task-technique matrix: An Alternative system for classifying research in mathematics education, The

School Science and Mathematics, Jan 1999 by Donoghue, Eileen F

This paper introduces a schema for classifying research in mathematics education that deepens and extends the Romberg/Bourne model. It employs dual hierarchical systems, one relating to the research task and the other to the research technique. The six task categories-descriptive, interpretive, prescriptive, analytic, synthetic, and theoretic-are defined with respect to breadth, depth, and scope of a study. The six technique categories-reflective, referential, evidential, clinical, empirical, and experimental-are defined in terms of utilization of resources and design intentions of the investigator. The pairing of task categories with technique categories permits the creation of a classification matrix for research in mathematics education. By examining the location of studies within this task-technique matrix, both producers and consumers of research in mathematics education can discern historical trends influencing current and future investigations.

It is not an exaggeration to say that trends at the research level in the biological and physical sciences anticipate and determine practice in the classroom and laboratory. Similarly in mathematics, research emphases during past decades are believed by some to have contributed to declines in interest and enrollment at the undergraduate and graduate levels. The distinguished Anglo-American mathematician Peter Hilton (1994) characterized the architecture of research in pure mathematics from 1950 through the 1980s as being largely vertical, leading to the construction of what he termed "needles of knowledge" rather than to integrated understanding. By needles of knowledge, Hilton referred to concentrations of effort that produced narrow pillars of understanding. The base of the pillar may be extremely limited. According to Hilton, the field of mathematics is now paying the price for excessive research specialization. The purpose of this report is to propose a bidimensional classification system and to review examples of mathematics education research in order to determine if similar specialization has occurred.

Previous Classifications of Research

Summarizing the work of Buswell and Judd ( 1925), Frobisher and Joy (1978), Smiler (1970), and Suydam (1968), Kilpatrick (1992) reported an almost exponential growth in mathematics education research during the same period described by Hilton. Perhaps this growth, too, has led to needles of knowledge with possibly the same damaging effects upon classroom practice and eventually upon student and teacher interest (see also Begle & Gibb, 1980). In order to determine the extent to which research in mathematics education has become "specialized," some means of classifying or categorizing research products is necessary. A classification scheme could assist research analysts in discerning trends, both historical and contemporary, and, ideally, in recommending productive future directions for research in mathematics education.

In the past, several schemas for classifying research have been proposed. Some were refinements of the basic research/applied research duality (Scandura, 1967), while others sought to relate these two categories on a continuum (Begle & Gibb, 1980; Hilgard, 1964). DeVault and Romberg (1967) described mathematics education research as a dynamical system in which entering characteristics and educational outcomes were mediated by content and instructional practice.

In the Review of Educational Research, Romberg (1969) departed from the journal' s previous practice of reporting mathematics education research using categories based on school levels by proposing the following eight categories: (1) association learning (students are recipients of information); (2) activity learning (students are active in acquiring knowledge); (3) problem solving; (4) mathematics teaching; (5) effectiveness of instructional programs; (6) association of learner characteristics with mathematical achievement; (7) attitudes toward mathematics; (8) evaluation and measurement. Studies in Category 5 were divided further into descriptive, comparative, experimental, and "other" studies.

According to Romberg, these categories were an extension of Bourne's (1966) task dichotomy that distinguished between associational studies and hypothesis testing studies. To aid in the selection of a research plan, Fox (1969) devised a matrix that paired a dimension identifying the time of interest to the investigator (past, present, future) with a dimension indicating the intention for the completed report (describe, compare, evaluate); however, the categories along each dimension were discrete entities with no explicit relationships among them. Moreover, this matrix was designed for planning a study, not for analyzing studies already completed. Schemas also have been devised that based the categories on variables such as teacher, learner, topic, materials, and instructional methods (Begle, 1979; Smiler, 1970). Johnson(1980a, b) used methodology as the basis for differentiating five categories for research: survey, experiment, case study, evaluation, and philosophicallhistorical.