Over the past few years, Asian students, especially students from Shanghai, China, always obtain extremely excellent performance in mathematics competitions (e.g., PISA, TIMSS, etc.) in comparison with their peers from other countries. A growing number of scholars hope to discover what can be learned from these high-scoring Asian education systems. In particular, the Department for Education (DfE) of the UK is adopting the Shanghai Mastery Pedagogy to improve the mathematics achievement of their students (Boylan et al., 2019).

        Distinguished from the English and other mathematic education practices, Shanghai whole-class interactive teaching aims to develop conceptual understanding and procedural fluency of students. Some big ideas, such as coherence, variation, representation, and structure, are promoted by mastery specialists (NCETM, 2017). Teaching is famous for its mathematically meaningful and coherent activities with well-designed models and examples systematically using variations. Actually, the characteristics mentioned above, especially the effective use of “variation”, are also noticeably emphasized in the exploration of Chinese mathematics teaching (Gu et al., 2004). Teaching with variation has almost become a common teaching routine for many Chinese mathematics teachers (Marton et al., 2004) and has been applied either consciously or intuitively for a long time in China (Li et al., 2011).

        The main research question of the study is: How do Chinese mathematics teachers make use of variation to foster student learning in their teaching? While there have been extensive studies on the effective use of variation in mathematics teaching, some gaps still exist in the following aspects. Firstly, most of the studies utilized one of the variation theories as the lens to analyze mathematics teaching in China (e.g., Qi et al., 2017; Mok, 2017; Häggström, 2008). However, insufficient attempts have been made to employ an integrated variation perspective based on several variation theories. Secondly, most of the existing studies adopted the approach of quasi-experimental or lesson study with intervention (e.g., Pang et al., 2017; Al-Murani, 2007; Kullberg, 2010). Nevertheless, very few studies adopted the naturalistic perspective to explore what actually happens in more authentic and diverse situations. Thirdly, due to the limited size of the research, some studies chose one or very few excellent public lessons or experimental lessons, even if a series of lessons were collected (e.g., Mok, 2017; Pang et al., 2016; Pang et al., 2017). The mathematics structures, relationships, and coherence within and between the sub-topics are not the major factor and draw little research attention, but they are actually the key ideas of Chinese pedagogy and the very essential platform for unfolding variation. Lastly, including the movement of the UK, most practices and studies were unfolded in a relatively primary or junior stage, while the senior-level mathematics knowledge and topics were less involved.

        The basis of the theoretical framework is the Variation Theory (VT) of Ference Marton. With the help of variation and invariance, students could “discern” the “critical aspects” of an “object of learning” with certain “patterns of variation” (Marton, 2015). The “critical aspect” in VT is considered identical with a dimension of variation (Pang & Ki, 2016). Watson and Mason (2005) further developed this concept with the term “dimension of possible variation”, associated with the notion of “range of permissible change” on the extension of Marton’s originally general notion. This extension captures the qualities of variation arising in mathematics (Mason et al., 2009) and better fits the nature of mathematics. Meanwhile, their concepts of example and example space are also elaborated in a mathematical manner. In addition, the analysis is also inspired by the Chinese theory “Bianshi Jiaoxue” (teaching with variation), which is developed by Chinese mathematics expert Gu Lingyuan (Gu et al., 2004).

To address the gaps mentioned previously, the current research aims to employ an integrated variation perspective based on several theories of variation to analyze the teaching of a mathematics topic over a series of around ten lessons in a naturalistic setting in Mainland China. Specifically speaking, the topic of function in the senior high school curriculum is chosen as the research target, which contains three consecutive sub-topics, namely power function, exponential function, and logarithmic function. The rich and complex mathematics relationships and connections (similar expressions, inverse relationships, etc.) between them enable the exploration of variation in an intertwined mathematics structure. The teachers participating in the study were six ordinary mathematics teachers in the local schools of three cities in China. The schools and teachers were chosen under the following criteria -- (1) following the national curriculum guide, (2) possessing high teaching standards, (3) being comparable between teachers (similar education background and teaching experience), (4) being comparable between classes (similar student achievements in mathematics). During the whole process of all lessons conducted in all classes, the video recording was used to collect the complete data, together with the semi-structured, qualitative classroom observations and field notes. Then, teachers were interviewed with the use of the technique of video-stimulated recall in the semi-structured approach. They were requested to discuss the reason for specific learning activities and their reflections on the incidents that happened during the lessons. Meanwhile, the issues observed by the researcher were further validated in the interviews. Student performances were collected by pre-test and post-test, the school’s mid-term and monthly tests. Furthermore, the survey to collect student-generated examples also provided the researcher with an effective approach to examine the example space of students. The data analysis was based on the integrated theoretical framework mentioned in the last section. Within each lesson, the analysis was carried out in detail in each teaching activity and example to explore how pedagogical actions enable students thoroughly experience the task and variation. After transcribing the video and audio recordings and calculating the test results, these data from different resources were aligned with and linked to the corresponding teaching activity. In accordance with the analysis, the comparisons were further conducted from various perspectives and layers, including the comparison within and across the teaching of different mathematics sub-topics of the same teacher and that of teachers in the same school and across schools.

