Showing papers in "International Journal of Computers for Mathematical Learning in 1998"

The complex process of converting tools into mathematical instruments: the case of calculators

Abstract: Transforming any tool into a mathematical instrument for students involves a complex ‘instrumentation’ process and does not necessarily lead to better mathematical understanding. Analysis of the constraints and potential of the artefact are necessary in order to point out the mathematical knowledge involved in using a calculator. Results of this analysis have an influence on the design of problem situations. Observations of students using graphic and symbolic calculators were analysed and categorised into profiles, illustrating that transforming the calculator into an efficient mathematical instrument varies from student to student, a factor which has to be included in the teaching process.

Game Design as an Interactive Learning Environment for Fostering Students' and Teachers' Mathematical Inquiry.

Abstract: Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments.

Network of Influences in an Implementation of a Mathematics Curriculum Innovation.

Abstract: There is widespread interest in reform in U.S. mathematics education, engendered by influences from comparative educational research (National Center for Education Statistics, 1996; Stigler, Lee and Stevenson, 1990) to position documents from prestigious organizations (National Council of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991; National Research Council, 1989). One route to such reform is the development and implementation of innovative curricula. We studied the adoption of one such innovation, emphasizing teachers’ social construction of knowledge and beliefs. The innovation was designed by two of us (Clements and Meredith, 1994; Clements and Sarama, 1995) as part of a research project funded by the National Science Foundation (NSF). 1 In this context, we developed a software environment with correlated curriculum materials. During the field testing phase of development, we worked with a school that was attempting to alter its mathematics program according to recent reform recommendations. Given the apparent convergence of goals of the school staff and the designers, we believed the adoption process began with a good chance of success. We planned to assess the effect of the innovation on the knowledge and beliefs of students and teachers. During the project, however, concerned and confused about what we observed, we altered the direction of the research study to examine, and attempt to positively affect, the implementation of the innovation.

Epistemological Change through Peer Apprenticeship Learning: From Rule-Based to Idea-Based Social Constructivism

Abstract: The thrust of peer apprenticeship learning is in how an individual's personal beliefs, disposition, and epistemology can be transformed through an “apprenticeship-like” learning process. This paper describes the peer apprenticeship learning situation between two students, Dom (age: 13) and Ming (age: 14), in distributed computer-mediated co-construction of mathematical meanings. Ming was initially procedural or rule-based in his problem solving methods, encountering numerous difficulties with his approaches. Dom, however, was constantly engaged in playing with ideas through conjecturing and perceiving mathematical relationships. In the initial stages of co-construction efforts, Dom and Ming were solving the problems rather independently as each appropriated a radically different epistemology of mathematics – Ming was rule-based whereas Dom was idea-based. In the cause of increasing difficulties, Ming soon recognized that his methods were inadequate and, after a considerable struggle, positioned himself in an “apprentice-like” manner in order to follow Dom's conceptualizations. Through monitoring of Dom's conceptualizations and with personal experimentations to concretize his understanding, Ming was gradually able to assimilate Dom's mathematical meaning perspectives. We depicted such a learning situation as peer apprenticeship learning. As a result of assimilating the disposition towards playing with ideas, Dom and Ming were able to engage in meaningful idea-based social constructivism.