Comparing Visual Representations of Fractions: Discrete Segments and Length Differences Influence Children’s Strategy Use

• In Event: Poster Session 05
  In Poster Session: PS 05 Topic - Education, Schooling

Abstract

Fractions are notoriously difficult for children to learn, but proficiency with fractions is key to later mathematics success (e.g., Booth, Newton & Twiss-Garrity, 2014; Siegler et al., 2012), positioning fraction education as a crucial target for improved curriculum. One representation commonly used in teaching fractions is a “tape diagram” (Figure 1). Although a promising tool, little is known about how characteristics (like length and segmentation) of tape diagrams affect how students approach problems that use them.
Discrete segments in visual representations of fractions lead children to compare them less accurately, perhaps by encouraging erroneous counting procedures (Boyer & Levine, 2012). Children also struggle when bars being compared have different lengths, with greater challenges as length difference increases (Boyer & Levine, 2012).
Although children’s accuracy patterns support inferences about how discreteness and length disparity influence strategy use, additional measures could provide further evidence of the mechanisms underlying these patterns.
With this aim, we analyzed children’s tape diagram magnitude comparisons using four metrics: reaction times, accuracy, eye movement patterns, and verbal strategy reports. Eye-tracking has been used to investigate what children fixate on in fraction problems (Huber, Mann, Nuerk, & Moeller, 2014; Obersteiner & Tumpek, 2016), which may reveal solution approaches. Verbal strategy reports provide a direct measure of strategy use. The combination of accuracy, reaction time, eye-tracking and strategy self-reports allows us to benefit from each measure’s advantages while mitigating its disadvantages, giving us greater confidence in our conclusions.
22 children (grades 4-6) participated. Each completed 48 magnitude comparisons with eye-tracking. 17 of the participants then completed 12 additional comparisons on paper, and explained aloud how they solved each problem. Interrater reliability for strategy reports was greater than 88% simple agreement for each strategy; disagreements were discussed and reconciled.
Participants were less accurate (OR = 0.08, χ2 = 8.11, p < 0.005) and slower (β = 2380.25, p < 0.01) on trials with discrete segments. They were also more accurate (OR = 12.58, χ2= 12.86, p < 0.001) and faster (β = -506.45, p < 0.05) on trials with same-length bars. Saccade counts and reaction times suggest counting strategies were responsible for reduced performance on discrete trials. Verbal reports revealed that participants were indeed more likely to use counting strategies (OR = 1148.26, χ2= 15.33, p < .0001) and to mention symbolic numbers (OR = 13.44, χ2 = 5.68, p = 0.02) on discrete items. Participants were less likely to use benchmark strategies on same-length trials (OR = 0.03, χ2 = 8.27 p = 0.004). Table 1 provides strategy descriptions and the proportion of participants who used each strategy at least once for a given problem type.
This study offers insights, supported by multiple converging measures, that features such as length differences and discrete segments influence children’s strategy use, and that children use strategies adaptively, given the affordances of the different diagrams. These findings have implications for fraction education. For example, educators could help children understand the relationships among different possible strategies and could demonstrate how to appropriately apply counting strategies.

Authors

• Victoria Jay, University of Wisconsin - Madison
  Presenting Author

• Vijay Marupudi, Georgia Institute of Technology
  Non-Presenting Author

• Andreas Obersteiner, Freiburg University of Education
  Non-Presenting Author

• Martha Wagner Alibali, University of Wisconsin - Madison
  Non-Presenting Author