An analysis of elementary school children's fractional knowledge depicted with circle, rectangle, and number line representations

It is now well known that fractions are difficult concepts to learn as well as to teach. Teachers usually use circular pies, rectangular shapes and number lines on the paper as teaching tools for fraction instruction. This article contributes to the field by investigating how the widely used three external graphical representations (i.e., circle, rectangle, number line) relate to students' fractional knowledge and vice versa. For understanding this situation, a test using three representations with the same fractional knowledge framed within Fractional Scheme Theory was developed. Six-hundred and fifty-six 4th and 5th grade US students took the test. A statistical analysis of six fractional Problem Types, each with three external graphical representations (a total of 18 problems) was conducted. The findings indicate that students showed similar performance in circle and rectangle items that required using partwhole fractional reasoning, but students' performance was significantly lower on the items with number line graphical representation across the Problem Types. In addition, regardless of the representation, their performance was lower on items requiring more advanced fractional thinking compared to part-whole reasoning. Possible reasons are discussed and suggestions for teaching fractions with graphical representations are presented.