U.S. G3 and G4 lessons: Formal Teaching

In the U.S., the AP (x) was formally introduced in both G3 and G4. In my project, the lessons included word problems that were similar to the Chinese Ping-pong ball and jump roping tasks described above. Teaching the basic properties through word problems is not common in the U.S. As such, these lessons provided great learning opportunities for students. As analyzed earlier, word problems with the structure of “groups of groups of objects” can be solved in two ways that can then be compared to generate an instance of the AP. In the U.S. lessons observed, the word problems however were approached in different ways with different purposes. Below I will discuss relevant lessons with an eye on how these learning opportunities may be maximized.

In the G3 classroom, the teacher followed the textbook sequence to first introduce the definition of the AP. She emphasized that the answers stayed the same “no matter where we are putting the parentheses” and “no matter how we group the numbers.” After this, she presented the sticker word problem, "'Chris and Katie each received 4smile stickers a week for 3 weeks. Flow many smile stickers did they earn altogether?” The teacher first sketched the problem situation with an array model (see Figure 5.8a).

A word problem used to teach AP of multiplication in a U.S. G3 classroom. Redrawn by Anjie Yang

Figure 5.8 A word problem used to teach AP of multiplication in a U.S. G3 classroom. Redrawn by Anjie Yang.

This representation clearly illustrates the problem structure. Indeed, I think it was insightful to stack the stickers in three rows to indicate three weeks. Depending on whether one treats the number of weeks or people as the biggest group, the problem structure may be viewed either as “3 groups of 2 groups of 4" (3 weeks with each having 2 people and each of whom receiving 4 stickers) or “2 groups of 3 groups of 4” (2 people with each having 3 weeks and in each week receiving 4 stickers).

After drawing, the teacher offered a number sentence 2x4x3 and suggested the class group the factors in different ways to find the answer (see Figure 5.8b). Below is the guidance:

T: So, there’s two people each receiving four stickers and it’s happening three times. So, we are multiplying 2 x 4 x 3 to see how many stickers they got altogether.

T: So, we have 2x4x3. So, we need to group these factors somehow. Raise your hand to tell me the way you want to group them first. Where should I put the parentheses? Flow should we solve it first?

SI: (Student suggests adding the parenthesis around 2 and 4, resulting in the left-most solution from Figure 5.8b)

T: Now if we group these factors a different way, we should get the same answer. So where should we put the parentheses now? Around what two numbers? What two factors?

S2: (Student suggests adding the parenthesis around 4 and 3, resulting in the right-most solution from Figure 5.8b)

T: So no matter how we grouped the factors we got the same answer. It didn’t matter where we put the parentheses. It didn’t matter how we grouped them we came to the same product.

In the above episode, two aspects are worthy of discussion. First, the numerical solution 2x4x3 was suggested by the teacher with little discussion on the meaning of each step. Indeed, some of the steps, as described in Figure 5.8b, cannot be referred to the word problem context based on the meaning of multiplication (ax b refers to “я groups of b"). For instance, in the solution of (2 x 4) x 3, the second step “8 x 3” cannot be explained because there were no 8 groups of 3 but 3 groups of 8 (consider each row of squares). In the solution of 2 x (4 x 3), the first step “4 x 3” also did not reflect the array model’s grouping. The lack of connections made between numerical and visual representations could be addressed by stressing the meaning of each step, as done in the Chinese lessons.

Second, the AP was directly told to the students at the beginning of the lesson and the word problem was then used as a pretext for computation in which the AP was applied. The above process neglects the opportunity' for students to make sense of the AP. To address this limitation, a teacher could reverse this representational sequence starting with the word problem, guiding students to solve this problem with two solutions (see Figure 5.8b), and asking students to compare the solutions to generate an instance of the AP. Essential features of this instance could then be articulated and further examples could be posed to enable generalizations, leading to a formal revealing of the AP.

In a similar vein, the G4 classroom began with a video provided by the textbook publisher, which started with the definition of the AP. Next, the video suggested two different story contexts to illustrate (4x2) x 3 = 4 x (2 x 3). For 4 x (2 x 3), it correctly suggested “4 groups of tambourines. Each group has 2 rows of 3 tambourines.” For (4 x 2) x 3, it suggested “3 groups of tambourines with 4 rows of 2 tambourines in each group,” which is problematic. This is because the story context indicates 3 groups of (4x2) tambourines which corresponds 3 x (4 x 2) rather than (4 x 2) x 3.

Moreover, as mentioned in the section about the CP(x), the U.S. G4 textbook contained a word problem with the equal groups structure to teach the CP: “There are 4 sections of seats in the playhouse theater. Each section has 7 groups of seats. Each group has 25 seats. How many seats are there in the theater?” In my view, this is a far better context for illustrating the AP than the CP. Perhaps to show the power of the AP in computation, one might want to modify this problem with the structure of 7 sections of 4 groups of 25 seats, which can be solved to generate an instance of the AP, (7 x 4) x 25 = 7 x (4 x 25).

In summary, I noticed that the U.S. G3 and G4 lessons contained promising word problem contexts that had the potential to meaningfully illustrate the AP. I also noticed powerful representations and attempts to explain number sentences with concrete contexts. However, in order to maximize the potential of these concrete contexts, teachers may incorporate insights from the Chinese lessons by resequencing the representations (from concrete to abstract, and from specific to general) and asking questions about the meaning of each solution step.

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