Teaching Additive Inverse through a Part-Whole Model
Imagine you are a first-grade teacher. How will you teach the following fact family to your students during their initial learning?
- 5+3= 8-3=
- 3+5= 8-5=
- 1 What is the big idea that you will expect students to understand?
- 2 What representations will you use during teaching?
- 3 Which deep questions will you ask during teaching?
In the above scenario, the big idea is the inverse relation between addition and subtraction (additive inverses). Readers may have thought of various representations (c.g., cubes, fact triangles, real-world situations) and different questions. However, how these representations are used with questions to facilitate student understanding is not that straightforward.
Below I share insights from relevant Chinese and U.S. lessons. Note that Chinese lessons appeared to have common structures while the U.S. lessons varied from each other.
Insights from Chinese Lessons
Figure 2.2 shows a worked example task in a Chinese first-grade textbook (Sun & Wang, 2012)2. This worked example is situated in a swimming pool context that indicates a part-whole model (five boys and three girls, or five children in the pool and three children out of the water).
During the first two years of my project, four different Chinese teachers taught this same lesson3. In the fourth year of the project, two of these teachers re-taught this lesson. Across the six lessons, the teachers approached the worked example in similar ways. All of the teachers unpacked the worked example through in-depth use of representations and deep questioning. The common teaching elements included asking students to:
- • Verbalize the mathematical information in the picture
- • Pose and solve addition and subtraction word problems
- • Explain the meaning of operations based on the picture
- • Compare the number sentences to learn the big ideas
Figure 2.2 A worked example in a Chinese textbook lesson. This G1 textbook image was taken from Sun & Wang (2012, v.l, p.58). Permission of citation was granted. © Jiang Su Phoenix Education Publishing.
The overall representational sequence of each lesson transitioned from concrete to abstract with the abstract being folded back to the concrete. In all lessons, the teachers asked sets of deep questions to elicit deep student explanations. Both aspects are core features of the TEPS approach. Below are elaborations of each element:
- 1 Verbalize the mathematical information in the picture. First, the teachers started with an engaging question before showing the swimming pool picture: “What sport do you like best in the summer?” or “Do you like swimming?” Next, the swimming pool picture was presented and the teachers asked students to verbalize what they saw in the picture. In several lessons, the teacher asked students to describe the “mathematical information” they saw. Note that at this stage, the tact family indicated at the bottom of Figure 2.2 was not presented to students.
- 2 Pose and solve addition and subtraction word problems. Most teachers then suggested that students pose a math problem based on the swimming pool picture. Students tended to create an addition story problem first: “There are five children swimming in the pool. Three more children join in. How many children are there now?”, “There are three children sitting on the side of a swimming pool. Inside the water, there are five children swimming. How many children are there in total?” I would like to point out that not all children have the same ability to pose a math problem as indicated by children’s struggles and peer input observed during the Chinese lessons. However, all Chinese teachers appeared to value such opportunities to develop children’s problem posing abilities, which is a critical aspect in mathematics education (Singer, Ellerton, & Cai, 2015).
After each story problem was posed, teachers would request that students suggest a number sentence to solve the problem. Teachers then recorded the number sentences on the board (5 + 3 = 8, 3 + 5 = 8). In a similar vein, two subtraction story problems were generated and solved by students, with the number sentences recorded by the teacher on the board (8 - 3 = 5, 8 - 5 = 3). This eventually resulted in a group of number sentences that formed a fact family. Variations included the following: In some classrooms, students suggested a subtraction problem first, which demanded the teachers’ flexibility in adjusting their lesson plans to accommodate these unanticipated responses; other classrooms created one addition word problem after which the teacher requested two different numerical solutions to represent it.
In all classrooms, the teachers asked students how they obtained their computational answers. Students commonly responded with their knowledge of number composition (e.g., 8 is 5 and 3; 3 and 5 makes 8). Note that prior to teaching addition, the Chinese G1 textbook introduces a unit named “composing and decomposing (
''A—j'a ),” which contains four lessons. The teacher’s guide explains that the knowledge of number composition sets a foundation for students’ learning of addition and subtraction as well as inverse relations. In my project, two Chinese G1 teachers purposefully reviewed this piece of knowledge before their new lessons.
- 3 Explain the meaning of operations based on the picture. After the number sentences were generated, all Chinese teachers asked students to explain the number sentences derived from the story problems. This included:
- • Explaining the meaning of a number sentence
- • Explaining the common operations between two number sentences
- • Explaining different operations among all four number sentences
First, teachers asked students to explain the meaning of an individual number sentence right after it was generated. Typical questions included “What does this number sentence mean?” and “Why did you use addition (or subtraction) to solve it?” Alternatively, some teachers asked students to explain each number sentence after the whole fact family was generated. Second, teachers asked students to explain the common operation between two number sentences, for example asking, “Why were both problems solved by addition (or both by subtraction)?” This type of comparison question drew students’ attention to the structural similarity between these problems and stressed the meaning of the indicated operation. Third, teachers asked students to explain the occurrence of different operations in the four number sentences: “Why were the first two problems solved by addition, but the last two problems were solved by subtraction?” This is also a comparison question that emphasizes the different meanings of operations and the structural differences between addition and subtraction. In all these cases, the abstract number sentences were folded back to the concrete real-world situations. Consequently, the meanings of the operations were stressed with sense-making.
4 Compare the number sentences to learn the big ideas. In addition to the explanations above, the teachers also asked students to compare the number sentences in order to identify the relatively abstract, yet fundamental, big ideas. All teachers asked students to compare the two addition sentences so that they could verbalize findings about the commutative property. Note that in China, the full terminology for this property is not introduced until the fourth grade (elaborated upon in Section 4.3). Some teachers also asked students to compare the two subtraction number sentences. Finally, all teachers asked students to compare the four number sentences to reveal the inverse relations.
Students’ explanations to teachers’ comparison questions revealed different levels of understanding. For instance, some students noticed that the same three numbers—3, 5, and 8—occurred in each number sentence. Some students noticed the location changes of the numbers (e.g., in addition “3 + 5 = 8,” 8 is at the end; in subtraction “8 - 5 = 3,” 8 is in the front). This form of observation indicates a surface-level understanding because it only focuses on numbers and their positions in the number sentence. Some Chinese teachers asked follow-up questions with a quantitative and structural focus. For instance, they asked why 3, 5, and 8 occurred in all four problems and what role each number played across the problems. This line of questioning prompted students to understand that regardless of the operations, “3” and “5” were parts (e.g., the part of the children who were out of the water and the part of the children who were in the water) and “8” was the whole (e.g., total children). Such comparisons have the potential to promote a further understanding of inverse relations based on the part-whole structure (e.g., part + part = whole, whole - part = part). This is particularly insightful because students often have difficulties conceptualizing parts and wholes when facing such tasks (e.g., see Figure 2.1 in Section 2.1).