Teaching Multiplicative Inverse through an Equal-groups Model
Scenario
Imagine you are a third-grade teacher and your students have learned the equal-groups meaning of multiplication. Now, you start teaching them division word problems as listed in the CCSS tabic (Tabic 3.1, top row, middle and right):
If 18 plums arc shared equally into If 18 plums are to be packed 6 to
3 bags, then how many plums will a bag, then how many bags are
be in each bag? needed?
How will you teach this lesson? Use the questions below to guide your planning:
- 1 What are the big ideas yon expect students to understand ?
- 2 What representations will you use during teaching?
- 3 Which deep questions will you ask during teaching?
Readers may agree that the teaching focus in the above scenario should be the two types of division (“group size unknown” and “number of groups unknown”) that can be linked back to students’ prior knowledge of multiplication (equal-groups model). Of course, after division is taught, there could be lessons that explicitly bring multiplication and division together. However, the scenario above shows that early in the process of teaching division, multiplicative inverses can be involved because division is the inverse operation of multiplication.
To teach the tasks in the scenario above, readers may have thought of asking students to sketch pictures or use cubes to represent the story problems. One may also ask students deep questions about the meaning of operations and the connections between both problems. Below I provide insights from similar lessons in U.S. and Chinese classrooms which contribute to the targeted approach of TEPS.
Insights from Chinese Lessons
The Chinese second-grade textbook presents a lesson involving word problems that are similar to the above Common Core tasks. This lesson focused on two types of division involving the number 7 because the prior lesson was about multiplication by 7. The worked example was situated in a real-world context of children making flowers with pictorial illustrations.
It stated, “A total of 28 flowers have been made.” A cartoon figure then posed two questions listed side by side:
Every 7 flowers can make one garland. If we make 4 garlands, how many
How many garlands can be made from flowers are in each garland?
these flowers?
The Chinese textbook then provided numerical solutions for each word problem (28 7 = □, and 28 -t 4 = □) and suggested finding answers
based on the multiplication Koujue (□ ife) represented on the very bottom of the page as “( ) seven twenty-eight,” and “four ( ) twenty-eight,” respectively. Note that Koujue are ordered sets of three numbers, in which the final number is the product of the first two. Given that multiplication Koujue are a powerful and traditional Chinese way to master multiplication facts and that multiplication facts present a great challenge to many U.S. students, I will briefly introduce how Chinese students learn multiplication Koujue before describing how Chinese teachers teach the two division problems similar to the scenario.
Multiplication Koujue. As mentioned above, prior to this division lesson, the textbook presents a lesson about multiplication by 7. The worked example was situated in a real-world activity about making boats using triangles. Students were prompted to consider the following question: If it takes 7 triangles to make a boat, how many triangles are needed to make 1 boat? 2 boats? ...7 boats? This is a multiplication problem based on the equal groups model. Students were expected to first solve this problem and then record their answers in a table. In our videotaped lesson, a teacher replaced this boat scenario by asking students to figure out the number of characters in an Ancient Chinese poem. This poem was arranged neatly in four rows with seven characters in each row. In this sense, this poem was an array model that served the same purpose of introducing the Koujue about 7. The teacher asked students how many characters were in one row, two rows, and so on. After solving each of the above sub-problems, students were guided toward multiplication Koujue as listed in Table 3.3. Students were then expected to memorize the Koujue so that the information could be flexibly retrieved for later use.
Let’s take a close look at the Koujue, “one seven seven, two seven fourteen, three seven twenty-one...” In this list, the second number is kept
Table 3.3 Introducing Multiplication Koujue about 7 in a Chinese Lesson
Meaning |
Number sentence |
Koujue |
1 group of 7 |
1x7 = 7 |
One seven seven |
Adding 2 groups of 7 |
2 x 7 = 14 |
Two seven fourteen |
Adding 3 groups of 7 |
3x7 = 14 |
Three seven twenty-one |
the same (always 7), while the first number is increased (until 7) so that all possible multiplication facts about 7 are accounted for.^{5} Two points need to be made. First, I want to reiterate that multiplication Koujue are taught in Chinese classrooms with conceptual underpinnings because they were generated from real-world contexts based on the meaning of multiplication. This contradicts a stereotype that the Chinese multiplication Koujue is purely based on rote memorization. Second, Koujue is somewhat similar to skip counting which takes place in many U.S. classrooms. The difference between Koujue and skip counting is that in U.S. classrooms, students only count the products (seven, fourteen, twenty-one...) while the Chinese students count out the two factors (2, 7) along with the associated product (14). It seems to me that the latter is more effective because it allows students to retrieve complete multiplication facts (e.g., 2 x 7 = 14, and 7x2 = 14). Given that skip counting already takes place in many U.S. classrooms, I believe that multiplication Koujue are likely both learnable by U.S. students, and preferable to the memorization of multiplication tables that traditionally occurs in the U.S. As will be seen in the Chinese lesson below, multiplication Koujue play a critical role in the way Chinese students learn both multiplication and division.
