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In this study, we examined the ability of grades 1-8 pre-service teachers (PSTs) to engage in thinking about patterns, relationships, and functional rules. Using the algebraic habit of mind Building Rules to Represent Functions as a framework, we examined whether and how well our PSTs (n = 18) used seven features of this habit of mind: organize information, predict patterns, chunk information, use different representations, describe a rule, describe change, and justify a rule. The data revealed distinct and significant differences among our PSTs' demonstrated abilities to justify a rule, organize information, predict a pattern, and describe a rule. Their ability to justify a rule was the weakest of the seven features, and it was correlated with the ability to predict patterns and the ability to chunk information. Our analysis reveals the complexity of the relationships among the seven features and provides a tentative grouping of features that together might support one another. Attention to these relationships has the potential to enhance instruction aimed at helping students learn to analyse problem situations that focus on writing rules to represent functional relationships.

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers' ability to use the features of the algebraic thinking (AT) habit of mind "Building Rules to Represent Functions" and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers' own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers' ability to recognize and interpret students' facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students' written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks' potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers' own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students' written work. Implications for mathematics teacher education are discussed.

In this study, we used problem posing as a measure of the effect of middle-school curriculum on students' learning in high school. Students who had used a standards-based curriculum in middle school performed equally well or better in high school than students who had used more traditional curricula. The findings from this study not only show evidence of strengths one might expect of students who used the standards-based reform curriculum but also bolster the feasibility and validity of problem posing as a measure of curriculum effect on student learning. In addition, the findings of this study demonstrate the usefulness of employing a qualitative rubric to assess different characteristics of students' responses to the posing tasks. Instructional and methodological implications of this study, as well as future directions for research, are discussed.

Assessed the effectiveness of instruction in promoting sixth-grade students' conceptual understanding of the concept of averages using a leveling model and an open-ended problem-solving approach. Found that students were more capable of using the leveling approach and applying formal averaging algorithm to solve problems than using the open-ended approach. (JPB)

Students in grades three to seven were given tests on which problems were presented in either verbal or telegraphic format. Only in grade six was a significant difference found, favoring the verbal format. Problems with conventional syntax appeared to be easier to interpret than those with bare bones syntax. (MNS)

Eight problems in each of three formats were administered to 854 students in grades three through seven. Readers of high ability chose correct operations more often than low-ability readers. The drawn format was easier than the other two formats; its advantage was greater for low-ability readers than high-ability readers. (MNS)

The need of learning-disabled students for more than extensive practice is discussed. The rigid thinking and error reinforcement that may result are noted, and a solution involving the analysis of students' needs is described in some detail. (MNS)

• Curriculum matters for teachers' implementing high-level instructional tasks. • The higher cognitive demand of tasks implemented, the higher students' growth. • Increases in cognitive demand enhance growth rates in math learning. • Controlling for the cognitive demand, student growth on problem solving is similar. • Controlling for the cognitive demand, CMP students have a lower computation growth. This paper compares the longitudinal effect of instructional tasks on algebra learning that used a Standards -based curriculum [Connected Mathematics Project (CMP)] to that of classrooms that used a traditional curriculum (non-CMP). CMP was developed based on the National Council of Teachers of Mathematics (NCTM) Standards and can be characterized as a problem-based curriculum. CMP teachers were more than three times as likely to implement high-level instructional tasks than non-CMP teachers. Increases in the cognitive demand were associated with enhanced growth rates in problem solving, computation, and equation solving. Notably, when controlling for the cognitive demand of the instructional tasks, the advantage of the CMP curriculum over the non-CMP curricula on students' growth in problem solving disappeared. However, non-CMP curricula had an advantage on students' growth over the CMP curriculum after the cognitive demand of the instructional tasks was controlled. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we present the results from a longitudinal examination of the impact of a Standards-based or reform mathematics curriculum (called CMP) and traditional mathematics curricula (called non-CMP) on students’ learning of algebra using various outcome measures. Findings include the following: (1) students did not sacrifice basic mathematical skills if they are taught using a Standards-based or reform mathematics curriculum like CMP; (2) African American students experienced greater gain in symbol manipulation when they used a traditional curriculum; (3) the use of either the CMP or a non-CMP curriculum improved the mathematics achievement of all students, including students of color; (4) the use of CMP contributed to significantly higher problem-solving growth for all ethnic groups; and (5) a high level of conceptual emphasis in a classroom improved the students’ ability to represent problem situations. (However, the level of conceptual emphasis bears no relation to students’ problem solving or symbol manipulation skills.) [Copyright &y& Elsevier]