Terezinha Nunes Carraher

Culture, Arithmetic and Mathematical Models

Three sources of data on Mathematical knowledge are analyzed here in detail, leading to four main conclusions. First, reasoning principles underlying written and oral Mathematics appear to be the same. Second, there are diverse ways of understanding and using mathematical concepts which depend upon the cultural conditions under which Mathematics is practised. Third, schools transmit culturally perfected mathematical tools (such as numeration systems, algorithms, and formulas) and tend to make students good model-users but are perhaps carrying out this function at the expense of meaning. Fourth, Mathematics learning in daily life produces meaningful procedures which may, however, be of restricted applicability.

Negotiating the results of mathematical computations

Abstract Psychologists have traditionally treated concepts (and conceptual structures) as the basic source of organization for knowledge disregarding the role played by the personal and social goals implicit in situations in which knowledge is acquired. These goals perhaps define ‘the rules of the game’ and have a much more important place in determining the organization of knowledge than has been hitherto acknowledged. An analysis is presented of implicit goals in mathematics lessons and situations in which mathematics is used outside school in an attempt to show that the loss of meaning observed when students solve mathematical problems is consistent with learning in this situation, while mathematical problems solving outside school creates goals that result in the preservation of meaning and in different ways of doing mathematics.

UNDERSTANDING THE ALPHABETIC SYSTEM

Alphabetic writing is a conventional system of representation which allows for the generation of not previously learned combinations of elements into meaningful units. Thus its acquisition must be conceived as going beyond response learning and involving knowledge of the rules of the system, even if only intuitive. In an alphabetic system, the rules that relate speech to writing are based upon a phonemic analysis of the word, and not upon meaning, as in ideographic writing, nor upon syllabic analysis, as in a syllabic system. Alphabetic writing is thus a system which probably requires some degree of awareness of linguistic form.