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, 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 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.
Computer-Intensive Algebra and Students' Conceptual Knowledge of Functions
Brian R. O'Callaghan
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.
Open and Closed Mathematics: Student Experiences and Understandings
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.
The Construction of the Social Context of Mathematics Classrooms: A Sociolinguistic Analysis
Bill Atweh, Robert E. Bleicher, Tom J. Cooper
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.
Children's Problem Posing Within Formal and Informal Contexts
Lyn D. English
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.