Teaching Multiplicative Inverse through an Array Model
Imagine you are a third- or fourth-grade teacher and your students have learned the equal-groups meaning of multiplication. Now, you’re planning to teach them the array model using the following group of word problems that are listed in the CCSS table:
- • There are 3 rows of apples with 6 apples in each row. How many apples are there?
- • If 18 apples are arranged into 3 equal rows, how many apples will be in each row?
- • If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?
How will you teach this lesson? Use the questions below to guide your planning:
- 1 What are the big ideas you expect students to understand ?
- 2 What representations will you use during teaching?
- 3 Which deep questions will you ask during teaching?
Readers may have determined that the big idea illustrated by this group of word problems is the eoncept of multiplicative inverse. As for representations, one may first draw a picture to illustrate the story situation (three rows of apples with six in each row). It is also possible to use cubes or other manipulatives or drawings to represent the apples. All these representations serve to help students visualize the problem structure of the array model.
In terms of deep questions, one may help students understand the array model by tapping into their prior knowledge of the equal groups model (see Section 3.3). In other words, one may engage students in discussion regarding why multiplication or division is used for each problem. Given that this is a group of inverse word problems, teachers may also ask comparison questions to elicit students’ understanding of inverse relations.
Note that the array model is closely related to the equal groups model. In this scenario, if one views “6 apples in each row” as one group, they will then see 3 groups. As noted in the footnotes of the Common Core Table, this is the easiest form of the array model. A more difficult version might describe the array as “3 rows and 6 columns.” The lessons in this section will use arrays that are similar to the second variety.
An additional note is that array models are more popular in the U.S. than in Chinese lessons. In the earlier section where I introduced multiplication Koujue, the Chinese teacher replaced an equal group model suggested by the textbook with an array model (a Chinese poem); yet initial learning of multiplication and division in China is rarely based on arrays. For this reason, this chapter will only discuss insights from the U.S. classrooms. However, I will briefly discuss opportunities to incorporate the general Chinese lesson insights to better illustrate the TEPS approach.
Insights from U.S. Lessons
The following examples are drawn from observations of one G3 and one G4 classroom. In both lessons, the array models were used to teach the inverse relationship between multiplication and division. Even though the array models in both lessons did not make use of real-world contexts, both teachers raised insightful questions that either stressed the meaning of operations or multiplicative inverse relations.
The U.S. G3 lesson. In the third-grade lesson, the teacher asked students to make 2 rows of 4 using either cubes or by drawing. She then asked the students to write two multiplication and two division equations for that array on their slate. Two students were then called to the board to share their equations based on the drawing of a 2-by-4 array. The teacher then led a class discussion on the set of equations below:
T: Why do we start both division equations with the number 8?
SI: Because it’s the highest number.
T: Okay, it’s the highest number. And why do we have to start the division equations with the biggest number?
SI: Because you can’t do 4 divided by 2 equals 8 or 2 divided by 4 equals 8. T: Right, because what does divide mean?
SI: Um, split into certain amounts.
T: Good. So if divide means to split into equal groups, we have to start with everything, the biggest number. We have to always start with the biggest number when we’re doing fact families and we’re doing division equations because division means break them up into groups, so we have to start with everything before we break them up into groups.
In the above excerpt, the teacher initiated the conversation with a deep question “Why do we start both division equations with the number 8?”. One student responded, “Because it’s the highest number.” This is a common type of response that students tend to provide. In fact, Ma (1999) reported that some U.S. teachers in their interviews stressed that the first number of a division equation should always be the biggest number. Ma pointed out that this statement is not always mathematically true and has the potential to cause student misconceptions and future learning difficulties. Of course, in the case of whole number division, it happens to be true that the first number is always the biggest but when learning fractions, this docs not need be the case (e.g., Vi - 4). What’s important to note is that there is no rule that requires the largest number to be placed at the beginning of a division problem. In the above classroom conversation, the teacher acknowledged the student’s observation but was not satisfied with this response, as indicated by her follow-up question, “Why do we have to start the division equations with the biggest number?” This is a great question to direct the students’ attention to the meaning of division. Since the student again tried to justify the original claim using computation, the teacher provided another hint in the form of the third deep question, “Because what does divide mean?” With this guidance, the class discussion shifted direction toward the meaning of division and the teacher summarized at the end that division means splitting everything into groups. Perhaps, instead of using the word of “everything,” the teacher could have introduced the notion of "total", or better, "product." Overall, I believe this teacher’s emphasis on the meaning of division in the above excerpt is a much deeper explanation than simply emphasizing the rule “because it is the highest number.” Looking back, I see the teacher elicited deep explanations of division by asking the following:
- • A question about what numerical choice was made
- • A question about the justification for that numerical choice
- • A question about how the primary concept shaped that numerical choice
The teacher also employed a similar instructional style to discuss the role of “8” in the multiplication equations derived from the equal groups model. The above discussion illustrates one path toward prompting deep classroom conversion, that is, focusing on the meaning of operations beyond computational procedures. This focus is a common method observed in Chinese classrooms in the project. In fact, Chinese classrooms would probably use this conversation as a jumping off point for comparisons that elicit students’ explanations of the inverse relationship between multiplication and division.
