Teaching the Distributive Property
Imagine you are a third- or fourth-grade teacher and you are aware that students in elementary school should learn the big idea of the distributive property (DP), ax(b+c) = axb+axc. This big idea may be introduced to students first informally and then formally. To proceed, you may consider the following questions based on either formal or informal teaching of this property:
- 1 What examples will you use to teach the DP?
- 2 What representations will you use during teaching?
- 3 What deep questions will you ask during teaching?
Readers may use an example like 13x7 = (10 + 3)x7=10x7 + 3x7to teach the DP. As with the other properties, the teaching goal would probably focus on either sense-making or a specific application of this property. Depending on the goal, one may choose to situate the teaching in either concrete or abstract contexts (e.g., array model, story problems, computation). Regardless of the context, one may ask deep questions that facilitate connection-making between different representations or shift students’ attention to the underlying property. Of course, depending on students’ learning level, one could focus on developing students’ informal or formal understanding of this property.
During my project, there were similarities and differences observed from the U.S. and Chinese lessons. For instance, both countries had lessons that involved computational strategies (e.g., the equations above). The strategy of breaking up a factor was a major component of the U.S. lessons; however, Chinese lessons initially presented the DP in contexts of problem solving to aid sense-making. In addition, Chinese lessons paid greater attention to using the DP in the opposite direction, which was essentially missing from the U.S. lessons.
In this section, I will present the cross-cultural insights for teaching the DP. I will discuss both formal and informal teaching in the Chinese and U.S. lessons, which will be separated into either problem solving or computational contexts. As with the other topics, I will focus on teachers’ representation uses and deep questioning during the teaching of worked examples. Together, these instructional insights will illustrate the targeted approach of TEPS for teaching the DP.
Insights from Chinese lessons
Formal teaching of the DP occurs in fourth grade in China. However, Chinese students are exposed to informal use of this property as early as second grade when multiplication Koujue are introduced. Across the observed lessons, the common approach was to emphasize meaning-making through teachers’ deep questions about representational connections.
Informal Teaching of the DP
In this section, I will introduce Chinese computation and problem-solving lessons that informally teach the DP. The computation lessons involved learning the basic multiplication facts (Koujue) and two-digit multiplication algorithm. For problem-solving lessons, there were different types of word problems that increased in complexity across grades. Regardless of the lesson focus and context, each Chinese lesson used worked examples that were situated in real-world contexts.
Computation: Multiplication Facts in G2. Chinese students start learning the multiplication facts (called Koujue) in G2. Earlier in Section
3.3, I introduced how multiplication Koujue about 7 was introduced in a real-world context as part of both the textbook and an actual lesson. In my project, we also videotaped lessons on the Koujue for 6s and 9s. Throughout these lessons, there were discussions about how the students could memorize the harder Koujue through recall of easier ones. For instance, after learning the 6s Koujue, one student pointed out that “Four six twenty-four”5 is hard to remember. The teacher then launched into the following class discussion:
T: How to remember four six is twenty-four? Who has a good strategy? SI feels that this one is particularly hard. Are there any other Koujuc that may help her remember this one?
This open-ended question elicited three different strategies:
S2: Two six twelve. And, four 6s contains two “two six twelves.”
S3: Three six eighteen. And, three 6s and one 6 are four 6s. So, 18 + 6 = 24.
S4: You can consider “four six” as “five six.” Five six thirty! And, subtracting one six, that gives you “four six twenty-four.”
In the above explanations, students broke the factor of “4” into either 2 + 2, 3 + 1 or 5 - 1, in a way that was similar to the break apart strategy that was observed during the U.S. lessons (elaborated upon later in this section). One noteworthy feature of the Chinese students’ explanations was that they were based on the equal groups meaning of multiplication. For instance, four 6s (or four groups of 6) were considered as 2 groups of “two 6s” (2x6 + 2x6), or 3 groups of 6s plus one 6 (3 x 6 + 1 x 6) or 5 groups of 6s subtracting one 6(5x6-lx6). Reasoning about such computational strategies based on the meaning of multiplication was common among Chinese students (see Section 5.1, student response to the DP items).
