Formal Teaching of the CP of Addition in G4
Chinese students formally learn the CP of addition (along with the AP of addition) in G4 as part of a lesson titled Properties of Addition. In my project, two G4 teachers taught this same lesson with similar approaches. Both lessons situated the worked example in the real-world context of jump roping (see Figure 4.2). Both teachers asked a set of similar deep questions to guide students’ exploration. Together, their teaching approaches exemplify the development of students’ algebraic thinking via thoughtfully considered representations and deep questions. In this section, I introduce the teaching of CP through this example task from one of the G4 lessons.
For fourth graders, this one-step word problem might be too easy. It only took 3 minutes for the class to discuss its two solutions (28 + 17 = 45 and 17 + 28 = 45). After this, the teacher asked, “Through the computation, what relationship did we find between these two number sentences?” Students suggested 28 + 17=17 + 28, which is an instance of the CP. The teacher then asked the class to observe the number sentences and share
Figure 4.2 Worked example used to teach CP of addition in a Chinese G4 lesson. This G4 textbook image was taken from Sun & Wang (2014, v.2, p.55), Permission of citation was granted. © Jiang Su Phoenix Education Publishing.
what this equation was meant to express. Based on student responses, the teacher then revealed the pre-written statement, “Switching the positions of two addends, the sum will not change. ” Although it might seem like this revelation would imply the end of the discussion of the example task, the teacher changed the at the end of the above italic statement into a and spent another 13 minutes prompting students to explore the reasonableness of this conclusion. These discussions were important for infusing the lesson with algebraic thinking such as generalization, reasoning, and proof. Below is a relevant episode:
T: Is this true? I just added a (what)?
Ss: A question mark.
T: Why did I add the question mark? You please.
SI: I think that we only have one case. It is unclear whether the other problems will share the same pattern.
T: Give her a round of applause.
T: True statements should be generalizable, right?
Ss: Right.
T: You just have one example today, is it enough to support your conclusion?
Ss: Not yet.
T: What does this indicate?
Ss: Pose more examples.
T: More examples. Are you willing to pose?
Ss: Yes
Next, the class worked to find more typical examples. Students came up with various examples including fractions, decimals, and cases with special numbers (e.g., square numbers). The teacher then asked the class if they could exhaust all of the possible examples. Students realized that there were infinite examples and, as such, the teacher suggested adding an ellipsis “...” to the end of their example list. However, this still did not help the class reach a conclusion about whether they could officially delete the question mark. To resolve this dilemma, the teacher asked if they could find any counterexamples to disprove this conclusion.
T: Then, we can do another thing. Can we try to find counterexamples? If you switch the positions of two numbers, their sums are different. Are there such examples? When we add two numbers, if we switch their positions, their sums are different. Can you find such examples? Think about it.
SI: I think there are some. If you add a multiplier...
T: Just adding two numbers.
SI: Just adding two numbers. I think there might be no such examples. T: Not “might be.” Are there, or arc there not?
SI: There must be none.
T: Yes or no, on earth?
Ss: No.
T: Now I give everyone 20 seconds to think about this. Can you find two such numbers that if you switch their positions, the sums will be different. Nobody talk. Just think quietly.
T: Okay, you please.
S2: I think it is impossible. This is because when you add two numbers, even if you switch their positions, you still add these same two numbers. How can you get a different answer?
T: Reasonable?
Ss: Yes
T: See, he says impossible, not true.
S3: When you switch the position of the two numbers, the sum should remain the same. This is because the sum is indeed a combination of these two numbers.
T: So even if you change their positions, the sum is still a combination of these two numbers, right? So, will the sums in the end change?
Ss: No.
T: We have made further sense based on our real-life, is this right? Okay.
Now what we can do to the question mark in the rule we found?
Ss: Remove it.
