Student Response to the AP (×) Items
Two items were used to assess the students’ understanding of the AP of multiplication. One was a non-contextual item asking students to compute (3 x 25) x 4 in an effective way and then explain why their strategy worked.
Some students (both U.S. and Chinese) simply followed the order of operations by multiplying 3 x 25 = 75 first and then highlighted the role of parentheses in their explanation. Such responses did not recognize the AP (level 1). Other students (level 2) chose to multiply 25 x 4 first, which implicitly applied the AP. However, they simply stated that the purpose was to make computation easier by making a 100 but not why they were allowed to do so. There were also students who made statements about the number properties in loose ways. For instance, one U.S. student who computed 25 x4 first explained, “My strategy was to change the place of the numbers.” Although this explanation touched upon the existence of number properties, it is fundamentally inaccurate since the number position was not, in fact, changed. This type of statement might indicate a conflation between the CP and the AP. Finally, there were some Chinese, but no U.S., students who provided explanations that explicitly acknowledged the AP by naming or describing the property7 or providing the formula (level 3).
The second AP (x) item was a contextual task (word problem) with solutions given. Students were expected to compare the two strategies to identify the AP:
Mr. Levin's students are tasting foods grown in rainforests. He put 5 pieces of mango on each plate and put 2 plates on each table. There are 3 tables. How many pieces of mango are there?
John solved it with: (3x2) x 5 Mary solved it with: 3 x (2 x 5)
Both are correct. Compare the two strategies, what do you find?
For some students (level 1), there was no evidence that they had even a vague understanding of the AP. Instead, their explanations were purely computational or general:
SI: I found that the results are the same, (a Chinese example)
S2: There are 30 pieces of mango. Both of them are correct and both strategies add up to 30 (a U.S. example).
S3: They are making the simple strategies to find it out (a U.S. example).
Other students (level 2) compared the two solutions and identified differences in surface features (e.g., parenthesis location and different number combinations) of the numerical sentences:
SI: One is to first compute 3x2 while the other is to first compute 2 x
5. (a Chinese example).
S2: Both are multiplication. The locations of parentheses are different;
but the answers are the same, (a Chinese example)
S3: The parentheses are combining 2 different numbers, (a U.S. example) S4: I did 3x2x5 =30 pieces of mango. Both strategies are correct; but the parentheses are moved, (a U.S. example)
S5: One strategy does 3x2 then x 5 and the other strategy does 2x5 then x 3, that is the comparison, (a U.S. example)
An interesting observation at level 2 was that some Chinese students attempted to explain the meaning of each step of the numerical solution by referring back to the story context. This allowed them to reason upon why both solutions resulted in the same answers. Such reasoning was missing from the U.S. student responses. Consider a typical Chinese example translated:
I found that Xiaoming first computed that there are a total of six plates and then found a total of 30 mangos; Xiaofang first computed that there arc 10 mangos on each table and then a total of 30 mangos. They got the same answers. 1
Finally, there were some Chinese G4 students who explicitly identified the AP of multiplication after comparing both solutions (level 3). This level of understanding did not occur in the U.S. students’ responses. Typical examples included:
SI: (3 x 2) x 5 = 3 x (2 x 5) uses the AP of multiplication.
S2: I found (a x b) x c = a x (b x c)
S3: I found that when multiplying three numbers, one may either first multiply the first two numbers or first multiply the latter two numbers, the product remains the same.
A few U.S. students attempted to point out the big idea. However, their responses indicate conflation between the AP and CP of multiplication. One U.S. student stated, “Both of the strategies will have the same product and all they did was changed the number order. Just like the commutative property.”
Student Response to the DP items
There were two non-contextual tasks that involved the DP. The first item asked the students to compute in the most effective way while the second item asked students to evaluate the given computational strategies. Explanations were required for both items. Note that the evaluation item was similar to an earlier task about the CP, which had been adapted from a U.S. textbook. Below are the items:
(Computation task) Please use effective strategies to solve: 102 x 7 and 98 x 7+2x7. Show your strategy and explain why it works (see Figure 5.1). (Evaluation task) To solve 8x6, Mary thought: 3 x 6 = 18, 5 x 6 = 30, 18 + 30 = 48; John thought: 10 x 6 = 60, 2 x 6 =12, 60- 12 = 48. Explain why each strategy works.
