 # Teaching Additive Inverse through a Comparison Model

Scenario

A second-grade teacher blogged about the six big feelings she experienced—frustration, self-doubt, anxiety, anger, terror, and remorse—when teaching comparison problems using a tape diagram (Vanduzer, 2017). The problem she had taught stated, “Shawn and James had a contest to see who could jump farther. Shawn jumped 75 centimeters. James jumped 23 more centimeters than Shawn. How far did James jump?” To help students draw the tape diagram, the teacher first acted out the problem by jumping Shawn’s distance. She then led the class in measuring 23 centimeters to show James’ jump. However, this act-it-out process did not prove helpful. In particular, a student refused to draw the diagram and became quite disruptive, causing the teacher’s six big feelings.

If you were the second-grade teacher in the above scenario, how might you teach this comparison problem with tape diagrams? Moreover, how might you convert this single problem to its inverse word problems? Could you also use tape diagrams to model these two new problems? And, how may these diagrams be used to teach additive inverses through a comparison model?

Readers may come up with a group of inverse word problems with corresponding tape diagrams similar to the ones listed in Table 2.4 below. The first problem from Table 2.4 is the task attempted by the teacher from the Scenario, which was to find the bigger quantity (75 + 23 = 98). The other two problems are tasks derived from the initial problem: One is to find the difference (98 - 75 = 23), and the other is to find the smaller quantity (98 - 23 = 75). Together, the solutions to these three problems show inverse relations between addition and subtraction. Among these three problems, the second problem (find the difference) is comparatively easier than the other two (find the bigger or smaller quantities). Nevertheless, all three problems share the same comparison structure, which is harder to manipulate than the part-whole model discussed in Section 2.3.

In Table 2.4, the representation sequence for each problem (word problem, tape diagram, numerical solution) makes use of concreteness fading. During this process, the tape diagram plays a transitional role to fade the concrete into the abstract. Tape diagrams are widely used in East Asian countries because they powerfully illustrate problem structures and quantitative relationships (Ding, 2018; Ding, Chen, & Hassler, 2019; Murata, 2008; Ng & Lee, 2009). However, the model is relatively new in the U.S.

Table 2.4 Inverse Word Problems with Tape Diagrams in the Scenario

 Word Problem Tape Diagram Numerical Solution 1 Shawn and James had a contest to see who could jump farther. Shawn jumped 75 centimeters. James jumped 23 more centimeters than Shawn. How far did James jump? (Find the bigger quantity.) 75 + 23 = 98 (cm) 2 Shawn and James had a contest to see who could jump farther. Shawn jumped 75 centimeters. James jumped 98 centimeters. How much farther did James jump than Shawn? (Find the difference.) 98 - 75 = 23 (cm) 3 Shawn and James had a contest to see who could jump farther. James jumped 98 centimeters. Shawn jumped 23 centimeters less than Shawn. How far did Shawn jump? (Find the smaller quantity.) 98 - 23 = 75 (cm)

(NGAC & CCSSO, 2010) and is both schematic and non-transparent in nature. Thus, it may cause classroom struggles if not used appropriately (see the Scenario). In fact, the potential challenges are compounded when tape diagrams are used to model comparison word problems. As explained in Section 2.2 (knowledge brief), comparison models are related to but harder than part-whole models. In the next section, I will share insights from U.S. and Chinese classrooms about how a teacher might present comparison problems with tape diagrams and how comparison problems can be used to develop students’ understanding of inverse relations.

## Insights from Chinese Lessons

In year 1 of my project, two Chinese expert teachers taught lessons on comparisons. The worked example in both lessons involved a pair of problems where students were asked to find either the bigger or the smaller quantity. The original textbook example was situated in a context about making paper flowers with illustrative pictures: Ting made 11 flowers. Hua made three more than Ting. Ring made three less than Ting. Students were first expected to pose questions about this example. The textbook then suggested the questions (a) Who made the most? Who made the fewest? (b) How many flowers did Hua make? and (c) How many flowers did Ping make ? The latter two questions were the target of the lesson and were similar in structure to “Jump” problems (1) and (3) listed in Table 2.4.

Both teachers modified the worked example context to better fit their students’ interests. In Class 1, the teacher changed the problem context to “displaying sticks”: Ting placed 11 sticks. Hua placed three more than Ting. Ping placed three less than Ting. The teacher then selected two questions posed by students: (1) How many sticks did Hua place? (2) How many sticks did Ping place?. The teacher in Class 2 changed the context to a self-designed gamc/activity that compared her students’ favorite numbers with her own.