This study provides an in-depth and extensive understanding of the effective use of variation in Chinese mathematics practices through the lens of a Chinese researcher. By employing an integrated variation perspective, the current research contributes to the development and refinement of theoretical frameworks and better fits the nature of mathematics learning and teaching. The analysis of variation can thus be unfolded comprehensively. The lessons conducted in a naturalistic setting enable the investigator to explore the authentic and rich teaching designs in the Chinese mainland without being limited by the existing theories. The thick descriptions and detailed interpretations allow readers to generalize and improve their research and teaching practices. Furthermore, special attention has been paid to the mathematics structures and relationships within and between the sub-topics, allowing a more systematic and intertwined perspective of variation. Based on the preliminary analysis, several teachers thoroughly used the connections between sub-topics to achieve transfer and coherence. From the perspective of variation, the same dimension(s) of variation was opened up in different sub-topics. For example, teachers constructed a similar routine to teach the properties, such as domain, range, monotonicity, parity, etc., of every type of function in a coherent way. Meantime, the different types of functions can be viewed as various values of the dimension of variation of function. Furthermore, the concept of “exponential function” was linked to its easily-confused concept of “power function”. The comparison between them highlighted the critical aspect of the independent variable (varied in each function) and also showed the same requirements of the critical aspects of coefficient and constant. The teaching of logarithmic function based on its inverse relationship with the exponential function enabled students to understand the mathematical essence of associated critical aspects. Therefore, it is meaningful to analyze the use of variation in a more comprehensive manner.

Boylan, M., Wolstenholme, C., Maxwell, B., Demack, S., Jay, T., Reaney, S., & Adams, G. (2019). Longitudinal evaluation of the Mathematics Teacher Exchange: China-England-Final Report. Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: perspectives from insiders (pp. 309–347). Singapore: World Scientific. Li, J., Peng, A., & Song, N. (2011). Teaching algebraic equations with variation in Chinese classroom. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 529–556). New York, NY: Springer. Kullberg, A., Watson, A., & Mason, J. (2009). Variation within, and covariation between, representations. In Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 433-440). Thessaloniki: PME. Marton, F. (2015). Necessary conditions of learning. London: Routledge. Marton, F., Runesson, U., & Tsui, A. (2004). The space for learning. In F. Marton & A. Tsui (Eds.), Classroom discourse and the space for learning (pp. 3–40). Mahwah, NJ: Lawrence Erlbaum Associates Inc. Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10-32. Mok, I. A. C. (2017). Teaching Algebra through Variations: Contrast, Generalization, Fusion, and Separation. In Huang, R., & Li, Y. (Eds.), Teaching and Learning Mathematics through Variation: Confucian heritage meets western theories (pp. 187-205). Rotterdam, The Netherlands: Sense Publishers. Pang, M. F., & Ki, W. W. (2016). Revisiting the Idea of “Critical Aspects”. Scandinavian Journal of Educational Research, 60(3), 323-336. Pang, M. F., Marton, F., Bao, J. S., & Ki, W. W. (2016). Teaching to add three-digit numbers in Hong Kong and Shanghai: illustration of differences in the systematic use of variation and invariance. ZDM, 48(4), 455-470. Qi, C., Wang, R., Mok, I. A. C., & Huang, D. (2017). Teaching the Formula of Perfect Square through Bianshi Teaching. In Huang, R., & Li, Y. (Eds.), Teaching and Learning Mathematics through Variation: Confucian heritage meets western theories (pp. 127-150). Rotterdam, The Netherlands: Sense Publishers. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Erlbaum.