The early construction of mathematical meanings:Learning positional representation of numbers.

The purpose of this chapter is to discuss several critical issues in early mathematics education, particularly the use of manipulatives to teach positional representation of numbers. For our prehistoric ancestors, the representation of numbers larger than the ones that could be counted on their fingers and toes represented a significant challenge. This problem was solved by a brilliant insight of using position to represent and distinguish large numbers. Some tools have been extremely useful in performing math operations on large numbers. In the history, the abacus and the pascaline played a part in implementing and improving the system of positional representation of numbers. These tools can also play a significant role in helping young children understand the role of position. This chapter discusses and provides examples of how teachers can facilitate young children acquiring strategies for using these tools to represent numbers and to make arithmetical operations. A point worth noting is that the use of these tools is not intrinsically helpful in promoting children's learning. Their effectiveness depends largely on how thoughtfully and intentionally they are used, shaping classroom practices. In other words one simply cannot give children a set of blocks or any other tool and expect that learning is produced from access to those tools. Teachers must think carefully about what it is they want children to do and how they might benefit and then structure their environment and guide children so that these outcomes are likely to be achieved.