Knowledge in Brief
The Meaning of Multiplication
Even though there are different views on what a multiplication number sentence (e.g., 3x6) means across cultures, I argue that a consistent use of the meaning of multiplication, at least within a culture, plays an important role for mathematical reasoning. According to mathematics convention in the U.S., “я x IT is defined as “я groups of b" (Beckmann & Izsak, 2015). The Common Core uses the same definition. For instance, the thirdgrade standards expeet students to “interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each” (NGAC & CCSSO, 2010, p.23). Schwartz (2008) also argued for the importance of applying this basic meaning of multiplication with consistency. According to this researcher, if a teacher sometimes refers to 3 x 2 as 3 groups of 2 but other times as 2 groups of 3, mathematics will become spurious and meaningless.^{4} During my research, I have noticed that many U.S. teachers and students are inconsistent when applying the meaning of multiplication. One may argue that such a focus on the definition is made moot by the fact that the commutative property tells us that 3 x 2 = 2 x 3. However, as explained in the next section, the illustration of why this property works is still based on consistent uses of the same multiplication definition with the array model.
As a side note, in Chinese tradition, multiplication was defined in the opposite way—“я x IT is defined as “b groups of a"—and this meaning was consistently applied during reasoning. However, sinee the Chinese math reform in 2001, the definition of multiplication has been broadened to include both approaches (“я x IT defined as “я groups of IT or “b groups of я”) to decrease student learning pressure. Although this relaxation of the definition may provide some reduction in student cognitive load, it also can result in confused mathematical reasoning that would be avoided by using clear definitions. Indeed, many mathematicians in China have argued against this change, though the details of that debate are beyond the scope of this book. The relaxation of the definition of multiplication will be seen in some Chinese video clips in later sections.
Structure of Multiplication Problems
There are three major multiplication structures suggested by the Common Core: equal groups, array/area, and comparison (see Table 3.1, NGAC & CCSSO, 2010). Table 3.2 illustrates these problem structures using diagrams. Among these three models, the equal groups model is the most basic one because the problem situation contains several equal groups of
Unknown Product 
Group Size Unknown (“How many in each group?” Division) 
Number of Groups Unknown (“How many groups?” Division) 

Equal Groups 
3 x 6 = ? There are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether? 
3 x ? = 18, and 183 = ? If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? 
? x 6 = 18, and 18  6 = ? If 18 plums are to be packed 6 to a bag, then how many bags are needed? Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? 
Arrays, Area 
There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example. What is the area of a 3 cm by 6 cm rectangle? 
If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example. A rectangle has area 18 square centimeters. If one side is 3cm long, how long is a side next to it? 
If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Area example. A rectangle has area 18 square centimeters. If one side is 6cm long, how long is a side next to it? 
Compare 
A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Measurement example. A rubber band is 6cm long. How long will the rubber band be when it is stretched to be 3 times as long? 
A red hat costs $ 18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first? 
A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? 
General 
a xb = ? 
a x ? = p, and p  a = ? 
? xb =p, and p  b = ? 
Note: This table is taken from the Common Core, Glossary Table 2 (NGAC & CCSSO, 2010, p.89).
Table 3.2 Illustration of the Multiplicative Problem Structures
Model 
Context 
Qiiantity 
Illustration of Problem Structure 
Equal Groups 
Bags of plums 
# of bags (3) # of plums in each bag (6) # of total plums (18) Measurement: # of pieces of strings (3) length of each string (6 in) total length (18 in)  
Length of strings 

Arrays/ Area 
Apples 
# of rows (3) # of apples in each row (6) # of total apples (18) Measurement: One side: 3 cm Another side: 6 cm Area: 18 cm  
Rectangle 

Compare 
Hats 
Blue hat ($6) Red hat (3 times) Cost of red hat ($18) Measurement: At first (6 cm) Stretched (3 times) Length of the stretched (18 cm)  
Rubber band 
objects which can be traced back to repeated addition. These objects can either be discrete (e.g., three bags of plums with six plums in each bag) or continuous (e.g., three strings that are each six inches long; see Table 3.1). In such problems, the three quantities involved correspond to the number of groups, the group size, and the total (see Table 3.2).
The second model is the array/area model. When discrete objects are arranged in rows and columns, they indicate an array model (e.g., three rows of apples with six apples in each row; see Tables 3.1 and 3.2 ). Since each row or each column can be viewed as one group, the array model can be linked back to the equal groups model. Based on the meaning of multiplication, 3 rows of 6 are represented as 3 x 6. However, if we turn the array around or view each column as a group, we will then see 6 rows of 3, which should be represented as 6x3. This offers a visualization of the commutative property (elaborated upon in Chapter 5) by showing that the arrays for 3x6 and 6x3 contain the same number of objects in a different orientation (3 x 6 = 6 x 3). Note that when we push the discrete objects together without gaps and overlaps, the array model becomes a continuous area model (see Table 3.2). As such, even ifwe multiply the rows and columns of an array without considering their order, the meaning of multiplication (a
groups of b being represented as ax b) still applies. This holds true for the area model, with length and width replacing rows and columns.
The most challenging model in multiplication is the comparison model. Similar to the additive comparison model (see Section 2.4), it requires one to compare a bigger quantity with a smaller quantity. The relationship between both quantities (how many times bigger or smaller one quantity is) is called a “multiple.” The key to understanding this type of problem lies in the concept of “times,” which corresponds to the “number of groups” in the equal groups model. When we say, “A is 3 times as much as B,” we view В as a referent (or one group, one copy, one unit, or the “group size” as in the equal groups model). A then contains 3 groups of the same thing as B. As such, A = 3 x B, similar to the “product” in the equal groups model. In this sense, an understanding of comparison problems cannot be separate from an understanding of the equal groups model.
Similar to the CCSS additive table (see Table 2.1 in Chapter 2), the above multiplicative table contains sets of inverse word problems because the three word problems in each row involve the same three quantities, although each has a different quantity as the unknown. For instance, with regard to the plums problem (equal groups), the multiplication problem is to find the total (unknown product), which can be changed to two division problems: one is to find how many in each group (or “group size unknown”) and the other is to find how many groups (or “number of groups unknown”). Consequently, the arithmetic solutions to these word problems (3x6=18, 18r3 = 6, 18t6 = 3) form a fact family. These different sets of inverse word problems each provide opportunities to teach multiplicative inverses based on the three different models (Equal groups, Arrays/area, and Compare).
Types of Multiplicative Inverses
The inverse relationship between multiplication and division can be developed through both contextual and noncontextual tasks in a manner similar to additive inverses (see Section 2.2). Contextual tasks include fact families in context (see Figure 3.1 for an example) and inverse word problems (see Tables 3.1 and 3.2). Noncontextual tasks are numerical or computational in nature (e.g., writing a fact family or related facts based on a set of given numbers, using multiplication to compute/check for division, or finding missing numbers). For instance, to solve 28 r 4 = ( ), a student may recall that 4 x (7) = 28. Given that there was a lengthy discussion about the types of additive inverse tasks in Chapter 2, I will not elaborate much on multiplicative inverse variations.
With the necessary knowledge about the meaning of multiplication, the structure of multiplication problems, and the types of multiplicative inverses in place, I now turn to discussing the teaching of multiplicative inverses through the models of equalgroups, array/area, and comparisons, respectively.