# Knowledge in Brief

Two pieces of mathematical knowledge are necessary for teaching the additive inverse concept in a deep, meaningful way: the types of inverse tasks, and the structure of additive problems.

As indicated in Section 2.1, the literature (e.g., Baroody, 1987; Carpenter et al., 2003; Nunes et ah, 2009; Torbeyns et ah, 2009) suggests types of inverse tasks including (1) Fact families or related tacts, (2) Inverse word problems, (3) Using inverses to compute, (4) Missing numbers, and (5) Using inverses to check. Each type contains contextual and/or non-con- textual tasks. Elaboration follows.

• 1 Fact families and {2) Inverse word problems. These two types of problems share similarities while also having several differences (Carpenter et ah, 2003). A tact family refers to a group of related number facts involving a single operation and its inverse. An example of a tact family is 2 + 3 = 5, 3 + 2 = 5, 5 - 2 = 3, and 5-3 = 2. Students may initially learn two related tacts instead of four: 2 + 3 = 5,5-2 = 3.A tact family may be derived from a set of three related numbers or a concrete context such as the examples in Figure 2.1. Similarly, inverse word problems are a group of related word problems whose solutions form a tact family. The balloon tasks in Section 2.1 are such an example. Note that in tact families and inverse word problems, the coexistence of addition and subtraction may decrease student cognitive load and enable students to identify the relationship between inverse operations. This is easier than the application problems (as discussed below) where students need to recall inverse relations to solve new problems.
• 3 Using inverses to compute, (4) Missing numbers, and (5) Using inverses to check. These types of tasks require a student to solve or check a computational problem by thinking of its related fact. For instance, to compute 81 - 79 = ( ), students were expected to think of its related addition tact, 79 + (2) = 81 (Torbeyns et ah, 2009). A relevant but slightly harder computation problem is to find the missing number. In such tasks, students need to find the value of an unknown on the left side of the equal sign. An example is () - 79 = 2.

An effective strategy to solve this problem is to use its related tact,

79 + 2 = (81). This inverse-based strategy can also be used for checking computational answers. For instance, to check whether 81 - 79 = 2 is correct, one can use 79 + 2 = 81 to confirm.

Note that in the above computational tasks there is a risk that students will only learn numerical manipulation without concurrent conceptual understanding. As such, it could be effective if teachers situate students’ initial learning of the computational strategies in a real-world context. Real-world situations provide rich contextual information that may activate students’ familiar experiences and prior knowledge (Resnick & Omanson, 1987). For instance, to understand why ( ) - 79 = 2 can be solved using its inverse operation 79 + 2, a teacher could create a real-world situation: “There are some apples on a table. I took away 79. Now two apples are left on the table. Flow many apples were initially on the table?” This real- world context describes a “decreasing” situation that suggests a subtraction equation, ( ) - 79 = 2 (Nunes et al., 2009). To find the computational answer, students can be guided to see that if they put the 79 apples back together with the two leftover apples, they will obtain the initial number of apples. This is why they could solve the subtraction problem using its inverse operation, 79 + 2 = 81.

To help students make sense of additive inverse relations, a teacher must also understand typical structures of additive problems. Table 2.1 is taken from the Common Core State Standards (NGAC & CCSSO, 2010) and summarizes the common structures of additive problems. As seen from Table 2.1, each row contains a group of word problems that share the same structure but have a different quantity serving as the unknown. As a result, each row forms a group of inverse word problems undergirded by related quantitative relationships (e.g., Part + Part = Whole, Whole - Part = Part; Bigger - Smaller = Difference, Smaller + Difference = Bigger, Bigger - Difference = Smaller). These quantitative relationships permeate the process of solving both arithmetic and algebraic word problems (Choo, et al., 2009). Thus, helping students understand the quantitative relations and the inverse relations within each problem structure will likely deepen students’ understanding of both arithmetic and algebra (Cai & Moyer, 2008; Carpenter et al., 1999, 2003).

