In my last blog, we discussed how the student needs time to imagine counters, or use something that can stand in the place of counters, so the child will gain enough experiences to make just the numeral meaningful. How can we encourage students to do this?
Let’s imagine a child has the goal of figuring out how many counters there are in two piles when combined. The child is allowed to count out 5 counters, then count another 4 counters in a separate pile, and is asked how many counters in all or in the two groups if combined. The child only has to imagine the two piles as one pile and then be able to count all the counters to find the total. So what would happen if we covered one of the two piles?
What we have seen in the students with whom we work and the research we use to inform that work is that students will find something that can take the place of the hidden counters. If you cover the pile of five counters, the child might hold up five fingers or remember the domino pattern for five and count that instead of the counters that are hidden. If they have just counted the counters, the child might even try to remember what the pile looked like and count over the covered pile!
Imagining the counters that are no longer visible is the first step in gaining an understanding of what a numeral means. It is like trying to remember the names of your students without looking at your seating chart. You may mess up and you might have to look back at the seating chart, but if you always read from a seating chart or from name tags on the children’s desk you never meet the goal of just knowing their names. Taking away access to the seating chart creates a need to find another way to remember their names, just like taking away access to the counters creates a need to find something to count instead. Over time, the child depends on less and less concrete things to take the place of the hidden counters.
Eventually, students will not need to count the hidden pile at all. Next time we will look at what we can change in our tasks to help students get to that place.