Teaching the Associative Property of Addition
Scenario
Imagine you are a first- or second-grade teacher and you are aware that students in elementary school should learn the big idea of the AP of Addition, (a + b) + c = a + {b + c). How may this big idea be initially introduced to students? What if you teach 3rd or 4th grade: to what extent may 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 AP of addition ?
- 2 What representations will you use during teaching?
- 3 What deep questions will you ask during teaching?
For an informal teaching of the AP of addition, readers might recall the make a ten to add strategy [e.g., 2 + 6 + 4 = 2 + (6 + 4)] (see Section 4.2.2). This strategy can also be used to add two numbers such as 9 + 4. In this case, students could break 4 into 1 and 3, and then add (9 + 1) + 3 in a way that makes a 10. To help students make sense of this strategy, readers might choose to situate the example task in a story situation and use a ten-frame to illustrate the “make a ten” process. Readers may ask further questions such as “Why did the make a ten to add strategy result in the same answer as adding from left to right?” For a formal introduction to the AP, readers may build on the above examples such as (2 + 6) + 4 = 2 + (6 + 4) and reveal the name of this property.
Although the above instructional components are important, video observations from my project indicate that U.S. and Chinese lessons have very different representation usage and teacher questioning techniques. In this section, I will discuss the observed teaching methods, which is expected to provide insights on how the AP can be taught in ways that develop children’s algebraic thinking. Given that the sequences in which the U.S. and Chinese lessons introduced the AP (first informally and then formally) and the lesson topics matched closely, I will begin by introducing informal teaching of the AP with insights drawn from both Chinese and U.S. lessons, in that order. I will then do the same for the formal teaching of the AP. Taken together, this section will illustrate the insights of the targeted approach, TEPS.
Informal Teaching of AP of Addition
Insights from Chinese Lessons
We videotaped two Chinese lesson topics that involved informal teaching of the AP in Gl. One was titled “9 + something” (9JnA), which focused on the make a ten to add strategy. The other was titled “two-digit + one digit numbers” ( —fi'fl ) that focused on the standard algorithm
of addition.
Make a Ten to Add Strategy. Two Chinese teachers (T1 and T2) taught the same lesson of “9 + something” in both years 2 and 4 of my project, resulting in a total of four lessons examined for this topic. Across these lessons, the teachers used the same worked example provided by the textbook. This example was situated in a story context where a monkey saw nine red apples inside a box and four green apples outside the box (see Figure 4.3). The cartoon figure raised a question, “Flow many are in total?” This example problem could be solved by 9 + 4. Students could compute it in different ways; yet, making a ten to add was the targeted strategy of this lesson.
To prepare students for this lesson, both teachers used review tasks such as “10 + something” as well as number combinations whose sums add up to 10. After students solved a group of tasks such as 10 +1,10 + 2, 10 + 9, 10 + 5, the teachers, acting proud and surprised, asked “Why did you solve these problems so quickly?” This elicited the students’ explanation that “10 + something is ten-something” ^{2} (“ЮЙП/ЬЙ^^р~t"A ”), which enabled students to see the power of adding single digits to 10. After students were equipped with the necessary prior knowledge, the teachers (in both years) spent a long chunk of time discussing the monkey and apple worked example. Table 4.3 summarizes the key activities, along with deep questions and representation uses from all four lessons about “9 + something.” Note that the sequence of activities may be slightly different across classrooms. Elaboration follows.
Figure 4.3 The story situation in the Chinese worked example about 9+ something. This Gl textbook image was taken from the Sun & Wang (2012, v.l, p.88). Permission of citation was granted. © Jiang Su Phoenix Education Publishing.
