Abstract
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 part-whole 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.
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Notes
This written assessment test was constructed by the author and supported by an interdisciplinary research team that conducted a bigger project (of which this test was also part) at Human Computer Interaction Institute at Carnegie Mellon University.
Thanks to Profs. Margaret Smith and Kathy Crammer, retired teacher Jay Raspat for their valuable feedback and to the project members at Carnegie Mellon University: Vincent Aleven, Nikol Rummel and Martina Rau for their contributions and support.
Vincent Aleven and Nikol Rummel were the principal investigators of this US National Science Foundation funded project. I was the post-doctoral researcher on this interdisciplinary project responsible for the construction of the mathematics test and the mathematical content of the Tutor.
Martina Rau, who was then the graduate student in the Fractions Tutor Project, conducted the statistical tests.
Each fractional number was presented with each representation using three forms of the test for all the Problem Types.
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Acknowledgments
This work was supported by the United States National Science Foundation, # DRL-0910010. Thanks are due to the students, teachers, and school principals who participated in this study. This article is based on the presentation made at the 35th Conference of the International Group for the Psychology of Mathematics Education. Ankara, Turkey.
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Tunç-Pekkan, Z. An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations. Educ Stud Math 89, 419–441 (2015). https://doi.org/10.1007/s10649-015-9606-2
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DOI: https://doi.org/10.1007/s10649-015-9606-2