Modeling the Properties
To illustrate the basic properties, the Common Core calls for one to “illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.” (NGAC & CCSSO, 2010, p.29). Given that the array model is more straightforward in illustrating the CP and DP than the АР, I will discuss its use with the former two properties first, followed by the AP. In addition, given that the array model is built on the equal groups meaning of multiplication (see Section 3.2), I will illustrate how the model of each property can be linked to the equal groups meaning.
The CP of Multiplication. An array model provides a great opportunity to model the CP. This is analogous to the Common Core suggested apple task from Section 3.2: There are 3 rows of apples with 6 apples in each row, how many apples are there? (see Table 3.1). Recall, we can view this apple array as 3 groups of 6 (see Table 3.2). Based on the U.S. convention of multiplication (a groups of b being written as a x b), we would represent it as 3 x 6 = 18. When rotating this array 90 degrees, we will obtain 6 rows of apples with 3 in each row. Again, based on the meaning of multiplication, we would represent it as 6 x 3 = 18. Since both solutions find the same total of apples, these two equations are equivalent and thus we generate 3 x 6 = 6 x 3 as an illustration of the CP. Note that even though 3 x 6 and 6x3 have the same numerical value, the corresponding situations they modeled were indeed different.
The DP of Multiplication. We can illustrate the DP using an array or area model that has been broken into two parts. Imagine the apple array from Table 3.2 is broken into two parts: 3 by 2, and 3 by 4. We can figure out the total number of apples in two ways. One method involves first figuring out the total number of apples in each row (2 + 4) and then the total number of apples (3 rows of “2 + 4”). Consequently, we generate an equation that solves the problem, 3 x (2 + 4) = 18. The second method involves first figuring out the number of apples in each part (3 rows of 2, 3 rows of 4, respectively) and then the total number of apples, resulting in the solution of 3x2 + 3x4=18. The above two solutions can be compared to illustrate an instance of the DP, 3x(2 + 4) = 3x2 + 3x4.
Figure 3.3 from Section 3.4 shows an example of the area model. The length of an area model was broken into 10 and 5. Thus, the total area was solved in two ways: (a) Finding the total length (10 + 5) and then multiplying it by width (15x6) and (b) finding each subarea (10 x 6 and 5x6) and then adding to find the total area. When these two solutions are compared in an equation, (10 + 5)x6=10x6 + 5x6, it will illustrate the DP.
An additional note is, except for the area/array model, an equal group model can also effectively illustrate the DP. The perimeter item in Section
5.1 is one example. In that item, one may first find the 2 lengths and 2 widths, respectively, and then the total perimeter, resulting in a solution, (2 x 118 + 2 x 82). Alternatively, one may first find the pair of length and width and then the total perimeter that contains 2 pairs, resulting in the second solution, 2x(118 + 82). A comparison of these two solutions generates an instance of the DP, 2 x 118 + 2x 82 = 2x(118 + 82).
The AP of Multiplication. Of the three properties that involve multiplication, the AP is the hardest to illustrate. The CP and DP only involve multiplication at one level while the AP involves two hierarchical levels. Given that, the volume model can be used to illustrate the AP. Flowever, the meaning of multiplication is often misstated in situations where it is only being applied once (see Section 3.2). Since the volume models apply this easily misconstrued idea twice, they may not always be accurate representations of the AP. For instance, in a prior study (Ding, Li, & Capraro, 2013), I noted both correct and incorrect examples in existing books. Beckmann (2008) accurately illustrates how a volume model (4 by 2 by 3) may be broken up in two ways to illustrate the AP: four groups of (2 x 3) blocks and (4x2) groups of three blocks. However, an NRC (2001) example inaccurately suggests that four groups of (3 x 5) can be represented as (3 x 5) x 4 rather than 4 x (3 x 5). In tact, since volume models are expected for students in G5 (NGAC & CCSSO, 2010), I will not elaborate on how to use this model to illustrate the AP. Instead, I will discuss an alternative representation that uses the equal groups model, which may support elementary teachers in introducing the AP meaningfully prior to G5.
Figure 5.2 illustrates a task involving the equal groups model. In this task, there are three tables of two plates each containing five mangos. This task was identified from a U.S. textbook (Ding, 2016) and was used in our project test (see Section 5.1). There are two different ways to find the total number of mangos. In one method, we could first find the total number of plates (3x2 = 6) and then the total number of mangos (6x5 = 30), resulting in the first solution, (3 x 2) x 5 = 30. Alternatively, we could first find the number of mangos on each table (2x5 = 10) and then the total number of mangos (3 x 10 = 30), resulting in the second solution, 3 x (2x5) = 30. Note that both methods follow the U.S. convention of the basic meaning of multiplication, a groups of b is represented as a x b. A comparison of these two solutions would generate an instance of the AP, (3 x 2) x 5 = 3 x (2 x 5).
Figure 5.2 Illustration of AP using the Mango problem. Created by Meixia Ding.
In summary, regardless of which property is used, each can be illustrated in a word problem context. Students can be guided to solve the problem in two ways, each of which illustrates one side of the property. The key challenge for teachers when representing the properties is to be comfortable with the reasoning behind the meaning of multiplication and to apply it consistently across steps. If this is done, then a comparison of the two solutions will then generate an instance of the targeted property.
Cultural Differences in “How” and “When”
Even though the illustration of each of the properties is rooted in the meaning of multiplication, there is a cross-cultural difference in how multiplication is defined in the U.S. and in China. Thus, it is important to highlight this difference so readers can make the best use of insights gleaned from cross-cultural lessons. Recall (see Section 3.2), in the U.S. “a groups of b” is represented as ax b, whereas educational reform efforts in China loosely redefined “a groups of b” as either a x b or b x a. My personal stance is that this loosened definition diminishes students’ reasoning opportunities. In the following sections, it is important to note that “how” the basic properties are defined in textbooks is partially affected by how loosely the meaning of multiplication is applied in China (two-ways) versus in the U.S. (one-way).
Another cross-cultural difference is “when” each country formally introduces the basic properties of operations. In China, all properties are formally presented in G4 in the same unit titled “Properties of Operations.” The CP and AP of multiplication are arranged in one lesson with each property having one worked example. The DP is presented in two lessons one for introduction and the other for application of the DP. Prior to G4, Chinese textbooks also contain lessons or tasks that implicitly infuse these properties, but do not name them. For instance, the standard algorithms for multiplication (e.g., two-digit x one-digit; two-digit x two-digit; three- digit x two-digit) are all introduced before the formal introduction of the basic properties, including the DP. Thus, the lessons that introduce the standard algorithms only informally teach the properties. Nevertheless, vertical connections are clearly seen in the lessons across grades. In the U.S., there are different textbook series used across classrooms, but they generally introduce multiplication properties in G3, and then revisit them in G4. The array model plays a critical role across both grades, though the connections between informal and formal teaching of the basic properties are sometimes unclear. Interestingly, when teaching the standard algorithms for multi-digit multiplication, the U.S. textbooks introduce the partial product algorithms which involve extended use of the DP and the array model (elaborated upon later). Such instances do not appear in Chinese textbooks that formally introduce standard, but not alternative, algorithms.
Despite the cultural differences, lessons from both countries shed light on the teaching of these properties. Overall, the Chinese lessons consistently stress sense-making mainly through the equal groups model (even though there is a loose definition of multiplication). The U.S. lessons widely use the array model in a manner that might provide students with reasoning opportunities and various ways to learn multiplication. In this sense, an integration of cross-cultural lesson insights seems to be critical for supporting students to their learning potential.