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

Abstract

Examining how informal knowledge systems change after formal instruction is imperative to understanding learning processes and conceptual development and to implementing effective educational practices. We used network analyses to determine how the organization of informal knowledge about multidigit numbers in kindergartners ( N = 279; mean age = 5.76 years, SD = 0.55; 135 females) supports and is transformed by a year of in-school formal instruction. The results show that in kindergarten, piecemeal knowledge about the surface properties of reading and writing multidigit numbers and the use of base-10 units to determine large quantities are strongly associated with each other and connected in a stringlike manner to other emerging skills. After a year of instruction, each skill becomes connected to the “hub” abilities of reading and writing multidigit numbers, which also become strongly connected to more advanced knowledge of base-10 principles. These findings provide new insights into how partial knowledge provides the backbone on which explicit principles are learned.

Place value learning in kindergarten.

“…For example, on Which Number Has?, children were asked to identify which of three multidigit numbers had six tens. As noted above, these approximate and syntactic measures have indicated separable latent structures in kindergarten (Bower et al, 2022;Mix et al, 2022). We therefore treated these latent structures separately during data analysis and present the methods for each subtask within the same groupings here.…”

“…Thus, whereas better developed approximate understanding may be helpful, it cannot be the endpoint of learning, and previous research indicates it is not. By first grade, perhaps because of exposure to formal place value instruction or to accumulated informal experiences with multidigit numbers, the approximate and syntactic factors merge into a unitary place value factor that includes both kinds of measures (Mix et al, 2022), and performance on becomes more strongly interrelated and coherent (Bower et al, 2022). Presumably, this shift occurs because performance on both task types is increasingly based on a strong underlying understanding of base-ten syntax-for some children at least.…”

“…As noted above, tasks designed to measure place value understanding do not necessarily measure the same knowledge components. Prior research using factor analysis (Mix et al, 2022) and network clustering (Bower et al, 2022) identified two separate but related factors in kindergarten using six of these tasks. Three tasks—expanded notation, base-ten counting, and which number has?—all significantly loaded onto one factor characterized by exact knowledge of base-ten syntax—that is, the understanding that multidigit numbers may be decomposed into individual places that stand for base-ten units and counts of these units (e.g., “253” = [2 × 100] + [5 × 10] + [3 × 1]).…”

Teaching Place Value Concepts and Their Application Using the Concrete-Representational-Abstract Integrated Sequence

“…Learning and understanding our number system, or place value, is a critical benchmark in mathematics (Mix et al, 2017). Our number system is constructed with the digits zero through nine, which stand for an infinite number of quantities, and these digits are arranged in multiples of 10 from left to right with each place to the left being 10 times higher than the digit to the right (Bower et al, 2022). Bower et al (2022) provided an example such as 456 meaning [(4 × 100) + (5 × 10) + (6 × 1)].…”

“…Our number system is constructed with the digits zero through nine, which stand for an infinite number of quantities, and these digits are arranged in multiples of 10 from left to right with each place to the left being 10 times higher than the digit to the right (Bower et al, 2022). Bower et al (2022) provided an example such as 456 meaning [(4 × 100) + (5 × 10) + (6 × 1)]. Mix et al (2022) refer to this as count + unit syntax, meaning that 456 is four (count) hundreds (unit), five (count) tens (unit), and six (count) ones (unit).…”