Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction (original) (raw)

1993, Educational Studies in Mathematics

LEVEL OF ACTING AND REFLECTING IN MATHEMATICS INSTRUCTION ~s'nlhcr. Mathematics instruction contains two conflicting demands: on the one hand, the demand for economical efficiency and for well-developed "motorways" and, on the other hand, the demand that pupils should investigate and discover for themselves and have the freedom to "pave" their own ways. It is argued that tasks with a certain richness and quality offer some steps towards a construefive handling of this dilemma. The author tries to develop some properties of powerful tasks and to sketch the structure and philosophy of one concrete system of powerful tasks for the concept of angle. The main part of this paper presents seven examples of powerful tasks: five from the system of tasks and two with regard to pupils' working with 2-D-graphic systems. The construction of powerful tasks is viewed as a valuable contribution to bringing the theory and practice of mathematics education closer together. BACKGROUND Discussing the importance of tasks in mathematics education has a long tradition. A detailed analysis of the so-called Task Didactics (Aufgabendidaktik), which is one of the marked features of Traditional Mathematics, is given by Lenn6 (1969). Traditional Mathematics was the leading stream of mathematics education in Germany (and in a similar way in Austria) until the middle of this century, and then was progressively displaced by New Mathematics (which in turn is being pushed back more and more). Task Didactics is characterized by a partition of the mathematical subject-matter into specific areas (e.g., fractions, percentages, triangles, quadrilaterals). Each area is determined by a special type of task which is systematically treated, progressing from simple to more complex tasks (combination of simple tasks). Cross-connections (e.g., regarding fundamental ideas or structures) are not worked out in detail. In general, the teacher teaches theories and methods and the pupils have to apply them by solving tasks. How much has this situation changed? Recent empirical research studies, like those of Bromme (1986) or Clark and Yinger (1987), show that even nowadays mathematics teachers plan and organize their instruction on a large scale with the help of tasks. Research with regard to tasks takes different directions. There are many contributions to general considerations about tasks, for example: Wittmann (1984) views teaching units as the integrating core of mathematics education, incorporating mathematical, pedagogical, psychological, and practical aspects in a natural