Linguistic Facilitation of Understanding of Percentages

• In Event: 1-082 - Beyond Magnitude: Multiple Interpretations of Rational Numbers

Integrative Statement

The importance of mastering rational numbers and the failure to do so by many students have stimulated a large volume of research on fractions and decimals (e.g., Behr et al., 1984; Desmet et al., 2010; Siegler et al., 2011). However, little is known about how people understand percentages (Tian & Siegler, 2017). This paper presents two experiments examining children’s and pre-service teachers’ knowledge of percentages, “percent of a whole” problems in particular. We hypothesized that phrasing the problems in an everyday linguistic context (e.g., “35% of 78”) leads to better performance than presenting the problems with the multiplication sign (e.g., 35% × 78). Moreover, we examined whether the influence of phrasing on performance, if any, was specific to percentages or general to all rational number notations.

In Experiment 1, we asked 6th and 8th graders to judge whether the answer to a multiplication problem involving a whole number and a rational number would be greater than the whole number. Children were randomly assigned to three conditions, one for each rational number notation (decimals, fractions, and percentages). Each child completed problems both with the “of” expression and with the multiplication sign. Results showed that the “of” phrasing led to higher accuracy on percentage problems (F (1, 73) = 9.46, p = .003, η_p^2 = .11), though not on problems with decimals or fractions

In Experiment 2, we examined whether the advantage of the “of” expression in understanding percentage multiplication would extend to multiplication estimation. A previous pilot study showed that middle school students had much difficulty with percentage multiplication estimation, which led us to doubt whether pre-service teachers had such knowledge to teach their future students. Small groups of pre-service teachers were asked to estimate answers to multiplication problems involving a whole number and a percentage. Each group was randomly assigned to an “of” condition, in which problems were presented with the “of” expression, or a “multiplication-sign” condition, in which problems were presented with the multiplication sign. Results showed that pre-service teachers’ estimates were more accurate with the “of” expression, as compared to with the multiplication sign (β = 0.68, SE = 0.28, t (76.74) = 2.34, p = .02).

Finding a percent of a whole number requires converting the problem into an arithmetic operation and then solving the arithmetic problem. This conversion process might introduce errors if an arithmetic operation other than multiplication is chosen. However, in the current study, children and pre-service teachers demonstrated better understanding of percentage multiplication when the problem was presented with the “of” expression than with the multiplication sign. It is likely that children and pre-service teachers had knowledge about “percent of a whole”, but did not spontaneously apply that knowledge to problems presented in the context of the multiplication sign. The knowledge of “percent of a whole” might be developed from daily life, as phrases like “percent of” are commonly used. Educational implications of these findings will be discussed.

Authors

• Jing Tian, Teachers College, Columbia University
  Presenting Author

• Robert Siegler, Professor
  Non-Presenting Author