Investigating math students' ability to conceptualize fractions as two numerals having a single value

William Carroll Fanning, University of Texas at El Paso


It is no secret that math students of all ages have misconceptions about fractions. Multiple studies have shown that math students invariably see fractions as two separate and unrelated numbers, the numerator and the denominator, rather than a single number with specific value. This study is designed to investigate students’ interpretations of a fraction in terms of having a single value in relation to its two numbers, the numerator and the denominator. For the purpose of this study, the investigator developed and pilot-tested seven items that have the potential to uncover whether students attend to the value of a fraction when asked to reason about fractions in various situations. Twenty students were interviewed using these seven items. Twelve developmental algebra students at a college and eight 7th grade students from a private school were interviewed, each for approximately thirty minutes and the conversations were audio-recorded. Over 200 student responses were analyzed and categorized. Nearly 50 subcategories were identified. These subcategories suggested five progressive levels of understanding of fractions: Level 1—Fraction as merely two independent numerals almost unrelated; Level 2—Fraction as two independent numerals related as a part-whole concept; Level 3 —Procedural conversion of fraction without attending to value; Level 4—Fraction as having a general sense of value of fraction with some connection to the two numbers; Level 5—Fraction as a single identifiable value with connection to the two numbers. There are some differences between the college student responses and the 7th grade student responses. The 7th graders have an overall higher level of understanding and they fared better in four of the seven items. The college students outperformed the 7th graders on only one item. Perhaps this study will inspire further research on conceptual challenges students face in reasoning with a fraction as a conceptual entity that simultaneously has two numbers but a single value, which quantifies the multiplicative relation between the numerator and denominator.

Subject Area

Mathematics education

Recommended Citation

Fanning, William Carroll, "Investigating math students' ability to conceptualize fractions as two numerals having a single value" (2016). ETD Collection for University of Texas, El Paso. AAI10118205.