Abstract:

Equivalent fractions are fractions with the same numerical value, and the concept of which is built on the development of the relative-amount concept and multiplicative thinking. Previous studies have found that young children can solve non-symbolic equivalent fraction problems in an intuitive, global way, yet they often incorrectly make absolute-value judgments on problems involving discrete quantities. Boyer et al’s (2008) study suggested that it was due to an overextension of numerical equivalence concepts to proportional equivalence problems, but they did not make further analysis on the qualitative differences of concepts on equivalent fractions at lower grades of elementary school, nor did they explore the development process of the concept, while it is of great significance for guiding the early mathematical instruction. Our study examined first- through third-grade students’ operative thinking levels, and summarized the developmental pattern of the concept. Moreover, based on the principle of the zone of proximal development, two interventional experiments were carried out to improve first- and second-graders’ conceptual levels of equivalent fraction. Study 1 examined the operative thinking levels of first- through third-graders by employing the orange juice concentration matching task and analyzing their accurate rates and strategies in continuous, discrete and blended conditions, and on these grounds, we proposed a three-stage model for the development of equivalent fraction concept: the first stage is named the global-quantity concept, the second stage is the quantitative relative-amount concept, and the third stage is the formal concept of equivalent fraction. First-grade children, in the transitional period between the first to the second stage, have not yet formed the stable relative-amount concept; Second-grade children, in the transitional period between the second to the third stage, have developed a more mature relative-amount concept, but their multiplicative thinking has not yet developed. Study 2 verified the effectiveness of using children’s success on proportional problems involving continuous amounts to scaffold their performance on proportional problems involving discrete sets, which promoted the development of first-graders’ quantitative relative-amount concept. Study 3 brought the corresponding multiple-relationship between two dimensions in a certain situation to children’s attention, which improved second-graders’ levels of multiplicative thinking. In conclusion, this study proposed a three-stage model of the development of equivalent fraction concept, and further verified the model by two interventional experiments. The results proved the effectiveness of promoting first-graders’ quantitative relative-amount concept by using the intuitive global-quantity concept, as well as the significance of improving second-graders’ levels of multiplicative thinking by bringing the corresponding multiple-relationship between two dimensions to children’s attention.

Key words: equivalent fractions, relative-amount concept, multiplicative thinking, strategy, intervention