The U.S. G4 lesson. The fourth-grade lesson also contained a series of deep questions worth investigating. In this lesson, the teacher initially wrote
the pair of related facts 15x6 =_and 90 -r 6 =_on the board and
asked students to first solve and then share. Students came up with different strategies (see Figure 3.3, left) for multiplication including (a) breaking down 15 into 10 and 5 and then multiplying each number by 6, (b) breaking 6 into 4 and 2 and multiplying each number by 15, (c) drawing an array model with the side of 15 broken into 10 and 5, and (d) repeated addition (which was discouraged). Many students were also able to come up with the correct answer for 90 -r 6 = 15 because they noticed the division statement was a related fact to 15 x 6 = 90. The most impressive part of this lesson was that the teacher was not satisfied by simply obtaining the correct numerical answers. Instead, he raised further questions to draw students’ attention to the array model as a means to link multiplication and division.
Figure 3.3 A U.S. fourth-grade lesson involving an array model. Array for multiplication: Redrawn by Mohen Li. Array for division: Redrawn by Anjie Yang & Mohen Li.
T: So SI I want to use the array you have here that is so gorgeous to help us also figure out this 90 divided by 6. Where could we figure out this 90 on this array? So think about this division equation 90 divided by 6. Where could we see those numbers on this array?”
Students initial responses reflected that their understanding of the inverse relation was fairly limited. Typical responses included “just flip the multiplication equation around” and “I did the multiplication and division backwards.” In other words, students obtained 90 -r 6 = 15 mainly due to noticing that the related fact 15 x 6 = 90 was written beside it. The teacher, however, wanted students to understand how the elements of division and multiplication were related based on the array model.
T: Okay so we got 90 here as our answer and we could put 90 divided by 6 equals 15 here. But where are those numbers on this array? .... okay, so we know that inside of this whole big array there is how much?
With a few prompts, students figured out the area of this array was 90.
This allowed students to view 90 6 =_as a related task to_x 6 = 90.
The teacher further suggested using the array model to help figure out the missing number.
T: So then let’s even use this array here to figure that out. I am going to erase this side and erase this side (only leaving the line representing width of 6). We know we want 90, all we know is we’ve got 6 to count by, right? So what could we start off making an array to get us close to 90 with? We can do 6 by what? What is the easiest to always start off with?
After some discussion, students first reasoned that 6 x (10) = 60 (T drawing length of 10), which leaves an area of 30 needed to reach 90. They then suggested 6 x (5) = 30 (T continuously drawing length of 5), so that the two areas added up to 90. Based on the drawings, students then combined the two chosen lengths to obtain 15 as the missing factor
for_x 6 = 90 and thus the answer for 90 -r 6 = 15 (see Figure 3.3, right).
This G4 teacher’s use of the array model illustrates how visual representations can be used to help students make sense of numerical reasoning. This is different from the type of practice that draws an array to produce a fact family without clarifying how elements of multiplication and division are connected to the model. Of course, one may notice that the multiple computational strategies for breaking up a factor (either 15 or 6) and the strategy of breaking up an array are all undergirded by the same big idea— namely, the distributive property (see Section 5.5 for more discussion). This fact was not made explicit through class discussion and perhaps might be an area where this lesson could improve.
Possible Integration of the General Chinese Approach
Both U.S. lessons that used the array model provided insightful ways of stressing meaningful comprehension (e.g., the meaning of operations from the G3 lesson, or the connections between multiplication and division from the G4 lesson). However, as mentioned earlier, the array models from both lessons were presented in a relatively abstract manner that lacked real- world connections. As such, adding real-world contexts (e.g., the apple problem from the CCSS table, the arrangement of chairs in a theater, desks in a classroom, or the area of a rectangular playground) that can be modeled by the array models could have generated even richer classroom discussions. According to the literature (e.g., Gravemeijer & Michiel, 1999) and observations from Chinese lessons, real-world situations can serve to enhance opportunities for students’ initial learning by holding students’ attention and eliciting their personal experiences for sense-making. For instance, teachers could present an array and ask: Do you see any groups? What do they refer to? Students can also be guided to compare the word problems, the array models, and the problem solutions: How are they similar and different? How are they connected to each other? Through these conversations, a deep understanding of multiplication and division can be developed. Of course, real-world context should also be gradually removed to promote abstract thought. Once again, this aligns with the method I advocate throughout this book: concreteness fading.
Summary: Teaching Multiplicative Inverse (Array Model) through TEPS
In this section, I discussed the teaching of multiplicative inverses based on worked examples of an array model. Insights from the U.S. lessons with suggested modifications show the intended approach of TEPS. For representations, I suggest using the concreteness fading approach: Starting with a real-world context, modeling the story situation with an array model, and then solving the problem numerically. If possible, the abstract representations can be connected back to the array or the real-world context. For deep questions, I suggest that teachers focus on the meaning of operations (e.g., helping students see the “equal-groups” in the array) and foster explicit comparisons between multiplication and division based on various modes of representations.