Teaching the Commutative Property of Addition
Imagine you are a first- or second-grade teacher and you are aware that students in elementary school need to learn the big idea of the CP of Addition, a + b = b + a. How should this big idea be initially introduced to your students? What if you teach 3rd or 4th grade? To what extent would you prompt your students’ understanding of this big idea by that time? To proceed, you may consider the following questions:
- 1 What examples will you use to teach the CP of addition ?
- 2 What representations will you use during teaching?
- 3 What deep questions will you ask during teaching?
To teach the CP of addition, readers may use a pair of related addition facts such as 5 + 1 = 6 and 1 + 5 = 6, which may be linked to concrete or semi-concrete representations (e.g., story contexts, real-world objects, cubes, dominos or various drawings). To draw students’ attention to the big idea, one may ask questions that invite students to make comparisons. One may even formally introduce the name of this property at some stage of student learning. While these are all important instructional components, I would like to point out, based on video observations from my project, that specific representational arrangements and deep questions matter in how meaningfully students understand the big idea. Below, I will introduce cross-cultural insights drawn from the Chinese and U.S. lessons, which together illustrate the targeted approach ofTEPS.
Note that since Chinese textbooks present the CP of addition informally in Gl-2 and formally in G4, I will first introduce the informal teaching of this topic and then work toward formal lessons. Since U.S. textbooks formally introduce the CP at the very beginning of G1 and G2, I will reverse the process by starting with formal lessons, followed by informal teaching of the CP.
Insights from Chinese Lessons
Informal Teaching of the CP of Addition in G1 and G2
Related Addition Facts in Gl. The aforementioned swimming pool example introduced in Section 2.3 provides some brief insights about how the CP of addition is informally introduced in China during first grade. In that lesson, the objective was to teach additive inverse relations based on fact families about “8” (e.g., 5 + 3 = 8, 3 + 5 = 8, 8-3 = 5, 8-5 = 3). The textbook situated the worked example in the context of “one picture with four number sentences” (—HIH^). As mentioned earlier, we collected six videos of this same lesson. Across lessons, I found that the swimming pool context was primarily used to facilitate student sense-making of abstract ideas. Specifically, CP was taught informally during the discussion of two addition problems (e.g., 5 + 3 = 8, 3 + 5 = 8). Students would be invited to compare the number sentences, and the teachers would follow these comparisons with specific questions that oriented students’ attention to the key features of the CP.
In fact, before this swimming pool lesson, the Chinese textbook presented a lesson that introduced two related addition facts known as “one picture with two number sentences” (—0—^). During that lesson, the CP of addition was informally introduced for the first time. The worked example in that lesson was situated in the context of children planting trees and was solved in two ways: 5 + 1=6 and 1 + 5 = 6. In my project, two different Gl teachers taught this lesson in a similar way to the swimming pool lessons. Due to the lengthy discussion of the swimming pool example (see Section 2.3), I will only share the tree planting lessons. Table 4.1 summarizes the lessons of both teachers (T1 and T2), along with elaborations. 
Table 4.1 Deep Questions and Representations used in Chinese G1 Lessons that Informally Introduce CP
Typical Deep Questions
1 Problem posing and solving based on a real- world situation
(After students verbalized the mathematical information based in the picture)
Tl: Look at this picture, can you pose a question that can be solved with addition?
T2: Can you pose a question?
(The proposed problem was then solved numerically: 5 + 1= 6; 1 + 5 = 6)
Concrete to abstract
2 Explaining the meaning of the operation or number sentences
On meaning of the operation:
T1: Why did we use addition?
T2: Why did we list two addition number sentences?
On meaning of the number sentence:
Tl: What does 5+1=6 mean?
T2: Can you explain the meaning of each number sentence?
On the problem structure:
Tl: How many parts did we separate these children into?
T2: Observe carefully, we can separate the children in the picture into which two parts?
Abstract to concrete (fold back)
3 Comparing the number sentences to identify the patterns
Tl: Observe these two number sentences carefully, what are the similarities and differences between them?
T2: Observe these two number sentences, what do you find?
Follow up on “changes”:
Tl: Which two numbers are switched?
T2: Which and which are switched? Can you come up to point out which two numbers are opposite now?
Follow up on “non-changes”:
Tl: What about the results?
T2: What does not change?
4 Making sense of the patterns referring to the real-world situation
Tl: Why are the two numbers before and after the addition sign switched but the result “6” does not change?
T2: Why does this happen? Why are the two numbers switched with the results remained?
