Deep Questioning on Number Line Diagrams When Teaching Division

T2 taught a similar lesson including the discussion of the following word problem: There are 92 willow trees in a school. The number of willow trees is 4 times that of the poplar trees. How many poplar trees are in the school? The teacher first drew a number line to represent 92 willow trees. She then asked students to draw another line to represent the poplar trees. Based on student work, she selected three typical drawings (SI, S2, S3) for class discussions. First, she displayed the drawings of SI and S2 and asked students to compare and determine which drawing was correct. SI drew the willow tree number line longer (correct) while S2 drew the poplar tree number line longer (wrong). After discussion, the class rejected S2’s drawing. Next, the teacher asked the class to compare the drawings of SI and S3 (see Figure З.Б).

figure 3.5 Deep questioning on students’ drawings in a Chinese third-grade classroom. Redrawn by Meixia Ding.

As seen from Figure 3.5, both diagrams were correct. In each diagram, the top number line was meant to represent poplar trees (ffiffl) and the bottom number line was meant to represent willow trees (#Pt$). Note that S3 added dashes to the second number line while the SI did not. As such, the comparison this time focused on “which drawing was better.” This led to a class conversation around the function of the dashes that separated the willow tree line into 4 equal groups:

S: I think that the third one is better. Fie labeled the line to indicate the willow trees.

T: Fie did what to the willow trees?

S: Divided them into 4 even groups.

T: So poplar trees are?

S: One group.

T: One of the groups. Wonderful job! Let’s take a look at the first diagram. If you do not label the 4 even groups, what issue might we have? S: We might not know how many more times there are willow trees than poplar trees.

T: Or can we know the length of the line segment for the poplar tree?

S: No, we cannot.

T: In other words, we will have no way to figure out how long the line segment is, right? Any other opinions?

S: Then we cannot figure out the number of poplar trees.

T: So, what should we do to the willow trees?

S: We should divide the number of willow trees evenly into 4 groups.

T: Do you all agree?

Ss: Agree.

T: Excellent, so the poplar trees are ...?

Ss: One of the groups.

The above classroom discussion focused on the function of the dashed lines that separated the willow tree line into 4 equal groups with one of the groups representing the poplar trees. This is a similar teaching move to the discussion of the “dot” function in Section 2.4. It is also similar to T1 in this section who associated the concept of “times” with the meaning of “equal groups.” Given that the students’ drawings of the poplar tree line were based on the sentence “The number of willow trees is 4 times that of the poplar trees,” these discussions linked “4 times” to “4 groups.” These discussions led naturally to the numerical solution 92 -r 4 = 23 (willow trees). The teacher then asked students to further verbalize the meaning of this number sentence. Typical student responses indicated students’ understanding. For instance, “There are 92 willow trees. We divided the willow trees evenly into 4 groups, and one of the groups represents the number of poplar trees, so the number of poplar trees is 23.” Again, these explanations likely indicate students’ understanding of the quantitative relationship: Bigger v Multiple = Smaller.

After solving this problem, the teacher asked the students to check the solution (92 -r 4 = 23). Students were able to use both multiplication and division (23 x 4 = 92 and 92 v 23 = 4) to check the correctness of their answers. Per the teacher’s request, students also verbalized the real-world meanings for each of these number sentences (see below). These, along with the original word problem, form a group of inverse word problems that are similar to the CCSS example tasks presented in the scenario from the beginning of this section.

• • There are 23 poplar trees, and the number of willow trees is 4 times as many as the number of poplar trees, so 23 x 4 = 92. There are 92 willow trees.
• • There are 92 willow trees, and 23 poplar trees. The number of willow trees is 4 times that of the poplar trees.

In summary, T1 and T2’s teaching examples indicate how teachers may approach multiplicative comparison through deep questioning on representations that focus on the key concept of “times.” Both teachers also approached the word problems with an eye on opportunities to develop students’ inverse understanding.