When you hear the term partitioning, you might think about partitive division or partitioning a discrete set of objects, like dividing a dozen cookies among four people. Partitioning also applies to continuous intervals. An example would be the task of equally sharing a candy bar among 5 friends. The outcomes for how a child would solve this problem correspond to the child’s numerical stage in cognitive development, that is, the number operations that the child can take as given when approaching a math task. Partitioning is one of the operations and a description of it follows.
At the first numerical stage, even without counting, a child could correctly partition the 12 cookies by giving them out one at a time in succession. Such a child would accomplish the task and not even know how many cookies each person received without going back and counting. It would be quite another matter to ask a child at the first numerical stage to partition a continuous interval, like a candy bar, evenly among 5 people. Here, the child would probably begin cutting off pieces too small to be a fair share. As well, the child would not be able to track how many pieces were cut without recounting after each cut. The result would be one piece that is very disproportionate or too many pieces.
At the next numerical stage, a child would cut the pieces, but only hope that they are the same. The child at this stage doesn’t have the ability to think of taking one piece as a measure to mark off or duplicate to get the other pieces.
At the third numerical stage, the child could look at the whole candy bar and estimate a fair share. Then, using that as a measure, they would be able to duplicate shares to get the rest of the pieces. If the original estimate is off, the child could readjust before making the cuts.
Adults usually think that children’s inability to equally partition in a continuous bar is due to just undeveloped hand-eye coordination. Actually, it is due to undeveloped mathematical structures. What effect does all this have on the child’s ability to understand fractions? Fractions are not seen as measures or numbers within themselves, but as parts within a whole. It is the way a child can compensate, but it is very limiting. (Ulrich, 2014 – from Wisdome, vol. 4) (Steffe and Olive, 2010)
What can a concerned teacher or parent do to help spur the development that is needed? More to come…