Presented at ISEC 2000

Development Of Intervention Program In Mathematics In Regular Classes For Children With Low Early Mathematical Competence

Kavkler Marija and Magajna Lidija, Counselling Centre for Children, Adolescents and Parents, Ljubljana, Slovenia
Tancig Simona, Faculty of Education, University of Ljubljana, Slovenia

ABSTRACT

Arithmetics learning difficulties in school years can be influenced by an insufficient development of early mathematical competence. These children need special and intensive treatment in mathematics as early as possible.
The aim of the study is to develop effective program of interventions in mathematics for children in regular first grade classes with low early mathematical competence in preschool years.
A sample of 30 children aged seven years and five months which are low attainers on The Utrecht Early Mathematical Competence Test (Pennings et al., 1995) in preschool years, was divided in three groups of 10 children each. Two groups are experimental and one is control.
The first group of 10 children (E1 ) was included in a special program of intervention in math, based on realistic mathematics, conducted by the teacher in the classroom and by a special teacher once a week for 2 hours. The teachers of the children in the group E1 and the special educator attended training in realistic mathematics. The second group of 10 children (E2) was taught by their teachers in the classroom. The teachers of the second group attended the same training in realistic mathematics as the teachers in the first group. The third group of 10 children (Co) was presented with regular (traditional) teaching of mathematics without any special help in learning.

A set of tests was applied for assessing arithmetic knowledge, counting strategies and calculation strategies of all three groups of children at the beginning and at the end of the experiment.
The duration of the experiment was three months. During the experiment the teachers of both experimental groups (E1 and E2) introduced in regular classes elements of realistic mathematics (empty number line, bus problems, shopping problems, etc.).
Comparision of pretest and postest results on counting across all three groups shows that experimental group E1 made the greatest progress on correctness of counting and development of more sophisticated strategies.
One way analysis of variance of scores on the Sugerman's test showed significant progress of experimental group E1 and experimental group E2 on orally presented tasks of addition and subtraction.
Progress in achievement on written arithmetic tasks of addition is statistically significant for all three group.
Similarly, all three groups made progress in achievements on bus task. Progress in development of strategies is most obvious in experimental group E1.
Finally, results of the present study suggest that a program of intervention based on realistic mathematics in regular classes together with additional intensive and individualised program is the most beneficial for children with low attainment on early mathematical competence test in preschool years.It seems that such program, introduced as early as possible in schooling years, can be good prevention for later development of serious arithmetics learning difficulties.

INTRODUCTION

Children, who are low achievers in mathematics have important deficits in basic arithmetic knowledge and strategies. Their problems mathematics needs special professional attention because of the long-term negative effects on schooling and life-adaptation. Early and intensive help is needed.

The traditional practice in mathematics education in Slovenia puts more emphasize on consolidation of basic computational procedures, fact orientated learning and explicit teaching of strategies for mental calculation. Less attention is given to the growth of the processes of understanding and problem solving in children.

The Dutch approach of realistic mathematics teaching (RME) derives from a more complex and meaningful conceptualization of learning and is process oriented, focused on the growth of the students knowledge and understanding of mathematics. RME can be described by means of the following 5 characteristics (Treffers, 1987) : the use of contexts, the use of models, student's own productions and constructions, the interactive character of the teaching process and the intertwinement of various learning strands. The use of models, originating from context situations, as bridges to higher levels of understanding plays a key role.

Some important issues in recent development in this approach include implementation of RME teaching methods in classroom practice, more differentiated attention for specific groups of students and students with learning difficulties in mathematics.

AIMS

Arithmetics learning difficulties in school years can be influenced by an insufficient development of early mathematical competence. These children need special and intensive treatment in mathematics as early as possible.
The aim of the study is to develop effective program of interventions in mathematics for children in regular first grade classes with low early mathematical competence in preschool years.

METHOD

Subjects

The subjects of the study were 30 first grade children from regular schools (mean age of 7 years and 5 months). The 30 children were selected from 300 preschool children, tested in a previous study with the Utrecht Early Mathematical Competence Test (Pennings et al., 1995) and classified at the age of 6 years and 6 months as low attainers in mathematics ( Tancig, Kavkler, Magajna, 1999).

The sample, was divided in three groups of 10 children (3 boys and 7 girls in each group). Two groups were experimental and one served as a control group.

• The first group of 10 children (E1 ) was included in a special program of intervention in math, based on realistic mathematics, conducted by the child's own teacher in the classroom and by a special teacher in a small group once a week for 2 hours. The teachers of the children in the group E1 and the special educator attended a 10 hours distributed training in realistic mathematics.
• The second group of 10 children (E2) was taught by their teachers in the classroom. The classroom and the after-school teachers of the second group attended the same training in realistic mathematics as the teachers in the first group.
• The third group of 10 children (Co) was presented with regular (traditional) teaching of mathematics without any special help in learning.

Materials

A set of tests was applied for assessing arithmetic knowledge, counting strategies and calculation strategies of all three groups of children.