The practice of asking students to compare a group of related numbers sentences was used in all Koujuc lessons. Figure 5.9 (left) illustrates one such example that compared number sentences while informally incorporating the DP. In this lesson, students noticed that each of the number sentences had the same result of 28. The teacher then asked, “Why are these results the same?” “Did this happen accidently?” and “What are the relationships among these number sentences?” One student offered the explanation that, “3x7 = 21, and the second one is 4 x 7. And, 3 groups of 7 plus one more 7 is 4 groups of 7. That’s why the answers arc the same.” Such a response again shows that students were able to make sense of the connection based on the meaning of multiplication. The teacher further guided students to link the number sentences back to the pictorial context as indicated by the conversation in Figure 5.9 (right).
As was mentioned in Section 3.3, the textbook contained an equal group picture (involving flowers) that the teacher modified into the above array model. Such arrays were relatively rare in Chinese lessons but are a useful tool for demonstrating why 3x7 + 7 = 4x7. Note that this is an example of informally introducing the opposite direction of the DP (Ding & Li, 2010) because it combines 3 rows and 1 row into 4 rows, 3x7 + 1 x 7 = 4 x 7. This array model also provided the teacher an opportunity
Figure 5.9 An episode and board representation in a Chinese G2 lesson about Multiplication Koujue.
to infuse the CP as indicated by her follow-up question, why 4x7 and 7 x 4 were the same.
Computation: Two-digit Multiplication Algorithm in G3. Deep questions also appeared in multiplication lessons involving larger numbers. In a G3 lesson on two-digit multiplication, the worked example was situated in a story problem about 12 boxes of mini-pumpkins, each containing 24 pumpkins. Students were expected to learn the standard algorithm through this worked example. After the class solved 24 xl2 using the vertical computation method, the teacher encouraged the class to refer back to the pictorial context so that they could further analyze the meaning behind each step. Guided by the teachers’ set of questions, the class discussed what 24 and 12 represented in the problem (24 mini-pumpkins in each box and 12 boxes, respectively). Next, students figured out that the 12 boxes could be split into a small group of two boxes and a larger group of 10 boxes. Consequently, the two-box group contained 48 minipumpkins (2 x 24), and the 10-box group contained 240 mini-pumpkins (10 x 24). They concluded by discussing why they needed to add the two groups to obtain the total number of mini-pumpkins. The above discussion again grasped the meaning of multiplication and likely contributed to students’ implicit understanding of the DP, 24 x 12 = 24 x (2 + 10) = 2x 24 + 10x24.
In summary, Chinese teachers situated their computational worked examples (that implicitly involved the DP) in real-world contexts that were solved with number sentences. The abstract representations were often folded back to the real-world contexts for sense-making purposes. During this process, Chinese teachers tended to ask deep questions that focused on the meaning of each step and made comparisons between the concrete and abstract.
Figure 5.10 A translated Chinese G3 worked example that infused the DP. This G3 task was cited from Sun & Wang (2014, v.2, p.29).
Problem Solving: Word Problem involving Multiplicative Comparisons
in G3. In addition to computation, the Chinese third-grade textbook presented a problem-solving lesson that provided an opportunity to infuse the DP. The worked example was situated in a story context about shopping for two items. Figure 5.10 provides a translated description of this textbook example task.
Two G3 teachers taught this lesson. Even though their teaching styles were different—Tl’s lesson was more teacher-guided while T2’s was more student driven—both lessons had a similar representational sequence and contained many deep questions. Table 5.2 summarizes Tl’s discussion of the initial question: How much does it cost for a set of clothes?. Elaboration follows. 1
- 1 Verbalize the known and unknown quantities. As is usual for Chinese lessons, T1 asked the students to verbalize what information they could observe in the given problem. Her follow-up questions then asked students to articulate the given conditions (the known quantities) and the question to be solved (the unknown quantities).
- 2 Model the problem with number line diagrams. After the problem context was discussed, T1 suggested that students draw a number line diagram to represent the problem situation. Number lines are semi-concrete representations that facilitate concreteness fading while effectively illustrating the problem structure (Ding, Li et ah, 2019; Murata, 2008). During this process, T1 repeatedly asked questions that invited student input about how the diagram should be drawn (see Figure 5.11, left). For instance, after a student suggested drawing the number line that represented pants, she asked the class why they drew pants first. This elicited a deep explanation that the price of pants indicates “one unit” (—пШ). The teacher then made it explicit that when drawing a number line diagram, one should begin with a line that represents one unit. This teaching move might enhance students’ understanding of the general modeling method in addition to guiding them through the specifics of the given task. Similarly, when drawing the second number line to represent the cost of the coat, the teacher asked a question that drew students’ attention to the relationship between the two lines (The line for the coat was 3 times as long as the line for the pants). In the same vein, the teacher asked students where to label the question mark and what that vertical bracket meant (see Figure 5.11, left). These questions were a part of the teachers’ attempt to prompt students to analyze the quantitative relationship embedded in the number line diagram. Moreover, after the number
Table 5.2 Key Activities, Deep Questions, and Representations in Tl’s Lesson.