Tabic 4.3 Key Instructional Activities in the Lessons of “9 + Something'
Key Activity |
Typical Deep Qtiestions |
Representations |
1 Observe the picture and pose a problem |
Tl: Who knows what this picture tells us? T2: You guys did a great job. And, the monkey gets some apples from home as a treat for you! Look, how many are inside the box? (S: 9) How many are outside of the box? (S: 4) Can you pose a question solved with addition? .. .Who can say this problem completely? |
Concrete |
2 Explain the meaning of operation |
(After students suggested 9 + 4 as a solution) T1/T2: Why did you use addition? Tl: We also learned subtraction, why did you use addition? T2: What does 9 mean? What does 4 mean? |
Abstract to concrete |
3 Understand the process of making a ten to add |
(While discussing a ten-frame) T1/T2: Why did you move 1 [apple] from outside the box into the box? Why didn’t you move 2 or 3? (S: Over foil). Tl: What if the box has enough space? Will you put 2 in or will you still put 1 in? (While discussing the numerical expression) T1/T2: Why did you separate 4 into 1 and 3? Tl: I recall when we learned number composition, 4 can be separated into 1 and 3, 2 and 2. Would breaking it into 2 and 2 work here? Tl: 4 can be separated into 3 and 1. What else? (S: 2 and 2). Why didn’t you break it into 2 and 2? Why did you break it into 1 and 3?] |
Concrete and/ or abstract |
4 Verbalize making a ten to add completely |
Tl: Who can think aloud the fourth way completely? T2: Who can tell us the process again? How would you exactly figure out 9+4? |
Abstract |
5 Compare make a ten to add with other strategies |
T1/T2: You have thought of four ways [to find the answer]. Which way do you like best? Why? |
Abstract |
6 Compare alternative ways to make a ten |
Tl: As inspired by you guys, I want to ask, for 9+4, may I make the 4 a 10? Tl: Why do we need to separate the 9 into 3 and 6. Do we want to use 3 or 6? (Follow-up) Tl: Why is making the 9 a 10 better? T2: Do you prefer making the 9 or the 4 a 10? |
Abstract |
- 1 Observe the Picture and Pose a Problem. Both G1 teachers began by projecting the monkey and apple picture and asking students to verbalize what they saw. As introduced elsewhere, this is a common approach in Chinese classrooms from the very beginning of elementary school^{3}. The students were first guided to see nine red apples inside the box and four outside the box. The teachers then asked them to pose a question to be solved with addition. Moreover, the teachers encouraged them to form a complete statement using the story problem. Note that in the picture, the box (or apple container) functions as a ten-frame.
- 2 Explain the Meaning of an Operation. Students in each class suggested 9 + 4 as a solution for this word problem. Both teachers asked why they used addition. One teacher challenged the class by asking why they did not use subtraction given that they also learned subtraction. Another teacher asked for the meaning of 9 and 4, respectively. Note that the goal of this lesson was computation rather than problem- solving. Yet, both teachers asked questions to stress meaning-making during their teaching.
- 3 Understand the Process of Making a Ten to Add. Students in each class came up with different strategies to solve 9 + 4 (e.g., counting up, comparing it with 10 + 4, making a ten to add pictorially, physically, or mathematically). Each strategy was deemed valuable by the Chinese teachers. However, they focused discussion on making a ten to add, especially the correspondence between the concrete and abstract representations. Both teachers consistently asked a set of deep questions to promote students’ understanding of the process (as opposed to the result). Note that such discussions were missing from the parallel U.S. lessons in my project. For this reason, I will introduce the relevant Chinese lesson insights in detail. Below is a typical excerpt from T2’s lesson in which one student was sharing his strategy of using a ten-frame and counters that represented the red and green apples:
SI: My strategy is (pointing to the initial 9 red counters in the ten-frame) here arc nine counters, if we add one to the right side (pointing to the newly added green counter in the ten-frame) ...we will have 10 inside and 3 outside. Ten plus 3 equals 13 (see Figure 4.4 bottom right).
T2: Okay, I want to ask: (pointing to the newly added green counter) Why did you move this apple to the left side?
SI: Because this problem is not finding how many green apples or how many red apples but how many apples are in all.
T2: So, when you move this green apple here, what happened?
SI: Then the inside became a ten.
T2: So, it became a full ten. What about the right side?
