Journal of Numerical Cognition (JNC - PsychOpen)
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Developing a Rigorous Measure of the Pre-School Home Mathematics Environment
No full textChildren begin pre-school with varying levels of school readiness. Those children who enter pre-school with better foundational mathematics skills are more likely to succeed in school than those who do not. This initial variation in early mathematics suggests that experiences outside of the school setting, namely the home environment, may support learning and development. This study aims to systematically develop a comprehensive home mathematics environment questionnaire that reliably assesses the experiences of pre-school children (i.e., 3–5-year-olds) following recent recognised scale development and validation methods. Four studies were used to develop and validate the Pre-school Home Mathematics Questionnaire (PHMQ). Study 1 focused on 1) item generation through individual, in-depth interviews with parents of young children and 2) identifying previous questions from other home mathematics environment (HME) questionnaires to be incorporated into the PHMQ. Study 2 involved questionnaire refinement and was used to assess the psychometric properties of the new measure while addressing construct validity (i.e., factor structure and scale score reliability). Study 3 assessed content and criterion validity of the scale. Finally, Study 4 focused on construct validity through confirmatory factor analysis. Overall, the four studies demonstrate construct, content, and criterion validity. Hence, the newly developed PHMQ satisfies the American Psychological Association (APA) standards for psychometric adequacy
Making Sense of the Relation Between Number Sense and Math
No full textWhile several studies have shown that the performance on numerosity comparison tasks is related to individual differences in math abilities, others have failed to find such a link. These inconsistencies could be due to variations in which math was assessed, different stimulus generation protocols for the numerosity comparison task, or differences in inhibitory control. This within-subject study is a conceptual replication tapping into the relation between numerosity comparison, math, and inhibition in adults (N = 122). Three aspects of math ability were measured using standardized assessments: Arithmetic fluency, calculation, and applied problem solving skills. Participants’ inhibitory skills were measured using Stroop and Go/No-Go tasks with numerical and non-numerical stimuli. Finally, non-symbolic number sense was measured using two different versions of a numerosity comparison task that differed in the stimulus generation protocols (Panamath; Halberda, Mazzocco & Feigenson, 2008, https://doi.org/10.1038/nature07246; G&R, Gebuis & Reynvoet, 2011, https://doi.org/10.3758/s13428-011-0097-5). We find that performance on the Panamath task, but not the G&R task, related to measures of calculation and applied problem solving but not arithmetic fluency, even when controlling for inhibitory control. One possible explanation is that depending on the characteristics of the stimuli in the numerosity comparison task, the reliance on numerical and non-numerical information may vary and only when performance relies more on numerical representations, a relation with math achievement is found. Our findings help to explain prior mixed findings regarding the link between non-symbolic number sense and math and highlight the need to carefully consider variations in numerosity comparison tasks and math measures
Revisiting and Refining Relations Between Nonsymbolic Ratio Processing and Symbolic Math Achievement
No full textIn their 2016 Psych Science article, Matthews, Lewis and Hubbard (2016, https://doi.org/10.1177/0956797615617799) leveled a challenge against the prevailing theory that fractions—as opposed to whole numbers—are incompatible with humans’ primitive nonsymbolic number sense. Their ratio processing system (RPS) account holds that humans possess a primitive system that confers the ability to process nonysmbolic ratio magnitudes. Perhaps the most striking finding from Matthews et al. was that ratio processing ability predicted symbolic fractions knowledge and algebraic competence. The purpose of the current study was to replicate Matthews et al.’s novel results and to extend the study by including a control measure of fluid intelligence and an additional nonsymbolic magnitude format as predictors of multiple symbolic math outcomes. Ninety-nine college students completed three comparison tasks deciding which of two nonsymbolic ratios was numerically larger along with three simple magnitude comparison tasks in corresponding formats that served as controls. The formats included were lines, circles, and dots. We found that RPS acuity predicted fractions knowledge for three university math placement exam subtests when controlling for simple magnitude acuities and inhibitory control. However, this predictive power of the RPS measure appeared to stem primarily from acuity of the line-ratio format, and that predictive power was attenuated with the inclusion of fluid intelligence. These findings may help refine theories positing the RPS as a domain-specific foundation for building fractional knowledge and related higher mathematics
