Academic journal article The Spanish Journal of Psychology

The Degree of Abstraction in Solving Addition and Subtraction Problems

Academic journal article The Spanish Journal of Psychology

The Degree of Abstraction in Solving Addition and Subtraction Problems

Article excerpt

In this study, the incidence of the degree of abstraction in solving addition and subtraction problems with the unknown in the first term and in the result is analyzed. Ninety-six students from first grade to fourth grade in Primary Education (24 students per grade) solved arithmetic problems with objects, drawings, algorithms, and verbal problems. The participants were interviewed individually and all sessions were video-taped. The results indicate a different developmental pattern in achievement for each school grade depending on the levels of abstraction. The influence of the level of abstraction was significant, especially in first graders, and even more so in second graders, that is, at the developmental stage in which they start to learn these arithmetic tasks. Direct modeling strategies are observed more frequently at the concrete and pictorial level, counting strategies occur at all levels of abstraction, whereas numerical fact strategies are found at higher levels of abstraction.

Keywords: abstraction level, arithmetic problems, strategies, mathematical development

En este estudio se analiza la incidencia del grado de abstracción en la resolución de problemas de suma y resta con la incógnita en el primer término y en el resultado. Noventa y seis alumnos de primero a cuarto curso de Educación Primaria (24 escolares por curso) resuelven tareas aritméticas con objetos, dibujos, algoritmos y verbales. Los participantes se entrevistaron de manera individual y se registraron en vídeo todas las sesiones. Los resultados indican un patrón evolutivo diferente en el rendimiento para cada curso escolar según los niveles de abstracción. Resulta significativa la influencia del nivel de abstracción sobre todo en primero y más aún en segundo curso, es decir, en el momento evolutivo en que se inicia el aprendizaje de estas tareas aritméticas. Las estrategias modelado directo se manifiestan más en el nivel concreto y pictórico, las estrategias conteo ocurren en todos los niveles de abstracción, mientras que las estrategias hechos numéricos se encuentran en los niveles de mayor abstracción.

Palabras clave: nivel de abstracción, problemas aritméticos, estrategias, desarrollo matemático

The PISA report (Organisation for Economic Cooperation and Development/PISA, 2005) has once more revealed the low mathematical achievement of Spanish and Mexican students, making it necessary to increase the effort in the educational setting in general and in mathematical research in particular. From this perspective, this study represents an attempt to analyze the importance of the degree of abstraction when solving problems of adding and subtracting in the first four years of Primary Education.

According to the constructivist theoretical framework, children learn to solve arithmetic problems by elaborating cognitive strategies in an active and informal way, at early developmental stages. The degree of abstraction of the task may or may not be an important facilitating factor to solve arithmetic problems. We assume that children first learn at a concrete level, so as to subsequently develop their mathematical efficacy at more abstract and complex levels. Although there are many investigations on arithmetic problems in general, and, more specifically, on the semantic structure, the degree of difficulty, the developmental level, the strategies used, and the mistakes made by students (see, for example, Bermejo, 1990, 2004), there is a remarkable shortage of studies that directly address the degree of abstraction of such problems and its incidence in student's mathematical achievement.

Verbal problems of adding and subtracting are usually classified according to their semantic structure into four types: Change, Combination, Comparison, and Equaling (see the diverse classifications in: Bermejo, 1990; Bermejo, Lago, & Rodríguez, 1998; Bermejo, Lago, Rodríguez, Dopico, & Lozano, 2002; Carpenter & Moser, 1982, 1983; Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993; Riley & Greeno, 1988; Riley, Greeno, & Heller, 1983; Vergnaud, 1982). …

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