• AAZ - Automatization of arithmetic facts and procedures (Aubrey, C., Tancig, S., Magajna, L., Kavkler, M.,1998), a paper-pencil test was used to measure arithmetic knowledge for all basic artihmetic operations.
• Sugarman's test (1994) was administered for assessing calculation strategies for addition and subtraction. The instructions and the children's responses on the test are oral.
• A set of tasks was carried out for assessing counting strategies of the following types: counting out, counting backwards, sequential counting and flexible counting.
• Two simple real-life problems arithmetical problems with illustrations and oral instructions were used for assessing problem solving strategies.

During the initial and final testing the children were allowed but not encouraged to use various materials.

Procedure

The duration of the experiment was three months. Research started at the beginning of December and continued until the end of March.

The children of all three groups E1, E2 and Co were tested with the described set of tests for assessment of mathematical knowledge and strategies at the beginning and at the end of the experiment. During the experiment the teachers of both experimental groups (E1 and E2) introduced in regular classes elements of realistic mathematics (empty number line, bus problems, shopping problems, etc.).

The special teacher, working with the two small groups of 5 children in E1 included the same elements of RME as the classroom teachers with a more individualized approach. In E2 more everyday practice of RME methods was possible because also the after-school teachers attended the RME training.

ANALYSIS

Frequency test for each variable was obtained for each group at initial and final testing. Contingency tables for mathematical strategies were produced and qualitative analysis of data was done. The progress of the groups from the initial to the final testing was analized by ANOVA.

RESULTS

Table 1. FREQUENCY OF CORRECT ANSWERS ON DIFFERENT COUNTING TASKS BEFORE AND AFTER TRAINING

Table 2. ADDITION AND SUBTRACTION (SUGARMAN )

Table 3. SUBTRACTION: TYPES OF STRATEGIES ON SUGARMAN'S TEST (number of observations)

Table 4. AAZ: AUTOMATIZATION OF ARITHMETIC FACTS

CONCLUSIONS

• Comparision of pretest and postest results on counting across all three groups shows that experimental group E1 made the greatest progress on correctness of counting and development of more sophisticated strategies - as seen from the presented example of counting backwards. Observed progress of experimental group E1 is likely to be attributed to the special training of counting (during the program of intervention) for the E1 group. Both the experimental group E2 and the control group Co have less emphasis on counting skills. Since counting significantly influences the development of number concept and formation of basic arithmetic skills, special attention has to be paid to these two, specially at the beginning of schooling.
• One way analysis of variance of scores on the Sugerman's test showed significant progress of experimental group E1 and experimental group E2 on orally presented tasks of addition and subtraction (FE1 =22.93 , pE1 =0.0001 ; FE2 = 21.02 , pE2 =0.000 ). In both groups emphasis was on realistic mathematics which lead to better understanding of arithmetics operations. Greater progress is also seen in experimental groups E1 and E2 in strategies they used. Many children without any arithmetic strategy at the beginning of experiment made substantial progress after training.
• Progress in achievement on written arithmetic tasks of addition of the AAZ test is statistically significant for all three groups (FE1 =13.66 , pE1 =0.002 ; FE2 = 37.68 , pE2 =0.000 ; FCo =5.27 , pCo =0.03). But only both experimental groups E1 and E2 made also significant progress on subtraction tests. · Similarly, all three groups made progress in achievements on bus task. However, both experimental groups E1 and E2 made the greatest progress. Progress in development of strategies is most obvious in experimental group E1.
• Finally, results of the present study suggest that a program of intervention based on realistic mathematics in regular classes together with additional intensive and individualised program is the most beneficial for children with low attainment on early mathematical competence test in preschool years. It seems that such program, introduced as early as possible in schooling years, can be good prevention for later development of serious arithmetics learning difficulties.

LITERATURE

Aubrey, C., Tancig,S., Magajna, L., Kavkler, M.: Getting it right from the start?: the influence of early school entry on later achievement in mathematics. 8th European Conference on Quality of Early Childhood Education (EECERA). Santiago de Compostela (Spain), 2-5 September 1998.

Aubrey, C., Tancig, S., Magajna, L., Kavkler, M. (1998). Development of Numeracy in England and Slovenia. ECER (European Conference on Educational Research). Ljubljana, 17. do 20. september.

Aubrey, C., Tancig, S., Magajna, L., Kavkler, M. (1997). The Development of Children's Mental Methods of Calculation: How Does It All Add Up? Paper presented at the 7th European Conference for Research on Learning and Instruction. Abstracts. August 26-30, 1997, Athens-Greece.

Gravemeijer, K.P.E. (1991). Developing Realistic Mathematic Education. CD-B Press.

Streefland, L. (ed.) (1991). Realistic Mathematics Education in Primary School, CD-B Press.

Tancig, S., Kavkler, M., Magajna, L. (1999). The Development of Early Numeracy in Slovenia.

Treffers, A. (1987). Three Dimensions. A Model of Goal and Theory Description in Mathematics - The Wiscobas project. Dordrecht: Reidel.

Index