Scale and the evolutionarily based approximate number system: an exploratory study

ABSTRACT

Crosscutting concepts such as scale, proportion, and quantity are recognised by U.S. science standards as a potential vehicle for students to integrate their scientific and mathematical knowledge; yet, U.S. students and adults trail their international peers in scale and measurement estimation. Culturally based knowledge of scale such as measurement units may be built on evolutionarily-based systems of number such as the approximate number system (ANS), which processes approximate representations of numerical magnitude. ANS is related to mathematical achievement in pre-school and early elementary students, but there is little research on ANS among older students or in science-related areas such as scale. Here, we investigate the relationship between ANS precision in public school U.S. seventh graders and their accuracy estimating the length of standard units of measurement in SI and U.S. customary units. We also explored the relationship between ANS and science and mathematics achievement. Accuracy estimating the metre was positively and significantly related to ANS precision. Mathematics achievement, science achievement, and accuracy estimating other units were not significantly related to ANS. We thus suggest that ANS precision may be related to mathematics understanding beyond arithmetic, beyond the early school years, and to the crosscutting concepts of scale, proportion, and quantity.     

The Role of Mediators in the Development of Longitudinal Mathematics Achievement Associations

Despite research demonstrating a strong association between early and later mathematics achievement, few studies have investigated mediators of this association. Using longitudinal data (n = 1,362), this study tested the extent to which mathematics self‐concepts, school placement, executive functioning, and proficiency in fractions and division account for the association between mathematics achievement in first grade and at age 15. As hypothesized, a strong longitudinal association between first‐grade and adolescent mathematics achievement was present (β = .36) even after controlling for a host of background characteristics, including cognitive skills and reading ability. The mediators accounted for 39% of this association, with mathematics self‐concept, gifted and talented placement, and knowledge of fractions and division serving as significant mediators.     

Construction play and cognitive skills associated with the development of mathematical abilities in 7-year-old children

Construction play is thought to develop logico-mathematical skills, however the underlying mechanisms have not been defined. In order to fill this gap, this study looks at the relationship between Lego construction ability, cognitive abilities and mathematical performance in 7-year-old, Year 2 primary school children (N = 66). While studies have focused on the relationship between mathematics performance and verbal memory, there are limited studies focussing on visuospatial memory. We tested both visuospatial and verbal working memory and short term memory, as well as non-verbal intelligence. Mathematical performance was measured through the WIAT-II numerical operations, and the word reading subtest was used as a control variable. We used a Lego construction task paradigm based on four task variables found to systematically increase construction task difficulty. The results suggest that Lego construction ability is positively related to mathematics performance, and visuospatial memory fully mediates this relationship. Future work of an intervention study using Lego construction training to develop visuospatial memory, which in turn may improve mathematics performance, is suggested.     

Relational language influences young children’s number relation skills

Relational language is thought to influence mathematical skills. This study examines the association between relational language and number relation skills—knowledge of cardinal, ordinal, and spatial principles—among 104 U.S. kindergartners (5.9 years; 44% boys; 37% White, 25% Black, 14% Asian, 24% other) in the 2017–2018 academic year. Controlling for general verbal knowledge, executive function, and counting and number identification skills, relational language predicted later number relation skills, specifically number line estimation, β = .30. Relational language did not differentially predict number line estimation performance in children with low or high number relation skills, likely due to the restricted ranges of data within subgroups. Number relation skills, specifically number line estimation and number ordering, may be a pathway between relational language and mathematical skills.     

Making Sense of Number Words and Arabic Digits: Does Order Count More?

The ability to choose the larger between two numbers reflects a mature understanding of the magnitude associated with numerical symbols. The present study explores how the knowledge of the number sequence and memory capacity (verbal and visuospatial) relate to number comparison skills while controlling for cardinal knowledge. Preschool children’s (N = 140, M_{age-in-months} = 58.9, range = 41–75) knowledge of the directional property of the counting list as well as the spatial mapping of digits on the visual line were assessed. The ability to order digits on the visual line mediated the relation between memory capacity and number comparison skills while controlling for cardinal knowledge. Beyond cardinality, the knowledge of the (spatial) order of numbers marks the understanding of the magnitude associated with numbers.     

Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia)

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.     

Emergence of the Link Between the Approximate Number System and Symbolic Math Ability

Experimentally manipulating Approximate Number System (ANS) precision has been found to influence children’s subsequent symbolic math performance. Here in three experiments (N = 160; 81 girls; 3–5 year old) we replicated this effect and examined its duration and developmental trajectory. We found that modulation of 5-year-olds’ ANS precision continued to affect their symbolic math performance after a 30-min delay. Furthermore, our cross-sectional investigation revealed that children 4.5 years and older experienced a significant transfer effect of ANS manipulation on math performance, whereas younger children showed no such transfer, despite experiencing significant changes in ANS precision. These findings support the existence of a causal link between nonverbal numerical approximation and symbolic math performance that first emerges during the preschool years.     

