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The origin of arithmetic skills: a phenomenographic approach
Gothenburg, (1987)
Аннотация
"If teachers knew about, and used, children's own ways of learning maths, the teaching of maths by adults would end up in the learning of maths by children". This polemic was the hypothesis which the present study took as its point of departure.
As an introduction, two apparently conflicting phenomena are discussed. On the one hand pupils who experience difficulties with mathematics, even at the end of their school life demonstrate that they lack the most basic concepts of the ten numbers which constitute the arithmetical system used throughout the Westernized world. At the same time, experiences with some 25 different first-grade classes suggest that in any group of a score of children about to start school 1/7-years-old in Sweden), one or more will seem to have already developed these concepts.
The investigation consists of two parts: an interview study with school starters, and a teaching experiment. The purpose of the interview study, which was carried out before any formal teaching started in the years 1982 and 1983, was to answer the question: "what are the informal ways in which preschool children form concepts of the ten basic numbers?" To answer this question 105 school starters were given simple word problems, and were asked to describe the strategies they used to solve them. Their answers were tape recorded and transcribed word-by-word.
The study is an example of phenomenographic didactics. The children's own strategies, and the conceptions these strategies were thought to express, formed categories of description. These were used as a guide in the teaching experiment, in which 39 of the children who were interviewed in 1982 took part, and in which the other children interviewed in the same year formed a control group.
Twenty-four children had demonstrated that at the time they started school they had neither established the pre-concepts, nor acquired the counting skills, which they would need to be able to follow traditional maths teaching in grade 1. These children, 11 in the experimental group and 13 in the control group, were re-interviewed two years later. The results of these interviews showed that while at least nine of the thirteen children in the control classes had still not formed the concepts of the ten basic numbers, all the children in the experimental classes had done so and could also apply this understanding in more complex problem-solving.
The results of this investigation tend to confirm the starting hypothesis:
- that, even in their preschool days, children develop without formal instruction the concepts they need to solve their own everyday problems, adequately and efficiently:
- and that if teachers use the children's own concepts as the starting-point for more-advanced thinking, then the teaching of mathematics will assuredly result in the learning of mathematics.
