Properties of Multiplication: CP, AP, and DP
In this chapter, I will move from the properties of addition to those of multiplication. Note that the distributive property (DP) is not strictly a property of multiplication because it describes how multiplication and addition interact. As explained in Chapter 4, these properties are the backbones for learning algebra and should be learned and understood in elementary school. This chapter is organized in the same way as Chapter
4. First, I will start with examples of typical student work on the targeted properties. Then, I will introduce the mathematical knowledge needed for teaching. Finally, I will share cross-cultural insights for teaching the CP and AP of multiplication as well as the DP, each of which illustrates TEPS.
A Glimpse of Student Work
To assess students’ understanding of the CP, AP, and DP, my project items contained both contextual and non-contextual tasks. As reported elsewhere (Ding et al., 2019), of the three properties of multiplication, the CP (x) was more familiar to U.S. students than either the AP or the DP. The majority of U.S. students lacked explicit understanding of these properties by the end of G4. In contrast, Chinese students demonstrated an increased understanding of all properties, especially the AP and the DP after formally learning them. Overall, as with the properties of addition (see Section 4.1), student work from both countries collectively suggested no understanding (level 1), implicit understanding (level 2), and explicit understanding (level 3).
Student Response to the CP (×) Items
Two items were used to assess students’ understanding of the CP of multiplication. One asked for an evaluation and justification of the following computational strategy:
When solving 3x28, Mary wrote it as x 3. Is this order correctl
Why? Many students agreed that this order was correct. However, their explanations indicate different levels of understanding as described below:
At level 1 (no understanding), students made reference to the vertical format of multiplication as always occurring with the larger number on the top. However, putting the larger number on the top also caused the order of two numbers to switch. This change is justified by the CP. In the above item, even though it specifically asked, “Is this order correct?”, this order-related issue did not draw the attention of level 1 students. Typical responses included:
SI: Yes, because the number with more digits should be placed above, (a Chinese example)
S2: Yes, because you always put the bigger number on top. (a U.S. example)
At level 2 (implicit understanding), there were students who noticed the “order” of numbers did not matter; however, their explanations were either vague or loose, which might suggest an implicit understanding of the CP:
S3: This order is correct because if you turn it around, they are the same thing (a Chinese example)
S4: Yes, it is since she just did it upside down, (a U.S. example)
S5: Yes, because any order is correct for stacking multiplication problems, (a U.S. example)
Finally, there were some Chinese students (but rarely U.S. students) who demonstrated explicit understanding (level 3) as evidenced by their identification of the CP as the justification for switching the order of the numbers:
S5: Yes, because you can use the commutative property, (a Chinese example) S6: Yes, 3x28 = 28x3. This applies the commutative property, (a Chinese example)
Looking back at students’ responses, there is a possibility that using the vertical format of multiplication was a distracting factor that drew students’ attention away from the CP of multiplication. However, students’ responses to our item may also suggest that while teachers stress a computational procedure (e.g., putting the big number on the top), students should also be challenged to understand why that format works. One cannot assume students understand the rationale behind the procedures by themselves.
In addition, although Chinese students demonstrated stronger understanding of all the properties than their U.S. counterparts, they performed relatively poorly on the following item about the CP of multiplication (Ding et al., 2019):
To solve 8x6, Kate thought: Since 6 x 8 = 48, 8 x 6 = 48. Please explain why this strategy works.
For this item, many Chinese students did not recognize the CP of multiplication. Perhaps Chinese students’ instant recall of the basic tacts based on the multiplication Koujue (see Section 3.3) minimized classroom opportunities to discuss computational strategies and their underlying reasoning. Regardless, Chinese students’ responses to this seemingly easy item calls for awareness of students’ sensitivity and flexibility with the basic properties.