Analogies to familiar numbers can help children project the magnitudes of numbers they rarely (if ever) encounter. Un-fortunately, children may require source retrieval cues to revive their inert knowledge. Here we investigated effects ofthese cues in the context of estimating the location of fractions on number lines. During training, thirty-nine 10-year-oldslearned to map the location of integers and fractions on equivalent number lines (e.g., 3:8::3/8:1), and at post-test weregiven the same fraction number-line problems either with (Cue group) or without (No Cue group) a cue to remember thelocation of integers. Accuracy of estimates increased from pretest to training for both groups. However, children who re-ceived source retrieval cues during post-test improved their accuracy more than children in the No Cue group. Our resultsprovide further evidence thateven when sources and targets have been successfully mappedfailures of source retrieval canprevent analogical inference.

Despite a heated debate regarding a cognitive mechanism of magnitude representation, little has been done to di-rectly compare numerical and non-numerical estimation and provide a unified account of the two processes. In the current study,we examined estimation of numerical and non-numerical quantities on a continuum using various psychophysical functions.Inconsistent with the proportion reasoning and measurement skills accounts, estimates of both numerical and non-numericalquantities were better predicted by the logarithmic-linear model than by cyclic power models. Furthermore, individual dif-ferences in the degree of logarithmic compression was highly correlated over tasks, whereas bias measures from competingmodels did not show such associations. These findings suggest that estimation of both numerical and non-numerical magnitudeis processed via shared representation systems that are logarithmically or linearly constructed.

In numerosity comparison, performance is faster, more accurate, and less noisy with the ratio of compared numbers.Whereas the ratio-dependency has been intensively studied in relation to internal noise, processes of numerosity com-parison that may increase internal noise have not been fully understood. In this paper, we propose a process theory thataccounts for non-numerical, visuo-spatial processes in numerosity comparison. Consistent with the theory, we found thatas required processes decreased, performance improved significantly, to the extent that there were no differences betweennon-symbolic and symbolic number comparison in reaction time, accuracy, and internal noise. The findings suggest thatcomparing numerosities requires multiple processes homogenizing ancillary stimulus dimensions and that the homoge-nization processes are the major source of fuzziness in approximate number comparison.

Language is often depicted as the sine qua non of mathematicalthinking, a view buttressed by findings of language-of-trainingeffects among bilinguals. These findings, however, have beenlimited to studies of arithmetic. Nothing is known about thepotential influence of language on the ability to learn rulesabout the relations among variables (e.g., algebra). To testwhether arithmetic and algebraic thinking differ, Chinese-English bilinguals were trained to solve arithmetic and algebraproblems in either Chinese or English and then tested on newand old problems in both languages. For arithmetic problems,solution times were always longer for English than Chinese; inboth languages, solution times dropped during training; aftertraining, solution times continued to drop for old problems, butreturned to pre-training levels for new problems. In contrast,for algebra problems, solution times did not differ acrosslanguage; solution times dropped during training; aftertraining, gains in speed were preserved for both old and newproblems. These findings suggest that the contribution oflanguage to mathematical thinking may be limited to the areasof mathematics that are learned by rote and not by rule.

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.

Representations of numerical value have been assessed usingbounded (e.g., 0-1000) and unbounded (e.g., 0-?) number-linetasks, with considerable debate regarding whether one or bothtasks elicit unique cognitive strategies (e.g., addition orsubtraction) and require unique cognitive models. To test this,we examined 86 5- to 9-year-olds' addition, subtraction, andestimation skill (bounded and unbounded). Against themeasurement-skills hypothesis, estimates were even morelogarithmic on unbounded than bounded number lines andwere better described by conventional log-linear models thanby alternative cognitive models. Moreover, logarithmic indexvalues reliably predicted arithmetic scores, whereas modelparameters of alternative models failed to do so. Resultssuggest that the logarithmic-to-linear shift theory provides aunified framework for numerical estimation with highdescriptive adequacy and yields uniquely accurate predictionsfor children’s early math proficiency.