Insights from U.S. Lessons

U.S. lessons that informally taught the AP of addition included lessons on making a ten to add (Gl, N = 4) and lessons on making tens or hundreds to add (G2, N = 3). Similar to the other topics, the major insight that I gleaned from these lessons was the use of the semi-concrete manipulative, the ten-frame with counters. Figure 4.6 shows typical examples from U.S. classrooms.

When teaching students to make a ten to add (e.g., 9 + 3, 9 + 6), some U.S. teachers used one ten-frame (similar to those in Chinese lessons, see Figure 4.4) while others used two ten-frames (see Figure 4.6a). The way of using two ten-frames at first glance seemed unnecessary; however, if we recall how the Chinese teachers discussed alternative ways to make a ten (e.g., either making 9 or 6 into a ten), the use of two ten-frames has the potential to allow students to attempt either of these methods depending on which ten-frame they choose to fill. Such options might help students make sense of deep questions about which method is easier, or preferable. Interestingly, when teaching two-digit addition (e.g., 37 + 24), I also observed a G2 teacher skillfully draw a ten frame around the base-ten blocks in the ones place (see Figure 4.6b). By making 7 a 10, the first addend 37 then becomes 40. Consequently, 37 + 24 was converted into 40 + 21.

By using manipulatives, the majority of U.S. teachers in this project provided clear instructions that would help students complete their tasks. For instance, the teacher in Figure 4.6a asked students how many counters they needed to move into a ten-frame in order to make a ten, how many counters would be left outside, what new equation was demonstrated by the ten- frame, and what answer the ten-frame was giving. The teacher also recorded the new equations on the board based on the resulting manipulatives.

With dear instructions on manipulatives, students in this class could use the ten-frame to obtain the correct answer for the initial number sentence (e.g., 9 + 3 = 12). However, the majority of students struggled to come up with a new equation (e.g., 10 + 2 = 12) by themselves. Below were typical instances:

T: (to SI) So 9 + 3 is not the same as 10 + 3, right? They can’t be the same. I want you to show me 9 + 3.

T: (to S2) You’re telling me that 8 + 4 = 12 is the same as 10 + 4 = 14? You are telling me they are the same. If the sum’s not the same, they can’t be the same.

Students’ struggles likely indicate that they were unclear about the rationale behind the changes in the manipulatives. This calls for teachers’ questions that draw students’ attention to the connections between the manipulatives and the numerical equations. This is because the goal of using manipulatives is to help students process mathematical information that may not be immediately obvious. Referring to the Chinese lesson insights (sec Section 4.4, Table 4.3), the teacher could ask the following questions when students moved “1” counter into the ten-frame: Where did this “1” come from? Why did we move “1” but not “2” counters in? What happened to the number sentence of 9 + 3 when we moved the counter around? Specifically, what happened to 3 and what happened to 9? The teacher could also encourage students to recite their reasoning process based on the numerical representation (sec Table 4.3, activity 4). The above deep questions may prompt students to make effortful thinking on the to-be-learned method, making a ten to add.

There are a few other opportunities in the U.S. lessons in which Chinese lesson insights (see Table 4.3) can be incorporated to better support student learning. First, instead of being completely dependent on the ten- frame and counters throughout a lesson, the U.S. lessons may start with a real-world situation, which may be faded out through concreteness lading. There were a few occasions in U.S. lessons where a word problem was incorporated into a lesson. However, the word problem situation was not actively utilized as a sense-making tool.

Second, some students in the U.S. classrooms appeared to not know why they needed to learn the making a ten to add strategy. In fact, even after they learned this strategy, some students still did not understand why this strategy is useful. Below is an example conversation:

T: Does it make it easier to make a 10 and then count up?

S: No.

T: No, you don’t think so? Which way do you think is easier? It doesn’t matter if you don’t think it’s easier. It’s a strategy. Some strategies are easy and some strategies arc hard.

To better motivate students, a teacher may incorporate some Chinese lesson insights. For instance, a teacher may begin with review tasks (e.g., 10 + something) that enable students to feel the power of making a ten to add. Additionally, when discussing worked examples, a teacher could ask students to solve the problem (e.g., 9 + 4) using different strategies, which should be compared with making a ten to add for its efficiency.

Third, when given two addends, U.S. teachers tended to tell students to make the big number a 10 without clarifying the reasons. This can be enhanced by incorporating Chinese lesson insights on allowing students to make a 10 based on their preferences (turning 9 into a 10 or 6 into a 10) and then asking them, “Which way do you like better? Why?” (see Table 4.3, activity 6). As mentioned above, some U.S. teachers used two ten-frames when teaching a basic computation (see Figure 4.6a), which could have served as a tool to foster such a discussion. In fact, in one G2 lesson, I observed that a teacher discussed both ways of making tens to add and then neatly recorded them both on the board (see Figure 4.7). This teacher, however, did not invite students to make comparisons between these two methods. The deep questions listed in Table 4.3 could have been incorporated into such a teaching moment to make it even more fruitful.

A final observation is about the large variation within and across the U.S. lessons. There were often different sorts of worked examples (e.g., 9 + something, 8 + something, 7 + something) within a single lesson. Each was quickly discussed, and they were not necessarily arranged in any logical order. No comparisons were made across these examples to highlight the underlying idea. This form of presentation is arguably less effective than those focusing on connections among the strategies. In addition, there was a large variation across U.S. teachers’ written records about numerical transformations (see, Figures 17 and 18). This is also different from the Chinese lessons where the processes of number composition and decomposition were recorded consistently across teachers and grades (see Figures 15 and 16).

A U.S. G2 lesson that discussed two ways of making tens. Redrawn by Anjie Yang

Figure 4.7 A U.S. G2 lesson that discussed two ways of making tens. Redrawn by Anjie Yang.

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