 Relational Thinking: What’s the Difference?

• # Relational Thinking: What’s the Difference?

Instructional activities designed to encourage relational thinking in primary-grades classrooms can give students advantages when they reason about subtraction.

How much is 41 – 39? How about 100 – 3? Which of those computations was easier for you to do? It so happens that first graders are much more likely to solve 100 – 3 correctly than 41 – 39. Likewise, second graders are much more likely to solve 100 – 3 correctly than 201 – 199. Our data (Schoen et al. 2016) suggest that the latter problems are more difficult for students to solve correctly, because many students’ understanding of subtraction is limited by thinking about the operation only as take-away or by using a default procedure, such as the standard subtraction algorithm in the United States. In this article, we argue the importance of students learning to reason flexibly about subtraction. We highlight a useful but often-ignored way of reasoning, and we offer suggestions for teaching about subtraction.

In a 2014 study (see fig. 1), we found that of the majority of second graders who attempted to use the standard U.S. subtraction algorithm to compute 201 – 199, only 29 percent were successful (see table 1).  Some students obtained such answers as 198 as a result of “subtracting up” errors and, unfortunately, failed to notice the unreasonableness of such a large answer. Furthermore, even when these approaches were successful, the correct answer could have been obtained much more easily by counting forward from 199 to 201 or counting backward from 201 to 199.

We found similar results in first graders’ attempts to solve 41 – 39 = ___. Many students struggled with this problem, despite being able to correctly solve problems like 100 – 3 = ___. For many adults, by contrast, problems like 201 – 199 = ___ and 41 – 39 = ___ can easily be solved mentally. A characteristic that can make problems like these easy is that the numbers are close together. However, this characteristic is only useful if one thinks about subtraction as asking such questions as “How far apart are 41 and 39?” or “What would I have to add to 39 to get 41?” If, instead, one asks, “How much is left if I take 39 away from 41?” one has much work to do to find the answer and many potential pitfalls along the way.

Take-away subtraction

We found that students in both grade levels relied heavily on the take-away meaning for subtraction. For problems such as 100 – 3 = ___, thinking in terms of take-away served many students well. One popular strategy was to start from 100 and count down (99, 98, 97), often using fingers. For problems like 100 – 3, this way of reasoning is advantageous, because the subtrahend is small; the student must take away only 3. In 201 – 199, by contrast, the difference is small, but the subtrahend is large. In the latter situations, thinking about subtraction as take-away can be highly inefficient, whereas thinking of the difference as the distance between the given numbers is advantageous.

It is not a new idea that that there are limitations to the take-away meaning of subtraction. Gibb (1954) pointed this out more than sixty years ago in a paper that discussed different types of subtraction word problems and various strategies that children use to solve them. Her points are still valid and relevant today. When initially learning about the operation, students in the United States learn to associate subtraction with taking away, or removing, objects from a set. This meaning for subtraction is consistent with separate-result-unknown problems, such as this:

Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? (Carpenter et al. 1999)

Many teachers value the take-away meaning, especially when it comes to introducing subtraction initially (e.g., Maples 1959; Page 1994). In fact, teachers often orally read the subtraction symbol as “take away” (e.g., reading 13 – 5 as “Thirteen take away five”). The take-away meaning is certainly important and useful, especially when students are first learning about subtraction. However, equating subtraction with take-away is problematic, because many situations involving differences do not fit neatly with a take-away interpretation (Fuson 1986). For example, in a join-change-unknown problem, such as “Connie has 5 marbles. How many marbles does she need to have 13 altogether?” (Carpenter et al. 1999), the answer of 8 marbles can be obtained by computing 13 – 5; yet, nothing in the story is being taken away. If children are told to use subtraction to solve these kinds of problems—and yet learn that the sole meaning of subtraction is take-away—they may experience mathematics as not making sense.

Differences as distances

The point we wish to highlight is how powerful flexible reasoning about subtraction can be for students. Beyond the take-away meaning, many situations exist in which reasoning about differences as distances between numbers is helpful. Students who solve 201 – 199 by counting forward from 199 to 201 seem to be answering a different question than those who create a set of 201 objects and remove 199 objects from it. In the case of these numbers, “How far is 199 from 201?” seems to be a simpler question to answer than “How much is left if you take 199