Effectiveness of a Numeracy Intelligent Tutoring System in Kindergarten: A Conceptual Replication
No full textIntelligent Tutoring Systems are a genre of highly adaptive software providing individualized instruction. The current study was a conceptual replication of a previous randomized control trial that incorporated the intelligent tutoring system Native Numbers, a program designed for early numeracy instruction. As a conceptual replication, we kept the method of instruction, the demographics, the number of kindergarten classrooms (n = 3), and the same numeracy and intrinsic motivation screeners as the original study. We changed the time of year of instruction, changed the control group to a wait-control group, added a maintenance assessment for the first group of participants, and included a mathematical language assessment. Analysis of within- and between-group differences using repeated measures ANOVA indicated gains of numeracy were significant only after using Native Numbers (Partial Eta Square = 0.147). Results of intrinsic motivation and mathematical language were not significant. The effect size of numeracy achievement did not reach that of the original study (Partial Eta Square = 0.622). Here, we compared the two studies, discussed plausible reasons for differences in the magnitude of effect sizes, and provided suggestions for future research
The Causal Impact of Objective Numeracy on Judgments: Improving Numeracy via Symbolic and Non-Symbolic Approximate Arithmetic Training Yields More Consistent Risk Judgments
No full textPark and Brannon (2013, https://doi.org/10.1177/0956797613482944) found that practicing non-symbolic approximate arithmetic increased performance on an objective numeracy task, specifically symbolic arithmetic. Manipulating objective numeracy would be useful for many researchers, particularly those who wish to investigate causal effects of objective numeracy on performance. Objective numeracy has been linked to performance in multiple areas, such as judgment-and-decision-making (JDM) competence, but most existing studies are correlational. Here, we expanded upon Park and Brannon’s method to experimentally manipulate objective numeracy and we investigated whether numeracy’s link with JDM performance was causal. Experimental participants drawn from a diverse internet sample trained on approximate-arithmetic tasks whereas active control participants trained on a spatial working-memory task. Numeracy training followed a 2 × 2 design: Experimental participants quickly estimated the sum of OR difference between presented numeric stimuli, using symbolic numbers (i.e., Arabic numbers) OR non-symbolic numeric stimuli (i.e., dot arrays). We partially replicated Park and Brannon’s findings: The numeracy training improved objective-numeracy performance more than control training, but this improvement was evidenced by performance on the Objective Numeracy Scale, not the symbolic arithmetic task. Subsequently, we found that experimental participants also perceived risks more consistently than active control participants, and this risk-consistency benefit was mediated by objective numeracy. These results provide the first known experimental evidence of a causal link between objective numeracy and the consistency of risk judgments
Conceptual Replication and Extension of the Relation Between the Number Line Estimation Task and Mathematical Competence Across Seven Studies
No full textA recent meta-analysis demonstrated the overall correlation between the number line estimation (NLE) task and children’s mathematical competence was r = .44 (positively recoded), and this relation increased with age. The goal of the current study was to conceptually replicate and extend these results by further synthesizing this correlation utilizing studies not present in the meta-analysis. Across seven studies, 954 participants, ranging from 3 to 11 years old (Age M = 6.02 years, SD = 1.57), the overall estimation-competence correlations were similar to those of the meta-analysis and ranged from r = −.40 to −.35. The current conceptual replication demonstrated that the meta-analysis captured a stable overall relation between performance on the NLE task and mathematical competence. However, the current study failed to replicate the same moderation of age group presented in the meta-analysis. Furthermore, the current study extended results by assessing the stability and predictive validity of the NLE task while controlling for covariates. Results suggested that the NLE task demonstrated poor stability and predictive validity in the seven samples present in this study. Thus, although concurrent relations replicated, the differential age moderation, lack of stability, and lack of predictive validity in these studies require a more nuanced approach to understanding the utility of the NLE task. Future research should focus on understanding the connection between children’s developmental progression and NLE measurement before further investigating the predictive and diagnostic importance of the task for broader mathematical competence
Spacing Out! Manipulating Spatial Features in Mathematical Expressions Affects Performance
