Inverse Relations between Multiplication and Division
In this chapter, I will move from additive to multiplicative inverses. Similar to additive inverses, multiplicative inverses are one of the critical ideas that help to develop students’ algebraic thinking in early grades (Cai & Moyer, 2008; Carpenter et al., 2003). The links between Chapters 2 and 3 highlight the vertical connections that exist between relevant mathematics concepts over time. This chapter is organized in the same way as Chapter 2. First, I will start with examples of student work and introduce the necessary mathematical knowledge pieces in brief. Then, I will share instructional insights for teaching multiplicative inverses based on three models emphasized by the Common Core: equal-groups, arrays, and comparisons. All these insights are gleaned from the U.S. and die Chinese lessons, which together illustrate the approach of TEPS.
A Glimpse of Student Work
The project designed an instrument that contains eight items to assess students’ understanding of multiplicative inverses. Given that multiplication is taught in G2 in China and in G3 in the U.S., we invited Chinese G2-3 students and U.S. G3-4 students to take this test. Overall, we observed a similar pattern in student performance as with the additive tests (see Section 2.1). That is, the U.S. students demonstrated an increase in understanding over two years. On the other hand, the Chinese G2 students had a better initial understanding which improved over the first year; they continued to maintain that higher level of understanding in the following year. Similar to the additive instrument, we asked students to write fact families based on given contexts (see Figure 3.1). Most of the Chinese G2 students seemed to already know the basic multiplication problems and their performance reached a ceiling effect1. For instance, with regard to Ql, the correctness of Chinese G2 students in pre- and post-tests was 97.65% and 97.67%, respectively. This is different from the U.S. situation where the correctness rate for G3 students was 31.63% in the pretest and 77.60% in the posttest. Below are typical U.S. student mistakes (see Figure 3.1). The first task shows two plates of seven strawberries, and the second task shows 10 blocks arranged in a 2-by-5 array. Students were asked to write a fact family based on the given pictures.
Figure 3.1 Typical U.S. student mistakes in the “fact family in context” items.
In Figure 3.1, one can see a disconnect between the concrete and abstract representations. For instance, the first student (SI) wrote a division equation (2 -r 14 = 7) for the strawberry task, which indicates a lack of understanding about the meaning of division—sharing a total of 14 strawberries evenly between two plates. Furthermore, for the array model in task 2 above, the student could only grasp the meaning of the picture when using additive thinking (counting each row to obtain 5 and 5) and not when using multiplicative thinking (2 groups of 5). In addition to the disconnect between concrete and abstract representations, students’ awareness of the relationship between multiplication and division also appeared to be undeveloped. For instance, even though SI wrote a correct multiplication sentence (2 x 7) for the first task, one of the two generated division sentences was incorrect. S2 also did not seem to recognize the inverse relationships despite the context from the picture.
Another example from the instrument is a group of inverse word problems that involve the comparison model (elaborated in Section 3.2). Each problem is situated in a real-world context that compares Hillary and Geoff’s money spent.
a Hillary spent S9 on Christmas gifts for her family. Geoff spent 3 times as much as Hillary. How much did Geoff spend? b Hillary spent S9 on Christmas gifts for her family. Geoff spent S27.
How many times as much did Geoff spend? c Hillary spent some money on Christmas gifts for her family. Geoff spent 3 times as much as Hillary. If Geoff spent S27, how much did Hillary spend?
The first word problem is a multiplication problem and the next two are division problems. The solutions to these problems form a fact family: 3 x 9 = 27, 27 -r 9 = 3, and 27 -r 3 = 9. However, these comparison problems can present a challenge if students don’t understand the problem structure and cannot discern the embedded inverse relationship. In my project, the correctness rate on the pre-test for these three problems was
60%-80% for Chinese G2 students and 24%-45% for U.S. G3 students. After one year of learning, Chinese G2 students’ performance reached the ceiling effect (the correctness rate was 95%-97%). While U.S. G3 students’ understanding was greatly enhanced, there was also room for improvement (correctness rate was 65%-77%). In addition, the U.S. G4 student sample demonstrated better understanding for problem (a) than (b) and (c). The correctness rate for the three problems were 98%, 73%, and 69%, respectively. Many U.S. G4 students could obtain a correct answer for problem (a) using multiplication (9x3 = 27)2 or repeated addition (9 + 9 + 9 = 27). However, they overgeneralized this type of thinking to the other two problems. For instance, some students responded to problem (b) by adding 27 nine times or using 27 + 9 = 36. Likewise, they added 27 three times or added 27 to other numbers to solve problem (c). These responses indicate that many students couldn’t discern the existence of multiplicative inverses in the problems. Indeed, without a solid understanding of the concept of “times” and the associated quantitative relationships, multiplicative inverses are hard for students to grasp in a meaningful, structural way.
Students without a meaningful understanding of multiplicative inverses might not be able to automatically apply this relationship to solve new problems. Below is a computational problem in which students are expected to first solve a group of related facts and then explain how they get the division answer:
- 3x7=( )
- 21 + 7 = ( )
How did you get the answer for 21 -s- 7 = ( )
Students with inverse thinking should be able to use the multiplication fact to solve the related division fact without any extra computational effort. However, in my project, some students who attempted the problem ignored the multiplication fact 3x7 = 21. Rather, they used extra effort to figure out the division problem 21 -r 7 = ( ) listed right beneath the multiplication fact (Li et al., 2016)3. Typical explanations included:
SI: I made 3 groups and I put 7 in each group.
S2: I counted by 7s.
Based on our analysis of student work, we found that an understanding of multiplicative inverses does not occur automatically for students when they acquire early multiplication skills. Concrete pictorial/real-world situations often end up disconnected from numerical forms. This may result in an inert knowledge of multiplication facts that cannot be retrieved to solve new problems. To better develop students’ understanding of multiplicative inverses, teachers need to grasp several key knowledge points: the meaning of multiplication, the structure of multiplication problems, and the types of multiplicative inverses. I will discuss these elements in Section 3.2.