Insights from U.S. Lessons
U.S. lessons on the same topic provided alternatives for representation usage. Similar to what is reported in Section 2.3, in one observed lesson, the teacher used connected cubes (cube trains) to model a worked example situated in a real-world situation about Karen and Lily’s teeth lost (see Figure 2.5, Class 1). Again, I’d like to note that cubes are a representational tool that are widely used in the U.S. but rarely in Chinese classrooms. As seen from Figure 2.5, each of the cube trains were formed by discrete, yet connected, cubes that were used to represent individual teeth. The “discrete” nature of cubes may enable students to compare two quantities through one-to-one correspondence so as to accurately identify the “same as” part and the “more than” part contained in the bigger quantity. As discussed earlier, this point is key because it helps transform a difficult comparison problem into a relatively easy part-whole model. In this sense, the cube trains served as a transitional total linking a real-world situation to the numerical sentence (12 - 7 = 5). In addition, the connected cubes
Figure 2.5 Insights from representation uses gleaned from the U.S. lessons. Class 1: Redrawn by Mohen Li. Class 2: Redrawn by Anjie Yang.
can also be considered as pre-tapes that aid students in learning tape diagrams (Murata, 2008). In Class 1, the teacher went on to ask the class to compare the cube trains and circle the “more than” part. Again, note this “more than” part was created by the representation and was not inherent in the original situation. This was a great teaching move because it allowed students to use gesturing with the diagrams, which potentially contribute to students’ understanding of the quantitative relationships.
Note that after students circled the “5 more” teeth, the teacher did not ask follow-up questions. This lesson could be enhanced by incorporating deep questions about the cube train representation. Based on the Chinese lesson insights, one may consider the following questions:
- • We’ve circled the part of “5 more” from Karen’s teeth, but what is the leftover part? What does this part refer to?
- • Which two parts do Karen’s teeth contain?
- • To find how many more teeth Karen has than Lily, why do we take 7 from 12? What does the “7” mean? Can someone show it on the cube train?
These questions could draw students’ attentions to the fact that the bigger quantity (Karen’s 12 teeth) contains two parts: the part the same as Lily’s (7 teeth) and the part more than Lily’s (5 teeth). If one takes away the same as part from the bigger quantity, one can obtain the more than part, which is the difference between the two quantities. Such a reasoning process uses students’ prior knowledge of part-whole model to explain why subtraction is needed for this comparison problem. If the teacher wants to challenge her students further, she could even modify (or encourage students to modify) this task to its inverse problems (a) Karen lost 12 teeth. She lost 5 more than Lily. How many did Lily lose? (b) Lily lost 5
teeth. Karen lost 7 more than Lily. How many did Karen lose? This instructional technique is called variation (Huang & Li, 2012), which likely lends more opportunities to deepen students’ understanding of the comparison model including the additive inverse relations.
Another U.S. teacher taught a worked example involving comparison problems (see Figure 2.5, Class 2). This teacher started with a cube train that contained four white and five orange cubes. She then requested a fact family from students (4 + 5=9, 5 + 4 = 9, 9-5 = 4, and 9 - 4 = 5). Next, she erased the number “4” from each sentence, leaving a set of equations with “missing numbers” (see Figure 2.5). This led to her point that that even though a word problem could be solved by either addition or subtraction, a number sentence where the missing value is isolated on the right side of the equal sign (e.g., 9 - 5 = □) would be the easiest one to solve. To illustrate this point, the teacher created two story problems, one of which was a comparison problem: “Carol and Sara both collect Pokemon cards. Carol has 9 cards. She has 5 more cards than Sara. How many cards does Sara have?' The other was a part-whole problem: ''"'Nazar and Jalynn both have pretzels. If Nazar has 5 pretzels and together they have 9 pretzels, how many does Jalynn have?” For each word problem, the teacher drew a corresponding bar model to illustrate the problem structure (see Figure 2.5). She then asked students to pick an equation to solve the problem. The conclusion was that even though each problem could be solved by any equation in the fact family, 9 - 5 = □ was the easiest to use. Pointing to the two bar models on the board, the teacher also reminded students that they should consider whether the word problem was a part-whole model or a comparison problem.
