Required skill: understanding how to add two groups by counting the first group quickly, as described in my last post.
So far, we’ve talked about why “counting on”﹣ that is, the method of addition that involves counting up from the first number plus 1﹣is not actually obvious, and about how you can get started on teaching it. If you followed the last lesson, your child can now add 6 and 2 without counting every single item in the group of 6: rather, they can be reminded to “count quickly” and to say 1, 2, 3, 4, 5, 6 without actually pointing at each object.
What should you do next? After all, we don’t want your child to say “1, 2, 3, 4, 5, 6, 7, 8, 9” when adding 8 and 1 indefinitely, no matter how quickly they rattle off the first eight numbers. Eventually, we want to them to simply say that 8＋1 is 9!
The next step in building this skill is to work on fluency. Just because a child has understood an idea doesn’t mean that they have fully internalized it or made it their own. The best way to allow them to do so is to provide concrete situations where the skill is needed, and this is easiest to do using either games or concrete objects. Here are some suggestions:
- Play Addition War. This game works just like War, except that on each turn, a player puts down two cards instead of one. The winner is the person with the highest total. (And if there’s a tie, that means a war, which is always very exciting for the kids!)
Since cards have the numbers written on them, it tends to be fairly easy for kids to do “fast counting”: they can already see the numeral on the card, and therefore they already know how many dots are on the card and how high to count. Plus, there’s a new addition problem on every turn, allowing your child to get lots of practice.
You can also play other card games involving addition. For example, the game of blackjack was a surprising favorite in many of my classes.
- Play games in which you add dice. You can take almost any game standardly played with a single die and instead play it with two dice, where the total of the dice is considered to be the roll for that turn. In my homeschool classes, I usually used Chutes and Ladders, but lots of games would work.
This has the advantage of allowing your child to work on subitizing (rapid judgment of quantity) at the same time as counting on. Since you’ll be encouraging them not to count every single dot on the first die, they will need to learn to quickly figure out how many dots are on a die without counting.
In the same vein, any Tiny Polka Dot games involving addition can be used to both practice subitizing and counting on. In my classes, I used to set up games of Concentration in which the object was to flip over two cards that added to 10, but there are lots of other possibilities!
- Concrete situations which require adding. This can really be anything. “Look, I am showing 5 fingers on this hand and 3 on the other! How many is that in total?” “This bowl has 6 M&Ms and this one has 2. How many M&Ms do we have all together?” If a child has access to counters or is proficient with using their fingers, any addition question can be made into this form: if when asked to do 4＋3, the child shows 4 fingers and 3 fingers and counts them, this can be a good opportunity to remind them how to count on.
Note that this skill is best practiced when the first number is known and the second number is small. If the first number is not obvious, then counting from 1 may very well be the correct strategy, foiling your attempt to teach this shortcut. (If you have a bowl with some unknown number of M&Ms between 10 and 20, and a second bowl with 2 M&Ms, then your child will not be wrong to count from 1!) And I don’t tend to like “counting on” as a strategy when the second number is above 4 or 5, since it gets tedious and ineffective. When the second number is larger, there are better ways to add, as we’ll learn later on!
Another thing I want to emphasize is that this skill is best practiced in situations where there’s a concrete model to fall back on. If a child is having an off day or forgets the trick, they should not be made to feel like they no longer have a successful method for solving the problem: they should still be able to do the problem by simply counting from 1. After they do so, you can remind them how to do this quicker and ask them to do it again, but they should not feel like they did it “wrong.” After all, they didn’t! They were simply slightly less efficient than they may have been.
After a certain amount of practice, you will find that your child almost always uses the “fast counting” strategy instead of counting from 1. How long this takes depends on a lot of factors, such as how often you practice, the age of your child, and their overall level of mathematical aptitude. I recommend practicing this as often as possible to achieve fluency quickly. At that point, you will be ready for the last stage of “counting on,” in which a student can quickly and intuitively answer questions like 9＋1, while still understanding that they are doing the same thing as they do when they count all the items starting from 1. I’ll explain how to get there in my next post.
By the way, the feeling that what they are doing matches their intuition and mental model is precisely the thing that gives kids mathematical confidence. They aren’t learning something new: they are learning shortcuts for things they can already do! As we’ll see over the coming months, this feeling is key.
This is the third entry in a series of posts about counting on. Here are the first, second, and fourth entries in the series. These lay out precisely how I teach counting on, as well as my motivation for teaching it like this.