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

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

Apparently, no prior schema has been successful in capturing simultaneously the intention of the research and its methodology. Clearly, trends or fashions in research methodology can influence problem selection strongly and, hence, deserve equal consideration when charting the contemporary history of research in mathematics education.

A New Approach to Classification

This report proposes a bidimensional research classification scheme for mathematics education that considers problem selection and research methodology simultaneously. The schema (see Figure 1) includes on the horizontal axis six problem, or "task," categories and on the vertical axis six methodology, or "technique," categories. Each axis is ordered by the scope of evidence required. Thus, on the horizontal axis, descriptive tasks may require less evidential insight than theoretic tasks. In the sense of Bloom (1956), the sequence of tasks from description/interpretation through synthesis/theory is a nested sequence, each level incorporating aspects of previous levels. Similarly, the techniques along the vertical axis are partially nested and would be viewed by some as hierarchical. It is tempting to describe the task-technique scales as taxonomies.

The task categories on the horizontal axis are defined with respect to breadth, depth, and scope of the study. A descriptive study examines one specific problem or question, such as the problem solving strategy of "guess and check," or the frequency of errors of a certain kind in arithmetic computation, without consideration of how related topics may interact with the original problem. For example, the classical work of Thorndike on children's arithmetic errors would be considered descriptive. An interpretive study is a descriptive study that does consider effects within some cluster of problems related to the original problem. It interprets what is identified by a descriptive study in terms of broader curricular issues. For example, the relation of the problem solving strategy of guess and check to other aspects of the curriculum, such as calculator use, could be examined. In the prescriptive study, one or more questions are examined using the descriptive or interpretive mode, resulting in recommendations or rankings. In the case of the problem solving example, a prescriptive study may examine the effects of calculator use and estimation techniques on the guess-and-check strategy, producing some assessment of the relative importance of these related features and leading to recommendations for practice.

The analytic study decomposes a complex question into its componentparts, interpreting that problem in terms of related issues. It may include description and interpretation and, possibly, prescription. Polya's (1973) book on problem solving, How to Solve It, is an example of such a study. A synthetic study takes several results from the analytic level and relates them in ways that reveal insights that a single analytic study cannot. In one sense, the investigator has explored a larger nexus. For example, Polya's analysis of problem solving could be combined with levels of geometric reasoning to create a curricular sequence for geometric problem solving. The theoretic task category incorporates analysis and synthesis to relate or explain diverse phenomena. Piaget's theory of the stages of intellectual development would be placed in this category.