INTEGERS VERSUS FRACTIONS: A STUDY WITH EIGHTH GRADE STUDENTS

Aurora Gallardo

CINVESTAV – IPN

agallard@mail.cinvestav.mx

Rogelio Novoa

Universidad Autónoma del Estado de Morelos

In this research (Novoa, 2000), integers are examined in relation to fractions’ numerical domain, as they are taught at school. The study is based on Filloy’s (1990) methodological-theoretical framework through the development and integration of three components: models of teaching, models of cognitive processes and models of formal competence. We report on results derived from an experience involving 41 eighth grade students who answered an exploratory questionnaire. Nine students were selected to participate in individual clinical interviews.

Study’s Evidence. When localizing fractions on the number line, some miss-interpretations found were an integer plus a decimal fraction, or an integer plus a unitary fraction. Many students order the negative integers on the basis of their absolute value and when ordering unitary fractions they only consider the denominators. Some of them compare fractions taking only the numerators into account or they identify negative integers as positive unitary fractions having such numbers as denominators. In operations with integers, most students apply the law of signs (minus times minus becomes plus). Fixation on this law hinders subtraction operation. When doing a subtraction, the students can become aware of the order of integers. Nevertheless, when operating with fractions they focus attention on algorithms and miss which fraction is larger. In regard to problem solving, the students build their own syntax altering the algorithm for subtraction.

Study’s Implications. It is important that students realize that they are working with three different numerical domains: natural numbers, integers and fractions. They must learn to add and subtract integers before the law of signs is introduced.

References

Filloy, E. (1990). PME Algebra research. A working perspective. Fourteenth Conference Proceedings 1, México, pp. 1-33.

Novoa (2000). De los números naturales y quebrados positivos a los enteros y quebrados negativos. Un estudio con alumnos de segundo de secundaria. Tesis de Maestría. Universidad Autónoma de Morelos.