• 1 Teacher Chen’s favorite number is 45. Student A’s favorite number is 3 bigger than 45. What is this student’s favorite number?
• 2 Teacher Chen’s favorite number is 45. Student B’s favorite number is 35 smaller than 45. What is this student’s favorite number?

Despite differences in concrete detail, the two revised lessons share common features. In both classes, the word problems were completed with students’ input in the form of questions they posed or conditions they added. This element of problem posing is similar to the swimming pool example discussed in Section 2.3. In contrast to the swimming pool lesson, both teachers spent a significant amount of time engaging students in the process of modeling before they asked for numerical solutions. This may be due to the challenging nature of comparison problems. In Class 1, the teacher used a diagram where she arranged “sticks” in a line, as opposed to Class 2 where the teacher used tape diagrams (see Figure 2.4). When the discrete sticks are placed in a row, they can be viewed as “pre-tapes” (Murata, Figure 2.4 Co-construction of Linear Diagrams in Chinese Lessons. Redrawn by Mohen Li.

• 2008), a precursor for learning continuous tape diagrams. Regardless of the difference in their outward appearance, the representational sequences in both classes made use of the concreteness fading method; each started from a real-world context modeled by linear diagrams (Ding, Chen, & Hassler, 2019), which eventually led students toward numerical solutions. In addition, the modeling process in both classes suggested a focus on structural knowledge. Common elements included:
• • Involve students in progressive co-construction of diagrams
• • Ask deep questions about the diagrams to elicit deep explanations
• • Make comparisons between diagrams to deepen structural understanding
• 1 Involve students in progressive co-construction of diagrams. Both teachers took time to engage students in the process of co-constructing the diagrams. In Class 1, where the sticks were used as pre-tapes, the teacher asked a student to come to the front to assist in the modeling process, which was then gradually revealed by the teacher’s preplanned PowerPoint slides. The teacher began by asking, “How should we place the first row?” The student suggested the number of sticks mentioned in the problem. The teacher then asked, “For the second row, what’s first? What’s next?” The student suggested first placing the same number of sticks as the first row and then the extra sticks. Based on this suggestion, the teacher then asked the class, “What are the two parts contained in the second row of sticks?” Students were able to tell that the second row contained two parts—the “same as” part and the “more than” part. The teacher further illustrated this point by separating the second row into two boxes (see Figure 2.4, left), which eventually led to the numerical solutions.

In a similar vein, in Class 2, where the teacher used the tape diagrams, students were also invited to join the co-construction process. The teacher drew the first tape for both problems and then modeled how to draw the second tape for the first problem. Using her mouse cursor, the teacher simulated the drawing process by asking students where she should stop drawing. After the modeling process, she invited the class to complete the drawing for both problems. Each student then drew the second tape on a worksheet that was prepared by the teacher. The teacher walked around to collect typical student work for class discussion. Next, the class spent a significant amount of time discussing two of the student-produced tape diagrams (see Figure 2.4, right). This discussion, in turn, led to numerical solutions for both problems (elaborated on below).

The structure and the usage of modeling from the two lessons described above can be compared to the “Jump” problem from the opening Scenario. The teacher in the Scenario acted out the problem.

However, there was no instruction on how to draw the tape diagram, as opposed to how the Chinese teachers engaged students in co-construction of the diagrams. Thus, Chinese students might have had a lighter cognitive load than the students in the Scenario classroom, which could have enhanced the worked example effect (Renkl et al., 2004).

2 Ask deep questions about the diagrams to elicit deep explanations. During the process of co-constructing the diagrams, both Chinese teachers asked deep questions. In Class 1, after the comparison of the first and second rows, the teacher asked, “What are the two parts contained in the second row of sticks?” This question likely drew students’ attention to the key point of the lesson: The bigger quantity contains a “same as” part and a “more than” part. Note that these "parts" were produced by the representation, rather than existing in the original task. As mentioned earlier, the teacher also boxed the two parts within the second row to make this critical distinction even more visible for students (see Figure 2.4, left). If students have prior knowledge of the part-whole model (see Table 2.2 in Section 2.2), they should follow naturally that one can combine the “same as” part and the “more than” part to obtain the bigger quantity. Given that the “same as” part has the same value as “the smaller quantity,” it also makes sense that we can use “Smaller + Difference = Bigger.” In brief, the above deep question contributed to the lesson by transforming a comparison model, which has been noted for its difficulty, into a more familiar part-whole model.

