were a main reason for employing hierarchical linear models

### Estrella Johnson, Christine Andrews-Larson, Karen Keene, Kathleen Melhuish, Rachel Keller and Nicholas Fortune

Hierarchical Linear Model (HLM) 2 to determine the robustness of the effects of IOI and the potential interaction between IOI and gender. The appropriateness of a multilevel modeling approach for these data was determined by the sufficiency of the intraclass

### Gloria B. Barrett

Fitting least-squares lines to bivariate data is a topic traditionally discussed in introductory statistics courses, often in a unit of study that includes correlation. Recently, because calculators that graph bivariate data sets and compute regression equations have become widely available, this topic has also been included in many algebra and precalculus courses. After students learn about linear models, additional discussions may concern transforming data to achieve linearity so that students can find models of other functional forms. They can also use calculators or computers to fit curves without going through the reexpression process; this procedure would be more common in algebra or precalculus than in statistics classes.

### Richard Noss

This exploratory study aimed to investigate some elements of geometrical concepts that children learn through Logo programming. A test designed to probe children's conceptions of three components of length and angle was administered to 84 children who had learned Logo for one year and 92 who had not. The components of the concept of length were (a) length conservation, (b) length combination, and (c) length measurement. The components of angle were (a) right angle conservation, (b) angle conservation, and (c) angle measurement. Analysis of data using a linear modeling technique indicated a consistent trend toward a sex-treatment interaction.

### Berchie W. Holliday and Lauren R. Duff

Mathematics teachers understand that calculators have revolutionized the teaching of secondary school mathematics. After students have demonstrated their abilities to perform such computations without calculators, calculators can free students and teachers from performing redundant computations. Graphing calculators, in particular, free students from computing dependent values needed to construct line graphs, for example. But one problem is how to teach students to use a graphing calculator to plot, calculate, and graph linear equations of best fit from realworld data. Another problem is getting students to engage in the task and construct an increasingly useful conceptualization of linear modeling. In the beginning, teachers should, perhaps, provide direct instruction, followed by modeling how to enter and graph data sets efficiently.

### Larry Sowder

Despite the large number of studies devoted to discovery learning during the last decade (see, for example, Shulman & Keislar, 1966), little statistically supported data have appeared about how much information students require in order to form generahzations. No doubt many teachers, through classroom experience, know that pupils of different intelligence levels and at different grade levels are indeed capable of discovering, but there has also seemed to be a lack of quantitative informati0n about how performance differs among intelligence levels or among grade levels. The study reported here examined the performance of boys and girls from three intelligence levels at grades 4, 5, 6, and 7 on some numerical discovery tasks. In addition, a linear model was posited to see how well easily-obtained data would account for variability in performance.

### Douglas A. Grouws, James E. Tarr, Óscar Chávez, Ruthmae Sears, Victor M. Soria and Rukiye D. Taylan

This study examined the effect of 2 types of mathematics content organization on high school students' mathematics learning while taking account of curriculum implementation and student prior achievement. The study involved 2,161 students in 10 schools in 5 states. Within each school, approximately 1/2 of the students studied from an integrated curriculum (Course 1) and 1/2 studied from a subject-specific curriculum (Algebra 1). Hierarchical linear modeling with 3 levels showed that students who studied from the integrated curriculum were significantly advantaged over students who studied from a subject-specific curriculum on 3 end-of-year outcome measures: Test of Common Objectives, Problem Solving and Reasoning Test, and a standardized achievement test. Opportunity to learn and teaching experience were significant moderating factors.