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APPLICATIONS OF THE FILM IN MATHEMATICS
(f) Ability to make (and the habit of making) sound judgments with respect to practical quantitative problems.
(g) Disposition to extend one's sensitiveness to the quantitative as this occurs socially and to improve and exend one's ability to deal effectively with the quantitative when so encountered or discovered.
It is with the outcomes listed under the heading "Mathematical Understandings" that we shall be chiefly concerned in this discussion. During the period when the "drill theory" influenced arithmetic textbooks and methods of teaching, these outcomes were entirely neglected. Even at the present time we find many statements similar to the following by Douglass and Spitzer in the chapter, "The Importance of Teaching for Understanding," in the ForthyFifth Yearbook of the National Society for the Study of Education:2
"Yet, in spite of their demonstrable values, understandings have been neglected in the school, and they still are too often neglected, in favor of other learning outcomes, such as verbalism, barren factual information, and mechanical skills. In the foregoing discussion, attention is given to several factors which have produced this relative neglect, among them: an inadequate psychology of learning, over-reliance on textbooks, the tendency to teach by telling, the tremendous expansion of the curriculum, etc."
In many classrooms the textbook is the only type of material used in teaching arithmetic. Pupils often blindly imitate step-by-step procedures without understanding the numbers they are using and without thinking of the reasons fqr the steps they are imitating. When mathematical understandings are looked upon as important outcomes, the teaching process becomes one of proceeding from concrete meaningful experiences by a systematically planned series of steps to the use of abstractions and symbols. Then opportunities are provided for a return to
2 The Meets urment of Understanding, p. 25. Forty-Fifth Yearbook, Part I, of the Nat'l Society for the Study of Educ. Univ. of Chicago Press, Chicago, 1946.
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