A Mixed Methods Study of Chinese Students' Construction of Fraction Schemes: Extending the Written Test with Follow-Up Clinical Interviews
Understanding fractions is fundamental for expanding number knowledge from the whole number system to the rational number system. According to the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (2000), learning fractions is an important mathematical goal for students in grades three through five in the U.S. Moreover, the NCTM suggests that fraction instruction start in Pre-K and continue through 8th grade. At the same time, the Common Core State Standards for Mathematics (CCSSM) suggests that fraction instruction should occur from Grade 3 to 7. In contrast to the time spent on learning fractions in the U.S., students in China spend a relatively short time learning fractions (Zhang and Siegler, 2022). According to the Chinese national curriculum standards, the Chinese National Mathematics Curriculum Standards (CNMCS) for five-four system, the fundamental fraction concepts are taught in grades 3 and 5 only. However, Chinese students continue to have higher performance on fraction items in international assessments when compared with American students (Fan and Zhu, 2004). Consequently, over the last several years, researchers have investigated subject content knowledge and pedagogical content knowledge of Chinese in-service teachers and pre-service teachers via fraction division (e.g., Li and Huang, 2008; Ma, 1999). There are also studies exploring Chinese written curricula of fraction division (e. g., Li, Zhang, and Ma, 2009). Recently, a quantitative study from Norton, Wilkins, and Xu (2018) investigated the process of Chinese students' construction of fraction knowledge through the lens of fraction schemes, a model established by western scholars Steffe (2002) and his colleague Olive (Steffe and Olive, 2010). However, there is a lack of qualitative research that attempts to use fraction schemes as an explanatory framework to interpret the process of Chinese students' construction of fraction knowledge. The main purpose of this study was to investigate Chinese students' understanding of the fundamental fraction knowledge in terms of their understanding of the "fraction unit," referred to as a "unit fraction" in the U.S., using Steffe and Olive's (2010) fraction schemes as the conceptual framework. A sequential mixed methods design was used in this study. The design included two consecutive phases, namely a quantitative phase followed by a qualitative phase (Creswell and Plano Clark, 2011). During the quantitative phase, five hundred and thirty-four Chinese fourth and fifth grade students were administered an assessment. The quantitative data was first analyzed using a Cochran's Q test to determine if the Chinese participants in this study follow the same progression of fraction schemes as their American peers. Results indicate that the development of fractional schemes among Chinese 4th and 5th grade participants in this study is similar to their U.S. counterparts and the Chinese participants in Norton et al.'s (2018) study regardless of the curricula differences across countries or areas in the same country, the textbook differences, and the language differences. Next, two different analysis of variances (ANOVA), a three-way mixed ANOVA and a two-way repeated measures ANOVA were conducted. The three-way mixed ANOVA was used to inform the researcher as to the fraction schemes these students had constructed before the concept of fraction unit is formally introduced and after the concept of fraction unit is formally introduced. The results showed that the fraction knowledge of the students in this study developed from 4th grade to 5th grade. The analysis of clinical interview data confirmed this conclusion. The two-way repeated measures ANOVA was used to determine which model (i.e., linear, circular, or rectangular) is more or less problematic for Chinese students when solving fraction tasks. The results suggest that generally students' performance on linear model tasks was better than their performance on circular model tasks, but there was no statistically significant difference between performance on circular model and its corresponding rectangular model tasks. The results from the quantitative analyses were also used to screen students to form groups based on their highest available fraction scheme for a clinical interview in the second phase, the qualitative phase. In the qualitative phase, a clinical interview using a think-aloud method was used to gain insight into the role of students' conceptual understanding of the fraction unit in their construction of fraction knowledge. In this phase, students were asked to solve the tasks in the clinical interview protocol using the think aloud method. Two main findings were revealed analyzing the clinical interview data. First, a conceptual understanding of fraction units as well as a conceptual understanding of a unit whole play a critical role in the construction of Chinese students' fraction knowledge. Second, the lack of the understanding of a fraction unit as an iterable unit may be one of the reasons that obstructs students move from part-whole concept of fractions to the measurement concept of fractions. This study also demonstrates that a conceptual understanding of fraction units and the unit whole are a necessary condition for constructing of a conceptual understanding of fraction knowledge. Thus, implications of this study suggest that teachers not only should help students build a conceptual understanding of fraction units, but also need to confirm that students have constructed the concept of what the unit whole is before asking students to identify the fraction units for the referent whole. On the other hand, the tasks used in the present study only include continuous but not discrete wholes. Therefore, future research may focus on investigating how students identify fraction units and in what way the iterating operation could be used when students encounter a discrete whole.