Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter

Children’s multiplicative transformations of discrete and continuous quantities

“…However, a recent clinical interview study demonstrated that this aspect of children's thinking about physical quantity may itself be painstakingly constructed. Despite the apparent perceptual availability of the continuity of matter and length, the study found that many children between eight and twelve years of age do not yet possess a continuous model of matter or (in a pilot study) length (Smith et al, 2005). Such an understanding may be crucial to the construction of an understanding of rational number: all children who showed evidence of a discontinuous model of matter also did not yet understand that numbers could be divided infinitely, and all children who understood the infinite divisibility of number also showed evidence of a continuous model of matter (Smith et al, 2005).…”

“…An articulated model of fractions based on division is strongly related to middle school children's understanding of other aspects of rational number (Smith, Solomon, & Carey, 2005), and many researchers suggest that early intuitions about transformations of continuous physical amounts support later fraction learning (Resnick & Singer, 1993;Confrey, 1994;Moss & Case, 1999). But children's great difficulty in understanding fractions emphasizes the conceptual distance between intuitions about physical quantities and formal reasoning about rational numbers.…”

“…On the former view, making sense of rational number is especially difficult for children because their early concept of numbers -that numbers are what you get when you count -must be changed fundamentally when they are confronted with fractions (Smith et al, 2005;Gelman & Meck, 1992;Hartnett & Gelman, 1998). To understand rational number, children must come to form a new concept of number as infinitely divisible.…”

“…Many researchers have argued that learning about fractions requires conceptual change (Smith et al, 2005;Gelman, 1991;Gelman & Meck, 1992) rather than the enrichment of existing knowledge about numbers and quantities (Mix et al, 1999;Sophian, Geryantes, & Chang, 1997).…”

“…change are critical to children's later construction of an understanding of rational number (Smith, Solomon, & Carey, 2005), a famously difficult achievement of middle school math. It is often suggested that children's early intuitions about quantity transformations may support later learning about fractions, but these intuitions are often thought to rest on protoquantitative, nonnumerical notions of amount (Confrey, 1994;Mix, Levine, & Huttenlocher, 1999;Resnick & Singer, 1993;Resnick, 1992).…”