We’ve covered the critical pieces of developing counting with meaning in this post. What happens next? Enter additive thinking!
When counting gives way to reasoning, it is the beginning of additive thinking. The journey from “count everything” to “just know it” unfolds gradually over several years, and it looks something like this:
Children need to work within the numbers and quantities that they are comfortable to make this transition. The friendly territory in which to develop additive fluency is within 0-20. Over the course of their time building this fluency, students are also developing familiarity with the structure of our base ten number system and learning to work within its patterns to apply what they've developed though those smaller values.
Building that additive reasoning takes time and should happen in a predictably gradual way. Our Kindergarteners start by working within 5 (which is a really magical place to make surprisingly sophisticated mathematical observations), then within 10, and eventually in first and second grade within 20. A child who really knows the combinations within 5 has a solid foundation to stand on when they get to 10. The same goes for students who are moving from their 0-10s to their 0-20s. Skipping ahead before each piece in place is a bit like building the attic before the basement.
It is worth noting that while counting strategies can work...for any operation (which is why it is so easy for a student to get stuck in "the counting trap"), we're watching for and nudging toward the development of more sophisticated and nuanced mathematical reasoning. We want kids to notice things like 5 + 5 is always 10, and 6 is just one more than 5, and the fact that 9 is awfully close to 10 can be super helpful in a calculation. Kids need time and experience to play around with numbers and see these connections and relationships. This is the "growth zone" between reliance upon counting and developing full automaticity.
If you are curious about where your child is in this progression of additive reasoning, watch what they do when they add two small numbers together, then two slightly less small numbers. Do they count everything from one? Do they start from the bigger number and count on? Or do they just… know? All three are completely normal depending on age and experience, and they tell you something interesting about where a student is her journey.
In a future post, we're going to zoom in on the specific number relationships that get your child to that place of "just knowing." The "just knowing" piece for us adults is so instantaneous we are hardly cognizant of the process behind it, so this is your opportunity to watch it develop and support it at home!
[…] and allow for some interesting mathematical thinking. First, it moves children along the path from counting all to counting on. Counting all means counting from 1, so for that first 3/2 domino, or a roll of 3 and 2, a child […]