When children are direct modeling addition, they are generally using a counting all strategy: they count out the first set, count out the second set, and then count the whole, starting over from 1. Counting on, by contrast, is when you start with one of the addends and count on the other addend. For some problems, counting on is very similar to the action that would be direct modeled, for other problems, children need to have a more abstract understanding of the problem to use a counting on strategy.
Counting on from first vs counting on from higher
Join, Result Unknown problems are particularly well suited to a counting on strategy. The action in a JRU problem is basically a counting on action: there is some amount to begin with, and then you add in some more--so if you count on by the change amount in a JRU problem, you're just adding on the amount of the change. One of the big steps in abstraction is when, in solving a JRU problem, a child moves from a counting on from first strategy to a counting on from highest strategy. For example, in the problem:
Tricia had 4 butterfly stickers. Her friend gave her 7 more butterfly stickers. How many butterfly stickers does Tricia have now?
The action suggests starting with the first number: 4 and counting on 7 more:
She starts with 4...5, 6, 7, 8, 9, 10, 11. She has 11 stickers.
It's a lot more efficient, however, to start with the larger number and count on:
7...8, 9, 10, 11. She has 11 stickers.
That makes the numbers easier, but it no longer follows the order of the word problem. Being able see that you can get the right answer without following the order of the problem is a big step in abstraction, and in understanding addition in a more general way.
"Rote counting on" vs "counting on to solve problems"
One way of looking at counting on is just the counting process. Instead of starting with 1, you start with 5, or 8 or 10 and you count up. This is a rote counting exercise that can be called counting on, and it is a good preparation for figuring out/learning how to count on. This rote counting on skill is in the Common Core Math Standards for Kindergarten.
The other way of looking at counting on is that it's a strategy for solving problems, as in the example of Tricia's stickers. This is often something that is sometimes learned and taught in Kindergarten, and sometimes not until first grade. Most children will master counting on as a strategy for solving addition problems in first grade, though a few won't make the strategy switch until later. Counting on as a strategy for solving addition problems is in the Common Core Math Standards for 1st grade.
What does counting on look like?
The most common way of counting on involves putting up a finger for each number counted after the start number; in this case the sum is the end number. For example, to find 6+3, a child would count:
Some children may keep track of how many they have counted on in other ways, but this is the basic model for counting on.
CGI and appropriate problem types
The word problem type that is most strongly associated with counting on is Join, Result Unknown. Children are most often start solving problems by counting on for JRU problems where the start number is fairly large, and the change number is quite small--so that it would be inconvenient to count all, and it is easy to imaging counting on just one or two. This makes JRU problems particularly good to ask if you think children are ready to figure out/learn the counting on strategy. Once they have started using counting on as a problem solving strategy, children will usually solve most addition problems that way, so they will also use counting on to solve Part-Part-Whole, Whole Unknown problems, and Compare, Compared Quantity Unknown Problems (of the more...than... type). It's also notable that JRU problems are so closely associated with counting on, that even if children might use more efficient strategies for other problems, they may go back to using counting on for JRU problems.
To teach or not to teach
There are different philosophies about whether and how much to teach counting on. The CGI researchers found that children will generally come up with counting on as a problem solving strategy if they are given plenty of practice solving problems at appropriate levels for them. Other researchers (Rathmell, et al) have looked at effective ways to teach children how to count on. As a teacher, you will be the curriculum expert in your classroom, and you'll decide what strategies you want to try with your students.
Some teaching strategies that everyone supports are:
Some strategies that are more directive (moving from less to more directive)
Next, we'll look at some specific examples of how you might structure a lesson to teach counting on as a strategy to a class or group of students.
Sample lessons and activites:
Sample lesson: The basics. A set of word problems with discussion and prompting
[Prior knowledge: solving simple word problems by direct modeling, and rote counting on]
The teacher poses a word problem (JRU, large start, small change), and asks the children to solve it:
Matthew had 12 paper airplanes. He made 2 more paper airplanes. How many paper airplanes does he have now?
The teacher waits while children solve the problem, and then invites 2-3 children to share how they solved the problem.
The teacher highlights a counting on solution:
I'd like you to think about how Ellen solved the problem, she started with 12 and then counted the two more paper airplanes: 13 (raise a finger) 14 (raise a second finger). So she there were 14 in all. I have another problem for you to think about, and I'd like you to see if you can solve it Ellen's way:
Janet had 9 crackers. Her mom gave her 2 more crackers. How many crackers does Janet have now?
The teacher waits while children solve this problem, and then invites children to share a counting on solution:
Who can show how to solve this problem by counting on the way Janet did?
[This is the end of the lesson, and the class goes on to the next part of math time/circle time/whatever. The teacher repeats the lesson with different word problems for the next several days. The teacher continues to use some JRU problems, but also asks PPW-WU and CQU problems. The teacher continues to use problems where one number (usually the first) is quite large and the other is quite small--usually less than 5. If children are confident with counting that high, the start numbers could be any number less than 100. Counting on lessons are continued for a few weeks until children are quite comfortable using counting on]
Sample lesson: Counting books. Mouse Count
Mouse count is an example of a counting book where a certain number of things are added on a page. In mouse count, a snake adds several mice to its jar at a time, so at the end of one page, there may be 3 mice in the jar, and then on the next page, 4 more mice are added to the jar. A sticky note on the picture of the jar with 3 mice, with the number 3 on it can provide a starting number for the whole group to count on from 3 to find how many mice are in the jar after 4 more are added. There are also other counting books that offer opportunities to start with a given number and count on some more.
