When children perform fraction arithmetic, they generate a variety of solutions. In this study, we extended this research toadults. We report that adults performance is best for addition and subtraction, worse for division, and is susceptible to thesame kinds of strategy errors observed in 6th grade children. Specifically, solvers common strategy errors involved main-taining the values of fractions with common denominators even when that strategy was not appropriate. We also presenttwo other findings that were not observed in children. First, adults applied an incorrect division algorithm; they incorrectlyinverted the first, rather than the second operand in fraction division problems. Second, adults applied reduction proce-dures for fraction multiplication and division in order to simplify numerator-denominator pairs during fraction arithmetic.Our results suggest that strategy selection was cued by identifying common fraction components within problems.

People report solving division problems by mentally recastingdivision problems as multiplication (e.g., 72 ÷ 8 à 8 × [?] =72). Mediation of division by multiplication occurs mainly onlarger problems. Eye tracking data was used to determinewhether patterns of gaze durations on division problemsprovided support for mediation. Adults solved divisionproblems in two formats: traditional (e.g., 72 ÷ 8 = [ ]) andrecasted (e.g. 72 = 8 × [ ]). Processing of individual problemelements was compared across formats. Results providesupport for mediation. Processing patterns for traditionally-formatted problems were more similar to those for traditionaldivision in earlier work (72 ÷ 8) whereas problems in recastedformat (72 = 8 × [ ]) were more similar to patterns foundwhen participants solved multiplication problems (e.g., 8 × 9).These findings provide a novel source of support fordifferential processing of problems across presentationformats.

Math anxiety is a common correlate of math performance. However, the causal direction of this relation is unclear.Research with young students is limited but critical for determining patterns of development. Students (N = 147) completedmath measures (i.e., number comparison, arithmetic fluency, and math problem solving) and math anxiety assessmentstwice, first in grade 2 (Mage = 7 years:10 months) and then a year later in grade 3. Correlational analysis revealed thatmath anxiety is related only to arithmetic fluency, related to other types of math performance. Cross-lagged analyses wereconducted to evaluate causal relations between math anxiety and arithmetic fluency. These analyses showed that arithmeticfluency in grade 2 predicted change in math anxiety from grade 2 to grade 3, however, math anxiety in grade 2 did notpredict the change in arithmetic fluency from grade 2 to grade 3. These results suggest that math anxiety may be the resultof poor math performance.

Finger gnosis, the ability to mentally represent one’s fingers, predicts numeracy in children (Penner-Wilger et al.,2007, 2009) and adults (Penner-Wilger et al., 2014, 2015). It has been argued that the relation may reflect visuo-spatial memory,rather than finger gnosis ability per se. This rival hypothesis was not supported in adults (Penner-Wilger et al., 2015), but herewe examined it in children, using both a novel set of Grade 1 participants (N = 119) and a separate set previously reportedon (Penner-Wilger et al., 2007, 2009; N=146). In multiple regressions, for each set of participants, finger gnosis significantlypredicted numeracy skills, measured using the KeyMath Numeration subtest. Moreover, the relation between finger gnosis andnumeracy held for both sets of participants, despite controlling for visuo-spatial memory, measured using a Corsi-block test.These findings suggest that the relation between finger gnosis and numeracy is robust and does not reflect visuo-spatial memory.