Of course, the commutative property is an axiom and cannot be disproved. Nevertheless, in comparison with simply receiving this axiom, the fourth graders who participated in this level of exploration and sense-making would likely develop algebraic thinking. Based on these discussions, the class agreed to remove the question mark and accept the statement. The teacher then suggested that students find other ways to represent this verbal statement. Students came up with representations using letters (a + b = b + a) and shapes (0+A=A+0). The name of the “commutative property of addition” (ЙПЙЗСЙ#) was only formally revealed after the concept had been thoroughly discussed. After some brief practice with the CP, the teacher further linked this property to students’ prior knowledge: “In fact, this property has already been used in our previous learning. Do you know when you used it?” Students recalled the computation checks they learned in G2. Table 4.2 summarizes the key activities, typical deep questions, and representation uses from this lesson.
Let’s take a closer look at the major activities in the G4 lesson as listed in Table 4.2. In comparison with the G1 lessons (swimming pool/ tree planting) that informally introduced the CP, this G4 lesson clearly showed vertical connections in Chinese lessons across grades. First, the G4 lesson was situated in a real-world context, which generated a numerical instance for discussion (activity #1) in a way that was similar to the G1 lessons. As such, the representational sequence consistently
Table 4.2 A Summary of a G4 Lesson that Formally Taught the CP of Addition
Key Activity |
Typical Deep Questions |
Representations |
1 Solve a story problem to discover the CP |
T: Observing the two number sentences, what do you want to say? |
Concrete to abstract (specific to general) |
2 Question about the conclusion resulted from one example |
T: Why did I add the question mark? |
Abstract (general) |
3 Pose more examples |
T: How many more examples can you pose? |
Abstract (general to specific) |
T: Can we exhaust all of the examples? |
||
4 Pose counterexamples |
T: Can we try to find counterexamples? |
Abstract (general to specific) |
5 Reveal the name of CP and represent it with letters and shapes |
T: Except for using words to describe this rule, are there any simpler ways to represent it? |
Abstract (specific to general) |
6 Link CP to students’ prior knowledge |
T: Do you know when you used it (in our previous learning)? |
Abstract (general to specific) |
demonstrated the concreteness fading approach introduced in this book. Second, the G4 lessons went much deeper into discussing the CP than the G1 lessons by including a follow-up exploration of the “trueness” of students’ observations (activities #2-4). During this exploration stage, even though the representations are mainly abstract, I noticed that they first went from general to specific in finding examples and counterexamples. The specific was then linked back to general because the CP was formally revealed (activity #5). Furthermore, the CP was related back to students’ previous knowledge about checking. This then linked the general back to specific (activity #6). The teacher’s switch from general to specific and back provided ample opportunities to develop students’ algebraic thinking.
During the above exploration, the teacher posed a list of deep questions. For instance, “Can we exhaust all of the examples?” was meant to help students see that by posing examples, one cannot prove the legitimacy of a statement because examples can never be exhausted. Very often, teachers in U.S. classrooms misuse the concept of “proof’ by making requests like, “Can you prove it?” as proxies for the request, “Can you give me a numerical example?”. Such an instructional exercise is not helpful, and, in fact, might be harmful for students’ future algebraic development and proof comprehension. This mindset serves to plant students firmly in the realm of specific computational solutions, rather than gradually easing them toward the generalizations necessary for algebraic thinking. In addition, the deep question, “Can you find a counterexample?” likely exposed students to a new method of proving, namely, disproof by counterexample.
Given that it is only a fourth-grade class, this level of depth was impressive. Overall, I would argue that this teacher’s deep questions likely contributed to students’ development of the following mathematical, especially algebraic, thinking skills:
One example cannot justify the legitimacy of a general statement.
Examples cannot serve the purpose of proof because they cannot be
exhausted.
A counterexample can be used to disprove a statement.
A general statement can be represented in different ways.
Finally, I would like to point out that even though the depth of this G4 class was considerable, the conversation between the teacher and students came naturally. As seen from the dialogues above, the teacher always posed purposeful questions to orient the discussion while students offered suggestions for exploration, which, in turn, led to further discussions and discoveries. Overall, the representation uses and deep questions in this Chinese G4 lesson illustrate TEPS.