The Mental Number Line
Without counting, humans and other animals can estimate the size of arbitrarily large sets, albeit with decreasing precision. What representational resources underlie this ability to estimate?
One resource underlying estimation is a mental number line, which represents numeric values by associating symbols (e.g., number words and Arabic numerals) with spatial quantities. The mental number line has a number of interesting properties, which my colleagues and I have been examining developmentally.
One consistent finding about the mental number line is that early representations of numerical quantity initially increase as a logarithmic function of actual numeric value (much like the slide rule pictured above) before later increasing as a linear function of numeric value. This "logarithmic-to-linear shift" has been observed across a wide-range of numeric intervals, including 1-9 (Opfer et al., 2010), 0-20 (Young, Marciani, & Opfer, 2011), 0-100 (Siegler & Booth, 2004), 0-1,000 (Opfer & Siegler, 2007; Opfer & Thompson, 2008; Siegler & Opfer, 2003; Thompson & Opfer, 2008), and 0-10,000 (Thompson & Opfer, 2010). This shift is not only evident on number line tasks, but is also evident in children's categorization of numbers as "small," "medium," and "large" (Opfer & Thompson, 2008), in their estimation of measurements (Booth & Siegler, 2006), and in their estimation of set sizes (Booth & Siegler, 2006). Moreover, correlational and experimental studies show connections among these forms of numerical estimation, as well as connections to estimating magnitudes of fractions (Opfer & DeVries, 2008; Thompson & Opfer, 2008), arithmetic learning (Booth & Siegler, 2008), and memory for numbers (Thompson & Opfer, 2011; Young, Thompson, & Opfer, 2011).
To aid other researchers in exploring this finding, I wrote a tutorial explaining how Siegler and I initially used conventional, written number lines to characterize adults' and children's mental number lines.