Based on close observations of two 4-year-old children responding to their parents' requests for quantitative comparisons, we offer a "participationist" account of the origins and development of numerical thinking, one that portrays numbers as a product rather than a pregiven object of human communication. In parallel, we propose a participationist reinterpretation of relevant past research, most of which has been guided by the metaphor of learning-as-acquisition (of mental schemes, of concepts, etc.). The point of departure for our analyses is the assumption that thinking can be usefully conceptualized as a special case of activity of communicating and that learning arithmetic can be thought of as a development of a specialized discourse. We claim that the development of this discourse involves the ability to see as "the same" things that, so far, could only be seen as different. In the longer run, this ability will lead to the objectification of the discourse-that is, to the use of number words as if th...

/pdf/why-cannot-children-see-as-the-same-what-grown-ups-cannot-1osxfg1v7f.pdf

Why Cannot Children See as the Same What Grown-Ups Cannot See as Different?— Early Numerical Thinking Revisited

Most children enter school with reasonably good problem-solving strategies (Carpenter, Moser, & Romberg 1982). A significant feature of these strategies is that they reflect a careful analysis of the problems to which they are applied. However, after several years many children abandon their analytic approach and solve problems by selecting a memorized algorithm based on a relatively superficial reading of the problem (National Assessment of Educational Progress [NAEP] 1983). Why is it that many children begin school with simple but sound problem-solving procedures, and then, as they receive formal mathematics instruction, replace them with shallow and sometimes meaningless procedures? The question is fundamental. The answer is not entirely clear, but re-

Children's Mathematics Learning: The Struggle to Link Form and Understanding.

A model that describes the construction and execution of decimal computation procedures is presented. Our hypothesis is that students compute by relying solely on syntax-based rules; semantic knowledge has no effect on performance. To test the claim, a model is developed in which computation procedures are viewed as chains of component symbol manipulation rules. The model assumes that students acquire through instruction the individual rules that achieve subgoals in the computation process. The task for the procedural system is to select rules that satisfy each subgoal in sequence. The model specifies the rules of the system and identifies the syntactic features of the task that affect the selection of individual rules at each decision point. It then predicts the relative difficulty of decimal computation items and predicts the procedural flaw that will occur most frequently on each item. Written test and interview data are presented to test the predictions. Concluding comments discuss the nature of students' computation procedures, compare the model with other models of computation performance, and outline how the model might inform instruction. In this article, we present a model of how students compute with decimal numbers. The model consists of symbol manipulation rules that we believe are precisely the rules students acquire, store, and execute to compute with decimals. The purpose of the model is to describe the nature of students' computational skills and to demonstrate the extent to which students' computation performance is procedurally based. Our hypothesis is that by the time students reach upper elementary school their behavior on many mathematical tasks can be described in syntactic rather than semantic terms. Sufficient evidence has accumulated over the past 10 years to suggest that students' behavior on mathematical tasks changes in important ways as they

A Model of Students' Decimal Computation Procedures

60 children aged three, four and five years were given simple addition and subtraction problems in a variety of task forms. Overall, children showed a considerable degree of competence on these tasks: however, performance was significantly affected by age, social class, size of number involved and form of task presentation. Most children were successful when the tasks were embedded in concrete situations ‐‐ either real or hypothetical ‐‐ and when the numbers involved were small. A significant number of children also succeeded on large‐number versions of these embedded tasks. Very few children, however, succeeded when the task was phrased in the formal code of arithmetic ('what does one and two make?') irrespective of the size of number involved. The educational implications of these findings are discussed.

Can Preschool Children Add and Subtract

Summary This paper reports on an examination of a number of maths books originally undertaken as part of a contribution to a maths diploma course for primary teachers. The books cover the age range 3–13 and were published between 1970 and 1978. It was discovered that mathematical and scientific skills become increasingly defined as masculine as the pupils move through junior and middle school. Men feature predominantly in these materials throughout the age range and there are few models of adult females operating in the world of mathematics. The representation of girls becomes more marginal and they almost disappear from the scene towards the upper limit of the age range. The fading presence of girls in these books parallels the decline in girls' involvement and achievement in maths between infant school and GCE O and A level examinations.

Girls and boys in primary maths books

Early Mathematical Experiences

This paper gives an account of an experiment involving 176 children of 6 and 7 years and their difficulties with subtraction problems. By using two screening questions, four levels of attainment were established and two teaching programmes were then devised to help the children forward to their next level. The results show that it is possible to assess a level of understanding in subtraction quickly and accurately and that, through the teaching programmes, pre-requisite skills needed to establish a sound foundation for subtraction can be taught. The experiment indicates, however, that an attempt to symbolise too soon is a danger to be avoided.

A Subtraction Experiment with Six and Seven Year Old Children.

This paper arose from research on numerical errors by 6 and 7 year-olds particularly in subtraction. The present investigation covers four categories: 11 year-olds: school-leavers: student nursery nurses: bus conductor trainees. The central questions were whether earlier errors recurred at these later stages and how far feelings towards the subject (sometimes formed when young) affected performance. The sample was small (N=93) but the results at least suggest questions and directions for further work. In each category, for example, over 25 per cent got a very basic subtraction ‘sum’ wrong. This indicates that some young children are asked to do sums which are beyond their comprehension and the resulting confusion is difficult to remedy later.

An investigation into subtraction

Addition: From Concrete to Symbols.

Pre-school pre-mathematics

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