Enquadramento de números racionais em intervalos de racionais: uma investigação com alunos do ensino fundamental - Lage, Luciana (2006)
The purpose of the present study was to investigate which mathematical concepts, properties, and procedures, as well as settings (in terms of numerical, graphic, or other types of representation), are used by students of a 7th-grade class, in a private school in the city of São Paulo, when challenged to find solutions for mathematical activities. The activities involved framing rational numbers on rational intervals and were designed by teachers of that school, based on a proposal by a member of the same group of Algebra Education, at Pontifícia Universidade Católica de São Paulo, to which the author is affiliated that was developed from activities originally devised by Régine Douady (1986). Observation in the classroom covered four sessions and was conducted in the light of the case-study methodology and on the notion of tool object dialectic of Douady (1984). The analyses prioritized students written and oral productions that took form while they experienced the process of solving the activities proposed. The productions revealed that the students made use of interplay of two, three, or four of the following settings: numerical, native language, algebraic, and geometric. Several strategies for solution were devised, in which the students utilized as mathematical tools chiefly the notions of positive number, even number (with possible flaws in meaning), rational number (with possible flaws in meaning), multiplication, arithmetical average, segment, and numerical intervals (though with different meanings among the students). Other relations used were to be greater than, to be less than, and to be a multiple of, as were the order relations to be greater than of equal to and to be less than or equal to