How can we teach counting up for subtraction?

Edward C. Rathmell
University of Northern Iowa

• Create a word problem where the part being subtracted is nearly as large as the whole (with a difference of 1 or 2).
• Give the students an opportunity to think about this problem.
• Then ask several students to share how they figured the problem out. One of them probably will explain how they counted up. If not, then tell the students that you heard a student solve it by counting up, then explain how they were thinking. For 8 - 6, think 6, … then count up to eight, …7, 8. That's two more, so the answer is 2.
• You may want to model the counting up thinking by showing the whole, then covering it and reminding the students how many there are, then subtract the part by removing all of them at the same time.
• Verbalize the thinking and ask the class to verbalize the thinking.
• Then ask the class to use counting up to solve another problem.

Using a similar procedure for a few minutes everyday for two or three weeks will help nearly all of the class learn to use counting up. Besides the routine described in the bullets above, sometimes do the following.

• Ask what subtraction problem they did and how it could be written as an equation.
• Ask students to start at 8 and count up 2 for the fact problem 10 - 8 and compare it to counting back 8 from 10. Reinforce the efficiency of counting up.
• Discuss when counting up can be used efficiently, that is, when the part being subtracted nearly as large as the whole.
• After the students can count up, a few minutes of practice on a regular basis for two or three weeks will enable them to solve counting up problems in about 3 seconds. If they also know the generalizations and can count back, they will be able to solve 51 of the 100 basic subtraction facts quickly.

Children who only think of subtraction as take away will have difficulty believing that you can count up to subtract. Take away and counting up are inconsistent in their minds. However, students who have learned about parts and wholes will understand that you just need to determine the other part. It does not matter whether you count up or count back. Another way that children can make sense of counting up as a reasonable way to subtract is to think of subtraction as a comparison situation where you are trying to find the difference between the numbers. Again, it does not matter whether you count back or count up to find that difference.

So, before you can expect success with this strategy, students should begin to understand that subtraction is more than just take away. There are two kinds of experiences that can help prepare students. The teacher can informally use the part-part-whole language as they work with addition and subtraction situations. The teachers can also present word problems and other problem solving situations that involve problem structures which are comparisons between numbers.

If helping children learn to count back is a step in the direction of helping them learn to use addition to help them solve subtraction facts, counting up is a leap in that direction. The most important teaching strategy is to hide the part that is left so the students cannot count what is left. This procedure can be used in situations where the number being subtracted is nearly as large as the whole.

Show a set of counters. Create a situation where they will be covered or hidden. Remove nearly all of the counters, all at the same time. 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.

Create other situations where the part that is left after subtracting is hidden. Remove nearly all the counters. Verify the answers by counting up as the hidden counters are shown. From this point, it is not a very large step to begin adding on one or two more to subtract.

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