SI: Three apples are left.
T2: Three are left. And, putting them together?
SI: 13!
Figure 4.4 Understanding the making a ten to add process in a Chinese G1 classroom. This The dark circles are red counters and the light circles are green counters. Redrawn by Mohen Li.
T2: (To the class) So, his method is to separate the green apples into what and what?
Ss: 1 and 3.
T2: Let me record his method (see Figure 4.4). So, he thought of it this way. (Pointing to “4”). The “4" green apples are separated into 1 and 3 (recording on the board). Then he first added 9 and 1 (gesturing and recording), which equals to 10 (recording, and then pointing back to the 10 counters). And, then? 10 plus 3 is?
S: 13.
In the above excerpt, the teacher asked a deep question, “Why did you move this apple to the left side?” This elicited the student explanation that he wanted to make a ten. Later, when recording the student’s hands-on process as numbers, she asked, “So, his method is to separate the green apples into what and what?” This question likely oriented students’ attention to the number decomposing process, that is, “4” was separated into 1 and 3, which illustrated “how” 9 + 4 was transformed into 10 + 3.
Interestingly, T1 also asked students why they moved one but not two apples into the box. One student responded that there was only one space left. This response was not satisfying for the teacher who pressed for a further explanation which highlighted the critical process of making a 10 to the class:
Tl: Why did you move 1 from outside to the box? Why didn’t you put 2 or 3 in?
S: No enough space! If you put two, they are going to hit the sky!
Tl: Who thought the same thing? Anybody have other thoughts? What if there are two spaces left in the box? Will you move two in? Or will you still put one in? I want to see who will still put one in even if there are two spaces? Raise your hands and tell me why.
S: Adding 1 more (to 9) is 10, 10 plus 3, you can immediately get the answer! Yet, 11 + 2 is relatively harder to compute.
Tl: Do you understand? It is not because of the space in the box....
Similar deep questions were asked with the numerical representations. For instance, both teachers asked students why they separated “4” into 1 and 3 but not 2 and 2 (see Figure 4.4):
T2: I have a question: Why did you separate 4 into 1 and 3? I recall that when we learned number composition, 4 can be separated into 1 and 3, and 2 and 2. Would breaking it into 2 and 2 work here? You please. S: If you break it into 2 and 2, 9 + 2 = 11, you cannot then get 10 and
3. Nine plus “1” equals 10, 10 plus 3 equals 13.
T2: In other words, we separated 1 out for what?
S: Making a 10!
T2: Excellent. We want to make a 10. So, I need to separate “1” out.
Don’t forget after you have separated 1, how many is left in 4?
S: 3.
The above questions—Why did you put one but not two apples in? And, why did you separate 4 into 1 and 3 but not 2 and 2—all targeted the same key information. These deep questions demanded that students recognize both the process and logic behind the making a ten to add strategy.
4 Verbalize Making a Ten to Add Completely. After the focused discussion about making a ten to add, both teachers stressed students’ complete verbalization of the process. The teachers first asked a few students to model their thought process aloud and then practice it in pairs, or with the teachers themselves. In the counter episode described above (see Figure 4.4), the teacher led the class in the following conversation:
T2: Let me ask another student to say this completely: How did you compute it?
SI: I computed it in this way. First, I separated 4 apples into 1 and 3, and then I added 9 and 1 that was separated from 4, that equals 10. Next, 10 plus 3 equals 13.
T2: Well said. Please sit down. Can you all say it? Please try it with me (Pointing to the numerical expression in Figure 4.4).
Ss: (in unison with T) Separate 4 into 1 and 3, first add 9 plus 1 which equals 10, and then 10 plus 3 equals 13.
Note that the above in unison response was based on class discussion. Reciting the thinking method in unison after discussion is quite common and unique to Chinese classrooms. While one may question the true understanding of all students during this process, it is plausible that it helps wrap up and reinforce the key ideas and explicitly bring the to-be-learned information to all students. Students who need more time to digest this information would also obtain support from their peers through this unison response.