1 Verbalize the known and unknown quantities
T: Who can state this problem
completely based on the information in the picture and the words?
T: Who can tell us which two conditions this problem tells us and what question it asks?
2 Model the problem situation with number line diagrams
T: If I ask you to draw a number line diagram to represent this problem, what will you draw first? (S: Pants)
Why do you all want to first draw the line segment for pants?
T: Whose price should we draw next?
T: Now we have the two eonditions of the problem. Where should we label the question mark?
- What does this vertical bracket mean?
T: Who can verbalize the conditions and question based on this number line diagram?
3 Solve the problem in two ways while promoting meaning-making
T: How much does a set of clothes eost in total? Please write the number sentence by yourself.
T: Who can tell us in her solution, what she solved first and then what?
T: Do we have different solutions?.,.Hm, his solution is different from the previous one. Who knows what he computed first?
Abstract to concrete (folding back)
Table 5.2 (Continued)
4 Compare the two solutions for similarities and differences
T: What about these two solutions? (Same). What is different?
- Who can say this completely? How are the computation processes different?
T: Okay, we just solved the same problem using two different ways. Who can use one equation to represent these two number sentences?
T: Who can analyze whether these two answers are the same based on the number sentences themselves?
Figure 5. II Modeling the problem situations using number line diagrams.
line diagram was generated, the teacher asked students to verbalize the conditions and question in the story problem in relation to the number line diagram. Alternating her teaching moves back and forth between the problem situation and number line diagrams likely contributed to students’ understanding of the underlying problem structure. Indeed, such coordination of deep questioning and number line diagrams was frequently seen in Chinese lessons (see Section 3.5).
3 Solve the problem in two ways while promoting meaning-making. After a
lengthy discussion of the number line, Tl asked students to solve this problem. Students then reported two numerical solutions which were documented on the board:6
- 48 x 3 = 144 (¥) 1+3 = 4
- 144 + 48 = 192 (¥) 48 x 4 = 192 (¥)
For each solution, the teacher asked deep questions about the meaning of the steps (e.g., What did she solve first? And then what?). Students were able to explain that they began the first solution by finding the price for a coat, whereas they began the second solution by determining how many total units had been sold. Below is a sample of the conversation about the second solution:
SI: 1 + 3 = 4. The price of a coat is three times of a pair of pants. Then, if we consider a pair of pants and a coat together, then the price of a coat is three times that of a pair of pants; then, buying one coat is equal to buying three pairs of pants. So, buying a set of clothes is equal to buying four pairs of pants.
T: (to the class) Do you get it?
Ss: Get it.
T: Flow many times more is the price of a coat than the price of a pair of pants?
Ss: Three times.
T: That is, buying a coat is equal to buying how many pairs of pants?
Ss: 3 pairs.
T: Thus, to buy a set of clothes is 1 + 3 = 4, which is equal to buying how many pairs of pants?
Ss: Four pairs.
T: This is a good explanation. Who can explain this from the perspective of multiples?
S2: Pants is one unit. A coat is three times of a pant, that is to say, a set of clothes contains four units.
T: Give him a round of applause! Excellent! He can explain it from the perspective of multiples. We can view the price of a pair of pants as what? (One unit.) A coat contains three units. So, the whole set equals to how many units?
Ss: Four units.
T: We can then use one unit 48 to multiply by 4, 48 x 4 = 192(¥).
In the above episode, the teacher not only asked students to explain their number sentence based on the story situation but also challenged them to explain it from the perspective of multiples. This example shows how the teacher promoted abstract reasoning that went beyond the initial concrete context.
4 Compare two solutions for similarities and differences. After the story problem was solved in two ways and both solutions were justified using the story context, the teacher asked her students to compare the solutions. Students pointed out that both solved the same problem but the computational processes were different. The teacher prompted them to articulate how the processes were different and whether they could use one equation to represent these two number sentences. This request resulted in an instance of the DP even though the name of this property was not revealed: 48x3 + 48 = 48 x(l + 3).