Sources of Individual Differences in Children's Understanding of Fractions

Longitudinal associations of domain‐general and numerical competencies with individual differences in children's understanding of fractions were investigated. Children (n = 163) were assessed at 6 years of age on domain‐general (nonverbal reasoning, language, attentive behavior, executive control, visual‐spatial memory) and numerical (number knowledge) competencies; at 7 years on whole‐number arithmetic computations and number line estimation; and at 10 years on fraction concepts. Mediation analyses controlling for general mathematics ability and general academic ability revealed that numerical and mathematical competencies were direct predictors of fraction concepts, whereas domain‐general competencies supported the acquisition of fraction concepts via whole‐number arithmetic computations or number line estimation. Results indicate multiple pathways to fraction competence.     

Supporting mathematics vocabulary instruction through mathematics curricula

Abstract

This study examines four of the most commonly-used core mathematics curricula in the USA for evidence of support for research-based instructional strategies for mathematics vocabulary in first and second grade. Content analyses of the teachers’ editions of two units for each grade level were analyzed per curriculum (n?=?16). Statistically significant differences among curricula were found for number of target words (range 6–51 per unit), level of difficulty of terms (basic to technical), and number of support strategies per word. Multiple means of representation varied in terms of verbal and non-verbal strategies for target terms. These differences indicate children are experiencing substantially different mathematics vocabulary learning opportunities, which may impact later mathematics achievement. Implications for practice, curriculum development, and future research are addressed.     

Early manifestations of intellectual performance: Evidence that genetic effects on later academic test performance are mediated through verbal performance in early childhood

Intellectual performance is highly heritable and robustly predicts lifelong health and success but the earliest manifestations of genetic effects on this asset are not well understood. This study examined whether early executive function (EF) or verbal performance mediate genetic influences on subsequent intellectual performance, in 561 U.S.-based adoptees (57% male) and their birth and adoptive parents (70% and 92% White, 13% and 4% African American, 7% and 2% Latinx, respectively), administered measures in 2003–2017. Genetic influences on children's academic performance at 7 years were mediated by verbal performance at 4.5 years (β = .22, 95% CI [0.08, 0.35], p = .002) and not via EF, indicating that verbal performance is an early manifestation of genetic propensity for intellectual performance.     

Groupitizing reflects conceptual developments in math cognition and inequities in math achievement from childhood through adolescence

Understanding the cognitive processes central to mathematical development is crucial to addressing systemic inequities in math achievement. We investigate the “Groupitizing” ability in 1209 third to eighth graders (mean age at first timepoint = 10.48, 586 girls, 39.16% Asian, 28.88% Hispanic/Latino, 18.51% White), a process that captures the ability to use grouping cues to access the exact value of a set. Groupitizing improves each year from late childhood to early adolescence (d = 3.29), is a central predictor of math achievement (beta weight = .30), is linked to conceptual processes in mathematics (minimum d = 0.69), and helps explain the dynamic between the ongoing development of non-symbolic number concepts, systemic educational inequities in school associated with SES, and mathematics achievement (minimum beta weight = .11) in ways that explicit symbolic measures may miss.     

Experience and maturation: The contribution of co-occurrence regularities in language to the development of semantic organization

With development knowledge becomes organized according to semantic links, including early-developing associative (e.g., juicy–apple) and gradually developing taxonomic links (e.g., apple–pear). Word co-occurrence regularities may foster these links: Associative links may form from direct co-occurrence (e.g., juicy–apple), and taxonomic links from shared co-occurrence (e.g., apple and pear co-occur with juicy). Four experiments (2017–2020) investigated this possibility with 4- to 8-year-olds (N = 148, 82 female) and adults (N = 116, 35 female) in a U.S. city with 58.6% White; 29.0% Black, and 5.8% Asian demographics. Results revealed earlier development of the abilities to form direct (ds > 0.536) than the abilities to form shared co-occurrence-based links (ds > 1.291). We argue that the asynchronous development of abilities to form co-occurrence-based links may explain developmental changes in semantic organization.     

The Interplay Between Multisensory Associative Learning and IQ in Children

This study assessed the developmental profile of unisensory and multisensory processes, and their contribution to children's intellectual abilities (8- and 11-year olds, N = 38, compared to adults, N = 19) using a simple audiovisual detection task and three incidental associative learning tasks with different sensory signals: visual-verbal with pseudowords, novel audiovisual, and visual-visual. The level of immaturity throughout childhood was dependent on both, the sensory signal type and the task. Associative learning was significantly enhanced with verbal sounds, compared to novel audiovisual and unisensory visual learning. Visual-verbal learning was also the best predictor of children's general intellectual abilities. The results demonstrate a separate developmental trajectory for visual and verbal multisensory processes and independent contributions to the development of cognitive abilities throughout childhood.     

