Handbook of Research Design in Mathematics and Science Education

Anthony Edward Kelly, Richard A. Lesh, Anthony Edward Kelly, Richard A. Lesh

• Publisher: Routledge
• Year: 2012
• ISBN:
  9781135705824
• Edition: 1
• Topic: Education
• Subtopic: Education General

The Handbook of Research Design in Mathematics and Science Education is based on results from an NSF-supported project (REC 9450510) aimed at clarifying the nature of principles that govern the effective use of emerging new research designs in mathematics and science education. A primary goal is to describe several of the most important types of research designs that:
* have been pioneered recently by mathematics and science educators;
* have distinctive characteristics when they are used in projects that focus on mathematics and science education; and
* have proven to be especially productive for investigating the kinds of complex, interacting, and adapting systems that underlie the development of mathematics or science students and teachers, or for the development, dissemination, and implementation of innovative programs of mathematics or science instruction. The volume emphasizes research designs that are intended to radically increase the relevance of research to practice, often by involving practitioners in the identification and formulation of the problems to be addressed or in other key roles in the research process. Examples of such research designs include teaching experiments, clinical interviews, analyses of videotapes, action research studies, ethnographic observations, software development studies (or curricula development studies, more generally), and computer modeling studies. This book's second goal is to begin discussions about the nature of appropriate and productive criteria for assessing (and increasing) the quality of research proposals, projects, or publications that are based on the preceding kind of research designs. A final objective is to describe such guidelines in forms that will be useful to graduate students and others who are novices to the fields of mathematics or science education research. The NSF-supported project from which this book developed involved a series of mini conferences in which leading researchers in mathematics and science education developed detailed specifications for the book, and planned and revised chapters to be included. Chapters were also field tested and revised during a series of doctoral research seminars that were sponsored by the University of Wisconsin's OERI-supported National Center for Improving Student Learning and Achievement in Mathematics and Science. In these seminars, computer-based videoconferencing and www-based discussion groups were used to create interactions in which authors of potential chapters served as "guest discussion leaders" responding to questions and comments from doctoral students and faculty members representing more than a dozen leading research universities throughout the USA and abroad. A Web site with additional resource materials related to this book can be found at http://www.soe.purdue.edu/smsc/lesh/ This internet site includes directions for enrolling in seminars, participating in ongoing discussion groups, and submitting or downloading resources which range from videotapes and transcripts, to assessment instruments or theory-based software, to publications or data samples related to the research designs being discussed.

V
Clinical Methods

19
A Scientific Perspective on Structured, Task-Based Interviews in Mathematics Education Research

Gerald A.Goldin
Rutgers, the State University of New Jersey

This chapter considers one methodological aspect of qualitative research—the use of structured, task-based interviews in observing and interpreting mathematical behavior. Several scientific issues and their implications are discussed briefly, including: (a) the reproducibility, comparability, and reliability of observations; (b) the generalizability of research findings; (c) the importance of mathematical content and structures; (d) the role of cognitive theory in designing and interpreting interviews; and, (e) the interplay among task and contextual variables. In evaluating task-based interview methods scientifically, I argue against some claims that have been advanced in the name of epistemological schools of thought ranging from radical positivism and behaviorism to radical constructivism, social constructivism, and postmodernism. Finally, some broadly applicable principles and techniques are proposed for improving the quality of task-based interview research.

The perspective offered here is that of a physical scientist as well as a mathematics educator who has been involved in empirical and theoretical research on mathematical problem solving for 25 years. I maintain that sound principles of scientific investigation, as developed and applied in modern science, should be applied to this endeavor too. This should never be done dogmatically or automatically. However, it should be done rigorously, paying careful attention to the reasoning behind the application of the methodological ideas of science. Although I have learned much from the research of others who hold different views, I remain entirely unconvinced by the arguments that are advanced occasionally, claiming that scientific methods of inquiry are inadequate for, or irrelevant to, the study of human psychosocial activities such as teaching and learning mathematics and mathematical problem solving. Because one purpose of this book is to consider quality standards for qualitative research methods, it is essential to consider the fundamental scientific issues.

My experiences with task-based interview methodology originated and evolved through a series of studies of individual mathematical problem solving by elementary school, high school, and college students and adults, conducted in collaboration with my students (Bodner & Goldin, 1991a, 1991b; DeBellis & Goldin, 1991; Goldin, 1985; Goldin & Landis, 1985, 1986; Goldin & Luger, 1975; Goidin & Waters, 1982; Luger, 1980; Waters, 1980). Most recently, members of a group of investigators that I led at Rutgers University have been analyzing and interpreting the results of a series of five task-based interviews in elementary school mathematics. We created these interviews as part of a longitudinal study of individual children’s mathematical development (Goldin, 1993a; Goldin, DeBellis, DeWindt-King, Passantino, & Zang, 1993). Between 1992 and 1994, structured interviews were conducted with an initial group of 22 third- and fourth-grade children, 19 of whom completed the full series. Partial results have been reported (DeBellis, 1996; DeBellis & Goldin, 1993, 1997; Goldin & Passantino, 1996; Zang, 1994, 1995). The development of interview scripts for this series was guided by the views described in this chapter; in turn, my views were influenced by insights gained during the study.