Back to the Chinese lesson on division. Now I return to the Chinese lesson on division by 7. In particular, I describe in detail one typical lesson that included the following steps:
- • Verbalize the mathematical information
- • Solve and explain each problem
- • Compare the similarities and differences between the two problems
- • Change division problems to multiplication problems ^{[1]}
how the teacher’s questions focused students’ attention on the meaning of division and the quantitative relationships:
T: Let’s look at the math expression first. 28 - 7. Why do we use division to solve this problem? Who can explain it?
S: It told us the total and the number in each group, so we use division.
T: To find out the number of garlands means to find out what?
S: To find out the number of groups, so we use division to solve it.
T: Who can tell us, what is the quantitative relationship equation in this problem? The number of garlands equals what?
S: You can find the number of garlands using “the total - the number in each group = the number of groups.”
T: What exactly happens in this problem?
S: The total number of flowers - the number of flowers in each garland = the number of garlands.
T: (To class): State what he just explained to yourselves. (Students talked.)
T: We can also think in this way: There are 28 flowers in total. Every 7 flowers makes one garland, meaning every 7 forms one group. Therefore, to find out the number of garlands is to find 28 contains? (S: How many 7s.) So we can write the math expression like that.
In the above excerpt, the teacher asked students to verbalize a quantitative relationship equation (ШжЗк&зК,). This is a common practice in Chinese classrooms. The teacher not only requested a broad quantitative relationship equation (“the total - the number in each group = the number of groups”) but also asked students to map it to the actual story context (“the total number of flowers - the number of flowers in each garland = the number of garlands”). After that, the teacher had students derive the abstract relationship from the concrete context again—that is, how many 7s are in 28. Using a similar approach, the class then discussed and verbalized the quantitative relationship embedded in the second word problem (“the total number of flowers - four garlands = the number of flowers in each garland”). Note that in each problem, the teacher asked students to explain how they obtained the computational answer only after a thorough discussion of the quantitative relationships. The following dialogue is typical:
T: The answer to this problem is 4. Where does this 4 come from?
S: 4x7 = 28, so 28-7 = 4
T: If you use Koujue?
S: Four seven twenty-eight!
T: Okay, four seven twenty-eight. So the answer is 4.
Looking back, one might notice that the Chinese teacher spent less time on computational methods but more on developing the meanings of operations and quantitative relationships. This is a common feature in Chinese lessons observed in this project. It seems that students’ mastery of multiplication Koujue frees more class time for problem-solving.
3 Compare the similarities and differences between the two problems. Once
both problems were answered correctly, the Chinese teacher asked her students to further compare the two problems. Asking students to compare related problems has the potential to elicit deeper reasoning. The following excerpt serves as such an example:
T: Students, after completing these two problems, let’s compare them to see their differences.
SI: The first problem is to find the number in each group, and the second is to find the number of groups.
Ss: Disagree!
T: You, please.
S2: The first is to find the number of groups, and the second is to find the number in each group.
T: The first problem tells us that every 7 flowers form one garland, which is the number in each group, and we need to find out the number of groups. The second problem tells us the number in each group, and what do we need to find out? (Ss: The number of groups). This is the difference.
T: Let’s see whether there are similarities between these two problems.
S5: The total is 28.
T: We know the total for both problems. They are both 28. Some children just said that we all used division to solve them. Why did we use division to solve both?
S6: Because both have told us the total. So we use... (Ss: Use division.)
T: Agree or disagree?
