This paper is the result of a kind of personal intellectual odyssey that started nearly forty years ago. I wrote the first version of the paper in May of 1976 and circulated it to some friends but never published it. Since then, people have been kind enough from time to time to quote it and appreciate the importance of some of the ideas. Many have urged me to publish the paper. I have recently re-read the paper and decided that it contained many good ideas that are worth airing. This is a revised and substantially expanded version of that paper. Like the blind men palpitating the elephant, this paper will probably be read in very different ways by different readers. The mathematician is likely to say that it belongs in the domain of psychology; the psychologist will say that it deals with linguistics; the linguist will say that it deals with applied mathematics. They are each correct. There is a body of phenomena that this paper seeks to explore, i.e. the ability of people to make quantitative statements about their surroundings and to communicate those statements and their implications to others. These phenomena are real and worthy of attention. My interest in them grows out of many years of close observation of the performance of mathematical acts by people, both teachers and students, in both formal and informal settings. In this paper I attempt to analyze some of the ways in which people use quantities that are meaningful to them. This analysis is then used to shape the definition of a formal mathematics of quantities with referents in the world around us and to deriving the consequences and entailments of that formal mathematical structure. Implicit in this effort is an agenda for education and schools. Mathematics holds the position it does in the curriculum of our culture because we believe it to be a source of analytic tools that can help people negotiate with the social and physical world in which they find themselves. Of necessity this means that the mathematical abstractions of quantity, shape, space, pattern, function, arrangement, etc. that they deal with all have their counterparts in reality. I believe it behooves us to brings the abstractions and the realities that provoke their generation closer together. It is toward this end that this paper is devoted. The paper is presented on the site in four parts. Part I. how much & how many N.B. A paper entitled "Quantities, Units and Computing" by Marcus P. Foster [Computer Standards & Interfaces, 35, (2013) pp. 529-535] has recently come to my attention - it will probably be of interest. Judah L. Schwartz  mathMINDhabits by Judah L. Schwartz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |