Knowledge in Brief
Difference between CP and AP of Addition
As mentioned in Section 4.1, many students (including undergraduate students) tend to conflate CP and AP, explaining that they thought both properties referred to changing the order with the sums remaining. Although this is true to a point, the difference lies in changing the order of “what” in each property. Considering a, b, and c as any arbitrary numbers in a given number system, the CP of addition states that a + b = b + a while AP of addition states that (a + b) + c = a + (b + c). In other words, CP deals with the changing of the order of “numbers” while AP deals with changing the order of “operations” (Ding et al., 2019). Consider, for an example, 2 + 3. If we apply the CP to compute, we may switch the numbers around to obtain the equation: 2+3=3+
2. However, if we add 2 + 3 + 4 and apply the AP, we will not change the order of numbers. Rather, we may either operate on the first two numbers or the latter two numbers with the results staying the same: (2 + 3) + 4 = 2 + (3 + 4).
CP and AP are conflated at the elementary school level by a common practice named the “any which way” rule (R. Howe, personal communication, October 21, 2016), that is, “the answer is the same as long as the expression contains the same operands regardless of any ordering.” According to this rule, when students add 2 + 3 + 4, they can add these three numbers in any order (e.g., 2 + 4 + 3, 4 + 2 + 3, 3 + 4 + 2). Even though students can always obtain the same answer, opportunities to deepen understanding of the CP and AP are lost in the vague justification provided by the any which way rule. Indeed, the act of switching the latter two numbers to produce an equation (e.g.,2 + 3 + 4 = 2 + 4 + 3) includes multiple applications of the CP and the AP. The process is illustrated below:
2 + 3 + 4
= (2 + 3)+ 4 Order of operations = 2+ (3 + 4) AP of addition = 2+ (4 + 3) CP of addition = (2 + 4) + 3 AP of addition = 2 + 4 + 3 Order of operations
The procedures and properties behind the any which way rule is a topic that is beyond the scope of this book. However, the point here is that educators (and textbooks) may be compounding student confusion between the CP and AP when teaching methods that informally use both properties without noting their distinct purposes. Moreover, it is worth pointing out that standard algorithms for the operations implicitly make heavy use of the CP and AP. For example, 236 + 742 = (200 + 30 + 6) + (700 + 40 + 2) = (200 + 700) + (30 + 40) + (6 + 2) = 900 + 70 + 8 = 978. Teachers themselves should be clear on how the basic properties are employed during these processes.
Expectations from the Common Core
According to the Common Core State Standards (NGAC & CCSSO, 2010, p.15), students in first grade should be able to apply the CP and AP as strategies to add and subtract:
If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) In addition, to add 2 + 6 + 4, students should know the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property' of addition.)
It is unclear if the CCSS expects students to know the names of the properties in first grade. However, I have not found clear expectations in the CCSS for later elementary grades either. Nevertheless, one may agree that at the very least teachers should be clear that the computational strategies they demonstrate (e.g., turning around, making a ten) are undergirded by these basic properties. For instance, in the above example of 2 + 6 + 4, one employs the AP of addition to make a ten to add, that is, 2 + 6 + 4 = (2 + 6) + 4 = 2 + (6 + 4) = 2 + 10 = 12.
An understanding of these properties is a precondition for teaching arithmetic in depth, which will likely contribute to children’s algebraic thinking. For instance, the above method of making a ten to add method can be applied to addition with regrouping such as9 + 4 = 9 + (l + 3) = (9 + 1) + 3 = 10 + 3 = 13. This method can also be used with higher addition such as 59 + 4, 59 + 14, and 119 + 14, which in turn supports the learning of standard algorithms. For instance, using the AP, one may solve 59 + 4 by either of the following:
59 + 4 = 59 + (1 + 3) = (59 +1) + 3 = 60 + 3 = 63 59 + 4 = (50 + 9) + 4 = 50 + (9 + 4) = 50 + 13 = 63
The former uses the method of making tens to add while the latter is aligned with the standard algorithm that involves the regrouping process. Both are undergirded by the AP. Of course, related to the method of “making a ten,” there is an inverse process of “unmaking a ten” that is related to subtraction. For instance, to solve 63 - 4, one may decompose 63 into 50+13 and then take away 4 from 13. This process again uses the AP and is aligned with the standard algorithm, 63-4 = (50+ 13)-4 = 50 + (13 - 4) = 50 + 9 = 591. Overall, the make/unmake a ten process offers a great opportunity to deal with the associative property' informally, present the idea of equality, and lay a foundation for later formal learning of the AP of addition.
The above knowledge components should be mastered by elementary teachers in order to develop students’ algebraic thinking. In fact, many US first- and second-grade textbooks explicitly present the CP and AP of addition and then revisit the properties in later grades. As such, it is reasonable to expect students to formally grasp these properties in elementary school and use them for computation.
Cultural Differences in “When” to Teach
As previously mentioned, there is a cross-cultural difference in “when” the properties are formally introduced to students. The U.S. textbooks often formally present the CP and AP of addition as early as grade 1 or 2. In contrast, the Chinese textbook series infuse these properties informally in these earlier grades. Then, in grade 4, students formally learn these properties in an intensive manner during which they will be distinguished and practiced. Of course, the discussion of cross-cultural textbook differences goes beyond the scope of this book. However, knowing the existence of these cultural differences will help provide a context to the instructional differences that the readers will observe in later sections. In addition, textbook presentations from both countries indicate the necessity of developing children’s understanding of basic properties in both informal and formal contexts. The question then becomes, how can teachers grasp these opportunities to develop children’s understanding? The following sections provide insights for teaching the CP (Section 4.3) and the AP (Section 4.4) from both countries.