Formal Teaching of AP of Addition
Insights from Chinese Lessons
In my project, the same two G4 teachers who formally taught the CP (see Section 4.3) also taught the AP of addition as the 2nd part of their lessons. Note that the textbook lesson only contained one worked example situated in a context of children playing with two sub-questions. In other words, the example about the CP only used part of the initial story context (see Figure 4.2); yet, the example about the AP involved all three conditions and asked students to figure out how many children were jumping rope and kicking the shuttlecock (see Figure 4.8).
Both teachers approached this worked example in similar ways. In particular, the overall sequence of representation made use of concreteness fading and shifted focus from the specific to the general. There were times when the teachers folded the abstract back to the concrete to promote meaning-making. In addition, both teachers asked deep questions to prompt students’ thinking. Below are the main activities involved in the teaching of the worked example about the AP. Elaboration follows:
- • Solve the problem and explain multiple solutions
- • Compare the selected solutions to obtain an instance of AP
- • Pose or try more examples to verify the observations
- • Formally reveal the AP and represent it with letters
- • Compare the AP with CP
- 1 Solve the problem and explain multiple solutions. Both teachers asked students to solve this problem using as many methods as they could think of. In one class, a student reported all six permutations of the sum that could be found by rearranging the three numbers. For each solution, the teacher prompted the student to explain, asking, “Can you tell us how you thought of these solutions?” or “Can you explain what you found first and what you found next?” Both teachers then chose the solutions that could illustrate the AP for further discussions. Below is a typical excerpt:
T: Let’s grasp these two typical number sentences [On the board, it shows “(28 + 17) + 23” and “28 + (17 +23)”]. What does this first sentence mean? Can you explain it based on the story problem?
Figure 4.8 Worked example used to teach AP of addition in a Chinese G4 lesson. This G4 task was cited front Sun & Wang (2014, v.2, p.55).
S: First find the number of people jumping ropes and then add the number of people kicking shuttlecock.
T: Oh, first find out the number of people jumping ropes. The people jumping ropes plus the people kicking shuttlecocks equals the total. Continue, how about the second one?
S: The second is to first find the number of girls who attended this event, and then add the number of boys.
T: Okay, first find the number of girls, and then add the number of boys, which equals the total. (To the class) Is this all right?
Solving a two-step addition story problem is a piece of cake for Chinese fourth graders. However, the purpose of solving this worked example is to illustrate and make sense of the AP based on this story context. As such, both G4 teachers invited students to explain the meaning of both solutions. As seen from the above episode, the G4 students clearly explained each step of the number sentence in ways that folded the abstract representation back into the concrete situation.
2 Compare the selected solutions to obtain an instance of the AP. After students explained their solutions, both teachers invited them to compare the number sentences, resulting in an instance of the AP: (28 + 17) + 23 = 28 + (17 + 23). Furthermore, both teachers prompted students to compare the two number sentences focusing on computation process, “Just based on the number sentences and the computation process, how are they different?” and “Let’s look at these number sentences, what do you find?” Students in both classes noted that the order (position) of the three numbers remained the same, yet the order of operations (position of the parenthesis) changed. Based on students’ observations, both teachers concluded that “When adding three numbers, regardless of whether the first two numbers or the latter two numbers are added first, the answers remained the same.”
As a side note, in one class, the teacher chose three typical solutions, (17 + 23) + 28 = 28 + (17 +23) = (28 + 17) + 23. A comparison of the first two solutions enabled her class to revisit the CP that was somewhat hidden in this context. This is an interesting contrast to the to-be-reported U.S. lesson where the teacher told students that the AP was the property that dealt with parentheses. Clearly, the Chinese example here displays the inaccuracy of that statement since (17 + 23) + 28 = 28 + (17 + 23) makes use of parentheses but is an application of the CP not the AP. Student who try to identify the correct property by spotting parentheses could benefit from this example.
- 3 Try and pose more examples to verify the observations. Similar to the teaching of the CP (see Section 4.3), the above conclusions based on the instance of the AP was brought out for questioning. Both teachers asked the class whether the features observed from this group of number sentences held true. They then asked students to try more pairs of number sentences provided by the textbook. Students in one class also came up with their own examples that were either typical or special. And, several students stated in their own language that regardless of the parenthesis, one always adds the same three numbers.
- 4 Reveal the property and represent it with letters. Based on the above sense-making and verification process, the name of the AP was revealed to students. One teacher posed a further question, “This is called associative property. Why is it called ‘associative’? Think about it carefully. Here is a parenthesis.” Next, both teachers asked their students to use letters to represent the AP, (a + b) + c = a + (b + c), just as they did with CP of addition.
- 5 Compare the AP with the CP. After the AP was formally introduced, both teachers guided the class to look back on both properties introduced in this lesson. The dialogue below shows how one G4 teacher guided the class to differentiate between both properties (see Figure 4.9).
T: Let’s look at these two properties. One is the communitive property of addition. The other one is?
S: The associative property.
T: Commutative property of addition mainly refers to (switch his hands back and forth).
S: Switching the position [of numbers],
T: And, for the associative property of addition, does the position [of numbers] change?
T: But? (gesturing the parenthesis)
T: What happens to the order of operations?
S: It changes!
As indicated by the excerpt, it did not take much time to guide students
to explicitly compare the CP and AP. Yet, such a comparison is arguably
Figure 4.9 Explicit comparison between CP and AP. Redrawn by Anjie Yang.
critical as evidenced by the conflation between the CP and AP that exists even among undergraduate students (Larsen, 2010). In the above excerpt, it was made clear that the CP was to change the order of numbers while the AP was to change the order of operations. Later in the practice tasks, both classes discussed the textbook task, 75 + (48 + 25) = (75 + 25) + 48, which contained both CP and AP. Both teachers made sure that their classes understood how both properties were involved.