Sample lesson: Game/activity. Bears in a cave
Materials: bear counters and a large paper cup or appropriate cave substitute. The teacher puts 5 bears under the paper cup (or cave), and puts 2 bears outside the cup/cave. The teacher tells the class:
There are 5 bears inside the cave and two more out here, can you figure out how many bears there are in all?
The teacher waits while the children solve the problem, and then invites children to share how they solved the problem. The teacher chooses a counting on solution to highlight:
Did you notice how Timmy started with 5 because he knew there were 5 bears in the cave, and then he just counted the two more--so he counted 5, for the bears in the cave, 6 (touch a bear), 7 (touch the other bear)
I'm going to change the bears around, and I'd like you to see if you can use the counting on way the Timmy used to solve this one:
The teacher puts 8 bears in the cave and 3 outside
There are 8 bears in the cave right now. Can you all see the 3 bears that are outside the cave? There are 8 bears hiding in the cave, and 3 outside the cave. How many bears are there in all?
The teacher waits while the children solve the problem, and then invites a child to show how to solve it by counting on.
The teacher then explains that this will be the lesson for the day, and assigns children to work in partners. The teacher invites a child to come up and be her partner.
Lisa's going to be my partner, and we're going to show just what you're going to do, so Lisa, would you put some bears in the cave, and would you count carefully, so you know how many are in the cave?
Lisa puts 6 bears in the cave
OK, Lisa, now put some bears outside the cave
Lisa puts out 4 more bears
Lisa, tell me how many bears are in the cave, and how many bears are outside of the cave
Lisa tells the class that there are 6 bears in the cave, and there are 4 bears outside of the cave. The teacher models counting on.
So, I'm going to start with 6, because Lisa told me there are 6 bears in the cave. 6, 7 (holds up a finger), 8 (holds up another finger), 9 (holds up another finger), 10 (holds up a fourth finger). There are 10 bears in all.
Lisa, would you help me check if my answer is right?
Lisa and the teacher take the bears out of the cave, and Lisa counts them all to be sure that there are 10.
The teacher continues giving directions:
Next, we trade jobs, and I put out the bears, and Lisa counts on to figure out how many there are. When you are doing this with your partner, I want you to take turns putting the bears in the cave, and figuring our how many bears there are in all.
The teacher gives each pair of students a paper cup cave, and 10-12 bears, and children make up and solve bear problems for part of math time (perhaps 15 minutes--this activity can last as long as children are interested, and can become a choice time activity. Note that children can be given different numbers of bears to regulate the difficulty of the problems that are created).
Optionally, the class may meet to discuss the bear problems, and partners can show a problem and how they solved it.
Notice that the bears in a cave activity is designed so that the children can't see and directly count the first addend--the bears hiding in the cave--which reinforces the counting on strategy.
Sample lesson: Dino Math
You can watch a video of a kindergarten teacher teaching a lesson where children solve similar problems to the bears in a cave problems. The activity is perhaps slightly less well suited to teaching counting on than the bears in a cave activity, because the structure makes it easier for children to count all rather than counting on. The lesson shows a good example of a teacher modeling clearly for the children what they should do in the activity, and it also shows a way to integrate a recording sheet with a counting on activity.
Go to http://www.learner.org/resources/series32.html and click the VoD (video on demand) button next to 12. Dino Math to view the video. You need a fast internet connection to use the videos on this site, so if you only have dial up you will need to go to a library or similar place where you have access to cable or DSL. You may need to sign up for a free account to watch the video.
Valuable tips and important considerations:
Changing habits Children who are in the habit of solving problems by counting on, may need some motivation to find a new strategy worth trying. For those children who have the counting skills, but are stubbornly reluctant to count on, it helps to ask problems that are very easy by counting on, and almost impossibly hard by counting all. Following Rathmell's suggestions, I have had success with problems like:
Miranda had 43 butterfly stickers. Her friend gave her 1 more butterfly sticker. How many butterfly stickers does she have now?
Paul has 54 Pokemon cards. His friend gave him 2 more Pokemon cards. How many Pokemon cards does he have now?
Notice that in each case the start number is high enough that it doesn't even occur to you that you might want to count starting from 1. Notice also, that the second number--the change number--in each case is quite small: in the first problem it's 1, then 2, then I would go to 3, and then bounce around between 1 and 3 for another 5-6 problems, and I would probably never go higher than a change number of 4 or 5 (see next section). Also notice that for these first two problems, the counting on doesn't go across a decade number. After another several successful counting on solutions of this type, I would move to problems with numbers like 48+3 and 29+2 and 39+4, so that the children are practicing counting on across decade numbers.
Efficiency and number choice
One thing to keep in mind is that counting on is the first counting and thinking strategy children will learn, but there will be a lot more. Counting on is easy, fast and efficient if you're only counting on by a very small number (1, 2, or 3), and once you get to numbers larger than 5, counting on isn't going to be a particularly effective strategy. Some children do get stuck on counting on and don't go on to develop more efficient strategies, but in your planning as a teacher, you should be looking ahead to trying to prepare children for more efficient strategies too. Interspersing counting on problems with problems that are solved on 10-frames can be helpful for keeping strategy choices open and flexible.