No full textThe current study explores the effects of physical spacing within mathematical expressions on student performance. A total of 2,152 students in 5th-12th grade were randomly assigned to one of four conditions within an online problem set, with terms in algebraic expressions spaced 1) neutrally, with no spaces in the expression, 2) congruent with the order of precedence through grouping terms, 3) incongruent with the order of precedence, or 4) mixed, a combination of the previous conditions. Results show that students who viewed incongruent problems made more errors and had to solve more problems to complete the assignment than those who viewed congruent or neutrally spaced problems. Additionally, students who viewed problems with mixed spacing had to solve more problems to complete the assignment than students who viewed congruent problems. These findings suggest that viewing expressions with spacing that is incongruent with the order of precedence presents challenges for students. Overall, these results replicate prior research in perceptual learning in a natural homework environment and support the claim that physical spacing between terms does influence student performance on order of precedence problems
Linguistic Inversion and Numerical Estimation
No full textNumber line estimation (NLE) performance is usually believed to depend on the magnitudes of presented numerals, rather than on the particular digits instantiating those magnitudes. Recent research, however, shows that NLE placements differ considerably for target numerals with nearly identical magnitudes, but instantiated with different leftmost digits. Here we investigate whether this left digit effect may be due, in part, to the ordering of digits in number words. In English, the leftmost digit of an Arabic numeral is spoken first (“forty-one”), but Dutch number words are characterized by the inversion property: the rightmost digit of a two-digit number word is spoken first (“eenenveertig” – one and forty in Dutch). Participants (N = 40 Dutch-English bilinguals and N = 20 English-speaking monolinguals) completed a standard 0-100 NLE task. Target numerals were read aloud by an experimenter in either English or Dutch. Preregistered analyses revealed a strong left digit effect in monolingual English speakers’ estimates: e.g., 41 was placed more than two units to the right of 39. No left digit effect was observed among Dutch-English bilingual participants tested in either language. These findings are consistent with the idea that the order in which digits are spoken might influence multi-digit number processing, and suggests linguistic influences on numerical estimation performance
The Relationship Between Children’s Approximate Number Certainty and Symbolic Mathematics
No full textWhy do some children excel in mathematics while others struggle? A large body of work has shown positive correlations between children’s Approximate Number System (ANS) and school-taught symbolic mathematical skills, but the mechanism explaining this link remains unknown. One potential mediator of this relationship might be children’s numerical metacognition: children’s ability to evaluate how sure or unsure they are in understanding and manipulating numbers. While previous work has shown that children’s math abilities are uniquely predicted by symbolic numerical metacognition, we focus on the extent to which children’s non-symbolic/ANS numerical metacognition, in particular sensitivity to certainty, might be predictive of math ability, and might mediate the relationship between the ANS and symbolic math. A total of 72 children aged 4–6 years completed measures of ANS precision, ANS metacognition sensitivity, and the Test of Early Mathematical Ability (TEMA-3). Our results replicate many established findings in the literature, including the correlation between ANS precision and the TEMA-3, particularly on the Informal subtype questions. However, we did not find that ANS metacognition sensitivity was related to TEMA-3 performance, nor that it mediated the relationship between the ANS and the TEMA-3. These findings suggest either that metacognitive calibration may play a larger role than metacognitive sensitivity, or that metacognitive differences in the non-symbolic number perception do not robustly contribute to symbolic math performance
What Applying Growth Mixture Modeling Can Tell Us About Predictors of Number Line Estimation
No full textNumber line estimation tasks have been considered a good indicator of mathematical competency for many years and are traditionally analyzed by fitting individual regression curves to individual responders. We innovate on this technique by applying growth mixture modeling and compare it to traditional regression using a sample of 2nd graders (n = 325) who completed both 0–20 and 0–100 number line tasks. We explore the effects of gender, special education needs, and migration background. Using growth mixture modeling, more children were identified as logarithmic responders than were identified using regressions. Growth mixture modeling was able to identify the significant effects of gender on class membership for both tasks, and of special education needs for the 0–20 task. Overall, growth mixture modeling provided a more complete picture of individual response patterns than traditional regression techniques. We discuss the implications of these findings and provide recommendations for future researchers to use growth mixture modeling with future number line task analyses