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Philosophies of mathematics and mathematical education
This is a study investigating (1) the causes and events leading to the curriculum revisions which incorporated abstract philosophical issues into secondary mathematics, and (2) the consequences of these curriculum changes. Assessments are made of the recommendations of the Commission for Mathematics (1959) and other contemporaneous study groups such as the Bruner Woods Hole Conference (1959). To be able to train more mathematicians, scientists and engineers, the curriculum was changed to place great emphasis on structure, sets, logic, rigor and abstraction. These ideas were to permeate all grade levels of mathematics. By 1970 dissatisfaction with the "new math" was in evidence. Bruner, Thorn and Kline and others concluded that abstract reasoning and rigor for K-12 were optimistic and unreal; the predictions as to the great need for mathematicians, engineers and scientists has also proven to be a socially unjust error. Classical mathematics is devoid of philosophy; modern secondary level mathematics confuses mathematical topics with philosophical issues. Two principle investigations were undertaken: (1) To trace the evolution of the philosophy of mathematics from Pythagoras (570 BC) to Carnap and Tarski (1974), and to determine why so much reliance was placed upon philosophy by the curriculum developers. (2) To determine why all of the principle investigators of the 1960- 1965 study teams were from the universities and why they are predominantly the authors of mathematics texts for more than one hundred years. The outcome of the investigations show for the first investigative item that the concepts of Peano, Russell and Hilbert, philosophically destroyed by G8del in 1931 should have been sought out as the curriculum panaceas for the 1970's. Philosophers know that the Hilbert construction of geometry is not superior to that of Euclid. In many respects Hilbert is the poorer from the pedagogical viewpoint. Nonetheless Euclidean geometry gave way to an abstract set-theoretic approach. The entire curriculum was moved towards abstraction and mathematics despite evidence that philosophers had given up on these issues. Most had already conceded that all mathematics is synthetic a priori and thus must be borne of experience. As to the second investigative item, professors from universities dominate curriculum committees. Consequently they are predominantly the authors of secondary mathematics texts because secondary mathematics teachers are inadequately trained in mathematics and in those sciences dependent upon mathematics. See more in text.