Counting. Nothing in mathematics gives us more of a return on our investment.
In my previous blog, we briefly looked at beginning counters, who start with rote counting and then move on to count concrete items. To understand the background of what will follow, I recommend that you read the May 23rd, 2018 blog.
Before we continue, I will mention that the Common Core Kindergarten standards repeatedly refer to students using “OBJECTS or DRAWINGS” to solve problems in the OA and NBT domains. The availability of objects and drawings is critical, because beginning counters need concrete items to count, even if they are drawn and then counted. The payoff for all of their concrete counting experience comes when they begin to be able to count even when items are hidden!
Unfortunately, the Common Core 1st and 2nd Grade standards only briefly refer to objects or drawings and NOT AT ALL TO COUNTING HIDDEN ITEMS. We lost an investment opportunity when we left this out of the standards, which is evident by the many children who fail to conceptually transition from counting concrete items to adding abstractly.
Counting: Imagined Items – Counting All
The bridge between counting concrete items and “counting on” is when students begin to count imagined items in additive situations (we like to say addition, but it’s still counting). The students’ investment in counting lots of different concrete items (including items like objects, patterns, movements, sounds, touches, etc.) gives them resources to draw from as they begin to mentally create imagined items. If students have had few counting investments, they will have few resources to draw from, like this kindergartener:
As you might have noticed, she just guessed because she had no way to imagine 8 or 5\. On the other hand, students with lots of investments in counting different items begin to be able to imagine counting hidden items. After some investments in meaningful counting experiences, watch this same student three sessions later as she is able to imagine a dice 5 pattern and use it to help her count two collections, one of them hidden:
Students may also use previous experiences of counting fingers to help them count hidden items. In this next video, you may notice this kindergartner’s hesitancy to use or show his fingers at first. In fact, he guesses twice before taking the time to think the situation through carefully. When he decides on 9, he seems very confident and pulls his hands out of his pockets to show how he thought about it. As he demonstrates with his fingers, his previous experiences with finger counting help him to be able to count all (both hidden collections) in his imagination:
Students generally begin to imagine finger patterns or spatial patterns (referred to as “mental images” in K.OA.1) to help them count in additive situations because they are readily available.
Next time, we will look at what happens as students start to cash in on their investments of counting imagined items as they begin to count on.