Insights from U.S. Lessons: G3
One U.S. G3 teacher in the project formally introduced the AP of addition. Below, I first provide an overview of the lesson. I will then identify insightful components in this lesson, followed by suggestions on ways to incorporate the Chinese lesson ideas. In this lesson, the AP of addition was introduced together with the two other properties: the CP of addition and the identity property. The lesson went smoothly with all suggested textbook tasks covered. First, the teacher projected the textbook definition of the AP onto the board. It stated, “Associative property': The way in which the addends are grouped does not change the sum.” Along with this definition, there was a note, “Use parentheses ( ) to group numbers.” The teacher then read and elaborated on this definition with an example about adding 2 + 8 + 3 using the AP.
After the definition was introduced, the textbook provided a task asking students to use the AP to fill in the blanks of number sentences: (6 + 4) +
_= 6 + (4 + 3), and (7 + 2) + 5 = 7 + (_+ 5). The teacher guided the
students to first compute both sides of the equation to identify the missing number. She then told the class it did not matter where you put the parenthesis.
Next, there was an example situated in a word problem context: “Matthew saw 9 sailboats, 4 rowboats, and 6 canoes. How many boats did he see altogether?” The teacher projected this word problem on the board, which was already solved with (9 + 4) + 6. The teacher then stated:
T: This is going to show us the associative property. Grouping using those parentheses. Right? Remember these are the parentheses (Points to the parentheses and then reads the word problem). So, we see altogether, we know we need to add these three numbers up. The associative property is going to tell us it doesn’t matter how we group them with the parentheses. We will get the same answer.
In the end, the teacher projected a slide that involved both the CP and AP. On the left side of the slide, there were three groups of computation tasks. On the right side, there were a list of property' names (CP, AP, Identity). The teacher first guided students to find the sum for each computation task and then linked each example to the corresponding property'.
Insightful Components of the U.S. Lesson. Looking back, this lesson contained several components that had the potential to introduce the AP meaningfully. First, it contained varied tasks such as computation, word problems, and the formal definition of the AP, which shows different levels of abstraction. Second, the word problem task provided a real-world context that may aid students’ sense-making of the AP. Third, both the CP and the AP were presented, which provided an opportunity to differentiate between these two properties. All these components provide great learning opportunities for students.
A closer inspection of this lesson, however, suggests improvements so as to maximize student learning. In particular, the representational sequence, the approach to this word problem, and the differentiation between the CP and the AP could serve as discussion points that may incorporate some of the Chinese lesson insights.
Incorporate the Chinese Lesson Insights. Based on the existing U.S. lesson components, I suggest three modifications. First, the tasks should be resequenced to promote student involvement in exploring the property. As introduced above, the U.S. G3 lesson started with directly introducing the definition, which was abstract and general. It then proceeded to apply the AP for computation. Because of this organization, the students likely missed an opportunity for sense-making because the definition of the AP was directly told to them. Based on the Chinese lesson insights, I suggest starting the lesson with the word problem task (elaborated upon below), followed by the computation tasks for verification, which further leads to revealing the AP and its definition. This new task sequence would simultaneously engage the students in the co-constructing of the definition of the AP and follow the concreteness fading method that has been revisited in this book.
Second, I suggest spending more time solving the word problem to generate an authentic instance of the AP. The existing word problem in the G3 textbook—Matthew saw 9 sailboats, 4 rowboats, and 6 canoes. How many boats did he see altogether?—could have served as a problem-solving task to illustrate the AP. However, in this lesson, the teacher directly projected the solution (9 + 4) + 6 on the board and only asked students to apply the AP for computation. Taking the Chinese lesson insights, a teacher could encourage students to solve this word problem in multiple ways. The teacher could then select two typical solutions [e.g., (9 + 4) + 6 and 9 + (4 + 6)] and ask students to explain the meaning of each step. For instance, one may first find out the total number of sailboats and rowboats (9 + 4) and then the total boats. Alternatively, one may first find out the total number of rowboats and canoes (4 + 6) and then the total boats. Finally, the teacher could have students identify that both solutions solved the same problem, thus generating an instance of the AP: (9 + 4) + 6 = 9 + (4 + 6).
