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 five 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.

This article describes a research project that examined the effects of the Computer-Intensive Algebra (CIA) and traditional algebra curricula on students' understanding of the function concept. The foundation for the research is a proposed conceptual framework that describes function knowledge in terms of component competencies. The results indicated that the CIA students achieved a better overall understanding of functions and were better at the components of modeling, interpreting, and translating. No significant differences were found for reifying, which emerged as the most difficult component in the proposed function model. Further, the CIA students showed significant improvements in their attitudes toward mathematics, were less anxious about mathematics, and rated their class as more interesting. A higher percentage of students successfully completed the CIA course.

This paper reports on three-year case studies of two schools with alternative mathematical teaching approaches. One school used a traditional, textbook approach; the other used open-ended activities at all times. Using various forms of case study data, including observations, questionnaires, interviews, and quantitative assessments, I will show the ways in which the 2 approaches encouraged different forms of knowledge. Students who followed a traditional approach developed a procedural knowledge that was of limited use to them in unfamiliar situations. Students who learned mathematics in an open, project-based environment developed a conceptual understanding that provided them with advantages in a range of assessments and situations. The project students had been "apprenticed" into a system of thinking and using mathematics that helped them in both school and nonschool settings.

This study employed sociolinguistic perspectives developed by Michael Halliday to investigate the social context of two mathematics classrooms that differed in the socioeconomic backgrounds and genders of their students. Our analysis focused on the effects of these two factors on teacher perceptions of student needs and abilities and on how these perceptions shaped the discourse in the classroom. We argue that as mathematical knowledge is being constructed in the classroom, the student participants are being constructed by the teachers according to their ability in and need for different dialects of mathematics that have different values in our Western culture.

This study investigated the problem-posing abilities of third-grade children who displayed different profiles of achievement in number sense and novel problem solving. The study addressed (a) whether children recognize formal symbolism as representing a range of problem situations, (b) whether children generate a broader range of problem types for informal number situations, (c) how children from different achievement profiles respond to problem-posing activities in formal and informal contexts, and (d) whether children's participation in a problem-posing program leads to greater diversity in problems posed. Among the findings were children's difficulties in posing a range of problems in formal contexts, in contrast to informal contexts. Children from different achievement profiles displayed different response patterns, reflected in the balance of structural and operational complexity shown in their problems.