Insights from U.S. Lessons
Formal Teaching of the CP of Addition in G1-G3
In our project, the G1 lesson titled “Add in any order” was located in the first unit of the textbook (Lesson 1.6) and was fully devoted to introducing the CP. Similarly, at the beginning of the G2 textbook, there was a lesson named “Addition properties” in which both the CP and the AP were introduced. In the G3 textbook, there was once again a lesson titled, “Addition properties” that included the CP. Several instructional insights from the five videotaped lessons are elaborated on below.
Semi-Concrete Representations. One unique feature of the U.S. lessons teaching the CP was the use of semi-concrete representations such as cubes, counters, and dominos. Most of these lessons suggested that students build a cube train with two different colors to represent an addition fact, then literally turn the train around to generate the related addition fact conjectured by the CP. In the following classroom episode, the G1 teacher first demonstrated how to turn the cube train around and then suggested her students do the same at their desks:
T: Watch me. He has 2 + 3, right? Watch. Turn that so it’s 3 + 2. I’ll do it one more time and I’ll do it slow. I’m getting the 2 + 3 and turning it around to make 3 + 2. Turn it around, turn it around. 3 + 2. (While walking around and commenting on a student’s cube trains) I love it. I love it. 3 + 2, 3 + 2... Good job.
Hands-on opportunities that use semi-concrete representations may allow students to better learn the targeted concepts in ways that could be incorporated with the aforementioned Chinese lesson insights. In the Chinese lessons, the representational sequence flowed directly from concrete story situations (e.g., swimming pool, children planting trees) to number sentences. If the semi-concrete representations like cubes or counters can be used to model the story situations, some students, especially those who struggle with abstract mathematical concepts, may benefit more from the concreteness fading process. On the other hand, many U.S. lessons directly started with or were limited to semi-concrete representations. These lessons could have been enriched if the real-world contexts, like those in Chinese lessons, could be utilized as well. Overall, an integration of the U.S. and Chinese lesson insights suggests a representational sequence that is aligned with the concreteness fading approach: real-world contexts, semi-concrete representations, and then number sentences.
Explicit Comparison and Follow-Up Questions. Even though comparison questions did not occur frequently in the U.S. lessons on the CP, I observed two occasions that contained insights of this sort. In one lesson, the teacher wrapped up her lesson by asking students to explicitly compare two number sentences:
T: So if I asked you a question... How is 3 + 2 = 5 the same as 2 + 3 = 5.
How are they the same?
S: They both equal 5.
T: Oh, they both equal 5. Right. Anything else?
S: They have the same numbers.
T: They do have the same numbers...
T: How is 3 + 2 = 5 different than 2 + 3 = 5? How are they different?
S: Because they are turn around.
T: They’re turnaround. They’re just flipped.
In the above episode, the teacher asked students to compare two number sentences (3 + 2 = 5 and 2 + 3 = 5) and how they are the same and different. Such an explicit comparison was quite rare in the U.S. lessons even though it was widely seen in the Chinese videos. The literature shows that explicit comparison is critical for the development of children’s mathematical comprehension (Star & Rittle-Johnson, 2009). When two things are compared, students’ attention can be oriented toward the similarities and differences, which facilitates student encoding of the structural elements of the targeted concept. As seen in the above dialogue, some first graders were able to communicate, “They are turned around.” Perhaps, to promote an even clearer understanding, the teacher could have followed up with a question similar to the Chinese lessons, “which and which are turned around?” Such a prompt could serve to orient students’ attention to the essential feature of CP, that is, the two addends were switched, but the sums remained the same.
Another G1 lesson also contained an insightful instance of questioning. After the CP was introduced (you can add in any order), the teacher asked a deep follow-up question about why they could add in any order:
T: Why can you add in any order? Docs anybody know the answer yet?
Why can you add in any order, SI?
SI: (not sure)
T: Why can you turn the train around? Why can you turn the train around?
SI: Because, they’re both the same.
T: If you turn the train around, does the total answer change? Does your sum change if you turn the train around? No!
T: So why can you add in any order? Think, I’ll come back to you. S2. S2: If you switch it around, it’s the different color and different code, but the number still stays the same.
T: So it’s a different color and a different code but the number still stays the same. So, the addends switch around but the number still stays the same. So, let’s try to watch one more model.
In the above episode, the teacher prompted students by asking a followup deep question which came straight from her textbook, “Why can you add in any order?” The teacher suggested that the students turn the cube trains around to derive their responses. Students intuitively noticed that after the cube train was switched around, “it was the different color and different code” but the number of cubes stays the same. This example shows that deep questions can be structured around semi-concrete physical tools to facilitate students’ learning of concepts. This is different than an observation from a U.S. G3 classroom where a teacher asked, “What do you notice about those addition sentences?” and accepted the student response, “The answers are the same,” without any follow-up questions. Perhaps, to promote student understanding on a structural level, this teacher could have further oriented students’ attention to the part-whole structure embedded in the cube train.