A Close Look at a Chinese Lesson
Table 2.3 lists the key activities, typical questions, and representation uses from one teacher’s instruction of the swimming pool worked example. Note that due to the scope of this book, I will focus on worked examples rather than full lessons. However, Appendix A shares this teacher’s lesson plan (translated), which provides a rough sense of her full lesson.
Table 2.3 Unpacking a Worked Example for a Fact Family in a Chinese Lesson
Key Activity |
Typical Questions |
Representation |
1 Verbalize the picture. |
T: Do you like swimming, students? (Yes!) (Shows the swimming pool picture.) Look at this picture carefully. What mathematical information have you obtained? |
Concrete |
2 Pose and solve two addition word problems. |
T: Can you pose a question that is solved by addition? |
Concrete to abstract |
T: Okay, can you write a number sentence to solve her question? (5 + 3 = 8) |
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T: Tell us how you get this answer? (Chinese teachers expect students to apply their prior knowledge of “number composition of 8” to find the answer.) |
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T: Okay, what else? (Request for another addition word problem.) |
(Continued)
Table 2.3 (Continued)
Key Activity |
Typical Questions |
Representation |
3 Compare two addition number sentences. |
T: Observe these two number sentences on the board. What do you find? |
Abstract |
T: Which and which are switched? |
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T: What did not change? |
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4 Pose and solve two subtraction word problems. |
T: Using this same picture, can you pose a question that can be solved by subtraction? |
Concrete to abstract |
T: Who can write the number sentence? |
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T: Do you have other questions to ask other than this one? |
||
T: Can you list a number sentence for her question? |
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5 Compare two subtraction number sentences. |
T: Please observe these two subtraction number sentences carefully. Between these two, what do you find? |
Abstract |
6 Explain the number sentences based on the real-world situation. |
T: We just listed four number sentences based on this picture. Please pick one number sentence and explain its meaning in a group of four students. |
Abstract to concrete (fold back) |
7 Compare four number sentences to identify the connections. |
T: Now, please observe these four number sentences together carefully. See what you can find from these four number sentences. {Small group discussion followed by students’ sharing. In the end, Tsummarized students’ findings, as indicated below.). |
Abstract |
T: We just looked at one picture from different angles and listed how many number sentences? (S: 4) Among which two are? (S: Addition) and two are? (S: Subtraction). The two numbers in addition are switched, but the answers are? (S^T: Not changed). For subtraction, the number in the very front did not change, but die latter two numbers are? (S/T: Switched). And we also found the results in addition become what in subtraction ... the number in die very front, right? .. .To turn addition opposite... it becomes? (S/T: Subtraction). |
The above lesson elements in Table 2.3 demonstrate several features: (1) the teaching of the worked example was situated in a real-world context which was familiar to students; (2) the overall representation sequence was from concrete to abstract with the latter occasionally folding back; and (3) the teachers asked questions throughout the lesson, during which comparison was used to promote connection-making.
However, it should be noted that the last series of comparisons in Table 2.3 (key activity 7) did not fully utilize the potential of deep questioning. In the teacher's summary of the relations, she focused more on the location of the numbers while only referencing the part- whole relationship implicitly in another part of the lesson. Promisingly, in her re-teaching of this lesson in year 4, we noticed that she added the questions, “When do we use addition?” (S: Combining two parts into a whole) and “When do we use subtraction?” (S: Taking one part from the whole). She also explicitly illustrated the part-whole relationship using her self-made cards of “part, part, and whole.” Finally, she emphasized that regardless of addition or subtraction, both operations in the lesson were about the connections between two parts and a whole.
Insights from U.S. Lessons
U.S. lessons contributed additional insights. In my project, all U.S. lessons about fact families used connected cubes (also called “cube trains”), a unique tool which rarely occurred in the observed Chinese classrooms. Figure 2.3 (left) shows one example.
Given that cube manipulatives are semi-concrete, meaning they arc- less concrete than real-world situations but more concrete than number sentences, they can be used as a transition tool between real-world contexts and abstract equations. In the above U.S. lesson (Figure 2.3, left), the G2 teacher started her discussion with three number facts in a fact family (7 + 1=, 8 - 1=, an 8 - 7 =) and how one knows addition would help solve subtraction. She then asked students to use the cubes to generate the fourth number fact that is an addition (1+7 = 8). The representation uses could be enhanced by adding a real-world situation like the swimming pool context in the Chinese lesson, which can then be modeled with cubes, leading to the number sentences in the end. Of course, the Chinese lesson may also benefit from adding a semiconcrete representation like cubes when fading from the swimming pool context to number sentences. Taken together, the above suggested
Figure 2.3 Semi-concrete representations used in the U.S. classrooms. The image on the left has been altered to provide a zoomed-in view of the cubes. Redrawn by Mohen Li.
representational sequence of “real-world context - cube - number sentence” is aligned with the method of concreteness fading, which stresses a “gradual” transition from the concrete to the abstract. As reviewed earlier, this representational sequence is found most effective in supporting student learning (Fyfe et ah, 2015) and is recommended by the IES practice guide (Pashler et ah, 2007).
Figure 2.3 (right) shows another U.S. teacher who taught fact families using the following representations: a story-context about sandwiches, a bar diagram, and a fact family. This sequence also aligned with the con- creteness fading method. The bar diagram (also called tape diagram) is a powerful tool emphasized by the Common Core due to its ability to illustrate the problem structure (NGAC & CCSSO, 2010) and form a potential transition from the concrete to the abstract (Pashler et ah,
2007). Section 2.4 provides more discussion about the bar model or tape diagram.
With that in mind, both U.S. lessons could have been enhanced by integrating the deep follow-up questions that occurred in the Chinese lessons. For instance, comparison questions (see Table 2.3) could have been explicitly asked to orient students’ attention to the meaning of operations and inverse relations between addition and subtraction. I will also illustrate the point of asking deep questions about the diagrams in Section 2.4. Overall, it seems that an integration of U.S. and Chinese classroom insights provides concrete illustrations of opportunities to develop students’ algebraic thinking in elementary classrooms.
Summary: Teaching Additive Inverse (Part-Whole Model) through TEPS
In this section, I discussed the teaching of additive inverses through the part-whole model. The integrated insights from Chinese and U.S. lessons contribute to the targeted approach, TEPS. That is, teachers should engage students in the process of solving a worked example task through the use of representations and deep questioning. In terms of representation uses, teachers should situate the worked example in a real-world context which can be linked to abstract solutions through concreteness fading. During the modeling process, teachers should focus students’ attention to the part-whole structure (part + part = whole; whole - part - part). To promote structural understanding, teachers should also ask a set of deep questions, including the use of comparisons. The deep questions may include ones that target the meaning of addition and subtraction, the structural similarities and differences between relevant representations (concrete, abstract), and the undergirding idea of additive inverse relations.