Terezinha Nunes

Even the most elementary mathematical activities, such as counting, are carried out with the support of cultural tools: number words are cultural tools. In order to use cultural tools to solve mathematics problems, children have to rely on logical principles, which guide their problem solving activity. The aim of this presentation is to discuss how children's logic and cultural tools come to form a reasoning system, which can change either from progress in logic or form the use of new cultural mathematical tools. 

Three examples will be used to illustrate this point. The first one shows how children have to use counting in different ways to solve different types of problems. Knowing how to count is necessary but not sufficient for knowing how to use counting to solve problems. The counting tool, is this case, is under the control of the children's logic.

The second example will focus on counting systems and how their own structure influences how children progress in counting. Some oral number systems are more regular than others and children who learn to count using more regular systems understand the structure of the system earlier on and can use it more efficiently than children who learn to count using less regular systems.

The third example will focus on the translation across different mathematical tools. Under this theme the differences between oral and written arithmetic and also between representing a situation using fractions or ratios will be discussed.

Finally, it will be suggested that teaching mathematics involves helping children to develop reasoning systems where a sound logical understanding forms the basis for using different mathematical tools flexibly and for making connections between different tools.