How does the part-part-whole language help students learn subtraction facts?

Edward C. Rathmell, University of Northern Iowa

Children have much more difficulty learning subtraction facts than addition facts. This can be attributed to several things.
First, counting what is left to solve subtraction facts tends to put the focus on counting rather than on the relationship among the parts and the whole. By the time children have counted what is left, they are not thinking about how many there were in the whole or how many were taken away. The subtraction fact will not be learned until the three numbers, the parts and the whole, are related. Also, the subtraction generalizations and counting back do not help students with as many subtraction facts (only 37) as the addition generalization and counting on help with addition facts (64). Part of this difference is that counting up to subtract is more difficult because students do not understand the connection between counting up and taking way. Students also have much more difficulty counting back than they do counting up. Consequently, many students are not able to count back three easily. For these reasons, there are many more difficult subtraction facts than difficult addition facts.

Second, the language that students use for subtraction often is "take away." That language does not encourage students to think about how addition can help them. Students tend to think of subtraction as something entirely new that they have to learn.
Using part-part-whole language with subtraction facts can help students understand the relationship between addition and subtraction. The teacher should just naturally use that language for addition (the parts are 4 and 3, so the whole is 7). Then the teacher should continue using the same language for subtraction (the whole is 7 and one part is 3, so the other part is 4). The students will begin to understand they can use what they know about addition to help them with subtraction facts. This will help students identify number families, such as 3, 4, and 7, as number relationships that can help them solve either addition or subtraction fact problems.

The part-part-whole language also has some added benefits besides helping students learn subtraction facts. This language helps students become better problem solvers. Identifying the parts and the whole is helpful in many different problem-solving situations, but it is particularly helpful for word problems. Simply trying to represent a word problem as parts and the whole can help students decide which operation is appropriate to solve the problem. For example, if you know both parts, add to find the whole. If you know the whole and one part, subtract to find the other part.

The part-part-whole language even works for comparison problems. Comparison problems can be represented as one small set, one large set, and the difference between the two sets. The large set can be partitioned into the part that matches the small set and the difference. For comparison problems, if you know both parts of the large set, add to find the whole. If you know the whole and one part of the large set, subtract to find the other part, whether that be the part that matches the small set or the difference. A second grader explained this to the other students in one class where the part-part-whole language had been used informally by the teacher throughout the school year.

The part-part-whole language can also help students make sense out of the symbols that they write for addition and subtraction number sentences. The parts that are added are on either side of the plus sign. The whole is equal to the sum of these parts. For subtraction, the whole minus one part equals the other part. Students who have learned this connection to number sentences do not solve the problem 3 + ___ = 5 by writing an 8 in the blank. The structure of the number sentence does not detract from the parts and the whole.

The part-part-whole language also can help students identify relationships among many other numbers. If one fact is given, others can be derived from that, even for larger numbers. For example, students who have learned about parts and wholes, can solve problems like 35 - 16; 36 -16; 35 - 20; 16 + 18; etc, if they are provided the information that 19 + 16 = 35.