Deep Questioning on Manipulatives When Teaching Division
After the review problem about multiplication, the teacher sequentially presented the following two example tasks:
Example 1 Example 2
Row 1:10 sticks placed (in a Row 1: ?
picture) Row 2:20 sticks placed (in
Is 5 times as many as the words)
2nd row js 5 times as many as the
Row 2:? 1st row
Both examples focused on the same concept—finding the smaller quantity—but varied in one surface feature. In Example 1, the smaller quantity was in the second row; in Example 2, the smaller quantity was in the first row. During the teaching of Example 1, the teacher asked all students to display their sticks on their desks. The teacher herself displayed 10 sticks evenly on the board and then invited a student to come to the board to share his answer. This student placed “two sticks” in the second row, as was agreed upon by the class. The teacher then asked: “Elow can you let me see, at a glance, that she is correct? Can you think of a good way to let me see at a glance that she is correct?” Asking students to let her “see at a glance” that the first row now has 5 times as many as the second row invited students to manipulate the sticks further. As seen from Figure 3.4 (left), students carefully separated the 10 sticks in the first row (the bigger quantity) into 5 equal groups. In other words, students arranged 5 equal groups in the first row to illustrate it was “5 times” as many as the second row. This indeed explained the concept of “5 times” based on the equal groups meaning. The
Figure 3.4 Deep questioning on manipulatives in a Chinese third-grade classroom. Left: Redrawn by Anjie Yang. Right: Redrawn by Molten Li.
teacher further confirmed this action on her screen by boxing the 10 sticks into 5 groups (see Figure 3.4, right).
After this intense discussion that focused directly on the meaning of “5 times,” the class quickly came up with the numerical solution, 10 -r 5 = 2 (see Figure 3.4, right). Based on this, the teacher asked more deep questions to elicit further explanation:
T: Why do we use division? Why division? Please share your thinking. SI: To find out how many times “which” is as much as “which,” we use division.
T: Anything else? Who else wants to share? You, please.
S2: To find the two sticks in the second row, the only way is to use 10 divided by 5.
T: Ok, who else wants to share? Between the number of sticks in the first and second rows, which row did you view as “one group”, and which one as “five groups”?
S3: The second row is viewed as one group, and the first row is viewed as five groups.
T: Great! It asks us to figure out the number of sticks in the second row; that is, how many sticks are in one group, right? So, to figure out how many are in one group, we should use “10 divided by 5” which equals the number of sticks in one group, right? Okav, the number sentence is 10 -r 5 = 2.
In the above conversation, the first two students did not give convincing responses because they did not refer to the meaning of division. The teacher then provided a hint enabling students to realize that that the second row was viewed as one group and the first row contained five groups. She then explicitly concluded that finding the second row was to find one group and to find one group, they needed to use division. The above discussion explained why division was being used based on its basic meaning; it may also contribute to students’ understanding of the quantitative relationship: Bigger v Multiple = Smaller.
In a similar vein, while teaching the second example (knowing the 2nd row contained 20 sticks, which is 5 times as many as the 1st row), the teacher asked students to illustrate the concept of “5 times.” After the class used their manipulatives to figure out that the empty first row should contain 4 sticks, the teacher asked:
T: Flow may we check if what he said is correct? Please use the 20 sticks on your desk and separate them to check whether 4 is correct ... There are 20 sticks on the blackboard. What did you do with these 20 sticks so we know how many sticks should be put in the first row? What did you do?
A student then came to the board and carefully arranged the sticks of 4 to a group, clearly leaving 5 groups. The teacher asked further questions to help students link the concept of “5 times” and “5 equal groups.”
T: How many groups did you try to divide these 20 sticks evenly into? Ss: Five groups.
T: Why did you try to divide them evenly into 5 groups? Where did you see that you need to divide the second row into Б equal groups? Ok, you please share.
SI: Because the second row has 20 sticks which are five times as many as the first row.
T: Because the second row is 5 times as many as the first row. We just need to divide the second row evenly into 5 groups, right?
T: Now you can tell at a glance that the second row has how many groups? How many groups?
Ss: Five groups.
As seen from the above conversation, the teacher’s purpose in asking students to manipulate the sticks was to understand the concept of “5 times” rather than to determine that there should be 4 sticks in the first row. This emphasis on understanding a mathematical concept, rather than obtaining a computational answer, is an important distinction of Chinese lessons that is repeated from the teaching examples in previous sections. In this particular lesson, students actually called out the computational answer quickly. The teacher, however, expanded upon students’ responses by requesting a way to make it obvious that the proposed answer was correct. This elicited students’ follow-up explanations of the targeted concept, 5 times.