Equipartitioning Operations for Connected Numbers: Their Use and Interiorization
During her fourth grade, Melissa had been paired with another child who had constructed only the tacitly nested number sequence. The teacher of the two children geared her activities to the other child and Melissa essentially served as the other child’s interlocutor. The other child did not construct any fraction schemes during her fourth grade and, Melissa, whose role was to interact with the other child at her level, constructed at most the partitive fraction scheme. Melissa had constructed the explicitly nested number sequence, so we paired her with Joe in their fifth grade in order to investigate her constructive itinerary. Our purpose in the upcoming analysis is to investigate her construction of the iterative fraction scheme as well as the unit fraction composite scheme and compare her progress with that of Laura while she was in the fifth grade. Although Joe served as Melissa’s interlocutor, we investigate whether the schemes that he established during his fourth grade were permanent as well as any accommodations he might make in them. In what follows, we analyze 17 protocols that were selected from the teaching episodes that started on the 20th of October and proceeded on through the 4th of May of the children’s fifth grade.