Counting-on is one of the things I have come across in Les Steffe’s research that is crucial, but not necessarily an obvious goal to have for students. It would seem that if a student could count-on (ex: given the problem 6+5, would start at six and count-on five more rather than starting from 1 and counting all the way to 11), then the student likely sees the number six similarly to how we as adults see 6. As it turns out, they still have to move through some other steps before constructing what Steffe calls a composite unit. In order to understand what a composite unit is, I am first going to tell you how he recommends helping a child to construct this composite unit.
Steffe talks about first having the child segment their counting into, say, 3’s. 1,2,3… 4,5,6… 7,8,9… etc. The child can recognize that there are three elements, but can the child see how many threes they have segmented? For this to happen, the elements in three would need to be contained in a unit that is countable as one item. They need a mental structure that is similar to taking marbles and putting 3 in each bowl, and then counting the number of bowls. This mental ‘containing unit’ can make 3 elements a composite unit that can be counted as one thing. Now the student would be able to say 1,2,3, that’s one, 4,5,6, that’s two, 7,8,9, that’s 3, etc. Once the student develops this composite unit, they will see 3 as the elements 1,2,3, but also as one thing containing those elements.
This development plays an important role in a couple of areas mathematically. The more obvious one is multiplication, since the activities suggested seem multiplicative. But it turns out that the development of the composite unit first plays an important role in their additive reasoning, which is why the activities which help develop this unit are important prior to helping a child construct multiplicative reasoning.
After a child can count-on, the next goal should be for the child to be able to use their number sequence to keep track of counting-on. For instance, in the first problem I introduced, the child could hold 6 in their mind saying, “6, … 7 is one, 8 is two, 9 is three,” etc. They used their concept of 5 to keep track of their counting-on past 6. They no longer need fingers or blocks to keep track of ‘five more’. This occurs in children who have constructed a composite unit.
Students’ construction of a composite unit ends up playing a role in multiple layers of their mathematical understanding. I have already mentioned the role in addition and multiplication, but it turns out that it prepares the student for developing fractional knowledge as well. I will make this connection in my next blog, and read my December 13 blog post to learn about helping students develop a composite unit.