Teaching the Commutative Property of Multiplication
Scenario
Imagine you are a third- or fourth-grade teacher and you are aware that students in elementary school should learn the big idea of the CP of multiplication, a x b = b x a. How may this big idea be introduced to students first informally and then formally? To what extent may you prompt your students to understand this big idea at the elementary school level? To proceed, you may consider the following questions:
- 1 What examples will you use to teach the CP of multiplication ?
- 2 What representations will you use during teaching?
- 3 What deep questions will you ask during teaching?
To teach the CP of multiplication, readers may have thought of using an example such as 6 x 4 = 4 x 6 to discuss this property. One may use a representation such as an array or an area model to show it makes sense. One may even ask deep questions such as why the initial array shows 6x4 and why the turn-around array shows 4x6. This type of question demands an understanding of the basic meaning of multiplication (6 groups of 4 vs. 4 groups of 6). One may also ask students why turning around the arrays does not change the total amount. Alternatively, readers may have thought of using the equal groups model to teach the CP. First, one may represent six groups of four objects (no need to line them up) and then regroup them into four groups of six objects.
The above representations both make sense, though in my mind the array model illustrates the CP more elegantly. In addition, questions that focus on the meaning of multiplication are the key to facilitating sense-making. In my project, the cross-cultural videos suggest further insights during both informal and formal teachings of this property. Below, I will first introduce the Chinese lessons, followed by the U.S. lessons. Limitations observed from each side are informative for reconsidering better ways to teach this property to develop students’ algebraic thinking.
Insights from Chinese Lessons
Although Chinese lessons in general stress sense-making, the teaching of the CP does not follow the same lesson structure. This is mainly due to the aforementioned two-way definition of multiplication. Despite this limitation, the formal lessons consistently contained deep questions about the features of the CP. In addition, Chinese lessons prior to the formal introduction of the CP consistently utilized this property in the context of checking computations, similar to those on the CP of addition [or CP(+)]. Below, I introduce both the informal and formal teaching of the CP of multiplication [or CP(x)].
Informal Teaching of the CP (×)
Chinese students start learning multiplication in second grade. Due to the broadened two-way definition (see Sections 3.2 and 5.2), the CP is informally introduced to students when they are exposed to multiplication for the first time. The G2 textbook used in our project first introduces multiplication in a pictorial context: There are four desks with two computers on each desk. Students are asked to figure out how many computers are in all. Figure 5.3 illustrates the translated textbook page. The textbook suggests adding 4 groups of 2 to achieve a total of8:2 + 2 + 2 + 2 = 8. The textbook then introduces that adding 4 groups of 2 can also be represented using multiplication, written as either 4x2 = 8 or 2x4 = 8.
Even though this lesson does not mention the CP, the property in which switching the two multipliers does not change the answer is informally presented to Chinese students through the introduction of multiplication itself. Because multiplication is explained with a two-way definition, it is given that one can switch two numbers around to multiply. The fact that this switch is purely definitional means there is no reason to explore “why” such a switch is possible. As such, I argue that this broadened definition of multiplication likely diminishes students’ opportunity for reasoning.
Figure 5.3 The first lesson on multiplication in the Chinese G2 textbook (translated). This G2 task was cited from Sun & Wang (2014, v.l, p.21).
The lessons on multiplication that we videotaped for the project included pairs of multiplication facts in the same arbitrary way as described above. This contrasts with how some of the U.S. lessons approached the CP(x) though sense-making (to be elaborated upon later).
Formal Teaching of the CP (×)
As mentioned earlier, Chinese students formally learned the CP of multiplication (along with the AP of multiplication) in G4. These lessons were consistent with the general Chinese approach in which a single worked example of the CP was presented in a pictorial context: three groups of students are kicking a shuttlecock, with each group containing five students. To find out the total students (3 groups of 5), the class came up with two solutions: 3x5 and 5x3. According to the new multiplication definition in China, both were correct. As previously mentioned, since the two solutions were based on the given definition of multiplication, there was no need to reason about why 3 x 5 = 5 x 3 in the concrete context. In other words, the concrete representation was not used for sense-making in this case.