It is not my intention to describe the specifics of these studies here, but to focus on methodological suggestions and conclusions drawn in part from them. The chapter is organized as follows. The next section summarizes the meaning, importance, and limitations of task-based interview research in mathematics education. Here I try to explain the notion of structured interviews that are designed to investigate hypotheses using qualitative analyses of data, and offer some brief examples. The ideas presented carry forward and expand considerably on earlier deliberations about the measurement of mathematical problem solving outcomes (Cobb, 1986; Goldin, 1982, 1986; L. Hart, 1986; Lucas, et al., 1980) and the relation between cognitive theory and assessment (Goldin, 1992c). This is followed by a discussion of key scientific issues in connection with the methodology, and the case for explicit rejection of certain damaging conclusions derived from dismissive epistemological belief systems. The final section offers a preliminary set of broad, guiding principles and techniques for establishing and enhancing the quality of task-based interview research in the domain of mathematics.

TASK-BASED INTERVIEWS

Structured, task-based interviews for the study of mathematical behavior involve minimally a subject (the problem solver) and an interviewer (the clinician), interacting in relation to one or more tasks (questions, problems, or activities) introduced to the subject by the clinician in a preplanned way. The latter component justifies the term task-based, so that the subjects’ interactions are not merely with the interviewers, but with the task environments. Group interviews with two or more subjects fall also within the purview of this discussion, leading to the need to expand our interpretations of some of the ideas.

Normally, provision is made for observing and recording for later analysis what takes place during the interview, through audio- and/or videotaping, observers’ notes, and the subject’s work. Explicit provision is made too for contingencies that may occur as the interview proceeds, possibly by means of branching sequences of heuristic questions, hints, related problems in sequence, retrospective questions, or other interventions by the clinician. It is this explicit provision for contingencies, together with the attention to the sequence and structures of the tasks, that distinguishes the “structured” interviews discussed here from “unstructured” interviews, which may be limited to “free” problem solving (where no substantial assistance that would facilitate a solution is given by the clinician to the subject) or to the handling of contingencies on an improvisational basis. By analyzing verbal and nonverbal behavior or interactions, the researcher hopes to make inferences about the mathematical thinking, learning, and/or problem solving of the subjects. From these inferences, we hope to deepen our understanding of various aspects of mathematics education. We may aim to test one or more explicit hypotheses, using qualitative analyses of the data; we may seek merely to obtain descriptive reports about the subjects’ learning and/or problem solving; or we may hope to achieve an intermediate goal, such as refining or elaborating a conjecture.

Of course, the design of structured task-based interviews needs to take into account their research purposes. These may include (for example) exploratory investigation; refinement of observation, description, inference, or analysis techniques; development of constructs and conjectures; investigation or testing of advance hypotheses; and/or inquiry into the applicability of a model of teaching, learning, or problem solving. In addition the design is affected by the complexity of the phenomena in the system being investigated.

Task-based interviews can serve as research instruments for making systematic observations in the psychology of learning mathematics and solving mathematical problems. They also can be adapted as assessment tools for describing the subject’s knowledge and/or improving the practice of mathematics education (cf. R.B.Davis, 1984). The value of taskbased interviews for either of these purposes lies in the fact that they provide a structured mathematical environment that, to some extent, can be controlled. Mathematical tasks can be adjusted in wording, content, setting, sequence, and structure, based on express criteria and the outcomes of prior research. Interview contingencies can be decided explicitly and modified when appropriate. In comparison with conventional, paper-and-pencil test-based methods, taskbased interviews make it possible to focus research attention more directly on the subjects’ processes of addressing mathematical tasks, rather than just on the patterns of correct and incorrect answers in the results they produce. Thus, there is the possibility of delving into a variety of important topics more deeply than is possible by other experimental means—topics such as complex cognitions associated with learning mathematics, mechanisms of mathematical exploration and problem solving, relationships between problem solving and learning, relationships between affect and cognition, and so forth. A few examples may illustrate some of these ideas and their evolution.

During the 1950s and 1960s, many researchers investigated the use of strategies by problem solvers. These studies were consistent with the prevailing behavioral focus in psychology, considering a “strategy” to be essentially a pattern in behavior. Strategy scores were defined, based on the kinds of discrete choices made by subjects during problem solving. For example, Bruner, Goodnow, and Austin (1956) distinguished various sorts of “focusing” and “scanning” strategies in conjunctive concept identification tasks, whereas Dienes and Jeeves (1965, 1970) found “operator,” “pattern,” and “memory” strategies in card tasks that had the underlying structure of a mathematical group. The nature of the interviews was to pose problems where the spectrum of choices available at each point (i.e., the set of possible “behaviors”) was limited—for instance (depending on the task) to trying an exemplar, making a guess or conjecture, or choosing a card. The tasks and questions were highly structured in order to circumscribe the outcomes. Then, certain kinds of hypotheses could be investigated quantitatively: ways in which strategy scores might depend on task variables, subject var...

Table of contents

• Cover Page
• Half Title Page
• Title Page
• Copyright Page
• Contents
• Preface
• Part I: The Need to Address Priority Problems
• Part II: Reflecting on Instruments and Methods
• Part III: Teaching Experiments
• Part IV: Classroom-Based Research
• Part V: Clinical Methods
• Part VI: Curriculum Design As Research
• Part VII: Toward Assessment Design
• References
• Author Index
• Subject Index