Prospective Teachers’ Graphical Representations for Two Simultaneously Changing Quantities in Dynamic Functional Situations

This study aims to reveal how prospective teachers express the relationships between variables through graphical representations when interpreting dynamic functional situations involving two simultaneously changing quantities. 100 prospective middle school mathematics teachers participated to this case study. The data consisted of prospective teachers’ written responses to the task involving filling bottles with water and the graphs of volume as a function of height and clinical interviews were used to examine their covariational reasoning and graphing abilities. The findings showed that only six prospective teachers' graphical representations were correct for both dynamic functional situations. The most significant and common problems in the graphical representations were found such as (i) inability to coordinate slopes for linear relationships between variables, (ii) representing nonlinear relations of variables as linear relations, (iii) reversing the roles of dependent and independent variables, (iv) representing the relationship between variables as decreasing rather than increasing and (v) representing the relationships between variables to include more or less partitions to the graph than required by the given dynamic functional situation.