(art from Clip Art Etc)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. A counting strategies is different from a direct modeling strategy because it relies on keeping track of amounts by counting on and back (perhaps keeping track with fingers) rather than acting out the actions with counters or tally marks.
Counting on 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.
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).
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 kindergarten or first grade. Counting on as a strategy for solving addition problems is used as an example 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.
To teach or not to teach
There are different philosophies about whether and how much to teach counting on. The CGI researchers found that (most) 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 (less directive) 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.
It's helpful if children have practiced rote counting on (starting with a number and counting up from it)
It'e helpful if children have practiced finger flashes, so they associate amounts with a configuration of their fingers (it's easier to count on 7 using your fingers if you know how many seven fingers is). Finger flashes are when the teacher (or student leader) calls out a number, and everyone holds up that number with their fingers as quickly as they can.
(art from Clip Art Etc)
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.
Sample lesson: The basics. A set of word problems with discussion and highlighting the counting on strategy.
The teacher poses a word problem (JRU, large start, small change) to a group of children, 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: 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.
Adapted from http://thinkingwithnumbers.com/QuestionsAnswers/Question15.htm
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. If you are showing addition with manipulatives, and you want children to count on, you need to have the first addend hidden to make it less easy to start counting at 1.
Optional 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.
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 other representations can be helpful for keeping strategy choices open and flexible.
Counting back is the counting strategy most strongly associated with subtraction.
Counting back is a natural strategy to use in solving SRU: separate, result unknown problems. The idea of counting back is easy: count back for each item you are taking away, and that will take you back to the answer:
If I had 8 grapes, and I ate 2 of them, how many grapes would I have left?
I start with 8 grapes. I eat 1, and then I have 7. I eat another, and then I have 6. I have 6 grapes left: 8...7,6.
Counting back to subtract is significantly harder than counting on to add because we're all better at counting forwards than backwards, and we're all more likely to make mistakes when counting back than counting forward. One implication of this is that we'd really like children to gain other subtraction strategies as well besides counting backwards. Eventually, we want children to think of subtraction as a missing number addition problem, and be able to use addition strategies to help them solve the subtraction problems.
A good thing to practice with numbers is telling what number comes before a number. So, if you give a child the number 8 and ask what comes before 8, that's practice for counting back. Knowing what number comes before a given number is a skill that is used when solving problems by counting back.
To figure out 9-2 by counting back, most children will say
9...
8 (put up a finger)
7 (put up another finger)The answer is 7. In this case, the number counted is the number left after taking away the number of fingers that are up.
Some children will say:
9 (put up a finger--this means the 9 th item is taken away)
8 (put up another finge--this means the 8th item is taken away)The answer is 7. After the 9th and 8th counters are removed, the last counter left is the seventh counter. In this case, the number counted is the name of the counter represented by the finger that is put up or the object that is taken away.
Discussing a word problem, and highlighting a counting back strategy:
When introducing counting back, you should ask questions for which counting back is an efficient and a natural strategy. This means you want an SRU problem (to make the counting back strategy more natural) and a problem where you are taking away just a very small number (the subtrahend should be 1 or 2). For example:
Karen had 9 balloons. Two of her balloons popped. How many balloons does she have left?
If you ask a group of children to solve this problem, and ask how they solved it, it is very likely that some children will solve the problem by counting back, so this could start a lesson where you discuss and highlight a strategy, and then ask children to practice the strategy with a similar problem.
Sample activity/game: bears leaving the cave:
Put a known number of counters under a cup, and tell children how many are in the cup ("I have 11 counters under my cup" or "There are 11 bears in the cave").
Then remove a small number of counters 1 at a time, and count back to figure out how many are still under the cup ("I'm going to take some out, and I want to know how many are still under the cup." Take out a counter "10"; take out another counter "9". "When I take out 2 counters there are 9 left under the cup").--paraphrased from http://thinkingwithnumbers.com/QuestionsAnswers/Question20.htm
This is a specific example of a way to use manipulatives to encourage children to develop counting strategies instead of direct modeling. In general, to encourage children to develop counting strategies, you want to structure the problem so that one part of the manipulatives are hidden and can't be directly counted. In this case, you want to have the minuend (the start number) hidden, and to leave the difference (result) hidden so that the thing that is shown is the amount to count back
Counting back is actually pretty inefficient most of the time. When I use the word efficient, I mean that it can be reliably be used by most children in a short amount of time--less than 3 seconds. For most children, it's efficient to count back by 1 or 2, but when you get much larger than that, it becomes difficult and inefficient (it's sufficiently difficult and boring that it's what I do to fall asleep sometimes--I count backwards from 1000). So, for a problem to have counting back as one of it's most efficient strategies, the subtrahend should be 1 or 2.