In Table 2.1, the left column indicates four types of addition/subtrac- tion problems. In other words, inverse word problems can be learned through different types of additive word problems. The first two types (add to, take from) are naturally related since they both describe actions that are performed according to the same sequence: start, change, and result. In the literature, these two rows are designated as “change” problems. The next type (put together/take apart) differs from the “change” problems in that these questions involve two static parts instead of any actions. Some researchers designate these as “part-whole” or “combine” problems. In this row, the “total” is the “whole” while the two “addends” are the “parts.” Each of the two addends can serve as an unknown resulting in the corresponding subtraction problems. Regardless of their differences, the first three types of problems share an essential resemblance because each presents a relationship between parts and the whole. Thus, Choo, Flong, Mei, and Lim (2009), in introducing the Singapore Model Method, group the first three problem types together as “part-whole.”

Tabic 2.1 Common Structure of Additive Problems in the Common Core

 Result Unknown Change Unknown Start Unknown Add to Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ? Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5 Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5 Такс from Five apples were on the table. I ate two apples. How many apples are on the table now? 5 - 2 = ? Five applies were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 - ? = 3 Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? - 2 = 3 Total Unknown Addend Unknown Both Addends Unknown Put Together/ Take Apart Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ? Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? i 5, 5 - 3 = ? Grandma has five flowers. How many can she put in the red vase and how many in her blue vase? 5 = 0 +5,5 + 0 5 = 1+4, 5=4+1 5 = 2 + 3, 5 = 3 + 2 Compare Difference Unknown (“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5,5-2 = ? Bigger Unknown (Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ? Smaller Unknown (Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 - 3= ?, ? + 3 = 5

Source: This table is taken from the Common Core, Glossary, Table 1 (NGAC & CCSSO,

2010, p.88).

The part-whole model suggests the basic meaning of addition and subtraction. That is, addition refers to putting two parts together to find the whole while subtraction refers to taking away one part from the whole to obtain the other part.

The last problem type in the CCSS table is designated as the “compare” (or comparison) model. The comparison model is a more difficult variation of word problems in which bigger and smaller quantities are compared to each other rather than being combined to form a whole. In other words, the two given quantities are linked with a third quantity based on their relationships (more than, less than); yet, the smaller quantity does not belong to the bigger quantity. For instance, if one has five apples and eight oranges, one cannot “take away” five apples (smaller quantity) from eight oranges (bigger quantity) to obtain three oranges (difference). Since apples and oranges are different objects, children would be confused by apples being taken away from oranges. Likewise, it does not make sense to children how one can add three more oranges (difference) to five apples (smaller quantity) to obtain eight oranges (bigger quantity). It seems that children who have learned addition and subtraction based on the part- whole model might face challenges when making sense of comparison problems. Note that although the comparison model is more complex (Nunes et al., 2009), an understanding of this model still demands an understanding of the part-whole model. I will discuss this in Section 2.4.

Table 2.2 illustrates the structural similarities and differences of the word problems listed in the Common Core (see Table 2.1). These illustrations have varied names such as bar models or tape diagrams (Murata, 2008; NGAC 8c CCSSO, 2010) or Singapore models (Choo et ah, 2009). Cai and Moyer (2008) found that this type of model enables “pictorial equation solving,” which lays a foundation for students’ algebraic problem solving in Singapore textbooks (Ng 8c Lee, 2009). Returning to Table

2.1, each problem involves three basic quantities, any of which can serve as an unknown. Within each row, the inverse word problems can be illustrated with the same tape diagram (see Table 2.2). Flowever, across rows, the problem structures are changed and can be illustrated with different diagrams (see Table 2.2). For instance, the models for part-whole and comparison problems are distinct. An issue I noticed in classrooms is that when creating a group of word problems for the same fact family, teachers often allowed students to mix up the problem structures. I suggest that teachers encourage students to retain the same problem structure, which provides students an opportunity to make sense of inverse relations.

Now, with an understanding of the different types of inverse tasks and different problem structures, I will discuss how a teacher may approach an example task to teach inverse relations based on the part-whole and comparison models, respectively. Insights from the Chinese and U.S. lessons together illustrate the targeted approach of TEPS.

Tabic 2.2 Illustration of Additive Problem Structures

 Name in literature Name in ccss Context Quantity Illustration of structure Change Add to Bunny Initial bunnies (2) Bunnies joining in (3) Resulting bunnies (5) Take from Eating apples Initial apples (5) Apples eaten (2) Resulting apples (3) Combine (Part-Whole) Put together/ take apart Red and green apples Red apples (2) Green apples (3) Total apples (5) Compare (Comparison) Compare Lucy and Julie’s apples Lucy’s apples (2) Julie’s apples (5) Difference (3)