Insights from U.S. Lessons
The main insight from the U.S. classrooms was their use of the array model to illustrate the CP of multiplication. The discussions on arrays varied across classrooms. In addition, there were a number of great tasks associated with the CP of multiplication in G4 classrooms. However, differentiation between the CP and the AP could have been better clarified through comparison questions. Below, I share the main observations about informal and formal teachings of the CP in U.S. lessons.
Informal Teaching of the CP(×)
Insights from a U.S. G3 Lesson. One G3 teacher introduced the array model during a lesson in which the CP was tangentially discussed. The sequence of her representation included a story context, cube manipula- tives, array drawings, and number sentences. This structure serves as a
Figure 5.4 Representation uses in a G3 lesson on arrays. Redrawn by Mohen Li (b) and Anjie Yang (c).
strong example of how the concreteness fading approach can be used to
teach the CP with arrays (see Figure 5.4). Below is a list of key activities
the teacher used to teach the worked example. Elaboration follows.
- • Pose a story problem about arranging chairs
- • Discuss students’ cube arrangements
- • Discuss arrays using grid paper
- • Link arrays to number sentences ordered by the textbook
- 1 Pose a story problem about arranging chairs. This teacher started her lesson with a story context about arranging 12 chairs:
T: I have 12 blocks up here on my smartboard screen, and I have a problem to pose to you. I want you to imagine that our class is having a little production, a play if you will, up here at the front. And, there are 12 blocks to represent 12 chairs, and we would like to put these chairs in rows, the same number of chairs in each row. And I want us to start thinking of all the possible arrangements that we can put 12 chairs in, to watch the play.... How many rows, and how many chairs in each row?
2 Discuss students’ cube arrangements. Students were then given 10 minutes to use cube manipulatives to explore all the possibilities of how the chairs could be arranged. After this hands-on opportunity', the teacher used cubes on the smart board to share her observations with the class. Below is a typical episode of the teacher’s discussion:
T: I wanted to show you, when SI first put out her blocks, I noticed she used an array, 3 rows of 4, or the dimensions would be called 3-by-4. So, she had 3 rows, with 4 in each; 3 rows of chairs, 4 chairs in each row (see Figure 5.4b). Connection [sic] if you also had that arrangement. So, 3-by-4 (writes: 3 by 4).
T: I was really impressed with that because it was really quick, and I noticed right after that she did something where she did, she pulled these over (rearranges blocks on the smartboard), and did almost the reverse, where she had then 4 rows of 3, or the dimension 4-by-3 (writes: 4 by 3). And that gets us into the commutative property of multiplication, we’ll talk a little bit more about that.
The above episode contained several points of merit. First, the meaning of multiplication was used as a basis to discuss the array model. Recall that “я groups of b" in the U.S. is represented as “я x b." Aligned with this definition, the teacher explicitly referred to the 3-by-4 array as “3 rows of 4,” in clear contrast with the 4-by-3 array that was referred to as “4 rows of 3.” In other words, the teacher did not just focus on the two numerical dimensions of the array. Rather, she linked the array model to the equal groups meaning, indicating that each arrangement of the numerals had a distinct meaning. This is a critical teaching move that has the potential to allow students to engage in reasoning based on the one-way definition of multiplication. It is in direct contrast with the Chinese two-way definition of multiplication where such reasoning cannot be made.
The second feature of this lesson is the way the teacher linked the array model back to the story context. In the episode, the teacher stated, “So she had 3 rows, with 4 in each; 3 rows of chairs, 4 chairs in each row.” This made an explicit connection between the concrete and abstract. The subsequent practice task where 24 chairs were arranged is also noteworthy for how skillfully the teacher compared various arrangements (e.g., 12-by-2, 2-by-12,24-by-l, and l-by-24) and linked them to real-world situations in ways that allowed students to visualize the consequences of switching the order of the factors:
T: If you have 12 rows from front to back with 2 people, I’d hate to be sitting in the last row, wouldn’t you? And here we have 2-by-12, I think that’d be a bit nicer for us to sit in. And here you can see they had some more combinations to go. Could you imagine all 24 of us just sitting one after the other, all trying to watch a play in the front of the classroom? It would, like, go out the door, wouldn’t it? But it probably would be much more fair if it were 24 straight across.
