China G4 Lesson: Formal Teaching
As mentioned earlier, the AP of multiplication is formally introduced in G4 (together with the CP). Appendix C indicates one G4 teacher’s hill lesson plan. My discussion below, however, focuses on the teaching of the worked example about the AP. The worked example was situated in a word problem context similar to the Ping-pong ball problem in G3. It stated,
Huafcng Elementary School hosts a jump rope contest. Each class needs to send 23 students. Each grade has 5 classes. How many students will be sent for all 6 grades?
Consistent with the formal teaching of the other basic properties, the G4 lesson on the worked example contained two stages: (1) solve the problem in two ways to generate an instance of the AP and (2) discuss the instance, pose more examples, and formally reveal the AP. As elaborated below, the first stage of solving the problem was similar to the G3 lessons that emphasized meaning-making. The second stage went one-step further to promote generalization that led to the formal revealing of the AP. Throughout this process, the teachers’ deep questioning continuously played a critical role.
Solve the problem in two ways to generate an instance of the AP. The G4 teacher asked students to solve the problem using one condensed number sentence (as opposed to two separate steps). After students worked individually, two students were called to the board and shared the following solutions:
= 115x6 =23x30
= 690 (persons) =690 (persons)
The G4 teacher asked the students to explain their thought process. One student shared that she first found out how many people were in each grade and then used that to find the total number of people. Note that she did not add parentheses for the first step, because the multiplication order she suggested was assumed by the order of operations. The other student explained that she first found out how many classes existed before finding the total number of people. After the above discussion, the teacher guided the class to compare both solutions, resulting in 23 x 5 x 6 = 23 x (5 x 6). The teacher further suggested adding a “( )” to 23 x 5 x 6 to rewrite the sentence as (23 x 5) x 6, resulting in a clear instance of the AP, (23 x 5) x 6 = 23 x (5 x 6).
Two aspects of this discussion are worthy of note. First, the students in G4 demonstrated greater fluency in solving the problem and explaining their solutions than their peers in G3. For instance, the G4 students directly listed one condensed number sentence that contained two steps and completely verbalized their thinking process by linking each step back to the problem situation. Second, due to the two-way definition of multiplication in China, students in this G4 class could have generated other solutions such as 23 x (6 x 5) and (5 x 23) x 6. It is possible that the teacher purposefully chose the two solutions to generate an instance of the AP, (23 x 5) x 6 = 23 x (5 x 6). Such a teaching move is somewhat arbitrary, similar to the case of teaching the CP of multiplication in China (see Section 5.3).
Discuss the instance, request more examples, and reveal the AP formally. After the instance of AP, (23 x 5) x 6 = 23 x (5 x 6), was generated, the teacher projected three questions on the board and asked students to discuss them in small groups:
- 1 Observe this group of number sentences, what do you find?
- 2 Can you pose one more example to verify your finding?
- 3 Try to use letters to represent the rule you found.
The above three questions shifted students’ attention away from the word problem context to a more abstract level. First, students verbalized the essential features of the two numbers sentences on both sides of the equal sign. One student stated, “By observing these two number sentences, my finding is that the factors on both sides of the equation are the same. Yet, one is to first multiply the first two numbers while the other is to first multiply the latter two numbers. The answers are the same.” Such a student articulation was close to the narrative statement of the AP.
Next, the class posed further examples to verify their findings. Students came up with various examples that mirrored the original, as well as a student mistake that was acknowledged by the teacher as a non-example. Discussion of this non-example provided further opportunities for clarifying the essential features of the AP (e.g., The numbers on both sides are the same; the order of operations is changed). As seen in other Chinese lessons, posing more examples is common during the teaching of basic properties. This is because one cannot make generalizations based on only the single instance that results from the worked example. The process of generalization is an important path to develop students’ algebraic thinking (Kaput et ah, 2008).
Finally, the students used letters to represent their finding, (a x b) x c = a x (b x c). Given that they had already learned the AP of addition, they quickly suggested naming this finding the associative property of multiplication. Using the students’ input, the teacher revealed the lesson objective and made an explicit comparison between the CP and the AP of multiplication4:
T: Addition has the associative property, and multiplication also has the associative property! Okay, in today’s lesson, we studied the properties of multiplication. One is the CP of multiplication while the other is the AP of multiplication. We can use letters to represent both properties. The commonality between both properties is that the numbers at both sides of the equal sign are the same. The difference is the CP of multiplication changes the position of the multipliers while the AP of multiplication changes the order of operations.
Overall, the above teaching sequence is aligned with the common sequence occurring in other lessons that formally teaches the basic properties in China. That is, the worked example was first situated in a concrete story situation and then solved with an eye toward meaning-making. Two solutions were then compared to generate an instance of the targeted property. Students then posed more examples, which prompted further generalization. In the end, the property was represented and revealed formally along with its application for further practices. The above sequence aligns with concreteness fading. Throughout this process, deep questions were asked to promote representational connections and comparisons within and between the examples and non-examples.