Helen M. Doerr

A Modeling Perspective on Students' Mathematical Reasoning about Data.

A modeling approach to the teaching and learning of mathematics shifts the focus of the learning activity from finding a solution to a particular problem to creating a system of relationships that is generalizable and reusable. In this article, we discuss the nature of a sequence of tasks that can be used to elicit the development of such systems by middle school students. We report the results of our research with these tasks at two levels. First, we present a detailed analysis of the mathematical reasoning development of one small group of students across the sequence of tasks. Second, we provide a macrolevel analysis of the diversity of thinking patterns identified on two of the problem tasks where we incorporate data from multiple groups of students. Student reasoning about the relationships between and among quantities and their application in related situations is discussed. The results suggest that students were able to create generalizable and reusable systems or models for selecting, ranking, and weighting data. Furthermore, the extent of variations in the approaches that students took suggests that there are multiple paths for the development of ideas about ranking data for decision making.

What Knowledge Do Teachers Need for Teaching Mathematics Through Applications and Modelling

This paper begins by describing teachers’ knowledge as the creation and development of increasingly sophisticated models or ways of interpreting the tasks of teaching. One study illuminates several ways that pre-service teachers perceive the processes of modelling and the limits of their experiences with stochastic models. Results from a second study indicate that teachers need to have a broad and deep understanding of the diversity of approaches that students might take with modeling tasks. The second study also suggests a reversal in the usual roles of teachers and students by engaging students as evaluators of models.

Experiment, simulation and analysis: an integrated instructional approach to the concept of force

This paper develops a theoretical framework for a modeling process that integrates computer simulations with first‐hand experiments and computer‐based analysis tools. This modelling approach is then linked to the extensive body of research on students’ conceptions about force; it is argued that model building holds great potential to facilitate conceptual change through enabling students to make explicit and test their own hypotheses. Using this modeling approach, a curriculum unit that presented four tasks for investigating the forces involved in accelerated motion down an inclined plane was developed and implemented in a secondary school classroom. The results of studying a group of three students as they progressed through this unit illuminate the tradeoffs among experiment, simulation and analysis, while demonstrating that students are likely to pursue alternative hypotheses and diverse representations as they build a more complete model over an extended period of time.

Teachers’ Knowledge and the Teaching of Algebra

In this chapter an analysis of the research on teachers’ knowledge and practice and its development with respect to the teaching of algebra is presented. The chapter begins with a brief discussion of four dilemmas that were confronted during this analysis. The findings from research on teachers’ knowledge are reported in three areas: (a) teachers’ subject matter knowledge and pedagogical content knowledge, (b) teachers’ conceptualisations of algebra, and (c) teachers learning to become teachers of algebra. The chapter concludes with a discussion of critical issues and suggestions for further research.

Models and Modelling Perspectives on Teaching and Learning Mathematics in the Twenty-First Century

Research based on models and modelling perspectives (MMP) has shown that, in order for mathematical concepts and abilities to be useful beyond school, new levels and types of understandings are needed beyond those that have been emphasized in even the most innovative and future-oriented statements of curriculum standards. Similarly, teacher-level knowledge and abilities consists of a great deal more than the kind of beliefs, dispositions, and pedagogical content knowledge that have been emphasized in most past research on teacher development.

A Modeling Perspective on Interpreting Rates of Change in Context

Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students’ abilities to describe and interpret rates of change in the context of exponential decay. In this article, we elaborate the characteristics of the model development sequence and we examine how students interpreted and described non-constant rates of change in context. We provide evidence for how a focus on the context made visible students’ reasoning about rates of change, including difficulties related to the use of language when describing changes in the negative direction. We argue that context and the use of language, forefronted in a modeling approach, should play an important role in supporting the development of students’ reasoning about changing phenomena.

Bypass, augment, or integrate: How secondary mathematics teachers address the literacy demands of standards-based curriculum materials

This 3-year qualitative study examined how 26 teachers in four U.S. secondary schools addressed the literacy demands of curriculum materials based on standards from the National Council of Teachers of Mathematics. It was grounded in sociocultural perspectives that encourage study of language in local contexts, including classrooms, communities, and disciplines. The research question asked, “How do secondary mathematics teachers in a teacher–researcher collaboration understand and address the literacy demands of standards-based curriculum materials?” An interdisciplinary team of literacy and mathematics education researchers and graduate students worked with teachers to examine required curriculum materials, plan lessons, and share resources, while collecting qualitative data to capture teachers’ understandings and actions. Findings indicated that teachers responded in three ways to the materials’ literacy demands: (a) they used the materials selectively to bypass or reduce the literacy demands, (b) they augmented use of the materials with literacy support, or (c) they integrated use of the materials into long-term frameworks to develop mathematics and literacy learning simultaneously. The study suggests the value of using insights from mathematics and literacy to inform mathematics curriculum design and reminds us to take teachers’ perspectives toward such materials seriously, particularly when they collaborate to address issues about the materials’ use in their own school contexts.

Alternatives to Trajectories and Pathways to Describe Development in Modeling and Problem Solving

Much past research based on model eliciting activities (MEAs) has shown that it is possible to directly observe processes that enable students to develop progressively more productive ways of thinking about problem situations. In this paper, we introduce a class of MEAs that are designed to explicitly focus on the parallel and interacting development of systems of interpretation that occurs in realistic solutions to complex modeling tasks. When engaging with this class of activities, the model development that occurs is more like the interactions among evolving partial interpretations and primitive ways of thinking than the progression along a trajectory or pathway that refines a single model of the situation.

“It’s a Different Kind of Reading” Two Middle-Grade Teachers’ Evolving Perspectives on Reading in Mathematics

Through a qualitative analysis of data collected over four years of design-based research on the implementation of a reform-oriented mathematics curriculum, this study describes two sixth-grade teachers’ changing views of the role and place of reading in mathematics instruction. The findings reveal the evolution of the teachers’ perspective on mathematics instruction from one that did not include reading toward one in which reading was viewed as integral to students’ mathematics learning. The teachers’ views on reading in mathematics at the end of the project lend empirical support to theoretical propositions for a disciplinary literacy approach to mathematics instruction. At the same time, their views nuance such propositions by highlighting differences between the reading demands of school mathematics texts and those of disciplinary texts.

Teachers Learning from Their Teaching: The Case of Communicative Practices

Teachers’ learning about teaching mathematics from their practice is problematic to identify and analyze. Among many other things, teachers need to enact their knowledge of how students’ ideas might develop, what tasks might support that development and how to elicit the expression of students’ ideas. This enactment in practice potentially provides opportunities for teachers to learn about teaching mathematics. In this chapter, we seek to provide some insight into how this learning might occur. We begin by posing questions about the professional knowledge base for teaching and then examining data from a substantial longitudinal study by the first author. We focus on the learning of one of the teachers in the project about the role of communicative practices, including speaking, writing and reading, in students’ learning of mathematics. We conclude with a discussion of some theoretical implications for understanding teachers’ learning in practice and how it might contribute to the professional knowledge base for teaching mathematics.