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.
There are two typical ways that children count 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.
Problems for which counting back is efficient are pretty restrictive. Efficient in this context means that it can be reliably be used by most children in a short amount of time--3 seconds of less. 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. So, for a problem to have counting back as one of it's most efficient strategies, the subtrahend should be 1 or 2.
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 7 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. It's generally helpful to point out efficient strategies for solving problems, and counting back is one efficient strategy for solving this problem.
You can show this process with manipulatives by creating a situation where a known number of counters are hidden ("I have 11 counters under my cup") and then you 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").
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.