Definitions--Consensus or Confusion?.

Abstract

IN RECENT years, there has been considerable discussion and argument about the "new" or "modern" mathemat ics. A point on which everyone seems to agree, however, is that one of the charac teristics of the new math is greater preci sion in language, especially with defini tions. But, upon closer examination of the textbooks and manuals in current use, it becomes evident that there is indeed a great lack of precision and uniformity of definition. The many changes in the mathematics curriculum during the last decade were not designed by a single agency, but by a num ber of individuals and organizations who have produced new programs and text books at an increasingly rapid rate. These rapid changes have also produced some unfavorable effects. Many teachers were overwhelmed by the new math?they did not understand it and felt incompetent to teach it. To help them cope with the new programs, great investments have been made in in-service training. But imagine the frustration of one of these teachers when, after using one textbook series with the terms clearly defined, he reads another that defines the terms differently. Whose definitions should he accept? Logicians will say that it does not make any difference how a person defines his terms as long as he is consistent in their use. But elementary teachers and students are not logicians. Since these textbooks and manuals have been written by college teachers and high school teachers, it should be their concern and responsibility to bring about a uniformity of definitions. How many mathematics educators are aware of the variations in definitions? Following is a brief description of the extent to which writers disagree about some of the simple concepts.