Insights from the U.S. Lesson

As with other properties, formal instruction of the DP in the U.S. lessons generally started with its definition, followed by how the property could be applied for computation. Under this approach, the DP was directly told to students and there was no need for students to solve a problem to identify this property nor did they need to make any generalizations or abstraction because the most abstract version of the property had already been given to them. This is in direct contrast to the aforementioned Chinese approach that introduces the DP through problem solving in one lesson and then applies the DP to computation in the later lessons. Moreover, most U.S. lessons tended to treat the DP as more of a computational strategy than a basic property. In this sense, teachers focused entirely on the “distribution” aspect of the distributive property by presenting it as a tool meant to break numbers apart and then maneuver around parentheses. In fact, I noticed that some teacher language conveyed a possible misconception that the DP was equivalent to the breaking apart process (e.g., treating breaking 13 into 10 + 3 as the use of the DP). A typical statement I heard was, “This shows the distributive property, or breaking our numbers apart, in a different way.”

Despite the cross-cultural differences, several of my observed U.S. lessons from the project contributed insights into the teaching of the DP. First, the array/area model was widely used in the U.S. lessons both for formal and informal teaching. Very often, the array/area model was also broken apart in different ways, which provided opportunities for students to reason about various computational strategics. Second, some teachers asked students to add a story context to the array model in a way that enriched the representation use and potentially enabled sense-making. Along with these rich representations, U.S. students in some classrooms came up with insightful suggestions that brought up opportunities for introducing the DP. Below I elaborate on these insights.

The Array/Area Model: Breaking Apart in Different Ways

The array/area model was used widely and elegantly across the U.S. but not Chinese lessons in my project. Figure 3.3 in Section 3.4 shows an example of such an array model drawn on graphing paper on the smart board. Similarly, I observed a G3 teacher project a 4-by-6 array on the board and ask students how they could determine the product. The teacher recorded six different student strategies to break down the array and find the answer, including (a) count by 6s, (b) count by 4s, (c) count by 3s to 18 and then count by 2s to 6, (d) break the array into two sets of 2-by-6 (2x6=12,12 + 12 = 24), (e) break the array further into four sets of “2 x 3”, and (f) break the array into 3 groups of 8. The above strategies demonstrate the power of the array model in aiding class discussion and assisting students’ reasoning. The array model could be widely adapted by Chinese teachers during the teaching of multiplication facts. Earlier, I reported that a Chinese teacher who taught the multiplication Koujue for 7 revised the equal groups model (suggested by the textbook) to an array model. That was a great example of how array models could be integrated into Chinese lessons to maximize students’ learning.

I have also observed that U.S. lessons have some room to enhance the use of array models. Aligned with the textbook presentation, the above lesson only involved arrays while its follow-up lesson contained numbers. Although both lessons implicitly involved the DP, there lacked explicit connections between these two. For instance, in the follow-up lesson, the class came up with the following strategies for 8x6, (a) start with 4 x 12, and then solve it as 4 x 10 and 4 x 2, (b) start with 8 x 3 = 24, and then 24 + 24 = 48, and (c) start with 10x6 and then subtract 12. These strategies demonstrated students’ tremendous potential to learn the DP. To reach this potential, the teacher could have made explicit connections between the array model and numerical representations rather than keeping them separate. For instance, the teacher could have transformed the array-based strategies into number sentences in the first lesson and, conversely, she could have also linked the number strategies back to the array model in the second lesson. Moreover, I suggest asking a few deep questions about students’ strategies to reinforce these connections. For instance, when solving 8x6, students proposed 10x6 and then subtracting 12 as a solution. The teacher could have followed up by asking why this strategy worked. As seen from Chinese lessons, students could be guided to understand such a strategy based on the equal groups meaning (e.g., 8 rows of 6 is the same as taking 2 rows of 6 away from 10 rows of 6). In addition, after multiple strategies were proposed (in either the array or the numerical lesson), the teacher could have asked comparison questions to help students understand that these strategies share the common feature of breaking apart a factor to multiply. Addressing the undergirding patterns may contribute, either implicitly or explicitly, to students’ understanding of the DP.

In addition to the teaching of multiplication facts, the array model was also used to teach multiplication involving two- or three-digit numbers (e.g., 3 x 17, 5 x 143). Figure 3.3 indicates such an example. In another G4 classroom, the teacher explicitly asked students to use the DP to solve 3x17. Based on a 3-by-17 array, the class discussed different ways to break apart 17 (e.g., 10 + 7, 8 + 9, 11 + 6). The teacher then illustrated how to break 17 into 10 + 7 to multiply (similar to Figure 3.3). Next, she raised a great question, “Today I want us to sec if we can figure out, does this work all the time? And, what is the easiest way to break apart a number?” This is a deep question that could potentially elicit students’ understanding of the array model in two ways: (a) the various ways to break apart a factor to multiply always works due to the undergirding property of the DP and (b) among these various strategies, breaking apart a factor based on its place value (e.g., hundreds, tens, ones) is the most efficient and effective choice. Such an array model could illustrate the expanded notation, 3 x 17 = 3 x (10 + 7) = 3x 10 + 3x7, which undergirds the standard algorithms for vertical multiplication.

However, in the above lesson, the teacher did not press her students to explore her question using the example task 3x17. Instead, she moved on to a practice task with a similar problem, 6x13. Students in small groups came up with different strategies such as breaking 13 into 6 + 7 and 8 + 5. However, there was no discussion comparing these strategies for their commonalities. In this sense, her proposed question still did not get explored. The teacher could have done so by asking, “Why do these strategies all work?” and “What do these strategies have in common?” Such prompts may enable students to discover that the DP was the undergirding principle behind all of these strategies. Since U.S. lessons often directly introduce the DP to students, it is critical to link the varied computational strategies back to the underlying property so the big idea can be seen as truly meaningful.