Ss: Agree.
T: For number sentences, some students have said that in both sentences, we divided 28, right? What is the similarity in computation?
S7: Both used the same Koujue, “four seven twenty-eight.”
In the above dialogue, the teacher asked students to compare the two word problems for the differences (finding the number of groups vs. finding the group size) and similarities (both problems gave the “total” thus both required division to solve). The teacher then had students identify that the same computational method (a multiplication Koujue) was used for each. I argue that such comparisons have likely prompted rich mathematical connections (e.g., meaning of division, types of division, using multiplication to solve division) that go beyond finding computational answers only.
4 Change division to multiplication problems. Sinee the students had already determined the final answers to both word problems and compared their structures, one may expect that the teacher would be ready to conclude the class discussion of this worked example. However, the teacher took the discussion a step further by asking students to change these division problems to a multiplication word problem. The tactic of changing word problems to their inverse is related to the approaches of “variation” (Huang & Li, 2017) and problem posing (Singer et al., 2015), which often occur in Chinese lessons.
Students in this class came up with the multiplication word problem, “Every 7 flowers can make one garland. If we make 4 garlands, how many flowers do we need?” They also solved it with 4x7 = 28. The teacher then asked students why this problem was solved by multiplication and students were able to explain, “They told us the number of groups and the number in each group. Yet, they did not tell us the total. So, we should use multiplication.” The above discussion was quite similar to the aforementioned dialogues on why division was used to solve the example problems. It is important to highlight that each of the teacher’s questions had a structural purpose. Asking students why this newly created word problem was solved by multiplication prompted students to think deeply about the quantitative relationships. Together with the division problems they discussed, students would likely obtain a structural understanding of multiplicative inverses based on the following quantitative relationships:
number of groups x number in each group = total total -5- number in each group = number of groups total и- number of groups = number in each group
In the end, the teacher asked students how they found the computational answer for 4x7 = (28), which again brought up multiplication Koujue. The teacher then highlighted the power of Koujue, “Have you found that regardless of the two division sentences or the multiplication sentence, we used? (S: The same Koujue!). Excellent! It seems that Koujue is really useful.”
Looking baek, the above teaching example demonstrates several key features of representation usage and teacher questioning. For representations, the overall sequence moves from concrete to abstract. This sequence is similar to the other teaching examples found in Chapter 2 because the discussion of this worked example began situated in a real-world context about making flower garlands that was then solved numerically. However, the representational sequence was not limited to a linear progression from concrete to abstract. Instead, there was a strategy of “folding back” from the abstract to the concrete. For instance, when the class generated the numerical solutions, the teacher asked the students to explain the meaning of the number sentences based on the story situations; when the elass compared the number sentences, the teacher asked them to explain the similarities and the differences based on the story situations. This “folding back” strategy' can promote connection-making and provide flexibility beyond concreteness fading. With regard to deep questioning, it appears that this teacher had several purposes, including (a) asking the meaning of operations based on the real-world context, (b) asking students to compare numerical solutions with and without context, and (c) asking students to make variations by changing a problem to its inverse. These deep questions often demand comparisons among different representations, different solutions, or different problems. In other words, it appears that asking comparison questions (e.g., What is similar or different?) along with follow-up questions (e.g., Why arc they similar or different?) plays a key role in this teacher’s deep questioning.
- [1] Verbalize the mathematical information. As mentioned earlier, theworked example in the textbook contained a pair of word problems.Instead of presenting the Hill worked example at once, the teacher projected each sub-problem (without the solution) sequentially side byside. This strategy was supported by the cognitive load theory becauseit focused the students’ attention to the task in discussion (called “selective attention"). For each sub-problem, the teacher first asked studentsto verbalize the mathematical information they noticed from the pictureand the statement. She then asked students to verbalize the completeword problems so they could be discussed fully. This step was similar tothose in the Chinese swimming pool example described in Section 2.3. 2 Solve and explain each problem. Both word problems were quicklysolved by students using number sentences (28 -r 7 = 4; 28 -r 4 = 7).The speed of these responses was not surprising given that this lesson followed lessons on multiplication and division involving 1-6 andmultiplication by 7. What was most impressive about the lesson wasthe teacher’s use of deep questions. The following excerpt indicates