A milestone in cognitive development is understanding numerals to represent the exact number of discrete items in a set (i.e., the cardinal principle). This development has received much attention, but little is known about its relation to understanding numbers as measures of continuous quantity (e.g., “six blocks long” versus “six blocks”). To investigate this, 90 children were asked to complete two tasks: a give-a-number task, to assess cardinality knowledge, and a novel give-a-line task, to assess measurement knowledge. As expected, accuracy was greater on the give-a-number task than the give-a-line task. More unexpectedly, the quality of performance on the give-a-number task was as often negatively associated with quality of performance on the give-a-line task as it was positive correlated. Specifically, when asked to create a line N-blocks long, children who gave only approximately correct answers on the give-a-number task often outperformed children who gave exactly correct answers on the same task. These findings indicate that an approximate—and purportedly less mature—understanding of number possesses the hidden strength of being more flexible and suitable for measuring length.

There are ongoing debates whether logarithmic compression in number-to-space mapping reflects logarithmic encoding of large numbers (bias) or uncertainty about numeric value (noise). We tested these two hypotheses by disentangling the effect of bias and noise. When 80 adults and 80 4- to 7-year-olds were asked to estimate the number of dots on a number line, both children and adults were more logarithmic on 0-100 than 0-30 problems. Internal noise explained some of the variance in logarithmicity, but only for children. We then examined the wisdom of crowds effect by comparing accuracy of children's mean estimate with accuracy of each adult's estimates. As predicted, children as a crowd were not as accurate as individual adults, indicating that noise is not the only source of children's errors. Generally, increasing the size of a crowd also had a smaller effect on 0-100 than 0-30 problems, indicating that inaccuracy on 0-30 problems is likely due to noise. The present study provides evidence that bias and noise have an additive effect on logarithmic compression and that children's logarithmicity reflects bias in number representations, not just noise.

Political ideology leads educated adults–especially the highly numerate–to selectively reason about numbers that support their beliefs (“motivated numeracy”). We investigated whether supports that help children’s quantitative reasoning (number-lines) might also help political partisans. To test this, we asked 429 adults to interpret fictional data, in table or number-line format, about the effect of gun control on crime or the effect of a skin cream on rashes. We found data presented in number-line formats yielded greater accuracy than table formats controlling for numeracy skills (χ2 (1) = 21.88, p < .001), regardless of whether the true interpretation of data affirms, neutral to, or disaffirms participants’ political outlooks. Solving table problems after number-line problems yielded greater accuracy compared to solving table problems first (χ2 (1) = 4.78, p < .005), suggesting number-line practice is educational. Our research has important implications for communicating policy data and improving objectivity.

Whether different formats of numbers are represented by one or more systems across development is a subject of long-standing interest in the field of numerical cognition, with seemingly contradictory results. Here we examined numerical comparison to test a developmental theory that can reconcile these discrepancies. In Experiment 1, we found numerical understanding progresses through three continuous phases of association between numerical symbols and approximate sense of numerosity. In the youngest age group (prefluent phase), comparing numerals were slower than comparing dot arrays, but became similar (fluent phase) then faster (overlearning phase) with age. Because this developmental change occurred in the order of numeric range 1-9, followed by 10-99 and 100-999, multiple phases co-existed during childhood. Furthermore, results from Experiment 2 indicated that comparing different formats of numbers was affected by ratio even at the highest levels of proficiency, suggesting that the approximate number system is never fully replaced.

Learning a function from input-output pairs often follows exemplar or rule learning. Speed of exemplar learning and generality of rule learning were suggested to be promoted by visually aligning familiar and unfamiliar input-output exemplars. To test this, undergraduates (n=47) were randomly assigned to Full-, Partial- and No-Alignment groups. On each trial, students estimated fractions on number lines, and functions used to generate estimates were examined on 9 trial blocks. On pretest, estimates were (incorrectly) a linear function of denominators alone (Full 50%; Partial 50%; No 40%); on post-test, estimates were (correctly) a linear function of the whole fraction (88%; 44%; 47%). Virtually all change in the Full group occurred after training just two exemplars (75%). Also, regression to the denominator-only function differed across groups (0%; 38%; 33%). Finally, Full-Alignment group generalized to untrained problems more broadly than other groups. Findings demonstrated efficiency of visual alignment in function learning.