Elementary Students' Understanding of Inverse Relations in the United States and China

• In Event: Early Algebra in Elementary School: A Cross-Cultural Perspective

Abstract

Objectives

This study explores US and Chinese children’s strategies to solve additive inverse problems and their growth in understanding inverse relations as indicated by their strategies after one year of instruction.

Theoretical Framework

The inverse relation between addition and subtraction (additive inverses) is a critical topic in lower elementary grades (Baroody, 1987, 1999; Carpenter, Franke, & Levi, 2003). However, overwhelming evidence shows that children generally lack understanding of this relation (Baroody et al. 1983). Broody (1999) noted that children who solved two related operations (e.g., 9+3=12, 12-3=9) did not think that addition helped them solve subtraction. In addition, students could not spontaneously use addition to solve subtraction (Torbeyns, De Smedt, Ghesquière, & Verschaffel, 2009) or use addition to check subtraction (Baroody, 1987). Thus, instruction of inverse relations should target foundational pieces such as ‘part-whole’ schema (Canobi 2005; Risnick, 1983) through varied contexts over time (Bisanz, Watchorn, Piatt, & Sherman, 2009).

Methods

This cross-cultural study involves a total of 171 first and second grade Chinese students from four urban schools in China (NG1=81, NG2=90) and 113 US students from four urban schools (NG1=61, NG2=52). To analyze students’ strategies and reasoning process in solving additive inverse problems, five items that require explanations were selected from both a pre- and posttest given during the 2014-15 school year (see Figure 4). Each item was adapted from the literature (Broody, 1987, 1999; Nunes et al. 2009, Resnick, 1989, Torbeyns et al. 2009). Students’ strategies were coded based on inverse thinking, direct thinking, other conceptual strategies, and none/empty explanations for both correct and incorrect solutions (see Table 5 for examples).

Results

Both US and Chinese students made progress from pretests to posttests as indicated by the number of correct solutions (USpre=47%, USpst=65%; Chinapre=82%, Chinapst=86% see Table 6). Specifically, US students’ use of inverse-based strategies increased on average from 11% to 26% and Chinese use increased from 43% to 60%.

Students’ strategies across items revealed that for Q6 (81-79) few students employed inverse-based strategies (Table 6), which agrees with Torbeyns et al. (2009). However, for contextual problems (e.g., Q8a), students in both countries tended to use inverse-based strategies more frequently (Table 6). Perhaps contextual problems may better activate students’ part-whole schema, the key to understanding inverse relations (Risnick, 1983). An apparent cross-cultural difference found is that US students mainly used concrete drawings (e.g., circles) while Chinese students mainly wrote abstract number sentences (e.g., 6 + 2 = 8) to represent solutions.

Finally, US students demonstrated growth in using inverse-based strategies from grade 1 to 2 whereas Chinese students seemed to achieve sound understanding by the end of grade 1 ( see Figure 5). This most likely is due to curriculum differences; Chinese curriculum focuses on learning multiplication in grade 2.

Significance

Few studies have explored students’ understanding of inverse relations from a cross-cultural perspective. This study reveals current students’ understanding of inverse relations and suggests future research on teaching and learning additive inverses is needed.

Authors

• Xiaobao Li, Widener University

• Ryan Scott Hassler, Temple University

• Meixia Ding, Temple University