Chinese Students’ Hierarchical Understanding of Part-whole and Measure Subconstructs

Abstract

Fractions are a very important but difficult topic in elementary and secondary mathematics curricula around the world. The difficulty in teaching and learning fractions is usually caused by their various subconstructs. To facilitate learning of fractions, a test consisting of 20 items was developed to examine 310 sixth graders’ hierarchical understanding of the part-whole and measure subconstructs of fractions. Rasch analysis and cluster analysis were used to analyze the data. The results showed six distinguishable levels of understanding of the two subconstructs. Briefly, constructing fractions alone was the easiest to develop, followed by repartitioning/reconstructing and regarding fractions as numbers. The abilities involved at the highest level were successively partitioning, defining a unit interval freely and placing fractions on number lines. The results also indicated that most of the students were able to deal with part-whole subconstruct problems but still had difficulty handling measure subconstruct problems by the end of their elementary education. They could construct fractions/the whole, recognize the number notion of fractions and fractional unit and measure distances, but lacked enough ability to partition successively and place fractions on number lines. This study provides a clearer picture of hierarchy of understanding the two subconstructs and contributes to the corresponding development of instructional resources and the teaching/learning of fractions.