Insights from Computational Strategy Lessons: Applying the AP
Both Chinese and U.S. classes taught computational strategies that involved the AP. The commonality among these strategies lay in how the AP could serve to combine or break factor(s) for regrouping. In particular, teachers in both countries discussed the shortcut of adding a 0 when multiplying by tens. In addition, one U.S. teacher taught a lesson on the “halving and doubling” strategy. To illustrate this strategy, this teacher discussed the array model to show why manipulating factors did not change the result. Likewise, Chinese G4 lessons discussed various ways to apply the AP with the goal of making computation easier. For the sake of brevity, I will only focus on the topic, “multiplying by tens,” that was taught in classrooms in both countries. Note that the relevant U.S. lessons contained both formal and informal teaching of the AP while the corresponding Chinese lessons were from G3, when students had not formally learned the AP yet.
China G3 lesson: Informal Teaching
One Chinese G3 lesson discussed the algorithm for multiplying tens, and mentioned that adding a “0” to the end of a partial product was a helpful shortcut. The worked example of this lesson was situated in the story context, “Each soccer ball is ¥32. How much does it cost for 30 soccer balls?” Even though this was a word problem, the lesson focus was actually on computation rather than problem solving. To solve this problem, students posed 32 x 30. They then suggested multiplying 32 by 3 first and then adding a “0” to the end. Based on these responses, the teacher followed up with a set of deep questions, which were reinforced in later discussion using the vertical format of multiplication:
S: I covered “0” first. So, 32 x 3 is 96. Adding a “0” to 96 gives 960.
T: So, you computed the numbers before 0. And, 32 x 3 equals how much? S: 96.'
T: (To the class) She said adding a “0” to 96. Why do we need to add a 0? Why adding a 0?
S: Because she covered the 0 in the end. The “3” here is not a “3” but 3 tens.
T: (To the class) Agree?
T: The “3” here is not 3 ones but 3 tens. So, 32 x 3 = 96 (write it on the board). Here 96 in fact represents 96 what?
T: So, who knows what the second step is? We actually use 96 to multiply by what?
S: We multiply 96 by 10.
T: (Write 96 x 10 on the board) Equals?
T: (Complete writing “96 x 10 = 960”).
In the above episode, the teacher started with a deep question, “Why do we need to add a 0?” Students were able to link this shortcut to the concept of place value (3 tens and 96 tens). Very often, teachers would be satisfied with such an explanation and move away from the discussion. However, this teacher went further by asking questions that linked “96 tens” and “96 x 10.” Consequently, the above discussion illustrated how the AP undergirded the calculation shortcut: 32 x 30 = 32 x (3 x 10) = (32x3)xl0 = 96xl0 = 960. These questions, especially the follow-up ones, provided insight for developing students’ understanding of the AP informally.
Note that in both Chinese G3 lessons, teachers asked students to compare a group of related computational tasks like the following:
- 42 x4x5 32 x15x2 12x5x8
- 42 x 20 32 x 30 12 x40
Students found that the bottom number sentence was a convenient way to solve the top one. The teacher then stated, when encountering tasks like 42 x 4 x 5, one can either compute in a regular order (compute 42 x 4 first) or multiply the latter two numbers (4x5 = 20) to make computation easier. Such discussion not only allowed students to see the power of multiplying by tens but also foreshadowed the existence of the AP. After these discussions, both teachers asked students to pose more pairs of examples in this sort and to identify the preferred number sentence for each computation. Looking back, the above discussions contained a series of deep questions that involved comparisons, explanations, and example posing, all of which may serve as steppingstones for later formal learning of the AP.
U.S. G4 Lessons: Formal and Informal Teaching
Multiplying by tens was also taught by three U.S. G4 teachers in the project. Two teachers who used the same textbook explicitly discussed the AP while the third teacher, who used a different textbook, informally illustrated why this shortcut worked. These lessons presented insightful representations as well as opportunities to ask deep questions.
The two G4 teachers who taught the same textbook lesson presented a worked example that was situated in a story context that could be solved by the product of 30 x 20. As suggested in the textbook, teachers presented two strategies: using place value and using the AP. However, there were no connections made between these strategies. This is different from the aforementioned Chinese G3 lessons where the connection between these two was informally made.
The third G4 teacher who used a different textbook approached this topic through a pair of related problems (6 bags of 4 oranges; 6 boxes of 40 oranges). The class represented the first problem with an array model, 6 rows of dots with 4 in each row. For the second problem, a student suggested using one dot to represent every 10 oranges. Thus, 40 oranges could be draw with 4 dots. This led to students’ discovery of40 x 6 = 240:
SI: I pretty much did it the same by doing boxes but I did a key where one dot equals 10 oranges and then put 4 dots in each box. ...
T: So what he did here, does this represent what is happening in this problem and how so? What do you think?
S2: It does! And, it also helps you figure out the problem. One, because the way I figured it out was 4 x 6 = 24 and 40 is the same as 4 x 10. And since it is 40 x 6. And 24 x 10 and you get 240. And that is kind of showing you that you have the bags and you have 4 in each bag and you still have 4 in each box but each one is times 10.
The idea of “using each dot to represent 10 oranges” was insightful. Based on this idea, the same array model for 4x6 was used to model 40 x 6 except with each “dot” representing 10 oranges rather than just 1. This enabled students to notice that they could first solve 4 x 6 = 24 and then multiply 24 by 10 to find the total oranges. After both problems were solved, the teacher asked students to compare both problems and their responses, leading to the students’ discovery of the shortcut for adding a “0” when multiplying by tens.
While the array model and comparison questions were quite insightful, this lesson did not explicitly reveal the AP as an underlying idea. Indeed, based on student input in this lesson, I believe the G4 students were at the appropriate stage to understand that the AP was the underlying idea behind their methods. For instance, the teacher could have recorded the processes numerically for six bags of 40 oranges: 6 x 40 = 6x(4x 10) = (6x4)x 10, infusing the lesson with direct connections to the AP.
A final note: the two G4 teachers who used the same textbooks also employed a number line model suggested by the textbook. To represent 15 x 2 and 15 x 20, one number line was drawn to show 15 jumps of 2 units each while another represented 15 jumps of 20 units each. This model shared similarities with the array model above—that is, the same drawing was used to represent different scales of similar components (e.g., each jump represents either 2 or 20 units). Such a model also has the potential to illustrate the process that 15x 20 = 15x (2x10) = (15x2) x 10 = 300. However, both teachers followed the textbook presentation and thus focused exclusively on adding a “0” to the first product instead of exploring deeper connections to the AP, despite the fact that the fourth graders had already formally learned the AP.
Summary: Teaching the AP of Multiplication through TEPS
This section focused on the teaching of the AP of multiplication through either problem solving or computation. The former context provided opportunities to illustrate the AP while the latter provided opportunities to apply the AP for computation. Despite the two-way definition of multiplication that may have limited the process, Chinese lessons provided insights about how to ask deep questions to promote representational connections. This approach can be integrated into the U.S. word problem contexts to promote meaning-making. The U.S. lessons suggested unique semi-concrete models (e.g., arrays, number lines) that could help illustrate computational strategies in Chinese lessons to ensure student understanding. However, the AP could have been made more explicit in the U.S. lessons where computational strategies were discussed. Overall, an integration of cross-cultural insights suggests how the AP of multiplication can be taught through TEPS to develop student understanding that facilitates algebraic thinking.