Formal Teaching of the DP

The various instances of exposure to the DP in earlier grades served to prepare the Chinese students to formally learn the DP in the fourth grade. As stated in Section 5.2.4, the G4 textbook devotes two lessons to teaching the DP, including a lesson that introduces the property (lesson 1) and one that applies the property for computation (lesson 2). Although the project only videotaped the first lesson on the DP’s formal introduction, my prior textbook analysis of the second lesson (Ding & Li, 2010) found that Chinese computational lessons also stressed sense-making for the targeted strategies. For instance, in lesson 2, the worked example was to find the total cost of 102 sets of Chinese chess with a price of ¥32. Students were guided to first find the costs for 100 sets and 2 sets of Chinese chess, respectively, and then to find the total cost. This led to the computational strategy of breaking apart 102 to calculate 32 x 1027. Additionally, deep questions in the textbook (e.g., Is this way effective? What property of operation did you use?) highlighted the DP as the underlying property of the computational strategy (see more descriptions in Ding & Li, 2010). Given that this book is based on videotaped lessons, I will focus only on the initial, formal introduction of the DP (lesson 1). As to be described, this lesson contains many instructional insights that can be incorporated into the U.S. textbook and lessons.

While the worked example from the textbook was situated in a word problem with an equal groups structure (see Table 5.3, task 4), the Chinese G4 teacher started the lesson with five different tasks (including

Table 5.3 Five Review Tasks Used in the Formal Teaching of the DP.

Review Task Discussed

Resulting DP instance

1

2 4

x 1 3

(10 + 3) x 24 = 10 x 24 + 3 x24

2

Find the perimeter for this rectangle. 10m

(10 + 7) x 2 = 10x2 + 7x2

3

Find the area for this rectangle.

(15 + 10) x 8 = 15 x 8 + 10 x 8

4

The fourth grade of a school has 6 classes. The fifth grade has 4 classes. Each class gets 24 jumping ropes. Flow many jumping ropes do the fourth and fifth grade get altogether?

(6 + 4) x 24 = 6 x 24 + 4 x 24

5

(19 + 51) x 12 = 19 x 12 + 51 x 12

the textbook example). Tasks 1 and 5 were computational while tasks 2 through 4 were concrete context-based. The students agreed that all of these tasks were familiar to them from lessons they had seen in earlier grades. For instance, the first task, 24 x 13, was two-digit multiplication from G3. The fourth task about jumping rope was similar to the aforementioned G3 word problem about shopping with only slight structural difference. In addition, the number list (task 5) called for pairing numbers as seen from the previous example of informal teaching of the DP.

The teacher purposefully chose each task as a different circumstance that could generate an instance of the DP. For instance, after a student calculated 24 x 13 in task 1, the teacher asked follow-up questions to ensure that the class had a clear understanding that 13 could be viewed as 10 + 3. Consequently, the two partial products 10 x 24 and 3 x 24 were derived. The teacher then recorded this computational process on the board as (10 + 3) x 24 = 10 x 24 + 3 x 24, which generated an instance of the DP. The teacher then encouraged the students to solve the rest of the tasks in different ways. Students reported two solutions for each task and then compared them to generate corresponding instances of the DP (see Table 5.3, right).

As has been observed in nearly all the Chinese lessons in this book, the teacher asked the students to explain each of their solution steps. It should be noted that the approach to the perimeter problem (task 2) was similar to the NRC (2001) suggestion that a perimeter problem can be solved in multiple ways [e.g., L + L + W + W, 2L + 2W, or 2(L + W)] , which can be used to illustrate the DP. In addition, the area task (task 3) was similar to the kinds of array/area models that were frequently observed in U.S. classrooms (to be elaborated upon later). Overall, the above tasks used to formally introduce the DP demonstrated vertical connections to many lessons the students encountered across earlier grades. As will be reported below, these review tasks provided a foundation for the fourth graders to discover, make sense of, and formally learn the DP.

After the five DP instances were generated (Table 5.3, right), the teacher moved the lesson to the next level through the following activities:

  • • Observe and compare the equations to identify new patterns.
  • • Pose one more example of each kind to verify the observations.
  • • Represent findings with words, letters and shapes.
  • • Formally reveal the name of the DP.

The teacher first asked students to observe the five equations to see if they could identify any patterns that were new or noteworthy:

T: We have obtained the five equations on the blackboard. Look here. What do you find? Or what do you feel? Observe. Observe carefully. Why are the left and the right sides equal? Recall the Chinese saying, “Review the old to find the new.” What new information do you find? If you find something new, it indicates that you can teach others! What do you find?

When he noticed that some students did not seem able to discover the pattern, he adjusted his instruction by asking the class to pose an example similar to any of the review tasks and to explain why both sides of the equation were equivalent.

T: Okay, some students may need some guidance from the others... Please pose an example similar to this. Then, you need to explain why they (both sides) are the same using the situation we just recalled. ... No need to compute. You must explain how they are the same.

Students were then given time to first work independently and then discuss in small groups before reporting to the class. Two students shared their examples. Both examples (one about perimeter and the other a story problem) were examined to test whether the generated equations made sense. The teacher asked the class, “Can we exhaust examples like this?”

When the class admitted that they could not, they agreed to add to the list of the examples to indicate that it could continue on forever. This is a similar teaching move as the one when teaching the CP of addition (see Section 4.3).

Students’ explanations of their examples indicated that they were aware of the embedded property, even if they did not formally know the name of the DP yet. As such, the teacher prompted the class, “Can you represent this pattern? Who can try? Can we use words? Who can use words to do this first?” Several students described their observations, which were then explicitly restated as a summary by the teacher. The teacher further asked students if they could represent this statement in other ways. Students then suggested using shapes and letters to represent this property: (A + Q) x = ^x + )x or (a+b)xc=axc+bxc.

Next, the teacher prompted students to explain why (a + b) x c = a x c + b x c was a reasonable suggestion. This lifted students’ reasoning to a higher level of abstraction. Through collective discussion, the students were able to explain this formula based on the meaning of multiplication: The left side represents a + b groups of c, whereas the right side represents a groups of c plus b groups of c. It is worth mentioning that the students were also guided to reason about this formula from both directions (from left to right and vice versa). This serves as another instance where Chinese students were given explicit preparation to apply the opposite direction of the DP, a concept which was often overlooked by U.S. textbooks (Ding 8c Li, 2010) and poorly grasped by the U.S. students (Ding et ah, 2019). At the end of this worked example, the teacher revealed the name of the distributive property and requested students think about the meaning of the word of “distributive.”

One may notice that instructional activities described above were common across every formal Chinese lesson on the basic properties. First, concrete example tasks were solved in different ways that generated typical instances of the targeted property. Even though these tasks were often familiar to students (due to informal introduction in earlier grades), teachers still asked deep questions that focused on meaning-making and connections between concrete and abstract representations. Next, Chinese teachers tended to shift students’ attention to the structural features of the generated instances. To do so, they prompted students to discover, verbalize, confirm, and represent the identified patterns. During this process, representations further shifted from concrete to abstract and from specific to general, in alignment with the concreteness fading method. Meanwhile, teachers’ questions tended to facilitate student reasoning such as comparisons, example posing, generalization, and abstraction. The above forms of representation uses and teacher questioning provide insights on how TEPS can be used to teach the basic properties of operations.

 
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