Additionally, none of the small groups in this class chose to break 13 into 10 + 3 when they multiplied. Students could have been introduced to this effective strategy during the initial worked example of 3x17. The teacher could have asked students to try out varied strategies for breaking up 13 and then followed-up with the question, “Which strategy seemed to be most effective?” This may lead to students’ discovery that breaking a factor based on its place value to multiply will be most effective. In fact, elsewhere, the G4 lessons contained tasks that used base-ten blocks to represent one factor (e.g., 17 is represented with 1 ten and 7 ones), which were then displayed in an array (e.g., 3 rows of 17, broken into 3 rows of tens and 3 rows of 7s.). This innovative strategy of integrating arrays and base ten manipulatives could also have been used earlier when teaching the worked example.

In summary, the arca/array model plays a critical role for teaching multiplication facts and multi-digit multiplication in U.S. classrooms. This is a great model that enables students to visualize the partial products and the underlying DP. Given that Chinese lessons of this sort mainly contained discussions of story contexts and numerical solutions, the array model may be incorporated as an intermediate step to concreteness fading, which could benefit more students.

Adding Story Contexts for the Array Model

Another insight from the U.S. lessons is to enrich the array model with a story context. In one G4 classroom, the teacher started the worked example with a pair of computation tasks: 16 x 3 and 16 x 6. One student explained his strategy which the teacher then recorded as (16 x 3) + (16x3) = 16x6. The teacher then asked the class to explain why they agreed this strategy was true. After students’ elaboration, the teacher further prompted, “So my next question, you guys, is for some people to volunteer to share either a story context or a representation that would show why this is true.” One student used his existing array cards (8-by-4 arrays) to illustrate this observation that when one puts two 8x4 arrays together one would obtain a 16 x 4 array.

After this pair of arrays was discussed, the teacher further prompted, “What if this were a story, guys? Who can tell a story about this, that will help people understand?” To decrease students’ cognitive load, the teacher suggested sticking with 8x4 and 16 x 4 to pose a story as opposed to going back to the initial task. This request elicited a story problem posed by a student, “Oliver went to buy candy. He bought 8 bags with 4 candies in each. How many does Oliver have? How many would there be if there was .. .if there were 16 bags.” The teacher then made it clear that the result was, “if he has twice as many bags, he gets twice as much candy.”

The above lesson episode contained a shift of discussion from (16 x 3) + (16 x 3) = 16 x 6 to (8 x 4) + (8 x 4) = 16 x 4. However, both tasks informally involved the DP. The act of adjusting her teaching to incorporate student thinking is a strong demonstration of the teacher’s flexibility. During this process, the teacher posed a set of deep questions that asked students to explain why the break-apart strategy worked using numbers, arrays, and story contexts. Note that asking students to create a story problem for the arrays not only enriched the representation use of the lesson but also offered an opportunity for students’ problem posing. Perhaps, given that this was already a G4 lesson in which the students had demonstrated great thinking, the teacher could have explicitly pointed out that this strategy was undergirded by the DP. In addition, the teacher could have considered using concreteness fading as an alternative representational sequence that starts with a concrete story problem, then models it using arrays, and then concludes with numerical solutions.

Summary: Teaching the DP through TEPS

An integration of the instructional insights from the Chinese and U.S. lessons offers suggestions on how to teach the DP through TEPS. On one hand, Chinese worked examples are basically situated in concrete word problems that provide rich opportunities for class discussion. This can be incorporated into U.S. lessons where the DP is often used for abstract computation without contextual support. On the other hand, U.S. lessons widely use the array model, which could fit neatly into the concreteness fading structure of Chinese lessons. In the observed Chinese lessons, we have previously noted a few instances where the area/array models have been used, though such usage is relatively uncommon. Regardless of real-world contexts or schematic diagrams (e.g., arrays, number lines), it is important to shift students’ understanding from concrete to abstract and from specific to general. In terms of questioning, Chinese teachers constantly asked students to explain their strategies and solution steps by referring back to the equal groups meaning of multiplication. After multiple solutions were presented, they asked questions that initiated comparisons between different solutions and models. Such questions can be incorporated into the U.S. lessons where multiple solutions are already a great feature; yet the follow-up comparisons of these solutions are often missing. By integrating the above insights about representations and deep questions, teachers in both countries could potentially have more effective tools to teach basic properties like the DP.

Notes

• 1 In the Chinese instrument, we changed “John” and “Mary” in the item to typical Chinese names, “Xiaoming” and “Xiaofang.”
• 2 As mentioned earlier, we changed “John” and “Mary” to typical Chinese names, “Xiaoming” and “Xiaofang” in the Chinese instrument.
• 3 ¥2 represents 2Yuan.
• 4 This comparison step is not visible in her lesson plan in Appendix C.
• 5 The Koujue “Four six twenty-four” is taught based on the real-world context that 4 groups of 6 petals equals 24 petals. This Koujue indicates both 4 x 6=24 and 6x4 =24.
• 6 As mentioned earlier, multiplication in current Chinese textbook is defined two ways: “e groups of b" is represented as either a x bos b x a. So, 48 x 3 and 48 x 4 in the two solutions can represent 3 groups of 48 and 4 groups of 48, respectively.
• 7 The standard algorithm for three-digit numbers multiplied by two-digit numbers was introduced in an earlier unit before the formal introduction of the DP in Chinese 4^,h grade. Of course, this algorithm implicitly applies the DP. In the current example of 32 x 102, students were asked to use an efficient way to compute (1ВЙЧЗ‘7ГЙ1+3¥), which demands an explicit application of the DP.

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