Students tended to explain why these two number sentences were equivalent based on the story context. The teacher then prompted them to consider why this made sense based solely on the number sentence. The subsequent student explanations used the meaning of multiplication to explain that the left side was 3 groups of 48 plus one more 48 while the right side was also (1 + 3) groups of 48. Therefore, it made sense that the two sides of the equation were equal because they both represented 4 groups of 48. This is another example of how the teacher purposefully promoted student understanding that went beyond the concrete context of the initial problem.
T1 repeated this approach with increased student responsibility to discuss the second sub-question, “How much more does a coat cost than a pair of pants?” A similar number line diagram was drawn and presented on the right side of the initial diagram (see Figure 5.11, right). Note that the side-by-side presentation was found to be effective in promoting comparisons (Yakes & Star 2011). Based on the drawing, two solutions were suggested by students, which were then compared to generate another instance of the DP, 48 x 3 - 48 = 48 x (3 - 1). Furthermore, both observed G3 teachers guided their classes to make comparisons between the problem-solving process for both sub-questions (Each was solved in two ways; one found the sum while the other found the difference). The above ongoing comparisons on various representations (e.g., story problems, diagrams, number sentences) likely contributed to students’ later formal learning of the DP in G4.
Problem Solving: The Hypothetical Thinking Method in G4. Before students formally learned about the DP, there was a problem-solving lesson in which the DP was infused seamlessly. As will become clear, the way in which the DP was applied was not obvious at first glance. However, Chinese students skillfully applied a hypothetical thinking method that implicitly applied the opposite direction of the DP. In this method, students hypothesized about different ways to combine the groups from the problem (elaborated upon below). Figure 5.12 illustrates the G4 textbook example.
The class first discussed the regular solutions that were anticipated by the textbook: They found the cost of three Chinese chess sets (12 x 3 = 36)
Figure 5.12 A translated Chinese G4 worked example that enabled hypothetical thinking. This G4 task was cited from Sun & Wang (2014, v.l, p.70).
and four Go sets (15x4= 60), then added them together (36 + 60 = 96). These separate steps were also written into a single number sentence (12 x 3 + 15x4) which resulted in the answer, ¥96. Because the teacher encouraged students to come up with multiple solution methods, students suggested three other solutions that were not contained in the textbook:
- 1 Consider if there were only three Go sets, and add an extra one: (12 + 15) x 3 + 15,
- 2 Consider if there were four Chinese chess sets, and subtract an extra one: (12 + 15) x 4 - 12,
- 3 Consider if the Chinese chess set had the same price as the Go set and subtract the difference in price times the number of Go sets: 15 x (3 + 4) - 3 x 3.
Across these methods, students employed the hypothetical thinking method to pair numbers into groups. Below is the class discussion on the first hypothetical method:
S2: I bought three sets of Chinese chess and four sets of Go. I can view 4 as 3 + 1, then 12 + 15 = 27 and 27 becomes one group. I have three groups, so, multiply by 3. Since Go has four sets and we used three sets, we need to add 15, that is, 81 + 15 = 96(¥).
T: Who understands her? Please raise your hands. Okay, I will ask another student to explain her thoughts ...
S6: These four sets of Go can be separated into three sets and one set.
T: What she meant is, using one set of Go which is in red, let me use the red marker to represent it. (Sketching on the board). Assuming this is a set of Go, right? Using one set of Chinese chess to pair with? One set of Go. So, we can bundle them into what? (S: a group). A group. And, then? Continue.
S6: Three such groups.
T: We can make how many groups?
S: Three groups.
T: Yes, we can make three groups. So, we need to multiply by 3 because there are 3 groups. And then?
S6: Then, there are four sets of Go, it is more than three sets, it is four sets. So, we need to add one set of Go.
T: Okay, there is still one set of Go. We’ve made enough groups, yet, there is still one set of Go that was not made into a group. So, we need to add that set of Go. And we obtained this much of total cost, correct? Applause!
In the above episode, students tried to pair one set of Chinese chess and one set of Go into each group. Mathematically, their explanation could be expressed by the following thought process: 12x3 + 15x4 = 12x3 + 15 x3 + 15 = (12 + 15)x3 + 15. This process demanded that the students use a flexible understanding of the problem situation (e.g., viewing 4 sets of Go as 3 sets of Go and 1 more set) and an implicit use of the DP in the opposite direction, 12x3 + 15 x3 = (12 + 15)x3.To help other students understand this thinking method, the teacher asked clarifying questions and sketched the pairing-up idea on the board. Note that using the DP in the opposite direction was unfamiliar to most U.S. students who have primarily been exposed to breaking numbers apart to multiply, rather than merging them together (Ding & Li, 2010).