A Proposal for a Problem-Driven Mathematics Curriculum Framework

A framework for a problem-driven mathematics curriculum is proposed, grounded in the assumption that students learn mathematics while engaged in complex problem-solving activity. The framework is envisioned as a dynamic technologicallydriven multi-dimensional representation that can highlight the nature of the curriculum (e.g., revealing the relationship among modeling, conceptual, and procedural knowledge), can be used for programmatic, classroom and individual assessment, and can be easily revised to reflect ongoing changes in disciplinary knowledge development and important applications of mathematics. The discussion prompts ideas and questions for future development of the envisioned software needed to enact such a framework

Disciplinary Learning From an Authentic Engineering Context

This small-scale design study describes disciplinary learning in mathematical modeling and science from an authentic engineeringthemed module. Current research in tissue engineering served as source material for the module, including science content for readings and a mathematical modeling activity in which students work in small teams to design a model in response to a problem from a client. The design of the module was guided by well-established principles of model-eliciting activities (a special class of problem-solving activities deeply studied in mathematics education) and recently published implementation design principles, which emphasize the portability of model-eliciting activities to many classroom settings. Two mathematical modeling research questions were addressed: 1. What mathematical approaches did student-teams take when they designed mathematical models to evaluate the quality of blood vessel networks? and 2. What attributes of mature mathematical models were captured in the mathematical models that the student-teams designed? One science content research question was addressed: 1. Before and after the module, what aspects of angiogenesis did students describe when they were asked what they knew about the process of blood vessel growth from existing vessels? Participants who field-tested the module included high school students in a summer enrichment program and early college students enrolled in four general-studies mathematics courses. Data collected from participants included mathematical models produced by small teams of students, as well as students’ individual responses before and after the module to a prompt asking them what they knew about the process of new blood vessel growth from existing vessels. The data were analyzed for mathematical model type and science content by adopting methods of grounded theory, in which researchers suspend expectations about what should be in the data and, instead, allow for the emergence of patterns and trends. The mathematical models were further analyzed for mathematical maturity using an a priori coding scheme of attributes of a mathematical model. Analyses showed that student-teams created mathematical models of varying maturity using four different mathematical approaches, and comparisons of students’ responses to the science prompt showed students knew essentially nothing about angiogenesis before the module but described important aspects of angiogenesis after the module. These findings were used to set up an agenda for future research about the design of the module and the relationship between disciplinary learning and authentic engineering problems

Characterizations of Social-Based and Self-Based Contexts Associated With Students’ Awareness, Evaluation, and Regulation of Their Thinking During Small-Group Mathematical Modeling

Characterizations of Social-Based and Self-Based Contexts Associated With Students’ Awareness, Evaluation, and Regulation of Their Thinking During Small-Group Mathematical Modeling This exploratory study focused on characterizing problem-solving situations associated with spontaneous metacognitive activity. The results came from connected case studies of a group of 3 purposefully selected 9th-grade students working collaboratively on a series of 5 modeling problems. Students’ descriptions of their own thinking during small-group mathematical modeling, elicited during video-stimulated interviews, were analyzed to identify and characterize social-based and self-based contexts associated with metacognitive activity coded as awareness, regulatory, and evaluative. Three characterizations of the social-based contexts and 3 different characterizations of the self-based contexts emerged

The social- and self-based contexts associated with students’ awareness, evaluation and regulation of their thinking during small-group mathematical modeling

Characterizations of Social-Based and Self-Based Contexts Associated With Students’ Awareness, Evaluation, and Regulation of Their Thinking During Small-Group Mathematical Modeling This exploratory study focused on characterizing problem-solving situations associated with spontaneous metacognitive activity. The results came from connected case studies of a group of 3 purposefully selected 9th-grade students working collaboratively on a series of 5 modeling problems. Students’ descriptions of their own thinking during small-group mathematical modeling, elicited during video-stimulated interviews, were analyzed to identify and characterize social-based and self-based contexts associated with metacognitive activity coded as awareness, regulatory, and evaluative. Three characterizations of the social-based contexts and 3 different characterizations of the self-based contexts emerged