The above items called for students to use the DP either in the regular direction, (a+b) x с = ac + be, or the opposite direction, ae + be = (a + b) x c (Ding & Li, 2010). For instance, to compute 102 x 7, students should have first broken 102 apart into 100 and 2 and then applied the DP in the regular direction, 102 x 7 = (100 + 2) x 7 = 100 x 7 + 2 x 7. This is a common way of using DP in the U.S. textbooks (Ding & Li, 2010), which justifies the standard multiplication algorithm. In contrast, to solve 98 x 7 + 2 x 7, the DP needs to be used in the opposite way: by factoring 7 out, students can simplify computation by combining 98 and 2 into 100. Thus, 98 x7 + 2x7 = (98 + 2) x 7 = 100 x 7 = 700. Prior research indicates that opportunities to use the DP in the opposite direction are quite rare in the U.S. textbooks (Ding & Li, 2010). In my project, the student responses to “98 x 7 + 2 x 7” indicated a striking cross-cultural disparity. While 94% of the Chinese G4 students solved this problem by factoring out 7, only 6% of U.S. students did so. In fact, the majority of U.S. students used brute force calculation to compute both items based on the order of operations. This indicates a lack of understanding that the DP can be used bidirectionally (level 1).
With regard to those students who did solve these items in effective ways, some students simply described their computation procedures (often using the words “break down”) or stated that making a 100 served the purpose of making computation easier. However, they did not include an explanation of why they could do so (level 2).
SI: 100 x 7 = simple = 700; 2x7 = simple = 14; 700 + 14 = 714. (a U.S. example)
S2: It works because I split the problem into 100 x 7 = 700 + 2 x 7 = 14, which is easy to get 714. (a U.S. example)
S3: You can split 102 into 100 and 2, multiplying each of them by 7 to get 714. (a Chinese example).
S4: You can separate 102 into 100 and 2. It is easy to computelOO x 7. (a Chinese example)
Similar to the aforementioned observations with the AP item, some Chinese students attempted to make sense of their computation strategies
Figure 5.1 A typical level 2 Chinese student response to the DP item.
using the meaning of multiplication even though their understanding of DP was still implicit. For instance, some students argued that 102 groups of 7 could be viewed as 100 groups of 7 and 2 groups of 7. Likewise, they could combine 98 groups of 7 and 2 groups of 7 to get 100 groups of 7. Moreover, some explanations touched upon the features of the DP in ways that may have signaled their implicit understanding of the property. Figure 5.1 illustrates a Chinese example in which the student explained his solution to 102 x 7 using the meaning of equal groups. Fie also noticed the common factor of “7” in 98 x 7 + 2 x 7, which he multiplied by the sum of 98 and 2.
Only Chinese G4 students were observed explicitly identifying the DP as the reason behind their computational strategies (level 3). As with the cases for other properties, students’ explanations at this level either named the DP, described it in general language, or presented the formula, (a + b) x с = ac + be.
Student responses to the evaluation task (why do both strategies solve 8 x 6) indicated similar levels of understanding. At level 1, students attended to the results (e.g., all obtained 48). At level 3, students clearly mentioned the DP. Again, the most interesting observations occurred at level 2 reasoning. As was discussed earlier, some students attempted to reason using the meaning of multiplication. Below is a translation of one Chinese student’s response:
- 8x6 = 8 groups of 6 3x6 = 3 groups of 6 = 18 5x6 = 5 groups of 6 = 30
- 48 = 30 + 18 = 3 groups of 6 + 5 groups of 6 = 8 groups of 6.
In addition to the non-contextual items, we also provided one contextual item that expected students to recognize the DP. This item was adapted from NRC (2001) and encouraged students to solve a perimeter problem in multiple ways that could be made sense of using the DP:
The length of a rectangular playground is 118m and the width is 82m. What is the perimeter ?
John solved it with: 2 x 118 + 2 x 82 Mary solved it with: 2 x(118 + 82) Both are correct. Compare the two strategies, what do you find ?
Some students computed both solutions directly and reported that the answers were the same (level 1). Occasionally, some students would point out that one strategy was more convenient than the other. Such responses did not show their awareness of the DP:
SI: One strategy has much more work and one has a little less work (a U.S. example)
S2: I found that their methods are different but the answers are the same, (a Chinese example)
Students’ (level 3) explanations that explicitly named, described, or stated the DP in a formula only occurred among Chinese students. Finally, student responses that showed implicit understanding of the DP (level 2) consistently indicated a cultural difference in how the property was utilized. Many Chinese students analyzed the meaning of each step using the story situation. This observation was consistent with the earlier report that Chinese students reasoned about their computational strategies using the equal group meaning of multiplication. Below is a Chinese example translated:
I found that Xiaoming first computed the (total) length of the playground and then the (total) width of the playground; Xiaofang first computed “length + width” of the playground, and then find two “length + width.” Both answers are correct.2
In summary, students’ different levels of understanding may suggest a trajectory of student learning and a path to develop students’ understanding of these properties. In addition, cross-cultural differences in students’ understanding (e.g., Chinese students were the only ones who achieved a level 3 understanding) calls for the development of methods to help U.S. students achieve their potential. Moreover, Chinese students’ unique reasoning at level 2 of understanding (e.g., focusing on meaning-making) calls for exploration into how classroom instruction might play a role in facilitating students’ reasoning. Before exploring the classroom factors, I will provide brief background knowledge to situate the later descriptions of classroom teaching.