ADDITION: FROM CONCRETE TO SYMBOLS

ADDITION: FROM CONCRETE TO SYMBOLS

Introduction

The research report in this journal is concerned with the operation of addition. It investigates the possibility of assessing, and assisting, the readiness to cross the threshold from concrete to symbolism. This addition project grew out of previous research on difficulties experienced by 6 and 7 years old children in subtraction (Matthews, 1981, 1983) Synopsis

Laying sound mathematical foundation is of great importance for the future development of the subject. Young children need many concrete experiences before they are presented with the sophisticated tools of an abstract subject. As reported in Mathematics Counts (HMSO 1982),practical work is essential throughout the primary years). For most children, practical work provides the most effective means by which understanding mathematics and can develop. It is not babyish to work with practical materials while the need exists. As all children are different and learn at different rates, assessment of their achievements is vital. The objective of the research were to produce a method of indicating the level of understanding of individual children (aged 6 or 7) of the addition and equality symbols (+ and =) and their ability to use them and to devise a programme of activities to help those children found to be in need.A set of 3 simple screening questions was devised to find out which children need help. These were tested and refined using one-to-one interviews. A programme of activities was then developed. This was given to an experimental group of children at risk whose performance matched against that of a control group who were not involved in the experiment.

In an Inner London school a group of 43 children within the define age-range was selected for a pre-pilot experiment. These children were within 2 months of transferring to their junior school and were all expected to be conversant with the symbols +, =, , and -: some were even juggling with , presented them with the screening questions on a one-to-one basis and the children were assigned to one of the following categories:

i. Those not all ready for symbolism

ii. Those having some difficulty in coping with symbolism

iii. Those who were conversant, they needed no extra help.

A trial set of activities was devised and tried on them and the success rate seemed satisfactory. The main purpose was to discover whether the screening questions and activities were comprehensible and not treating to the children.

The programme of activities had been refined. It took account of Piagets work on the conservation of number and the childs ability to construct two equally numerous arrays from two arrays containing unequal numbers of objects (Piaget 1952). It also covered counting strategies (cf. Carpenter, Moser and Romberg 1982). In fact all possible pre-requisite skills and facts needed to facilitate the childs understanding of addition of two numbers were included in the programme. Concrete aids, hand calculators and paper and pencil were all employed as appropriate.The experimental group were taken through the programme individually, with weekly sessions of 10 to 15 minutes in duration within 8 weeks. The control group simply carried on with their normal school work.

In order to ensure that relational learning had in fact taken place, the experimental group was tested again after a period of between 5 and 7 weeks, with the following question asked.

i. What is 6 take away 3?

ii. What is 6 add 3?can you write it like sum?

iii. Tell me quickly what is 5 and 5.

iv. Now tell me what is 6 add 5.

v. Can you read this:

7

+ 3

Can you write the answer?

The problem is making sure that the children did not always connect the question with addition through the capture effect (Skemp, 1971). Question 2 is the original screening question that required written addition. Question 3 is a quick mental arithmetic calculation, followed quickly by question 4, to find out whether the children were using known facts or counting in ones. The final question was simply to check whether addition in a slightly different format was recognised.

Using Question 2 as the criterion for success, the result of this post-post test was that 27 children of 30 members of the experimental group had retained the learning that had taken place during the programme while the remaining three were still not quite sure of which symbols to use.

Conclusion

From the result shown the screening questions and the programme appear to have justified themselves. They are simple, practical and easy to administer they should help hard-pressed teachers when they are assessing individual performance. Assessment is an essential part of the work of all teachers, testing, whether written or practical, should never be an end in itself, (HMSO, 1982).Reference:Julia Matthews, Journal of Science and Mathematics Education in S.E. Asia, Addition: From Concrete to Symbols, June1989