- 5 Compare “Make a Ten” with Other Strategies. As introduced earlier, even though the making a ten to add strategy was the target of this lesson, students in both classes came up with different ways to compute 9 + 4 including counting on, memorizing this number fact, and comparing 9 + 4 with 10 + 4 (e.g., 9 is 1 less than 10, 10 + 4 = 14, so, 9 + 4 = 13). The teachers expressed appreciation for all strategies and recorded them on the board (e.g., see an example in Figure 4.4). Flowever, both teachers asked students to compare “make a ten” with the other strategies by asking “Which way do you like best? Why?” Arguably, such comparison questions can encourage students to make connections between the varied strategies and be aware that not all strategies are equally effective. Regardless of the student suggestions, the teachers eventually oriented their attention back to the targeted strategy, making a ten to add.
- 6 Compare Alternative Ways to Make a Ten. In both teachers’ classes (all four lessons), there were discussions of an alternative way to make a ten. That is, in addition to breaking 4 apart (1 and 3) to make 9 a 10, one may break 9 apart (3 and 6) to make 4 a 10. In one lesson, this alternative was suggested by a student; in another lesson, it was brought up by the teacher; and in the remaining two lessons, it was discussed as part of the class practice problems. Below is a typical excerpt:
T1: As inspired by you guvs, I want to ask, for 9 + 4, may I make 4 a 10? S: Yes!
T1: There was a student who just said, 4 plus 6 equals 10. I want to make 4 a 10, what can I do?
SI: Separate 9 into 3 and 6, give 6 to 4, 4 plus 6 equals 10, and 10 plus 3 equals 13.
Tl: Where did the 3 come from? Why is there a 3?
S2: 9 is separated into 6 and 3.
Tl: Okay, so the answer is also 13, right?
S: Yes.
Tl: Okay. Now, I want to ask, why 9 is separated into 3 and 6? Do we need 3 or 6?
S: We need 6 because only 6 plus 4 equals 10.
Clearly, both ways to make a ten are mathematically correct and apply the same method. Thus, discussions of the alternatives could contribute to students’ familiarity with and understanding of the making a ten to add strategy. However, the two ways are not equally convenient. As such, both teachers raised further deep questions by asking the students which way they preferred and why: “Do you prefer making 9 a 10 or making 4 a 10?” and “Why is making 9 a 10 better?” Students realized that giving “1” to 9 was an easier process than giviwng “6” to 4. Based on these discussions, Chinese teachers suggested that it would be more convenient to break the smaller number apart to make the bigger number a ten (in Chinese words, “ТсЬЙ. ШХШС"). The above discussion indicates that Chinese lessons focus on teaching students not only the strategies, but also how to analyze them and make their own choices about when and how they are most convenient. This is different from the parallel U.S. lessons where the teachers told students that they should always remember to make the bigger number a ten.
Looking back, most, if not all, of the above six steps (see Table 4.3) occurred during the teaching of worked examples. Every lesson started with a real-world situation, which was solved with addition (e.g., 9 + 4). To complete the computation, various strategies were suggested but making a ten to add was the focus of discussion. In particular, teachers made great efforts to help students understand the making a ten process with concrete and abstract representations, as well as the connections between them. Teachers’ specific choice of deep questions (e.g., why moving 1 apple but not 2) played an important role during this process. After this, both teachers stressed that students should fully verbalize the process of making a ten to add. Additionally, they encouraged students to make comparisons between making a ten to add and other computational strategies. They then pushed the comparisons a step further by contrasting alternative ways to make a ten. These comparisons arguably promoted students’ connection-making among various knowledge pieces. Because of the rich discussions contained in each worked example, neither Chinese lesson required more than one worked example before student practice.