The final meritorious point of this lesson was how the teacher briefly linked each pair of arrays (3-by-4 and 4-by-3) to the CP. Verbally, she mentioned that the 4-by-3 array was the reverse of the 3-by-4 array. She mirrored this by physically demonstrating how the student could rearrange the cubes to create the second version of the array.
Perhaps this visualization might have been more vivid if the teacher had rotated the first array 90 degrees in order to generate the second one. However, it is likely that the manner in which the cubes were displayed on the smart board did not allow for such a rotation. In this instance, it might have been more effective if the teacher had used physical cubes to demonstrate the rotation. Encouragingly, in her discussion of the 2-by-6 array, she mentioned that the cubes could be rotated 90 degrees to turn it into a 6-by-2 array.
Despite the merits described above, most of these insights were delivered to students directly by the teacher, rather than constructed by the students from the teacher’s deep questions. In subsequent interviews, the teacher explained that the idea of the CP was mentioned but quickly passed over because the focus of this lesson was to introduce arrays rather than the CP. However, even without identifying the CP by name, the teacher still could have asked deep questions that directed student attention to the features of this property. For instance, she could have asked students to compare and contrast the 3-by-4 array with the 4-by-3 array. Discussions of this sort are likely to contribute to students’ implicit understanding of the CP. This is because asking deep questions may elicit students’ self-explanations, which may enhance the worked example effect to promote students’ subsequent problem solving in new settings (Chi, 2000).
3 Discuss arrays using a grid paper. After the teacher finished describing some typical student work, she displayed a grid on the smart board to outline the corresponding arrays (see Figure 5.4c).
T: So, I actually want to show you on this grid another way of what you did. So, we heard there was 2-by-6, so 2, 3, 4, 5, 6... (T outlines a 2 x 6 rectangle on the grids)
T: And so we just said she did 6-by-2, so 3, 4, 5, 6, by 2... (T outlines a 6x2 rectangle on the grids)
T: And I’m actually going to label this “2 times 6” (writes: 2x6) and “6 times 2” (writes: 6 x 2).
T: Do you sec the relationship between them now?
T: Here we have 3-by-4. Erm, I’m sorry, we have 4-by-3, so 4 times 3 (writes: 4x3), and we have 3 groups of 4, we have 3 rows of 4, 3 by 4 (writes: 3 x 4).
In the above episode, it is worth mentioning the teacher’s rich representation uses. First, she used grid paper to draw the arrays in a way that changed the discrete model (separate cubes, see Figure 5.4b) into a continuous one (connected squares that form grids, see Figure 5.4c). In fact, if the lines between the grids were removed, the resulting figure would become an area model that would push the representation even further toward the abstract. Second, the teacher labeled the arrays with number sentences that matched their orientation. For instance, she labeled the 6-by-2 array as 6 x 2 and the 2-by-6 array as 2 x 6 to visualize how different notations have different representational meanings.
Perhaps the lesson could have been improved if the teacher had continuously promoted the connections between the array model (e.g., 2-by-6), the number sentences (2 x 6), and the equal group meaning (2 rows of 6) throughout the lesson. Although the teacher made such connections (e.g., We have 3 groups of 4, we have 3 rows of 4, 3-bv-4), there was a lack of questions encouraging students to verbalize these links. The teacher did ask the deep question, “Do you see the relationships?” when discussing the 6-by-2 and 2-by-6 arrays. Yet, she did not wait for her students to respond.
4 Link arrays to number sentences ordered by the textbook. Finally, the teacher projected the textbook page that contained every possible arrangement of the factors of 12 with corresponding number sentences, listed in order (see Figure 5.4d).
T: This poster is a picture of different possibilities, or ways, to arrange 12 chairs. And most of you came up with these different arrangements. .. .Then, it correlates, and it ties into the number of ways to make 12. Twelve is the same as saying 1 times 12, 12 times 1,2 times 6, 6 times 2, 3 times 4, and 4 times 3. All the possibilities.
In the above activity, the teacher used the textbook presentation as a summary of what students had explored. By organizing the list of number sentences, the class moved further than if their suggestions had been presented randomly.
