Carolyn A. Maher

Representing motion: an experiment in learning

Abstract In this paper we concentrate on how students make sense of motion. To understand the challenges the students actually faced, we focus on how learners work with models, representations, and their personal experience. The student subjects are participants in the Kenilworth longitudinal study by the third author. They work on a task, based on a set of 24 time-lapse photographs, to determine how a cat was moving as it progressed from walking to running. We document how learners work with a wide range of graphic presentations, including inscriptions, calculator-generated plots, drawings, and photographs. Analysis centers on how students reason, based on these representations, to make sense of the cat’s motion.

Using Videotapes to Study the Construction of Mathematical Knowledge by Individual Children Working in Groups

Videotaping small groups of students in a regular classroom environment makes it possible to study individual student cognitive growth in a social setting. The present report deals with student development of some new mathematical ideas over an extended period of time.

How far can you go with block towers

Abstract This report focuses on the development of combinatorial reasoning of a 14-year-old child, Stephanie, who is investigating binomial coefficients and combinations in relationship to the binomial expansion and the mapping of the binomial expansion to Pascals triangle. This research reports on Stephanies examination of patterns and symbolic representations of the coefficients in the binomial expansion using ideas from earlier explorations with towers in grades 3–5 to examine recursive processes and to explain the addition rule in Pascals triangle. This early work enabled her to build particular organization and classification schemes that she draws upon to explain her more abstract ideas.

Chapter 6: Building Representations of Children's Meanings

In Chapter 5 (Davis & Maher, this volume) we suggested the importance of paying attention to the fine detail of childrens ideas. We have given instances of videotaped episodes of children engaged in thinking about mathematics as they worked to construct a solution to a problem. An analysis of the childrens mathematical behavior has given us some insight into how they built up their representation of the problem. We have observed their attempts to connect this mental representation to the physical model that they also built, the picture that they drew, and their written symbolic statement of the problem situation. Having looked at children in Chapter 5, in the present chapter we shall look at how difficult is the teachers task of recognizing the actual ideas of students. Background We consider in this chapter a classroom episode in which a teachers representation of a problem situation is in conflict with that of her students, Brian. The teacher, a second-year participant in a teacher development project in mathematics, was attempting to implement her growing knowledge of content, childrens learning, and pedagogy in her classroom (Maher & Alston,1988). She began to include small cooperative group problem-solving explorations as a regular classroom activity, and she was integrating in her lessons problem tasks in which children were encouraged to build physical models to represent their solutions. The Brian and Scott episode described and analyzed in Chapter 5 was representative of her instruction at this stage of her participation in the project. The two fifth-grade students whom we saw in Chapter 5, Brian and Scott, worked together regularly as partners doing mathematics. Having considered the work and thought processes of these two students in the previous chapter, we now attempt to see their thinking from the perspective of their teacher. In fact, because we have the advantage of videotapes of Brian and Scott working together in earlier lessons, and we also have the notes written by the teacher after each of these lessons, we are able to study their thinking in close detail. From this we can gain added insight into their mathematical thinking precisely because we can watch it develop. (Of course, we also have the advantage of hindsight, and the opportunity to look at tapes over and over again, discuss them, look some more, discuss some more, and so on. This is very different from the situation that confronted the teacher when she was actually

From patterns to theories: conditions for conceptual change

Abstract This research builds on earlier work of the development of mathematical proof by young children. In this paper, we see the 9-year-old Stephanie extending earlier understanding of argument by cases to argument by mathematical induction, as she investigates, with other classmates, a particular theory.

Tracing fourth graders’ learning of fractions: early episodes from a year-long teaching experiment

Abstract Considerable and significant research 1 has been conducted focusing on what students know about fraction ideas. 2 However, much of the research looks at student understanding following some instruction. The research 3 reported in this paper differs from other studies in three significant ways: (1) the students were 9- and 10-year-old fourth graders; (2) they had not yet had an introduction to operational rules and definitions about fractions; and (3) the research design emphasized a student-centered approach. The research is based on the view that given particular conditions in which students are invited to work together and conduct thoughtful investigations with appropriate materials, they can build fundamental mathematical ideas.

Expanding Participation in Problem Solving in a Diverse Middle School Mathematics Classroom.

In this paper, we discuss our experiences with an after-school program in which we engaged middle-school students with low socioeconomic status from an urban community in mathematical problem solving. We document that these students participated in many aspects of problem solving, including the posing of problems, constructing justifications, developing and implementing problem-solving heuristics and strategies, and understanding and evaluating the solutions of others. We then delineate what aspects of our environment encouraged the students to take part in these activities, particularly emphasising the proactive role of the teacher, the tasks the students completed, and the social norms of our after-school sessions. Finally, we discuss the relationship between our study and the literature on equity research in mathematics education.

VMCAnalytic: Developing a Collaborative Video Analysis Tool for Education Faculty and Practicing Educators

This paper describes the genesis, design and prototype development of the VMCAnalytic, a repository-based video annotation and analysis tool for education. The VMCAnalytic is a flexible, extensible analytic tool that is unique in its integration into an open source repository architecture to transform a resource discovery environment into an interactive collaborative where practicing teachers and faculty researchers can analyze and annotate videos to support a range of needs from longitudinal research to improving individual teaching performance. This paper presents an overview of the design and functionality of the VMCAnalytic, which is a key component of the NSF-funded Video Mosaic Collaborative (VMC), together with a description of the underlying repository service architecture. The paper also describes the synergistic collaboration between digital library technologists and education researchers to build a research environment that can integrate with the VMCAnalytic tool to create a digital collaboration space. The prototype tool is available in as of January 2010 at the VMC website: www.video-mosaic.org.

The Longitudinal Study

Where do new ideas come from? Our view is that building new ideas is a process; new ideas come from old ideas that are revisited, reviewed, extended, and connected (Davis, 1984; Maher & Davis, 1995). Building new ideas also involves the retrieval and modification of representations of existing ideas. The representations that a learner builds for a mathematical idea or procedure can take different forms – physical objects or actions on objects, words, and symbols, for example. As the learner’s experience increases, old representations become elaborated, extended, and linked to new ones (Maher, 2008; Davis & Maher, 1997)

Learning to Reason in an Informal Math After-School Program

This research was conducted during an after-school partnership between a University and school district in an economically depressed, urban area. The school population consists of 99% African American and Latino students. During an the informal after-school math program, a group of 24 6th-grade students from a low socioeconomic community worked collaboratively on open-ended problems involving fractions. The students, in their problem solving discussions, coconstructed arguments and provided justifications for their solutions. In the process, they questioned, corrected, and built on each other’s ideas. This paper describes the types of student reasoning that emerged in the process of justifying solutions to the problems posed. It illustrates how the students’ arguments developed over time. The findings of this study indicate that, within an environment that invites exploration and collaboration, students can be engaged in defending their reasoning in both their small groups and within the larger community. In the process of justifying, they naturally build arguments that take the form of proof.