The development of place value concepts: Approximation before principles

Abstract

Place value concepts were measured longitudinally from kindergarten (2017) to first grade (2018) in a diverse sample (n = 279; M age = 5.76 years, SD = 0.55; 135 females; 41% Black, 38% White, 8% Asian, 12% Latino). Children completed three syntactic tasks that required an explicit understanding of base-10 symbols and three approximate tasks that could be completed without this explicit understanding. Approximate performance was significantly better in both age groups. A factor analysis confirmed that syntactic and approximate tasks tapped separate latent variables in kindergarten, but not in first grade. Path analyses indicated that only kindergarten approximate performance predicted overall first-grade place value understanding. These findings suggest that explicit understanding of base-10 principles develops from implicit, partial knowledge of multidigit numbers.

Spatial configuration of hundreds charts influences children's knowledge of base‐ten concepts

“…Further, the participants' place‐value knowledge did not improve from before the intervention to afterward, suggesting that solving problems with the hundreds chart is not related to growth in their knowledge of the principle of position. These findings align with recent research by Mix et al (2022), who found that explicit syntactic knowledge of place value reflects a different underlying construct from what they called “approximate” place‐value knowledge, defined as knowledge of implicit, informal rules for tasks such as number comparison and transcoding. For example, knowing that 543 is larger than 345 because the name for the former begins with “five” and not “three” is not the same as knowing that the 5 in the third position from the right means five groups of 100.…”

“…Place‐value knowledge is a specific type of number knowledge that entails interpreting the quantitative meaning of the position of a digit in a multidigit numeral. Although related to unit coordination, such as when composing and decomposing units in arithmetic algorithms (MacDonald et al, 2018; Thanheiser, 2009), place‐value understanding entails explicit knowledge of a “representational syntax” (Mix et al, 2022) that is defined by the principle of position (Cheung & Ansari, 2021). The principle of position establishes that each place in a numeral has a specific and distinct value determined by a power of 10 (i.e., 10 0 in the right‐most place; 10 1 in the second place from the right; 10 2 in the third place from the right, etc.…”

A Network Analysis of Children’s Emerging Place-Value Concepts

“…This secondary data analysis used previously published data from a study that tracked performance on a range of place-value measures from kindergarten to first grade (full details of the data collection are available in the article by Mix et al, 2022). Children were tested late in the spring in kindergarten and first grade.…”

“…A sensitivity test was conducted in G*Power (Version 3.1; Faul et al, 2009) before data collection for the larger study to determine what sample size would be adequate to detect a medium effect based on regression models (Mix et al, 2022). The results indicated that a sample size of 120 was adequate to detect a medium effect (i.e., Cohen’s f 2 = .11; Cohen, 1988) with the following parameters: an α of .05, power of .80, a sample size of 120, and six predictors.…”

“…Thus, early knowledge about reading and writing multidigit numbers appears to yield approximate knowledge about places and their relative magnitudes. Finally, this early approximate and incomplete knowledge about multidigit numbers has been shown to predict later success in learning count + unit syntax in school (Mix et al, 2022).…”