Applying van Hiele’s Levels to Basic Research on the Difficulty Factors behind Understanding Functions

Functions is considered an important mathematical literacy concept within the Organization for Economic Cooperation and Development’s (OECD’s) Programme for International Student Assessment (PISA), and it has been shown that Japanese junior high school students are experiencing problems understanding functions. This paper examines the difficulty factors behind the understanding of functions by referring to van Hiele’s theory of learning levels. This paper focuses on the prototypical stages for understanding functions from the perspective of the mathematical concept process model of gradual understanding: ‘[I] Extract a variate from a phenomenon and [II] Relate the 2 extracted variates’. The subjects of the study were junior high school students, who completed a questionnaire. The results of the analysis of the questionnaire responses found that for a certain number of students, concept formation for stages [I] and [II] was lacking, and that the situation was not necessarily improving as the class progressed, thus, suggesting that this may be a difficulty factor that affects the understanding of functions.