Properties of Addition: CP and AP
From this chapter on, I will shift the discussion from inverse relations to the basic properties of operations. I will discuss the properties of addition in this chapter, followed by properties of multiplication in the next chapter. This chapter focuses on two properties: the commutative property (CP) and associative property (AP) of addition. As explained in Chapter 1, the basic properties of operations undergird algebraic equation solving and are the backbone of algebraic understanding (Bruner, 1977; Flowe, 2014; YVu, 2009). Students’ understanding of these ideas can be developed through arithmetic learning in elementary school. This chapter is organized in the same way as earlier chapters. First, I will provide examples of typical student work that demonstrate different levels of student understanding. Then, I will briefly introduce the necessary mathematical knowledge for teaching. Finally, I will share cross-cultural instructional insights for teaching the CP and AP of addition, each of which illustrates TEPS.
A Glimpse of Student Work
The project designed an instrument including 10 tasks (14 subtasks) to assess students’ (G1-G4) understanding of the basic properties of operations, including the CP and AP of addition. Although students’ performance with basic properties in both countries appeared less satisfying than inverse relations, Chinese students demonstrated knowledge growth over several years while the U.S. students did not show a clear pattern (see detailed statistical analysis in Ding, Li, Flassler, & Barnett, 2019). Overall, it seems that by grade 4, most of the U.S. students have not developed formal understanding of these properties while many Chinese students have. Indeed, my project data indicate that there were different levels of student understanding for each property, which may show a learning trajectory that is informative for classroom teaching.
The CP of Addition
Among all of the properties, the CP of addition appears to be the most familiar to children in both countries in my project. However, student responses indicate different levels of understanding ranging from informal usage with specific number sentences to formal statements about the general property. Young children may have no understanding of the CP (level
1) or their understanding may be implicit. However, students’ implicit understanding (level 2) can be developed into explicit understanding (level 3) over time. For example, the following task—If you know 7+5 = 12 does that help you solve 5 + 7? Why?—elicited different levels of responses. Below I present example responses at levels 2 and 3.
Level 2: Implicit understanding. While most students in grades 1-4 agreed that this would be helpful, some students limited their explanation to this specific task itself. Typical responses were:
SI: Yes, because 7 + 5 is the same exact thing as 5 + 7 (U.S. example)
S2: Yes, it is helpful because 5 + 7 = 7 + 5 (Chinese example)
S3: Yes, because it is flipped around and it has the same total (U.S. example)
All students above noticed the relationship between 7 + 5 and 5 + 7. SI thought that 7 + 5 and 5 + 7 were “the same exact thing.” Even though this language is not accurate, the student was vaguely aware of the similarity between these two number sentences. S2 used the equal sign to connect the two number sentences, showing that both sentences have the same value. S3 was more detailed by noticing the two number sentences were “flipped around” but they still had the same total value. All these responses suggested students’ awareness of CP but it is likely limited to specific cases.
Level 3: Explicit understanding. Other students demonstrated a general familiarity with the CP by either using a formula to explain their reasoning or directly pointing out the undergirding property. Typical responses include:
S4: Yes, a + b = b + a (Chinese example)
S5: Yes, because it is communative [sic] property. You will get the same answer on both of them (U.S. example)
Note that there were many more Chinese students who demonstrated formal understanding of the CP by naming it the “commutative property of addition Many U.S. students, including fourth graders,
called the CP the “turn-around property.” This may be due to what they had learned from the textbooks or classrooms.
The AP of Addition
The AP is often mistaken by students as being the same thing as the CP. The conflation of these two properties was reported in the literature even at the undergraduate level (Larsen, 2010; Zaslavsky & Pelcd, 1996) and found with our U.S. student sample. Moreover, my project data indicated that U.S. students generally lacked the ability to apply the AP to make computation easier. In fact, while many Chinese fourth graders could recognize the AP that undergirds their computation strategy, very few U.S. counterparts could do so (Ding et al., 2019). Consider the following two items, in which students were asked to use efficient strategies to solve the problems and explain why their strategies worked.
(7 +19) +1 2 + (98 + 17)
To efficiently solve these tasks, students ought to first add 19 + 1 or 2 + 98 to create a multiple of 10. The underlying property behind this strategy is the AP. In my project, students who were able to obtain the correct computational answers demonstrated varied levels of understanding of the AP.
Level 1: No understanding. Students computed each task using order of operations; explanations were descriptions of computational procedures that did not make use of the AP. Below arc typical explanations:
SI: I first computed the numbers inside the parenthesis. (Chinese example) S2: I added 98 plus 17 because in order to add 2,1 have to add 98 and 17 first. Then I added 98 and 17 to 2. (U.S. example).
Level 2: Implicit understanding. Students used the anticipated strategy to solve the tasks; however, their rationale for adding 19 and 1 or 2 and 98 first was to obtain a multiple of 10 that would make their computation easier. It is unclear whether the students knew why they could add 19 and 1 (or 2 and 98) first. Figure 4.1 illustrates typical student work, including both U.S. and Chinese examples. Again, note that Chinese students only formally learn the AP of addition in fourth grade.
Figure 4.1 Typical level 2 student responses to the AP of addition items.
Level 3: Explicit understanding. At this level, students were not only able to apply the AP to solve the tasks, but also able to explicitly identify this property as the undergirding reason for their strategies. Our data analysis indicates that this level only occurred with some Chinese fourth graders who formally learned this property (Ding et al., 2019). These students explained that their strategies were “according to the associative property of addition." However, even though U.S. students were formally exposed to this property in grade 2 and revisited it in grade 3, there were no observed occurrences where they demonstrated this level of understanding.
The different levels of responses to the CP and AP items indicate that student understanding of the properties of addition can progress from implicit to explicit. Overall, the U.S. children observed in our study did not tend to explicitly identify when their strategies made use of the basic properties even though they had formally learned the properties several years before their Chinese counterparts.