A Model and Modelling Perspective on the Role of Small Group Learning

The problem solving process in most twenty-first century contexts involves teams working on problem situations. It involves partitioning a complex situation into parts that can be addressed by specialists. It involves communicating information in forms that are meaningful for other specialists and suitable for their tools. It involves planning, monitoring and assessing intermediate results. In other words, the primary activity is team-based mathematical modeling. While the use of small groups in mathematics classes has been encouraged for over 20 years, research has focused on the achivement of individuals (as measured by standardized tests) and social benefits (such as self-esteem, social acceptance) (Davidson, 1990). These areas of emphasis and research have been confined to a frame of reference of school mathematics, whereas the needs of the current work place go beyond these interests. For example, schools are interested in the achievement of individual children, whereas business and industry are more interested in the products formed by teams of experts–with little interest in the contributions of the individual. The focus of this chapter reaches beyond the common mathematics classroom context to consider the cognitive development of groups as they work on well-crafted model-eliciting problems. The focus is on the kinds of tasks and processes valued in work place team problem solving. The chapter begins by teasing out the social aspects embedded in the six design principles for construct-eliciting (i.e., model-eliciting) problems described by Lesh, Hoover, Hole, Kelly and Post, (2000). The tasks require collaborative effort, leading to a discussion about how working in small groups amplifies the mathematical power of the individuals in the group. Since the products produced by the groups represent a collaborative effort, the cognitive development of the "group-as-a-unit" is important to consider. This discussion focuses on the role of perspective-taking and the internalization of once external group processes that take place in small group modeling episodes. Then individual cognitive development is examined with respect to "local" model development–i.e. the development of a construct within a model-eliciting activity. This is in contrast to traditional views, which are age-related, and unconnected to the local problem solving episode. Finally, selected classroom issues in management and implementation guidelines are given, based on the experiences of the authors when we have implemented collaborative problem solving with both children and adults

A Models and Modeling Perspective on the Role of Small Group Learning Activities

The problem solving process in most twenty-first century contexts involves teams working on problem situations. It involves partitioning a complex situation into parts that can be addressed by specialists. It involves communicating information in forms that are meaningful for other specialists and suitable for their tools. It involves planning, monitoring and assessing intermediate results. In other words, the primary activity is team-based mathematical modeling. While the use of small groups in mathematics classes has been encouraged for over 20 years, research has focused on the achivement of individuals (as measured by standardized tests) and social benefits (such as self-esteem, social acceptance) (Davidson, 1990). These areas of emphasis and research have been confined to a frame of reference of school mathematics, whereas the needs of the current work place go beyond these interests. For example, schools are interested in the achievement of individual children, whereas business and industry are more interested in the products formed by teams of experts–with little interest in the contributions of the individual.\ud \ud The focus of this chapter reaches beyond the common mathematics classroom context to consider the cognitive development of groups as they work on well-crafted model-eliciting problems. The focus is on the kinds of tasks and processes valued in work place team problem solving. The chapter begins by teasing out the social aspects embedded in the six design principles for construct-eliciting (i.e., model-eliciting) problems described by Lesh, Hoover, Hole, Kelly and Post, (2000). The tasks require collaborative effort, leading to a discussion about how working in small groups amplifies the mathematical power of the individuals in the group. Since the products produced by the groups represent a collaborative effort, the cognitive development of the "group-as-a-unit" is important to consider. This discussion focuses on the role of perspective-taking and the internalization of once external group processes that take place in small group modeling episodes. Then individual cognitive development is examined with respect to "local" model development–i.e. the development of a construct within a model-eliciting activity. This is in contrast to traditional views, which are age-related, and unconnected to the local problem solving episode. Finally, selected classroom issues in management and implementation guidelines are given, based on the experiences of the authors when we have implemented collaborative problem solving with both children and adults