Similarly, the teacher in Class 2 also asked deep questions that were relevant to the tape diagrams. As seen in Figure 2.4 (right), this teacher compared two groups of drawings and asked students to explain why the second pair of diagrams contained a “dot.” Including the dot served to divide the tape into the same two quantities (the “same as” part, the “more than” part) that were identified by Class

• 1. As seen in Table 2.5, the teacher’s question about the function of the “dot” elicited students’ deep explanations of this sort. In fact, the teacher also invited two students to come to the board to explain their thinking using a teaching stick (see Table 2.5 for one example). Acts of “pointing to” exact parts of a diagram are called “gestures,” which is viewed as embodied cognition (Alibali, Kita, & Young, 2000). Based on the discussions of the diagrams, the class came up with two numerical solutions without challenges.
• 3 Make comparisons between diagrams to deepen structural understanding. The last feature was the use of comparisons4, which is also discussed in Section 2.3. In Class 2, where the tape diagrams were used, the teacher used comparisons throughout the worked example instruction. From the beginning, the two example tasks were set side by side: one task was to find the bigger quantity while the other task was to find the smaller quantity. Consequently, the two corresponding

Table 2.5 Deep Questions Elieited Deep Explanations

 Time Class Conversation 8:34-8:47 (T in Class 2 pointed to the 2nd pair of diagrams, see Figure 2.4, right). T: (pointing to the diagram on the left) He put a little dot on this bar. Do you know what he is trying to show by this dot? (pointing to the diagram on the right) There is another dot here. Discuss it with your table-partner. 9:10- 11:22 T: Who wants to come up and share? SI: (a boy, pointing to the first diagram) From here (the starting point) to here (the dot), it means 45, and it is the same as Miss Chen’s favorite number. From here (the dot) to here (the end) means it is 3 bigger than Miss Chen’s favorite number, (pointing to the second diagram) From here to here, it means the favorite number of the student is ten, and from here to here means the part of Miss Chen’s number that is bigger than the student’s. T: How about the dot? SI: The dot means it is equal in length to Miss Chen’s color bar, and from here to here (empty part) means the part of Miss Chen’s number that is bigger than the student’s. T: Can you all understand him? Ss: Yes. T: Let’s give him a round of applause. T: Who else wants to come up and explain? S2: (a girl, pointing to the first diagram). The function of this dot is to compare with the colored bar of Miss Chen’s favorite number. (pointing to the dot) From here on to the end represents the extra 3. (pointing to the second diagram) From the beginning to the dot means 45. There is an empty part. The color bar part means 10. T: Do you get it? Ss: Yes.' T: There is one thing that I didn’t understand, (pointing to the second picture) What does the empty part represent? Ss: 35. T: Seems like we all got this. So, we need to draw the colored bar that has the same length as Miss Chen’s, which is 45. And then we added the “more than” part that is 3. For the second question, we first find the part that is equal to Miss Chen’s. How much smaller is the actual color bar? (Ss: 35).

The image was redrawn by Mohen Li.

tape diagrams were also presented side by side (see Figure 2.4). This enabled the teacher to ask questions about the similarities and differences between the two diagrams (e.g., “Why did the second tape in both problems contain a dot?”; “Why was the second tape longer in the first diagram but shorter in the second diagram”). The above side-by-side comparison has the potential to contribute to students’ deep learning (Rittle-Johnson & Star, 2009; Star & Rittle-Johnson,

2009). In the end, when the two numerical solutions were generated, the teacher asked, “Why do they use addition for the first one, but subtraction for the second one?” Such a comparison question likely prompted students’ understanding of inverse relations.

In Class 1, where the pre-tapes (sticks) were used, the teacher taught the pair of examples sequentially. However, after she taught both worked examples and did two practice problems with the class, she displayed all four problems on one screen and asked students to compare the worked examples and the practice problems and to classify them into different categories. Students were able to classify them into two groups: the addition problems and the subtraction problems. The teacher further asked, “What kind of addition problems and what kind of subtraction problems?” This led to further clarification about when addition or subtraction is used to solve such problems. Once again, I argue that such instructional comparisons may contribute to students’ understanding of inverse relations. It is of interest to note that during the year 4 re-teaching, this teacher purposefully listed the two worked example questions—"How many did Hua make?” and “How many did Ping make?”—side by side. This enabled students to compare the use of addition and subtraction based on the corresponding diagrams.