Context-dependent variability in children’s conceptual models of division

• In Event: Poster Session 03
  In Poster Session: PS 03 Topic - Cognitive Processes

Abstract

Children’s strategy use is inherently variable (Siegler, 1996) and this variability is adaptive (Alibali & Sidney, 2015). We examined whether the context of children’s mathematical problem solving affects strategy variability. Prior research (Prather & Alibali, 2009) demonstrates that accuracy rates can differ across contexts. For example, in studies of fraction understanding, number lines often elicit more accurate reasoning than circles (Hamdan & Gunderson, 2017; Sidney et al., 2019). However, few studies examine variability among accurate strategies. In this study, we focused on whole number division.

Prior research suggests that children rely primarily on the partitive, or sharing, model of whole number division (e.g., Squire & Bryant, 2002). Partitive models are thought to be more intuitive than the alternative, quotative model of division (Correa et al., 1998, see Table). However, children may benefit from competency in multiple accurate division models (Jong & Magruder, 2014). In the current study, we examined intraindividual variability in division models across three problem-solving contexts. We expected that children's reliance on partitive models is a function of the story contexts often used in prior research, and that number line contexts would elicit quotative models of division (as in fraction division, Sidney et al., 2019). Thus, we examined children’s strategy use across and within three contexts: when using objects, stories, or number lines.

The study included 63 3rd- and 4th-grade children. Each child demonstrated four whole number division problems in each of three contexts, objects, stories and number lines (12 trials total), without feedback. Children’s behavior, verbal responses, and written work on each trial were coded to assess the strategy (partitive, quotative, or inaccurate).

Within-subject ANOVAs (IV: Context) on the percentage of partitive models used across trials and percentage of quotative models used across trials revealed that, across all children, different contexts did elicit different division strategies (see Figure). As expected, partitive strategies were more common in children’s story problems (45.9%) than in the object (33.2%) and number line (18.1%) contexts, F(2,124) = 14.64, p < .001. In contrast, quotative models were more common when children used number lines (39.7%) than in the object (24.6%) and story (10.7%) contexts, F(2,124) = 16.02, p < .001. Next, we examined intraindividual variability among the 59 children who used at least one accurate division strategy. The majority of children who used an accurate division strategy (81%; 48/59) used both partitive and quotative strategies across contexts. Furthermore, some children were variable within context: 17/59 (28.8%) of children used more than one strategy in the object context, 14/59 (23.7%) in the story context, and 12/59 (20.3%) in the number line context.

These findings demonstrate considerable intraindividual variability in children’s strategy use across and within context. Most children used both partitive and quotative strategies for whole number division, and different contexts reliably elicit different strategies. Given that number lines elicit quotative strategies in both whole number and fraction division (Sidney et al., 2019), this study provides further evidence of number lines as a key representation for uniting number concepts (Siegler et al., 2011).

Authors

• Pooja Gupta Sidney, University of Kentucky
  Presenting Author

• Jessica Blake, University of Kentucky
  Non-Presenting Author

• Julie Shirah, University of Kentucky
  Non-Presenting Author

• Amanda Kruczkowski, University of Kentucky
  Non-Presenting Author