Formal Teaching of CP(×)
During my project, I observed two G4 lessons that focused on formally introducing the CP (and the AP) as strategies for mental math with multiplication. Two insights were observed.
The Emphasis on Meaning of Multiplication. One of the G4 teachers started her lesson with a review of the CP using a video clip that had been provided by the textbook publisher. In this video clip, a man (M) and a woman (W) discussed the CP, which was simultaneously illustrated in a real-world context and with array images:
M: The commutative property states that you ean multiply two faetors in any order and get the same product.
M: 3 rows of 4 violins is 3 times 4, or 12 violins (image of an array).
M: Now look at 4 rows of 3 violins (image of an array).
W: How many violins are there in 4 rows of 3?
(Students wrote down their answer of 12).
M: 4 times 3 equals 12, so 3 times 4 equals 12 and 4 times 3 also equals 12.
The above episode is similar to the earlier context about arranging chairs. The tact that 3 x 4 = 12 and 4x3 = 12 was illustrated with the violins arranged in arrays of 3 rows of 4 or 4 rows of 3, respectively, served to distinguish the conceptual difference between the two number sentences. As argued previously, this is more meaningful than the Chinese lessons that introduce the CP based on the two-way definition of multiplication (e.g., 3 groups of 4 is represented as either 3x4=12 or 4 x 3 = 12).
Moreover, the teacher in this U.S. lesson explicitly emphasized the meaning of multiplication after the video discussion:
T: By the way, I just want to show you one thing before we move on. Notice when they did the models for this how 3 times 4 was 3 rows of 4? But 4 times 3 was 4 rows of 3. Now we know, because the commutative property tells us, that we ean turn around those facts, and we’re still going to get the same product. But please be aware, especially when they start asking you about matching equations to a picture or to a model, that the order of the numbers is important, beeause 3 times 4, you can really think of it as 3 rows, or 3 groups, of 4. But 4 times 3 is really 4 groups of 3, exactly, so you have to make sure that they match.
In the above episode, the teacher brought the meaning of multiplication to students’ attention even though the objective of the lesson was to teach the CP. Of course, the meaning of multiplication is defined arbitrarily. For instance, in Japan a x b is defined as “b groups of a” (Watanabe, 2003) while it is defined in the opposite direction in the U.S. (“a groups of b”). Regardless of the direction, the meaning of multiplication should be applied consistently as a foundation to illustrate and make sense of the basic properties of operations.
Tasks that Have a Potential to Differentiate the CP and AP. The
seeond insight observed from the formal teaching of the CP in the U.S. lessons was how the CP was applied to perform relatively complex computational tasks. Across both G4 lessons, the teachers discussed the following textbook task (Figure 5.5):
Figure 5.5 A textbook task discussed in U.S. G4 lessons. This G4 task was cited from Dixon, Larson, Burger, Sandoval-Martinez & Leinwand (2012, p. 107).
In this task, the CP was applied to switch 4x9 and 250 as a precursor to applying the AP. This task was quite similar to the instance generated from a Chinese G4 lesson discussed in Section 4.4, (17 + 23) + 28 = 28 + (17 +23) = (28 + 17) + 23. As argued earlier, tasks of this sort are helpful for correcting the common misconception that equalizes the AP with the use of parentheses. More discussion about the AP will be provided in the next section. Note that in both G4 lessons, the teachers went through the steps with the class without asking questions. The teachers could have incorporated deep questions such as how the CP and the AP were used in each step or how the two properties differed.
A Stray Observation: The Any Which Way Rule. While the U.S. lessons sometimes applied the CP with three numbers, there was a potential risk inherent in the any which way rule, a vague amalgam of properties that could lead to students’ conflation between the CP and AP. In both G4 lessons, the teachers discussed a textbook example that was situated in a story context with illustrations: There are 4 sections of seats in the playhouse theater. Each section has 7groups of seats. Eachgroup has 25 seats. How many seats are there in the theater? Both teachers guided the class to understand the problem situation, 4 sections of 7 groups of 25 seats, and generated the number sentence of 4 x 7 x 25. As consistent with the textbook suggestion, both teachers suggested that the students switch 7 and 25 using the CP, allowing them to multiply 4 and 25 first. Below is what the textbook suggested:
4x7x25 = 4x25x7 Commutative Property = 100x7 = 700
In this case, even though the positions of 25 and 7 were switched, the CP was not the only property involved. As indicated by the following reasoning processes, the AP is the other property' used. As such, when one justifies the above reasoning based on the any which way rule, there is a likelihood that students will conflate the CP with the AP.
- 4x7x25 = (4x7)x25 Order of operations = 4x(7 x25) AP = 4x(25x7) CP = (4x25)x7 AP = 100x7 = 700 or
- 4x7x25 = (4x7)x25 Order of operations = (7 x 4)x 25 CP = 7 x(4x25) AP = 7x100 = 700
As a final note, it should be mentioned that both G4 teachers spent time analyzing this story problem situation. However, the story context was only used as a pretext to generate the number sentence of 4 x 7 x 25. There was no discussion about the meaning of the sub-steps based on the story context (e.g., what does 4x7 mean?). Based on my observations, it is common for U.S. teachers to shift away from discussions of a story context to focus on computation strategies. This is especially true in a situation like this one where any way you multiply gets the same answer. Unless teachers believe that being able to relate the specific context of a problem to its numerical features has value, they are likely to ignore complicated contexts in favor of rules that are easily memorizable. In fact, I would consider the above story problem (4 sections of 7 groups of 25 seats) as a terrific example task for illustrating the AP (elaborated upon in the next section) rather than the CP because it has rich opportunities to express what each product means in context.
Summary: Teaching the CP of Multiplication through TEPS
In this section, I discussed informal and formal teaching of the CP(x) in both the U.S. and Chinese lessons. Cross-cultural differences were observed in the sample lessons. An integration of insights may shed light on how to better support students’ learning. The Chinese approach to introducing the CP was quite different from its usual style in that there was a noted lack of meaning-making, despite the existence of a story context. This is likely due to the two-way definition of multiplication that has been embraced as part of Chinese educational reform. In contrast, U.S. lessons stressed a one-way definition of multiplication that was expressed using the equal groups meaning. This allowed for reasoning using the array model when situated in concrete contexts (e.g., arranging chairs). The above sequence (from a real-world context to an array model and then to numerical solutions) is aligned with the concreteness fading method. In addition, the Chinese G4 lessons that formally introduced the CP contained sets of deep questions that likely promoted students’ generalization, reasoning, and proof skills. These deep questions could be easily incorporated into the U.S. lessons to promote conceptual depth. Overall, the cross-cultural insights in representation uses and deep questioning when teaching a worked example task illustrate how to teach the CP through TEPS.