Development of Decimal Magnitude Understanding and its Relation with Fraction Magnitude Understanding and Mathematics Achievement

• In Event: Connecting Fractions and Decimals

Abstract

Reasoning about numerical magnitudes is a foundational concept in mathematics learning (National Mathematics Advisory Panel, 2008). Most research examining the relation between magnitude understanding and mathematics achievement has focused on fraction and whole number magnitudes (e.g., Siegler, Thompson, & Schneider, 2011). The aim of the current study is to characterize the development of decimal understanding and examine its relation to the development of fraction understanding and later mathematics achievement. In particular, the current study contrasts common strategies for reasoning about magnitude, including holistic magnitude representation and rule-based reasoning (e.g., Schneider & Siegler, 2010).

Using a longitudinal design, 435 students were followed from third through sixth grade. Potential covariates (i.e., nonverbal reasoning, reading fluency, receptive vocabulary, working memory, attention, and whole number line estimation) were assessed in third grade. A fractions comparison task was given in the fall and spring of fourth grade, and a decimals comparisons task was given between these time points in the winter of fourth grade (prior to decimal instruction). In these tasks, students were presented with two decimals (or two fractions) and asked to identify which was larger, with blocks of items designed to capture different strategies or misconceptions. Mathematics achievement was assessed in the spring of sixth grade.

A 3-step latent class analysis identified three empirically distinct classes of response patterns for decimal magnitude understanding (step 1): children who were accurate (13%), children who had a partial understanding of place value but were negatively influenced by whole number properties (19%), and children who demonstrated both misunderstandings of place value and a whole number bias (68%). A separate latent class analysis on fraction magnitude understanding had parallel findings. Compared to children with misunderstandings (step 2), children with an accurate understanding performed better on whole number line estimation and receptive vocabulary and children with partial understandings performed better on nonverbal ability and fraction magnitude comparison. The third step found class membership also strongly predicted 6th grade mathematics achievement, with whole number and fraction magnitude understanding making unique contributions (and controlling for demographic and cognitive factors) as well as later fraction understanding (using a structural equation model).

That decimal and fraction magnitude understanding share a parallel structure of partial understanding, highlights the role of transitional knowledge in understanding how the overall magnitude of the representation depends on the location of the digits. Even though children with a partial understanding incorrectly apply some rule-based strategies, they are more likely to shift to a good understanding of later fraction magnitude compared to those who persist with whole number strategies. Findings suggest that children have multiple ways of reasoning about decimals and fractions, which include an analogue continuous representation as well as rule-based strategies that are strategically deployed based on task demands. That decimal, fraction, and whole number magnitude all independently predicted later mathematics achievement, suggest that reasoning about magnitude involves dissociable skills based on the form of the numerical representation and children's knowledge of those representations. Taken together, the findings support educational instruction that emphasizes decimal properties earlier.

Authors

• Ilyse Resnick, University of Canberra
  Presenting Author

• Luke Rinne, University of Delaware
  Non-Presenting Author

• Christina Areizaga Barbieri, University of Delaware
  Non-Presenting Author

• Nancy C Jordan, University of Delaware
  Non-Presenting Author