Philosophical and Epistemological Aspects of Mathematics and Their Interaction with Theory and Practice in Mathematics Education.
Hans‐Georg Steiner · for the learning of mathematics · 1987
Philosophical positions and epistemological . theoties related to mathematics, such as logicism, formalism, constructivism, structmalism, empiiicism, have always had a significant influence on the guiding ideas and leading piinciples in mathematics education.. This not only holds for cuniculum development and teaching methodology but also for theoretical work and empirical researCh related to the mathematical learning process. As Seymour PAPERT has pointed out in Mindstorms: BOURBAKI's theory of motheT-structure is learning theory Whether it is a good or bad one is another question. a similar way PLATO's and PROCL US' of EUCLID's elements with its idealistic ontology and its emphasis on the dialectic between analysis and synthesis being the capstone of the mathematical sciences comprises the elements of a mathematical pedagogy and of a theory of learning. [42], [50], [66] This phenomenon can be pursued thwugh the history of mathematics and its concomitant philosophy. Of course, the observation not only holds for global mathematical but also for epistemological views of particular parts or concepts of mathematics, such as the set-theoretical foundation of the function concept, the logical interpretation ofvatiables as place holders, the interpretation of geometry in terms of Felix KLEIN's Erlanger Program etc On the other hand, considering the inverse relation, what was said by Rene THOM in his 1972 Exeter speech is also true: In fact, whCther one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a of mathematics. Such philosophies may consist in a teacher's 'private opinion on the natw·e of mathematics and mathematical knowledge (often indirectly acquired in his own academic studies), and in his thoughts about how this is related to his teaching and to the learning of his students .. They are also inherent in didactical principles such as the spiral approach (BRUNER), the deep-end principle (DIENES: [16]), the operative principle (AEBLI), etc They underlie theories of stages in the learning process (PIA GET, VAN HIELE) or theories of transfer, and they also stand behind hypotheses of empirical studies on how children learn or faiL As Wi'S pointed out by Thomas S.. KUHN, each research paradigm has philosophical and ontological assumptions among its different components, an4 these are parts of the paradigm's bias, its strength and weakness .. Often they are a source of profound discrepancies between competing approaches and of controversies between research grtmps or individual researchers. , Recent developments in mathematics education show a new dynamics in the field. New and epistemological theories have entered the scene: the theory of epistemological obstacles (BACHELARD, BROUSSEAU( !OJ, CORNU, SIERPINSKA), a synthesis of Kuhnian theory dynamics and Piagetian genetic epistemology (v,GLASERSFELD, CAWTHRON & ROWELL) a new quasiempiricist of mathematics (LAKATOS [43], SNEED, JAHNKE), a complementarist view of mathematics (KUYK, OTTE [51]), the epistemology of microworlds and of the society of based on cggnitive studies within research on artificial intelligence (PAPER T, MINSKY} They claim to provide a better backgrnund for an investigation of pupils' real learning behaviour both from a cognitive and a social point of view. They also claim to give a more adequate foundation for the theory and practice of mathematics teaching .. Based on these developments, and supported by new empirical data, strong criticisms have been put forward especially against positions elaborated in close connection with the new-math philosophy DefiCiencies of this reform are being characterized by terms like HJourdain effect, Topaze effect, metacognitive shift, etc (BROUSSEAU [91]), and by exhibiting its one-sided mentalistic orientation guided by the assumption of the autonomous, universal, coherent, and homogeneous natures of both mathematical structures and the human mind Other failures are related to a purist or static view that neglects the applications which belong to mathematical theories and concepts as they do to physical theories, and neglects the representational, the social, the procedural and the processual dimensions of mathematics This criticism says that the stmctural concepts of mathe-