Qualitatively different approaches to simple arithmetic
This study explores the qualitative difference in performance between those who are more successful and those who are less successful in simple arithmetic. In the event that children are unable to retrieve a basic number combination the study identifies that there is a spectrum of performance between children who mainly use procedures, such as count-all in addition and take-away in subtraction, to those who handle simple arithmetic in a much more flexible way. Two independent studies are described The first contrasts the performances of children in simple arithmetic. It considers teacher selected pupils of different ability from within each year group from 7+ to 12+. It takes a series of snapshots of different groups of children and considers their responses to a series of simple number combinations. This first experiment shows qualitatively different thinking in which the less successful children are seen to focus more on the use of procedures and in the development of competence in utilising them. The more successful appear to have developed a flexible mode of thinking which is not only capable of stimulating their selection of more efficient procedures but, the procedures they select are then used in an efficient and competent way. However, the use of procedures amongst the more successful is seen to be only one of two alternative approaches that they use. The other approach involves the flexible use of mathematical objects, numbers, that are derived from encapsulated processes. The below-average children demonstrate little evidence of the flexible use of encapsulated processes. It is the ability of the more able children to demonstrate flexibility through the use of efficient procedures and/or the use of encapsulated processes that stimulates the development of the theory of procepts. This theory utilises the duality which is ambiguously inherent in arithmetical symbolism to establish a framework from which we may identify the notion of proceptual thinking. The second study considers the development of a group of children over a period of nearly a year. This study relates to aspects of the numerical component of the standardised tests in mathematics which form part of the National Curriculum. It provides the data which gives support to the theory and provides evidence to confirm the snap shots taken of children at the age of 7+ and 8+. It indicates that children who possess procedural competence may achieve the same level of attainment as those who display proceptual flexibility at one level of difficulty but they may not possess the appropriate mental tools to cope with the next. The evidence of the study supports the hypothesis that there is a qualitative difference in children's arithmetical thinking.