Despite the importance of understanding the relative magnitude of numbers, little is known about how this understanding develops. I will describe the results of a recent experiment that demonstrates a causal role of categorization in eliciting changes in numerical representations. The results indicated that kindergartners who were provided feedback that encouraged sorting the numbers between 0 and 100 into five equal size categories generated more linear estimates of numerical magnitude than kindergartners who were provided identical sorting instructions and practice but no feedback. Reasons for why improved categorization of numbers promotes linear representations and potential implications for instruction will be discussed.

Young children’s understanding of numerical magnitudes is closely related to their general math achievement, estimation skills, and arithmetic proficiency. Understanding of numerical magnitudes also is a core component of number sense, an ill-defined construct that nonetheless is widely viewed as crucial to success in mathematics and that is a key goal of mathematics instruction. Although existing data on the relation between mathematical proficiency and understanding of numerical magnitudes are correlational, they are consistent with the view that helping young children develop a better understanding of numerical magnitudes may lead to improved performance on a wide range of mathematical tasks. 

While a growing number of studies have described age-related changes in representations of numerical magnitude, few studies have explored how these changes occur. Studies of several types of estimation -- number line, measurement, and numerosity -- indicate that with age and experience, children move from logarithmic to linear representations of numerical magnitudes (e.g., Booth & Siegler, 2006). Studies of number line estimation can be used to illustrate the transition. On this task, children are asked to estimate the position of Arabic numerals on number lines that are empty except for a 0 at the left end and a larger number (typically either 100 or 1000) at the right end. On 0-100 number lines, kindergartners generate estimates consistent with a logarithmic representation, such that the estimates exaggerate differences among the magnitudes of smaller numbers and compress differences among larger ones. In contrast, second graders produce estimates consistent with a linear representation, such that their estimates neither exaggerate nor compress differences among numbers throughout the scale.

What processes are involved in the change from logarithmic to linear representations? What experiences are critical for the transition to occur? What role might general learning mechanisms, such as categorization, play in the acquisition of linear representations of numerical magnitude? Few studies have addressed these sorts of questions.

The present experiment was designed to test the hypothesis that subjective categorization of numbers (e.g., “really small”, “medium”, “really big”) plays a role in the acquisition of understanding of numerical magnitude. Specifically, it tested whether improvement in children’s categorization of numbers generalizes beyond a categorization task and leads to more successful performance on the number line estimation task as well.

We expected that training children to divide the numbers between 0-100 into five equal size categories would help children generate more accurate and more linear representations of numerical magnitude, as indexed by their number line estimates.

In the experiment, kindergartners were randomly assigned to a categorization training group or a control group. The same pretest and posttest was administered to all children and included a magnitude comparison (e.g., “Which is more 12 or 8?”), subjective categorization (e.g., “Is 12 a small, medium, or big number?”), and number line estimation task for the numbers between 0 and 100. In four intervening sessions, children in the categorization training group sorted numbers between 0-100 into really small, small, medium, big, and really big categories and completed triad tasks in which they identified the number that was a member of the same category as the standard; children in the control group were given identical experience except that they received no feedback on their performance.

Results from this experiment showed that the number line estimates of children who received feedback on their categorizations of numbers became dramatically more linear from pretest to posttest (Figure 1). On the pretest, the best fitting linear function accounted for 70% of the variance in the median estimates for each number; on the posttest, the best fitting linear function accounted for 96% of the variance. In contrast, the median estimates of children who had the same categorization experiences, but did not receive feedback, did not become more linear from pretest to posttest. The best fitting linear function accounted for 77% of the variance in the median estimates at pretest, versus 79% of the variance on the posttest. Hence, improvement in children’s categorization of numbers generalized beyond the categorization task and led to more successful performance on the number line estimation task as well.

The experiment demonstrated a causal role of categorization in eliciting changes in numerical representations. The results suggest that conditions that promote categorization of objects (e.g., comparison) might promote number concept development, which has potential implications for instruction. For instance, children might benefit from exposure to a wider range of numbers than may be typically used in preschool and kindergarten classrooms because expanding the size of numbers children encounter should, theoretically, promote the differentiation of their categories by encouraging comparisons between larger numbers.