The association between and development of school enjoyment and general knowledge

Learning environments can support the development of foundational knowledge and promote children's attitudes toward learning and school. This study explores the relation between school enjoyment and general knowledge from preschool (2016–2017) to kindergarten (2017–2018) in 1359 children (M_age = 55, 61 months, female = 50%; 58.5% Hispanic, 17% Black, 10% Asian, 10% White, 5% multiracial/other; linguistically diverse). Cross-lagged panel models showed significant bidirectional associations between preschool enjoyment and change in general knowledge from preschool to kindergarten with a standardized coefficient of β = .21 (p < .001) and associations between preschool general knowledge and change in enjoyment, β = .09 (p = .015). Exploratory analyses with teacher characteristics and demographic subgroup comparisons are discussed. These associations suggest the potential intervention strategy of promoting early school enjoyment to support broader academic development.     

Understanding heterogeneity in pathways between interparental conflict and children's involvement: The moderating role of affect-biased attention

The study examined the moderating role of children's affect-biased attention to angry, fearful, and sad adult faces in the link between interparental conflict and children's distinct forms of involvement. Participants included 243 preschool children (M_age = 4.60 years, 56% female) and their parents from racially (48% African American, 43% White) and socioeconomically (median annual household income = $36,000) diverse backgrounds. Data collection took place in the Northeastern United States (2010–2014). Utilizing a multi-method, multi-informant, longitudinal design, attention away from anger selectively amplified the link between interparental conflict and children's subsequent coercive involvement (β = −.15). Greater attention to fear potentiated the pathway between interparental conflict and children's later cautious (β = .14) and caregiving involvement (β = .15). Findings are interpreted in the context of environmental sensitivity models.     

Specific Preschool Executive Functions Predict Unique Aspects of Mathematics Development: A 3-Year Longitudinal Study

This study assessed the extent to which executive functions (EF), according to their factor structure in 5-year-olds (N = 244), influenced early quantity–number competencies, arithmetic fluency, and mathematics school achievement throughout first and second grades. A confirmatory factor analysis resulted in updating as a first, and inhibition and shifting as a combined second factor. In the structural equation model, updating significantly affected knowledge of the number word sequence, suggesting a facilitatory effect on basic encoding processes in numerical materials that can be learnt purely by rote. Shifting and inhibition significantly influenced quantity to number word linkages, indicating that these processes promote developing a profound understanding of numbers. These results show the supportive role of specific EF for specific aspects of a numerical foundation.     

Analogy construction versus analogy solution, and their influence on transfer

This study compares transfer performed by subjects trained to solve verbal analogies, with transfer by subjects trained to construct them. The first group (n = 57) received instruction in a strategy to solve verbal analogies and the second group (n = 66) was trained in strategies for constructing such analogies.Before and after intervention, all subjects received three analogical tasks: verbal, figural and numerical. The success rate was measured by an effectiveness measure. Even though both interventions improved performance, the construction group scored higher results in the numerical and figural analogy tasks. Following Sternberg's “componential sub-theory of intelligence”, the constructors' advantage is attributed primarily to higher-level activation of the three significant metacomponents: problem recognition, strategy planning and supervision in task performance.     

Maturation of consonant perception,but not vowel perception,predicts lexical skills at 12 months

Consonants and vowels differentially contribute to lexical acquisition. From 8 months on, infants' preferential reliance on consonants has been shown to predict their lexical outcome. Here, the predictive value of German-learning infants' (n = 58, 29 girls, 29 boys) trajectories of consonant and vowel perception, indicated by the electrophysiological mismatch response, across 2, 6, and 10 months for later lexical acquisition was studied. The consonant-perception trajectory from 2 to 6 months (β = −2.95) and 6 to 10 months (β = −.91), but not the vowel-perception trajectory, significantly predicted receptive vocabulary at 12 months. These results reveal an earlier predictive value of consonant perception for word learning than previously found, and a particular role of the longitudinal maturation of this skill in lexical acquisition.     

What Is the Long‐Run Impact of Learning Mathematics During Preschool?

The current study estimated the causal links between preschool mathematics learning and late elementary school mathematics achievement using variation in treatment assignment to an early mathematics intervention as an instrument for preschool mathematics change. Estimates indicate (n = 410) that a standard deviation of intervention‐produced change at age 4 is associated with a 0.24‐SD gain in achievement in late elementary school. This impact is approximately half the size of the association produced by correlational models relating later achievement to preschool math change, and is approximately 35% smaller than the effect reported by highly controlled ordinary least squares (OLS) regression models (Claessens et al., 2009; Watts et al., 2014 ) using national data sets. Implications for developmental theory and practice are discussed.     

“Why do dogs pant?”: Characteristics of parental explanations about science predict children's knowledge

Using a new method for examining parental explanations in a laboratory setting, the prompted explanation task, this study examines how characteristics of parental explanations about biology relate to children's knowledge. Parents (N = 148; M_age = 38; 84% female, 16% male; 67% having completed college) of children ages 7–10 (M_age = 8.92; 47% female, 53% male; 58% White, 9.5% Black, 9.5% Asian) provided answers to eight how and why questions about biology. Parents used a number of different approaches to address the questions, including providing more mechanistic responses to how questions and more teleological responses to why questions. The characteristics of parental explanations—most notably, how frequently parents provided correct responses—predicted children's performance on measures of verbal intelligence and biological knowledge. Additional exploratory analyses and implications for children's learning are discussed.