Counting up to is a useful counting strategy for subtraction, though it is not as natural an association as counting back.
Counting up to is a natural strategy to use in solving JCU: Join Change Unknown problems. Counting up to is similar to counting on. The difference is that if you are counting on, you know how many more to count, and if you are counting up to, you know where to stop counting. Counting up to looks something like this:
Karen made 7 pictures. How many more does she need to make to have 9 pictures?
I start with 7 pictures. I make some more... 8 (put up 1 finger), 9 (put up another finger). She needs to make 2 more (the number of fingers held up--the number of counts).
This is a lot like the counting up to JCU strategy that children do with manipulatives, with the difference that the first number (7) is not represented, only the number counted on is represented.
Counting up to to subtract is significantly easier than counting back to subtract because we're all better at counting forwards than backwards, and we're all more likely to make mistakes when counting back than counting forward. Most of us, however, think of subtraction as taking away, so for counting up to to be associated with subtraction problem types other than JCU, children need to make the connections between addition and subtraction (addressed in an upcoming lesson!). In the example of Karen's pictures, you can think of the problem numerically in two ways:
7+?=9
or
9-7=?
The more natural representation is the missing number addition, but it's not automatic that children who know how to count up to to solve JCU problems will apply that same strategy to other problems where it would numerically make sense.
When introducing counting up to, you should ask questions for which counting up to is an efficient and a natural strategy. This means you want a JCU problem (to make the counting up to strategy more natural) and a problem where the minuend and the subtrahend are close (the difference should be 1 or 2). The problem about Karen's pictures is a good example.
Show a set of counters and tell children how many there are. Hide the counters under the cup all at once ("11 bears went into the cave"). Remove nearly all of the counters, all at the same time ("9 bears came out of the cave"). Ask how many are still covered. If they can see nearly all of the whole, children often recognize that they only need one more or two more to make the whole. After a student suggests starting at the part and counting up to get the whole, try counting up as you show the hidden counters just to confirm that this strategy actually works..... Adapted (and mostly quoted) from http://thinkingwithnumbers.com/QuestionsAnswers/Question21.htm
The subtraction problems for which counting up to are ones where the minuend and the subtrahend are close together. That is the same as saying that the difference is small (the number of counts is small) but we usually talk about it as saying the minuend and the subtrahend are close because when you have a subtraction problem you know the minuend and subtrahend, but you don't know the difference until you solve the problem. Close in this case means 1 or 2 apart, maybe 3.
One thing to keep in mind about efficiency is that we want it to be easy enough that children, with a little practice, will be able to do the process without using their fingers--they will be able to keep track of the counts in their head without putting down their pencil.
When discussing efficiency of counting back and counting up to with children, it's nice to have a problem that's easy-ish to see in both ways: both as subtraction and as missing number addition. PPW-PU problems are good for this:
Sam had 9 notebooks. 6 were blue and the rest were red. How many red notebooks did he have?
It's pretty easy to think of having Sam's 9 notebooks, and taking away the 6 that are blue, and seeing how many are left. I'd do this by counting back from 9, holding up a finger with each count to keep track of taking away all 6 notebooks: 9....8 (1 finger), 7 (2 fingers), 6 (3 fingers), 5 (4 fingers), 4 (5 fingers), 3 (6 fingers). He has 3 red notebooks.
It's also pretty easy to think of this as a missing number addition: 6 blue notebooks + ? red notebooks = 9. Start with 6 and count up to 9 to find the number of red notebooks. 6... 7 (1 finger), 8 (2 fingers), 9 (3 fingers).
In this case, 6 and 9 are pretty close, and 6 is a pretty big number to be taking away, so here counting up to is a better strategy. The efficiency of the strategy is tied to the numbers you are computing with, not to the problem type.
Some children will use a counting strategy where you count back to, so for 9-6, they might count back from 9 to 6. This would look like: 9... 8 ( finger), 7 (2 fingers), 6 (3 fingers). This strategy is about as efficient as counting up to for numbers where the minuend and subtrahend are close, and some children find it more natural to do. There are two reasons why we would like children to get used to counting up to for subtraction: the first is that counting forwards is easier and less prone to error than counting back. The other reason is that it both requires and reinforces an understanding of the relationship between addition and subtraction, which leads to other more efficient subtraction strategies.
Counting up to subtract is a recommended strategy for coaching children who have certain math learning disabilities (Ronnit Bird--The Discalculia Toolkit)
