Academic journal article The Mathematics Enthusiast

A Proposal for a Problem-Driven Mathematics Curriculum Framework

Academic journal article The Mathematics Enthusiast

A Proposal for a Problem-Driven Mathematics Curriculum Framework

Article excerpt

Abstract: 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 technologically-driven 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.

Keywords: Problem-based Mathematics, Curriculum frameworks, Mathematical Modeling, Model-Eliciting Activities.

Introduction

Curriculum frameworks are commonly organized around categories of mathematical topics (e.g., number, geometry), such as in the new Common Core School Mathematics Standards (NGA & CCSSO, 2011) and the National Council of Teachers of Mathematics (NCTM) standards documents (1989, 2000) for the United States (U.S.). Oftentimes, to convey the nature of mathematics teaching and learning, the content topics are cross-referenced with other types of mathematical behaviors, such as the "process standards" (e.g., problem solving, reasoning and proof) of the NCTM documents, and the "practices" (e.g., model with mathematics, attend to precision) of the CCSSM document. Another approach is to formulate mathematics curriculum frameworks based on assumptions about learning mathematics, such as the Dutch curriculum framework described by van den Heuvel-Panhuizen (2003) (e.g., informal to formal, situated to generalized, individual to social). The developers of mathematics curriculum frameworks choose their organization and structure in order to communicate a mathematics curriculum to broad audiences (e.g., teachers, administrators, parents, students). The choices for content and the representation of curricula made by the framework developers, in turn, convey a distinctive perspective on mathematics curriculum, accompanied by inevitable (some intended, some unintended) consequences when users of the framework transform the represented curriculum into prescriptions for classroom experiences and assessment. A proposal for framing and representing a problem-driven mathematics curriculum is described in this article. The proposal envisions a framework that grows out of Lesh and colleague's work on models-and-modeling, which has focused on using modeling problems as sites for revealing and assessing students' thinking (e.g., Lesh, Cramer, Doerr, Post & Zawojewski, 2003), and more recently by Richard Lesh to teach data modeling (personal conversation, Dec. 21, 2012). The proposal also envisions a representational system that builds on a one originally posed by Lesh, Lamon, Gong and Post (1992), and is particularly poignant today because technology is now available that could carry out the proposal.

Why an Alternative Framework?

Assumptions about Curriculum Frameworks

Curriculum frameworks convey a view of mathematics learning to stake holders in education, influencing the full range of mathematics education activity-from implementation to assessment. For example, the two foundational NCTM curriculum documents (1989, 2000) contributed to a huge shift in views of mathematics curriculum in the U. S. Prior to the publication of these documents, schools, districts and state curriculum guides predominantly listed expected mathematical competencies by grade level, commonly referred to as scope and sequence documents. The NCTM standards documents introduced a process dimension (problem solving, reasoning, connections, communication) in addition to the common practice of describing mathematics competencies and performance expectations. Further, discussions about the mathematical processes and expected mathematical performances were embedded in the context of illustrative problems, teaching and learning scenarios, and ways of thinking about mathematics. …

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