The use of variation and connections in Chinese mathematics lessons

Abstract

The current research is dedicated to obtaining a more thorough understanding of the use of variation and connections in actual mathematics teaching practices in China, through a long period of engagement and using various methods of data collection. The research object is the topic of functions in the senior secondary school curriculum, which features rich and complex mathematics relationships and connections. The participants were six normal mathematics teachers at three schools located in three cities in China. Several theories and essential concepts are utilized inclusively in the analysis to reveal teaching practices in a more comprehensive manner. The study finds that the teachers provided fruitful examples to broaden the range of change of the exponent and to establish contrasts to address each critical aspect of the power function. For the graphs and properties, under the categorization approach, the discernment relied heavily on the selected examples and the manner in which they were presented, with the following three specific designs contributing to an improved understanding. First, the teachers provided at least two examples for each category. Second, in presenting these carefully selected examples, juxtaposed geometric representations were used to provide intuitive contrast and generalization. Third, examples were reused across the teaching of various critical aspects but were categorized differently based on different criteria. This categorization approach was again highlighted in the teaching of exponential and logarithmic functions. The properties were explored by presenting pairs of functions with carefully selected bases for contrast and generalization. Three important elements of the teaching were identified as being useful to connect the different types of functions and help students better grasp the mathematical essence. First, each lesson should not remain at the micro level of the critical aspect to be conveyed in that lesson; it should extend to the macro level of functional thinking. Focusing on learning an entire topic with a holistic approach allows for a more refined and precise “axis” for combing the learning of each sub-topic while preparing learners for future challenges. Second, teachers play an important role in making decisions on the selection and arrangement of tasks to guarantee the availability of necessary variations for discerning critical aspects and in presenting them differently during lessons. The presentation requires certain adjustments to be made according to the students’ responses and the explicit direction of the students’ attention to the connections between sub-topics, especially inherent and specific connections. The internal, inherent, and in-depth mathematical connections between specific types of functions play a more important role than the aforementioned general connection. Third, the fusion and combination of algebraic and geometric representations helps learners to discern critical aspects and establish connections between sub-topics. In conclusion, the underlying idea of the teaching design for Chinese mathematics is to view a topic with multiple sub-topics as a whole that contains interwoven connections and relationships. The sub-topics are like similar ropes spiraling up in a similar way, intertwining and interacting with each other and finally twisting into a thicker rope, with the in-depth mathematical connections as horizontal knots.