An analysis of inverse relations in the US and Chinese textbooks

Abstract

This study examines presentations of inverse relations in two U.S. elementary text series and one main Chinese text series. In general, the U.S. texts were found to resemble each other but to differ considerably from the Chinese text series in terms of types of tasks and representation uses across grades.

 

Theoretical Framework

Inverse relations between addition and subtraction and between multiplication and division are fundamental relationships in learning mathematics (e.g., Common Core State Standards Initiatives, 2010; Nunes, Bryant, & Watson, 2009; Piaget, 1952; Resnick, 1983; Zhou & Peverly, 2005). Without understanding these key relations, students are not able to develop essential additive or multiplicative reasoning skills (Nesher & Katriel, 1977; Nunes et al., 2009; Vergnard, 1988). An understanding of inverse relations can develop students’ reasoning abilities such as “if a+b=c, then a=c-b and b=c-a” and “if a×b=c, then a=c÷b for b≠0, and b=c÷a for a≠0” (Carpenter, Franke, & Levi, 2003), which also indicates a fundamental algebraic shift due to a structural perspective on number sentences (Nunes et al., 2009).

 

Prior studies reported that students who had learned strategies based on inverse relations could not spontaneously use them to solve new problems (Baroody, 1987; Torbeyns, De Smedt, Stassens, Ghesquière, & Verschaffel, 2009). Researchers (e.g., Torbeyns et al., 2009) pointed out that quality of instruction was a crucial factor that may hinder students’ development of mastery levels of understanding. These researchers called for interventions that may support meaningful and flexible learning. Empirical results also show that the learning of fundamental relations extends over a long period of school experience (Vergnard, 1988). However, prior studies lack exploration of how inverse relations can be developed across grades to reach a mastery level of understanding.

 

In this study, we explore instructional approaches that will allow students’ to develop adaptive expertise (Baroody, 2003) of inverse relations. In particular, we compare US and Chinese textbook presentations. Since little research has systematically studied inverse relations, findings from this study will contribute to not only the literature, but also to practices by offering alternatives that may be useful for instruction and textbook designs. In addition, our findings may be further developed into new interventions that involve instructional environments to support students’ learning of inverse relations.

Methods

The Institute of Education Sciences (IES) (Pashler et al., 2007) recommended instructional principles to support learning of key concepts and principles, including (a) interweaving worked examples and practice problems, (b) connecting concrete and abstract representations, and (c) spacing learning over time. Based on this framework, we ask:

1.  What types of examples and practices are used to learn and understand inverse relations? How are these tasks connected and changed over grades?

2.  What types of concrete and abstract representations are used to learn and understand inverse relations? How are types of representations connected and changed over time?

We explore these questions through a comparative textbook analysis. We selected one representative Chinese textbook series (grades 1-6), Jiang Su Educational Press textbook (JSEP) (Su & Wang, 2005). For comparison, we selected one traditional and one reform US textbook: Houghton Mifflin (HM, Greenes et al., 2005) and Everyday Mathematics (EM, University of Chicago School Mathematics Project, 2007). Comparative analysis may bring alternative findings. Examining textbooks across grades may systematically show how a mastery level of understanding can be reached.

Both quantitative and qualitative analysis will be used. All instances will be first coded and reliability checking with be conducted (at least 85%). These instances will be classified into computation and non-computation contexts and then into detailed problem types. Further, representation used in each instance will be classified as either concrete or abstract. Frequency and sequence of using types of tasks and representations will be analyzed across grades. The above analysis will be applied to both inverse relation cases with common features identified.

Partial Results and Conclusions

We have analyzed the Chinese textbook series. For US textbooks, we are still in the process of analysis. Below are partial results (mainly about Chinese textbooks):

The types of tasks

The US textbooks align well with current literature by offering the “fact family” as main contexts. Although Chinese textbooks also frequently use this type of problem, it also uses “fact family in context.” This problem type situates a fact family in a concrete situation which leads to “inverse word problems.” In addition, there were other types of word problems such as “separate problems linked by related situations.” Although these problems may not include the exact same quantities, a comparison of the sub-problems may focus students’ attention to the inverse “relationships” among quantities. Such an emphasis on relationships beyond quantities and numbers is highly recommended by Nunes et al. (2009). Table 1 and Figure 1 provide some details:

Table 1. Frequencies of each type of problem

 

Figure 1. Instances of inverse operations across grade

Note. “1(1)” denotes “grade 1(volume 1).”

 

The representation uses

Both US textbooks mainly presented inverse relations in computational contexts. Although HM presented the “fact family” under word problems, concrete situations were not used to help students make sense of inverse relations. EM consistently taught “fact triangles” where three numbers were arranged to form fact families. Although EM included problem solving contexts, these concrete situations were seldom used for teaching inverse relations. In contrast, the Chinese textbooks presented all worked examples in familiar concrete situations, helping students make sense of inverse relations. Another feature of representation uses of Chinese textbooks is the connections between concrete and symbolic representations, particularly, from concrete into abstract. Table 2 illustrates the distribution of Chinese textbooks’ representations.

 

Table 2. Distribution of representation uses

Educational Importance
This study brings alternatives that may advance current literature and practices, improving the procedural-based instructions of inverse relations (Baroody, 1987). The Chinese textbook approach also illustrates what is meant by “focused and coherent” textbook presentation. Since a common issue of US students’ mathematical learning is the lack of flexible understanding of key relations, our study focusing on inverse relations not only targets a significant topic but showcases approaches to developing  adaptive expertise of fundamental mathematical ideas.

Organization of the Session
We will present our paper in 15 minutes involving three authors. For two round-table discussions, in addition to answering audience’s questions, we will prepare questions seeking audience’s suggestions to improve this study and further this line of research.

 

References

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Common Core State Standards Initiative (2010). Common core state standards for mathematics.  Retrieved December 9, 2010 from http://www.corestandards.org/the-standards.

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