The insights gleaned from this part of instruction were the teacher’s use of multiple representations and discussions of rich mathematical ideas. The representations involved in this worked example included cubes, number sentences, real-world situations, and bar models. The mathematical ideas discussed were fact families, missing numbers, solving one problem using inverse operations, and different types of addition and subtraction problems (comparison model and part-whole model). Considering this lesson’s richness and intensiveness, it may be best utilized as a review after comparison and part-whole problems have both been taught.
However, two adjustments are needed if this is meant to be an initial lesson introducing comparison problems. First, the representational sequence could be changed to better align with the concreteness fading method. The lesson could start with the real-world context about Pokemon cards which could then be modeled with the cube train or bar models. These problems could then be further solved either arithmetically or algebraically, showing inverse relations. If a teacher wants to deepen students’ understanding, they can slightly change the Pokemon cards problem to a part-whole model similar to the Pretzels problem; however, I would suggest keeping the Pokemon card context consistent. After this problem is solved, the teacher could guide students to compare and contrast the two Pokemon problems, the corresponding diagrams, and the numerical solutions. This would allow students to see—even though both problems have similar story contexts with the same numerical solutions—that the problems have different structures. In fact, each word problem could be further changed to its inverse version which would then generate further classroom discussions. Second, as I have suggested for Class 1, this lesson would be improved through the incorporation of deep questions. Although the teacher of Class 2 demonstrated sound mathematical understanding throughout the lesson, she did the majority of the math work for her students (such as posing story problems and drawing the bar models). Her students would benefit more if she posed deep questions to engage students in this process and to elicit students’ deep explanations.
Back to the “Jump” Problem from the Opening Scenario
Returning to the “Jump” problem in the Scenario, one may incorporate the insights gleaned from the Chinese and U.S. lessons. Focusing on tasks (1) and (3) from Table 2.4, a teacher may present these two problems in one lesson. First, students can be guided to draw the tape diagrams for each task. If students lack prior knowledge of tape diagrams, connected cubes may be used as pre-tapes to establish this understanding. In this ease, numbers may be changed so cubes can be used to easily represent the tasks (e.g., Shawn jumped 2 feet, James jumped 3 more feet). Whether the teacher chooses to use the tape diagrams or connected cubes, critical questions should be asked to help students understand that the distance James jumped contained two parts: (a) the part the same as Shawn’s and (b) the part more than Shawn’s. Moreover, a teacher may ask students to compare the problem contexts, diagrams, and numerical solutions between the two tasks. The purpose of doing this is to understand why one problem is solved with addition while the other is solved with subtraction. More detailed discussion regarding how to teach the “Jump” problem can be found in Ding (2018).
Summary: Teaching Additive Inverse (Comparison Model) through TEPS
In this section, I discussed the teaching of additive inverses through the comparison model (see Table 2.1). For initial learning, the approach I suggested was TEPS. That is, one may first situate the worked examples in a pair of real-world contexts with the same story situations. Next, the story problems can then be modeled through semi-concrete representations (e.g., connected cubes, tape diagrams/bar models), culminating in numerical solutions. During the modeling process, teachers should help students grasp the key concept of the “same as” and the quantitative relationship: bigger quantity = same as part (smaller quantity) + more than part (difference). With this understanding, students can make sense of why addition is used to find the bigger quantity and subtraction is used to find either the smaller quantity or the difference. To help students see these connections, teachers’ deep questions should be used throughout the teaching of the worked examples. These questions may encourage students to compare and contrast (e.g., within and between representations) and elicit their deep explanations of the quantitative relationships, the meaning of operations, and the inverse relations embedded in comparison word problems.