Abstract to concrete (fold back)
sentences meant based on the story context. As emphasized in earlier chapters, these questions are deep because they target the meaning of operations (“addition” in this case) rather than requesting computational answers. Moreover, they asked questions that draw students’ attention to the part-whole structure of the concrete context. That is,
“the total” was separated into two parts: “the children who came in a line” and “the child who came with a stroller.”
3 Comparing the Number Sentences to Identify the Patterns. Next, both teachers asked students to compare the two addition number sentences (e.g., 5 + 1 = 6, 1+5 = 6) and report what they observed. Students tended to come up with loose observations of what the CP entailed (e.g., I found they are switched). Both teachers asked followup questions that prompted a more precise understanding of the specific features of the CP (e.g., Which two numbers are switched? What does not change?). These prompts likely oriented students’ attention to the essence of the CP even though the term was not mentioned at this time. Below is an example episode from Tl’s class:
T: Observe these two number sentences carefully. What arc the similarities and differences between them? Think about it and then share with your desk mate.
T: (after small group discussion) Who can tell us about your discovery? Okay, you.
SI: I found that regardless of the “+” in the middle, regardless 5 + 1 or 1 + 5, as long as the sign in the middle is the same, the answer in the end is the same.
T: Agree? Anybody wants to add anything? You can invite another student.
S2: I can add to it. The first number sentence and the second number sentence are switched, that is why the answer is still 6.
T: You just mentioned “switched.” Please be clear which and which are switched.
S2: 5 and 1.
T: He referred to the two numbers before or after the addition sign, right? (Yes). What happened to them?
T: Great word! The positions of the two addends are switched. What does not change?
S2: The result does not change.
T: Thanks to you two. Applause!
4 Making Sense of the Patterns Referring to the Real-World Situation. Both teachers asked further deep questions to promote students’ sense-making of the observed patterns. T1 asked students, “Why are the two numbers before and after the addition sign switched but the result ‘6’ does not change?” Some student responded that, “Combining 5 and 1, it is 6; combining 1 and 5, it is also 6.” The teacher then rephrased student responses and related this observation to the story context that contained two parts:
T: It seems that in order to know how many people came to plant trees, we can combine those who come in a line and those who come with a stroller; we can also combine those who come with stroller and those who come in a line. 5 + 1=6 and 1+5 = 6 are a pair of good friends!
In the above statement, the teacher folded students’ observation of the CP (Table 4.1, activity #3) back to the part-part-whole structure of story context. I contend that the early emphasis on the real world meaning of operations and number sentences (see Table 4.1, activity#2) sets a groundwork for students’ informal understanding of the CP (Table 4.1, activities #3 and #4). Overall, I noticed that the Chinese teachers’ deep questions during the G1 lessons promoted rich connections between concrete and abstract representations of the CP of addition.
Checking Addition in G2. Opportunities for informal teaching of the CP in Chinese G2 lessons were limited to checking the computation of two- digit and/or three-digit addition with regrouping. To learn these standard algorithms, the worked examples were usually situated in story problem contexts (e.g., donating books, student art exhibition). After the students learned the algorithms, they were expected to switch the two addends to re-compute to check for the correctness of their answers. Clearly, this is an informal application of the CP.
Some teachers went one step further to help students understand that CP was indeed the underlying reason behind the checking method. For instance, after a class discussed the worked example, they solved a group of practice tasks like the following:
- 313 + 605 = 918 87 + 252 = 339 943 + 154 = 1097
- 605 + 313 = 918 252 + 87 = 339 154 + 943 = 1097
The teacher then guided the class to observe and verbalize the patterns. Students noticed that the addends in these pairs of tasks were switched but the results remained the same. Based on these observations, the teacher linked it back to the checking method introduced earlier in the worked example:
T: Someone said this is somewhat like?
T: In fact, why could we check like that way? We could use this group of number sentences to figure it out. That is, when we switch the first and second addends, the results?
Ss: Do not change.
T: Yes, the results are exactly the same! That is why we can use this method for checking.
In summary, despite the fact that Chinese students have not been formally introduced to the term commutative property during Gl-2, they still have opportunities to informally learn the CP. During this process, teachers’ representational uses and deep questions play an important role in developing students’ understanding of this property'.
-  Problem Posing and Solving based on a Real-World Situation. Bothteachers projected the picture of children planting trees. Studentswere asked to observe and verbalize the mathematical information inthe picture. Next, they were challenged to pose a question based ontheir observations. As mentioned earlier, problem posing is a criticalthinking skill valued by the field (Singer ct al., 2015). 2 Explaining the Meaning of the Operation or Number Sentences. Theproposed word problem was then solved with two numerical solutions (5 + 1 = 6; 1 + 5 = 6). Both of the observed teachers askedtheir students to explain why they provided two addition number sentences (see Table 4.1). Furthermore, they asked what these number