Additional Observations from Practice Problems. Although this book focuses on the teaching of worked examples, I would like to share a few observations from the practice sections of these lessons, which may provide further insights on deepening students’ informal understanding of the AP (Appendix В shows T2’s lesson plan that shows the full lesson design.) First, during practice problems, both teachers continued asking deep questions that focused on the process of making a ten to add. For instance, when there was one little cube moved to a pile of nine in a picture, the teacher pointed to the numerical representation and asked, “Can you tell me which little cube this ‘Г refers to?” When students suggested making 9 a ten in order to add 9 + 7, the teacher asked, “Why didn’t you make 7 a ten?” When students suggested alternative ways to make a ten for 9 + 6, the teacher asked, “We have two ways to make a ten, do you prefer making 9 a ten or 6 a ten? Why?” All these questions helped reinforce what had been taught during the worked examples.
Second, the Chinese textbook provided a group of practice problems that involved 9, which was utilized in both teachers’ classes. After computation, both teachers asked students to observe the number sentences:
9 + 2 = 11 9+3=12 9 + 4 = 13 9+5=14
The teachers then asked the students, “What do you find?” Among the observations, students noticed that the ones digit of the sum was always “1” less than the number that was added to 9 (see the bolded above). The teachers then asked students, “Why is it the case and where did the ‘Г go?” This deep question elicited students’ deep explanation that the “1” was given to 9 to make a ten.
Third, the Chinese textbook also provided a group of practice problems like those below:
- 9+1+1= 9+1+8=
- 9+2= 9+9=
After students computed the problems, the teacher asked them, “Are there any connections between the tasks on the top and at the bottom?” Students noticed that both contained the same answers. The teacher further asked, “Why do they have the same answers?” Students noticed that if they added the latter two numbers in the first number sentence, they would obtain the other addend in the second number sentence (e.g., 1 + 1=2,1+8 = 9). The teacher further pointed out that the opposite was also true, since 2 could be separated into 1 and 1, and 9 could be separated into 1 and 8. The teacher even asked students to pose similar tasks of this sort. Arguably, this type of discussion can further contribute to students’ intuitive understanding of the AP.
Finally, at the end of the lesson, one Chinese teacher (Tl) asked, “For 9 plus something, we used the make a ten strategy to compute. What if it was 8 plus something or 7 plus something? Can we still make a ten?” Students confidently responded with “Yes!” According to the textbook, “8 (and 7) plus something” was the targeted content in the follow-up
Figure 4.5 Record of thinking process about two-digit and one-digit addition. Redrawn by Mohen Li.
lesson. As such, the above “what if’ question might have served to facilitate the transfer of students’ current learning to later lessons.
Standard Algorithm of Addition. As mentioned in Section 4.2, standard algorithms for multidigit arithmetic use the basic properties of operations extensively. In Chinese lessons, addition algorithms were mainly introduced by the end of second grade. However, the G1 textbook introduces two-digit plus one-digit addition with regrouping. Although this lesson does not present the vertical format, the addition process is the same as the standard algorithm. In my project, two teachers taught this lesson and the general approach to the worked example was consistent with what has been described in Table 4.3. Given that I have provided a lengthy description about the earlier lesson, I will only highlight two additional insights. First, even though the targeted concept is not the process of making a ten to add, I observed that a consistent format was maintained for recording the number composition process (see Figure 4.5). For instance, a sum for 24 + 6 was solved by separating the tens and ones digits of the double digit number (24 = 20 + 4) and then adding the two ones digits (4 + 6). Although it was never mentioned explicitly, this process did utilize the AP: 24 + 6 = (20 + 4) + 6 = 20 + (4 + 6). In my project, both Chinese teachers emphasized students’ verbalization of this thinking process based on the numerical format.
Second, even though alternative computation strategies were allowed (e.g., making a ten to add), students in this lesson were expected to master and directly utilize basic facts like 4 + 9 =13, which should have resulted from the earlier lesson about making a ten to add. In other words, even though making a ten to add is a powerful method, students were expected to move past it and retrieve the basic facts of addition in order to perform more complex addition.
Figure 4.6 Wide use of ten-frames in U.S. lessons. Redrawn by Mohen Li.