Pre-service Teachers' Ways of Thinking of Qualitative Graph Construction in a Continuous Covariation Situation

Abstract

In Middle School Mathematics 1 of the 2015 revised Korean mathematics curriculum, before the definition of a function, the task of constructing a qualitative graph of the relationship between two changing quantities has been newly proposed. In relation to the changes in the 2015 revised Korean mathematics curriculum, this study explored the reasoning of pre-service teachers revealed in the process of constructing a qualitative graph of the relationship between the two quantities that are continuously changing. Most pre-service teachers constructed graphs by focusing on the covariation than the correspondence on the relationship between the two quantities. Additionally, they successfully reasoned the global shape of the graph by forming the relationship between the two variables as a multiplicative object. However, many pre-service teachers showed difficulty in judging or not judging the differentiable at the point which changes the local shape of graph. Some pre-service teachers thought that the slope of the tangent at the inflection point was zero or invoked time even if it doesn"t require. The results in this study suggest that 1) in a continuous covariation situation, the formation of a multiplicative object such as a slope is a critical factor in the process of constructing the global shape of the graph, 2) smooth continuous covariation reasoning is a critical factor in the process of constructing the local shape of the graph, 3) covariation reasoning has a level difference in the process of constructing the global and local shape of the qualitative graph.

Keywords

• qualitative graph
• covariational reasoning
• ways of thinking
• smooth continuous covariation