Counting. Nothing in mathematics gives us more of a return on our investment.
In my first blog post in this series, we briefly looked at beginning counters, who start with rote counting and then move on to count concrete items. These are huge investments in their mathematical futures.
In the second post, I discussed and showed examples of what it looks like as students begin to count imagined items. This is the first payoff from all of the counting that they did with concrete items. They can imagine the concrete counting and use that to count “figuratively.”
In the third post, there was some talk and video showing how students begin to count their movements, and then begin to count by just saying number words.
Counting: Counting By Ones – Counting On
At this point, students begin to get huge returns on their investments of counting all by:
- Saying a number word sequence
- Counting concrete items
- Counting in the imagination
- Counting movements
- Counting number words.
The payoff? Counting on.
With enough experiences of counting from 1 in all those different ways, they begin to “just know” that they don’t have to count from 1 to 8 to know what “eight” means. They’ve done it plenty of times, and they can just start with 8 (first addend) and continue from there.
So if, for instance, a student encounters an additive situation where they need to join 8 and 3, they may say, “8…9, 10, 11” while keeping track of the 3 (second addend) on their fingers or with a spatial or temporal pattern. In the following video, pay close attention to the student’s fingers. You may see her counting to 3 on her fingers but never vocalizing her count. Because of the elapsed time, we know she “counted on” from 8.
Notice: SHE IS STILL COUNTING! Again, we normally call this addition, and that may well be addition. However, she still solves it by counting!
Next time, we will look at what happens as students’ investments of counting on by ones begin to pay off as they find that they can count their counts, leading to success with missing addend, subtractive, and multiplicative situations — STILL BY COUNTING!