Based on the generated instance, the teacher could orient students’ attention to the number sentences only, (9 + 4) + 6 = 9 + (4 + 6). Comparison questions such as, “How are these two number sentences computed differently?” could be asked. This may elicit student observations of different features of the AP. As seen from the Chinese lesson, students noticed features such that there were the same three numbers, that the order of numbers did not change, that the position of parenthesis was changed, and that the order of operation was changed. Teachers should always allow multiple students to verbalize their observations about the key features. Such articulation is different from the emphasis on the association between the AP and the key words of “regrouping” and “parenthesis.” As discussed earlier, the presence of parentheses does not necessarily mean that the AP is being applied. For instance, the aforementioned Chinese G4 example, (17 + 23) + 28 = 28 + (17 +23), indicates CP but not AP. In brief, if the above word problem context is well utilized, it can provide rich opportunities for problem solving and sense making of the AP (and possibly the CP as well). This is a different approach from simply applying the AP to obtain computational answers.
Third, I suggest making explicit comparisons of the AP and the CP. In the above U.S. G3 lesson, both properties were presented in the same slide and could have been compared. However, after students connected the examples and corresponding properties, the class moved away from this task. Taking insights from the Chinese lesson, a teacher could invite students to explicitly compare how the CP and the AP are different. It is important to use these examples to help students understand that both properties are about the change of order; yet, the CP is related to changing the order of numbers while the AP is about changing the order of operations.
Summary: Teaching the AP of Addition through TEPS
In this chapter, I reported insights and relevant observations from the Chinese and U.S. lessons on teaching the AP of addition. Similar to the earlier section about the CP, these cross-cultural lessons provide integrated insights on how to teach the AP through TEPS. For informal learning of the AP (in G1 or G2), topics such as making a ten (tens, hundreds) to add and the standard algorithm of addition (e.g., as seen from the Chinese lesson) involved this property implicitly. Across these topics, number composition plays a key role, which is undergirded by the AP. To help students understand the computational strategy (e.g., making a ten to add), teachers can start the new lesson with a real-world context, which may be modeled by a semi-concrete representation (e.g., ten-frame with counters) and then solved numerically in a way that makes use of concreteness fading. To facilitate students’ understanding, deep questions should be asked to orient students’ attention to what happened during the process and the connections made between manipulatives and numerical records. It is also important to encourage students to verbalize the reasoning process in a complete manner in order to prevent students from getting lost during the detailed procedures.
With regard to the formal teaching of the AP (in G3 or G4), I again suggest teachers situate the new lesson in a word problem context that can be solved in multiple ways. Teachers can then pick typical solutions that illustrate the AP for discussion. Once again, the concreteness fading approach should be applied so that the teacher can shift students’ attention from concrete to abstract and facilitate pattern-seeking and generalization. During this process, deep comparison questions should also be asked. For instance, teachers may ask how a pair of number sentences are the same or different, what the common features across a group of examples are, or how the AP is different from CP. Regardless of whether the AP is being taught formally or informally, TEPS—teaching a worked example through problem solving with good use of representations and deep questions—can be expected to contribute to students’ deep understanding of this property and the development of their problem solving skills and algebraic thinking abilities.
- 1 One may view “-4” as + (-4) and then apply (a + b) + c = a + (b + c).
- 2 In Chinese, 11 reads as “ten one,” 12 as “ten two,” and 13 as “ten three” and so on. Consequendy, “10 + 1 = 11” reads as “ten plus one equals ten one,” “10 + 2 = 12” as “ten plus two equals ten two,” and “10 + 3 = 13” as “ten plus three equals ten three.” This results in a pattern, “10 + something = ten-something.”
- 3 Kindergarten does not belong to elementary school in China.
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5 Properties of Multiplication