Looking at the Complexity of Two Young Children's Understanding of Number

This paper presents a qualitative study that investigated two third-grade students' understanding of number. The children were videotaped while they worked to record everything they knew about the number, 72. Their artifacts and conversations were then analyzed using the Pirie-Kieren dynamical theory for the growth of mathematical understanding as the theoretical framework. The results of the video analysis revealed the two students' understanding of natural numbers as being conceptually complex and existing in several different realms of the Pirie-Kieren model. Significant instances of "Primitive Knowing, Image Making, Image Having, Property Noticing, Formalizing, Observing", as well as how the children's understanding existed beyond "Don't Need" Boundaries are identified and examined in detail. Other features of the model--"Structuring, Inventising, Folding Back, and Connected Understanding"--are also explained and possible examples illustrating these kinds of mathematical thinking in relation to the two children's understanding of number are offered. (Contains 5 figures.)