Perhaps, the teacher could have integrated deep questioning to promote deep thinking. For instance, she could ask a question that draws students’ attention to the commonality across the three pairs of number sentences (e.g., 1x12 & 12x1; 2x6 & 6x2; 3x4 & 4x3, see Figure 5.4d), which would likely facilitate students’ awareness of the CP. In addition, the teacher might ask students whether this given number list could be re-arranged and, if so, why. Based on my experience with Chinese lessons, students could come up with a list in which the first factor increases. Such an alternative number list is a product of pattern-seeking and thus serves to develop students’ algebraic thinking.
In summary, the above G3 lesson on using arrays is a strong demonstration of how concreteness fading can be used with the array model and how the model can be linked to the equal groups meaning in a way that promotes sense-making. It also reveals that there is room in even the most thoughtfully structured U.S. lessons to incorporate deep questions to elicit students’ (rather than teachers’) deep explanations.
Possible Issues with Teaching Arrays. Even though the array model is a powerful tool for illustrating the CP, teaching this representation can become procedural when it is not based on the meaning of multiplication. Procedural lessons may cause students to memorize steps without understanding their actions. I observed such an issue in another G3 classroom in my project. Below is a typical episode:
T: Just like we did before, I look at my first number, my first factor, and I make that many rows. So, I’m going to use these (points to counters on the smartboard) to make three rows. 1, 2, 3. And I’m going to try to make them as neat as I can (placed three counters vertically in a column). Then I look at my second factor. The second number in my multiplication fact, 5. So I need to make sure I have five counters in each row (counts the first counter and adds four more counters to each row).
In the above episode, the teacher described the steps involved in creating an array. Since she began by arranging a single column, there was already one counter that was part of each row and she only added four more per row to complete her array. Technically, there was nothing wrong with this procedure. However, students’ follow-up responses in this lesson indicated that they had great difficulty understanding this technique. They did not seem to recognize that the three counters lined up vertically in a column were the initial counters for each of the three rows. They also seemed unsure as to why they should skip the first counter to add 4 more to each row, instead of adding 5 (as would seem to be indicated by the number sentence). The following episode provides such an example in which a student was attempting to make a 5 by 2 array:
T: Five rows. Good. 1,2, 3, 4, 5. Then you need 2 [in] each row.
(The student adds 2 more in each row, instead of adding 1 more to give the rows a total of 2 counters each)
T: So, I want you look. How many do you have in each row?
T: 3. Good. You only need 2. Remember, we made each row that was your first column.
Because the array process was described procedurally, the student in the above episode took each number in the array (5-by-2) as an individual instruction about how many counters to add. The counters that formed the initial column were part of a separate (and seemingly unrelated) step from the counters that were placed in each row. I noticed that the same issue occurred in other classes with other teachers, as well. The way that the two dimensions of the array were related to the equal groups meaning was only clear to the teacher and not the students. The teachers could have asked deep questions focusing on the meaning of the arrays and how they should relate to the number sentences.
Additionally, even though the CP had been explicitly discussed in the previous lesson, the above teacher drew an array for multiplication but overlooked that the array could also model the commutativity After the class drew a 6-by-3 array and recognized that it contained 18 counters, the teacher requested its turn-around fact without linking it to the array:
T: If 6 times 3 equals 18, who remembers from yesterday the turnaround fact or the commutative property? If we know this, what else do we know? Rubin?
S: 3 times 6 equals 18.
T: Good job. We have that commutative property or turn-around facts. Good job.
Although addressing the CP numerically has value, it ignores the fact that the model creates slightly different shapes for each of the two number sentences (6x3 = 18 and 3 x 6 = 18). By linking the turn-around fact to the turn-around array, the teacher could have also given students the chance to compare and contrast the features of the two multiplication facts and arrays. Such a comparison could potentially deepen students’ understanding of the basic property in a way that could be retrieved in future contexts.
In summary, while the array model is potentially a powerful tool for teaching the CP, the effectiveness of how teachers used the model varied across classrooms. On one hand, it can be used to elegantly illustrate important features of the CP. On the other hand, arrays can also be used as purely procedural tools for computation. Teachers’ deep questions play an important role in facilitating students’ understanding of this model. At least during students’ initial learning, it seems necessary to help students understand how the two dimensions of arrays are related to the basic meaning of multiplication. It is also critical to ask questions that promote comparisons between various related arrays.