Despite the above concern, the Chinese lessons contained many deep questions that likely promoted students’ algebraic thinking. Table 5.1 presents typical questions from one G4 lesson. These questions were similar to the ones that occurred in the CP(+) lesson (see Table 4.2), even though this lesson was taught by a different G4 teacher.
In Table 5.1, after the students solved the shuttlecock kicking story problem using 3 x 5 = 15 and 5x3 = 15, the teacher asked the class to compare both number sentences (activity #1). The students described their observations, noting both number sentences contained the same numbers, and the locations of the numbers were changed yet the products were still the same. The teacher then summarized students’ observations
Table 5.1 Deep Questions in a Chinese G4 Lesson that Formally Introduced the CP of Multiplication
Key Activity |
Typical Deep Questions |
Representation (level of abstraction) |
1 Compare two solutions to obtain an instance of CP(x) |
T: How are these two number sentences (3x5 = 15, 5x3= 15) the same and different? T: Observe this group of number sentences (3 x 5 = 5 x 3), what do you think? |
Specific |
2 Link to CP(+) and reveal CP(x) |
T: We have learned a similar property before. Which one might you think of? T: What does this number sentence (3 x 5 = 5 x 3) tell us? T: CP of multiplication! Let’s make a guess what the CP of multiplication is about? |
General |
3 Confirm CP(x) through posing more examples (and non-examples) |
T: Is this true? This is only our...? (S: guess!) So, what should we do to verify it? (One student posed various types of examples) T: Can any student pose a counter example? Switching the two numbers, the product is changed! Is there such an example? |
Specific |
4 Represent CP(x) using letters |
T: When we learned the CP of addition, we used letters a and b to represent the two addends. Now, let’s still use letters a and b to represent the two multipliers. Can you use letters to represent the CP of multiplication? |
General |
5 Link CP(x) to prior use of it |
T: Think about it, did we use the CP of multiplication before? Where did we use it ? |
Specific |
and asked which sign could be used to link these two number sentences. This resulted in an instance of the CP, 3 x 5 = 5 x 3. The teacher then went further, asking students to observe this group of number sentences to elaborate on what they found. One student stated that if one switches the position of two multipliers, the product will be the same.
Next, the teacher asked the students whether there was a similar property' that they had learned before (activity #2). Students thought of the CP of addition. They also suggested that 3 x 5 = 5 x 3 must therefore be an example of the CP of multiplication. Based on students’ reasoning, the teacher encouraged students to make a further guess, in their own words, about what the CP of multiplication might be:
SI: Two multipliers are not changed yet the locations of them are switched. The results are the same.
S2: In a multiplication sentence, if the two multipliers remain the same but their location is changed, the products will be the same.
Based on the students’ verbalizations, the teacher revealed the statement about the CP of multiplication: “When multiplying two numbers, the two multipliers can be switched, and the product remains the same.”
Interestingly, as with the CP of addition lesson (see Section 4.3), this G4 teacher also raised a question about the truthfulness of this conclusion (activity' #3). She then added a to the statement about the CP of multiplication and asked the students what they could do to verify this assumption. The students suggested posing more examples (e.g., 8 x 5 = 5 x 8, 8.75 x 1000 = 1000 x 8.75), which were acknowledged by the teacher. Then the teacher also encouraged them to find counter examples. Of course, no counterexamples existed. Based on the above discussion, the class agreed to remove the and reaffirmed their conclusion.
It was at this time that the teacher officially asked the class to name this property. The students affirmed their guess from earlier in the lesson that this must be the CP of multiplication. This teacher then asked the students to use letters to represent the CP of multiplication (activity #4). To conclude, she asked students if they had used this property before. Students quickly recalled that they used this property' to check multiplication in earlier grades (activity #5). Taken together, this set of deep questions seemed to shift students’ attention back and forth between specific and general, which arguably exercised their generalization, reasoning, and proof skills. Overall, I noticed that the teaching approach in this lesson was indicative of other Chinese lessons on the basic properties.