Understanding the Development of Children's Mathematical Thinking When Teaching

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Position Paper ? The Importance for Teachers of Understanding how Children's Mathematical Thinking Develops

"Why is innumeracy so widespread even among, otherwise educated people? The reasons, to be a little simplistic, are poor education, psychological blocks, and romantic misconceptions about the nature of mathematics." John Allen Paulos 1988

The body of knowledge of how children think mathematically has increased dramatically in the last thirty years. This has caused many changes in the way mathematics is presented to students. Until the 1960's, primary school arithmetic programmes in New Zealand focused on speed and accuracy in the four operations and their application. During the 1960's with the dawning of the space race, there was a belief that more mathematicians and scientists would be needed. This initiated many changes in the arithmetic/mathematics curriculum. The language of sets was included in the mathematics curriculum with less emphasis on skill and more on structure. Structured concrete materials were developed and the cuisenaire rods were introduced to primary classrooms. Furthermore algebra, geometry and statistics were given more emphasis with number and measurement.

After this space race focus, researchers world wide have in the last twenty years taught us a lot about children's understanding of numeracy and how they come to develop these ideas. Young-Loveridge (1989) drew attention to the fact that many of the understandings children had on entering school in New Zealand were not well matched to the curriculum and what they were taught. Baroody et al (1990) mentions the work of ten different researchers or research groups who assisted the understanding of children's mathematical thinking in the last twenty years. Young-Loveridge (1999) acknowledged that teachers in Australia and United States that were given a framework, were better able facilitate their student's learning. She valued the work of Fuson and Renick in the United States and has used their framework in her work. Also she uses Baroody's increasingly abstract models for building concepts of double-digit numbers.

During this period, the change from behaviourist thinking to a constructivist perception of learning has brought about fundamental changes in instruction practices. The belief that knowledge is actively created or invented and not passively received has changed the approach of mathematics facilitation. Students are encouraged to discuss and share their ideas and construct their own ways to solve increasingly complex mathematics tasks. The statement of the MSEB and National Council of Research in 1989 that 'in reality no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics,' has given rise to major changes in our understanding of how we research and facilitate mathematics. Researchers have used interviews and other qualitative research methods to find out how student's think mathematically and this has revolutionized our understanding of how children think mathematically. From this research we have evolved frameworks of how children gain competency over number.

The last decade has seen considerably more development in the western world in our understanding of numeracy and the development of number frameworks. The work of Askew et al (1997) found that highly effective teachers had a particular set of beliefs underpinning a series of classroom practices and this included what it meant to be numerate and which presentation and intervention strategies were effective. This work and emphasis has followed the Third International Mathematics and Science Study (TIMSS).

The TIMSS report is the largest, most comprehensive international study of mathematics and science achievement ever conducted. More than 500 000 students from as many as 41 countries participated in this study in 1994-1995. This report has considerable significance as the awareness grows of how mathematical and...
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