Inverse Relations between Addition and Subtraction
In this chapter, I will discuss the case of teaching additive inverse relations. A robust understanding of addition and subtraction is viewed as one of the pillars of first-grade mathematics (Howe, 2014). Additionally, Cai and Knuth (2008) pointed out that inverse relations are one of the ideas that can develop algebraic thinking such as solving equations in early grades. For instance, to introduce subtraction 3-1=2, students can be guided to think of 1 + ( ) =
3. In fact, the Common Core in the U.S expects students to master the additive inverse by second grade (NGAC & CCSSO, 2010). Below, I will start with a glimpse of student work, leading to a brief categorization of types of inverse tasks and the additive structure as summarized by the Common Core. Next, I will discuss the teaching of inverse relations based on two different additive structures, with reference to the relevant Chinese and U.S. lessons. Both cases will illustrate the observed approach, TEPS.
A Glimpse of Student Work
The project instrument for additive inverses contained eight items that included both contextual and non-contextual tasks (Li, Hassler, & Ding, 2016). These tasks were designed based on the literature suggestions (e.g., Baroody, 1987; Carpenter et al., 2003; Nunes et al., 2009; Torbeyns et al., 2009). The first and second graders in both countries responded to these items. One type of additive inverse task was to write a fact family based on a given concrete or semi-concrete picture (see Figure 2.1). Overall, the U.S. students in our project demonstrated increasing understanding from G1 to G2 (e.g., QU : G1 = 41.83%, G1 = 75.96%; G2 = 67.79%, G2[isi = 80.88%) while the Chinese students who had better initial understanding almost achieved the ceiling effect by the end of G1 and then maintained that level of understanding in G2 (e.g., Ql.. : G1 = 86.71%, G1 psi = 94.81%; G2prc = 93.90%, G2pst = 100%)*. Figure 2.1 presents example student mistakes that were more common in the U.S. classrooms.
In Figure 2.1, students were asked to “write a group of related number facts suggested by the picture.” While these concrete illustrations (goats, dominos) provide sense-making opportunities, student responses to Figure
2.1 showed a better understanding of addition than subtraction. The problem statement, “a group of related number facts,” did not help all students
Figure 2.1 Example student mistakes with fact family tasks.
produce a correct fact family which indicates a lack of awareness of additive inverses. It seems that students lacked the ability to make connections between concrete and abstract representations.
Another task was a group of related word problems, the solutions of which formed a fact family (also called “inverse word problems”):
a Peggy had 7 balloons. Richard had 4 balloons. How many more balloons did Peggy have than Richard? b Peggy had 3 more balloons than Richard. Richard had 4 balloons.
How many balloons did Peggy have? c Peggy had 7 balloons. She had 3 more balloons than Richard. How many balloons did Richard have?
A similar cross-cultural pattern was observed with student understanding of these items. By the end of G2, more than 95% Chinese students solved these problems correctly while only 67%-78% of U.S. students obtained the correct answers. Many U.S. children were unaware of the embedded inverse relations inherent to these problems. Some tended to use addition to solve all three problems (e.g., 7 + 4 = 11, 3 + 4 = 7, 7 + 3 = 10, respectively), likely due to seeing the keyword “more”. Even though some children also drew tallies or number lines as part of their solutions, these representations only helped them find the correct computational answers (e.g., jump 4 steps from 7 to reach 11) rather than understand the problem structure and relationship. In fact, this group of word problems has a more challenging problem structure than that of Figure 2.1. In Figure 2.1, the shared problem structure is part-whole while the structure of the above word problems is comparison (for more discussion, see Section 2.2). According to Nunes et al. (2009), comparison problems are harder than part-whole tasks because the former is about “relationships” that are more difficult to manipulate. Nevertheless, the above tasks share the same concrete context with related problem structures (Peggy - Richard = difference, Richard + difference = Peggy, Peggy - difference = Richard), which provides a great opportunity to help students make sense of the inverse relationship among these numerical facts: 7-4 = 3, 4 + 3 = 7, and 7-3 = 4.
Another item on the test instrument asked students to first solve a group of related facts and then to explain how they solved the subtraction item:
- 9 + 3 = ( )
- 12-3 = ( )
How did you get the answer for 12 - 3 = ( )r
A student with inverse understanding should be able to obtain the answer for subtraction using the related addition fact without extra computation. However, in my project, even though some students could figure out that 9 + 3 = (12), they did not use it to solve 12 - 3 = ( ) despite the fact that the subtraction statement was listed directly underneath the related addition statement. The following responses are typical:
SI: I was at 12,1 counted back 3 (A U.S. example).
S2: I first subtract 3 from 2, there was not enough, so I borrowed 1 and got 9 (A Chinese example).
In the above responses, the first student counted 3 down from 12. The second student described a procedure aligned with the standard subtraction algorithm. However, both answers indicated students’ lack of understanding of inverse relations.
The lack of understanding of inverse relations may also lead to inflexibility in solving problems. In my project, the task of 81 - 79 = ( ) was taken from the literature (Torbeyns et al., 2009). Researchers thought that the best way to solve this problem was indirect addition, either by thinking of 79 + (2) = 81 or counting “2" up from 79. However, many students from both countries did not use the conceptual shortcut of indirect addition. Rather, they used brute force calculation methods such as crossing out tallies or using the standard subtraction algorithm. Such responses indicate a lack of flexibility in using additive inverses to solve problems.
Overall, the above results indicate that students’ awareness of additive inverses may be challenging to develop. This may be due to their inability to connect concrete and abstract representations for sense-making, or perhaps their inability to transfer their existing inverse understanding to solving new problems. These learning challenges present various teaching opportunities. For instance, there are different problem structures for addition that may be used to promote sense-making, and there are varied inverse tasks that may be used to deepen students’ understanding of this relationship. In this chapter, I will introduce how to help students make sense of inverse relations during their initial learning through TEPS. Before presenting actual teaching examples, I will briefly summarize the necessary knowledge components that teachers should grasp in order to provide students with the richest possible learning opportunities.