Children's understanding of successive divisions in different contexts

To meet the demands of high-school mathematics, one must understand that mathematical symbols, beyond being used to represent events and situations of the external world (signifier function), are also meaningful by themselves (signified function). In four experiments it was investigated under which conditions and at which age level elementary-school children (grades 4-6) would become aware of this conflict between practical and theoretical considerations in mathematics. To encourage children to mention this conflict, they were presented with mathematical story problems describing familiar situations. These situations were constructed so that they would adhere to abstract mathematical concepts of infinite divisions and limit because they required successive divisions of quantities. Solving these problems resulted in a conflict between practical and mathematical considerations. Results indicated that students from grades 4 and 5 did not mention this conflict because either they neglected the mathematical perspective of the problems or their mathematical solution was based on severe misconceptions regarding rational numbers. Only in grade 6 had most subjects overcome these misconceptions. About 30% of these subjects mentioned the conflict between practical requirement and mathematical principles.