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Previous research suggests that the Approximate Number System (ANS) allows people to approximate the cardinality of a set. This ability to discern numerical quantities may explain how meaning becomes associated with number symbols. However, recently it has been argued that ANS representations are not directly numerical, but rather are formed by amalgamating perceptual features confounded with the set's cardinality. In this paper, we approach the question of whether ANS representations are numerical by studying the properties they have, rather than how they are formed. Across two pre-registered within-subjects studies, we measured 189 participants' ability to multiply the numbers between 2 and 8. Participants completed symbolic and nonsymbolic versions of the task. Results showed that participants succeeded at above-chance levels when multiplying nonsymbolic representations within the subitizing range (2-4) but were at chance levels when multiplying numbers within the ANS range (5-8). We conclude that, unlike Object Tracking System (OTS) representations, two ANS representations cannot be multiplied together. We suggest that investigating which numerical properties ANS representations possess may advance the debate over whether the ANS is a genuinely numerical system.

This study investigates how the approximate number system (ANS) and young children's symbolic skills jointly develop and interact. Specifically, the study aims at disentangling the directionality of the association between ANS acuity and a wide range of symbolic skills that reflect 4- to 5-year-olds' symbolic quantitative knowledge (enumeration skills, knowledge of the verbal count sequence, symbolic comparison skills, and single-digit arithmetic). After accounting for individual differences in several domain-general skills (visuospatial working memory, non-verbal reasoning, and phonological processing), path models on longitudinal data collected from 4-year-old children in Spain (N = 62) over one year revealed that earlier single-digit arithmetic and symbolic magnitude comparison skills predicted changes in ANS acuity over time. No contribution from earlier ANS to improvements in symbolic skills was found. Notably, the strength of the effect of visuospatial working memory on improvements in ANS acuity over time was like that of the auto-regressor -- the correlation between measures of ANS acuity across time points. Implications for extant theories on the nature of the associations between ANS and young children's symbolic skills are drawn.

Studies have demonstrated a significant correlation between the approximate number system (ANS) and early mathematical achievement. However, various explanations exist regarding the underlying cognitive mechanisms that underpin this association. The present study investigated whether the two hypotheses of inhibition control and visual form perception could explain the relationship between kindergarteners' ANS and mathematical achievements. In spring, data were collected from a sample of 180 kindergarteners (M[subscript age] = 5.77 years, SD = 0.59). A parallel mediation analysis was conducted to investigate the role of inhibition control and visual form perception in the link between ANS and math achievement. The results indicated an indirect effect of ANS on mathematical achievements through both inhibition control and visual form perception, with both mediators exhibiting a comparable magnitude. These findings highlight the significance of distinct cognitive abilities that can potentially enhance early mathematical advancement.

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.

Although interest in mathematics anxiety in young children (6-11 years) has increased recently, the results of previous studies on the role of mathematics anxiety in the relationship between number sense and math performance are inconsistent. The study's main objective was to establish whether math anxiety is a mediator or a moderator in the relationship between number line estimation (NLE) and mathematical outcomes. A cross-sectional, correlational design was used. A total of 234 Turkish public primary school children (third and fourth graders) participated in the study. The following hypotheses were tested: (1) math anxiety mediates the relationship between the approximate number system (ANS) (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency); (2) math anxiety moderates the relationship between ANS (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency). The results mostly suggest no mediation effect of math anxiety between NLE and math performance in young children. The results also suggest that increases in difficulty level in NLE (increases of errors in NLE) is related to more anxiety regarding mathematics. Furthermore, moderation analyses indicated that the acuity of mental representation of numbers in high math-anxious children is a key factor contributing to math performance. Developmental aspects of the obtained results are also discussed in the paper.

In this paper, the researchers report an experimental study on conceptual knowledge learning and application through embodied interactions, expecting that bodily movements would facilitate learning in a positive manner. The intervention was enabled by Unity3D and Kinect V2, and taught novice adult learners the concepts of and conversions between binary and decimal numbers. Fifty-three adult participants were recruited, and randomly assigned into the experimental and control groups. During the intervention, participants in the experimental group manipulated and interacted with the learning materials through their body movements enabled by Kinect; while those in the control group learned the same content but interacted using conventional mouse and keyboard. Pre- and posttests results indicated that embodied interactions facilitated conceptual learning for the participating adult learners. However, compared with the mouse interaction, the embodied interactions did not lead to significantly better knowledge retention and application results. The study outcome implies that embodied interactions do not necessarily lead to better learning performances over traditional mouse-based interactions, and the 'how' and 'how much' may play critical roles for such interventions. Suggestions and challenges in designing learning systems utilizing embodied interactions enabled by novel technologies are provided.

What comes to mind when one thinks about building? One may envision constructions with blocks or engineering activities. Yet, constructing and building a number system requires the same sort of imagination, creativity, and perseverance as building a block city or engaging in engineering design. We know that children invent their own notation for negative numbers and use creative reasoning with negative integers (Bishop et al., 2011). Building on that knowledge, the author describes a unique task, The Tree Problem, that supports children in the messy process of extending their number system from whole numbers to including negative integers. The Tree Problem facilitates opportunities for imagining negative integers. The task also supports children in creating their own ways of making sense of integers. After describing the task, the author illustrates how a fifth-grade student, without prior instruction with negative integers, used imagination and perseverance to build a number system with flexible representations of negative numbers.

This article describes students learning to build their own numbering system by recognizing and identifying patterns with interlocking cubes in different place values. The students used the Egyptian hieroglyphic numeral system in conjunction with this activity to connect learning in other subjects. Students used prior knowledge of place value to create a non-traditional representation of place value and quantity. The task utilized students' prior knowledge of place value and base values to deepen understanding of the exchange of numbers and geometrical representations of numbers.

This study seeks to analyse the awareness of the pre-service teachers on the counting methods, systems and tools used in the prehistoric method and the Ancient period and to examine the distribution of this awareness by gender. A total of 42 sophomore-level students studying at a university in the Western Black Sea region, Turkey, participated in this exploratory case study. The data were obtained through a form consisting of 6 questions, one of which is open-ended, after the 14-week course of history of mathematics. The data collection tool included questions on the counting methods used in the pre-historic period and the Ancient Egyptian, Ancient Roman, Babylonian, Ancient Greek and Mayan number systems. The data were analysed through descriptive analysis and content analysis. The findings indicated that the pre-service teachers most reported the methods of tallying, tying a knot, token, circular disc. Also, the question on the Ancient Egyptian number system was answered correctly by all pre-service teachers, the lowest performance was observed in the question on the Mayan number system. Analysis of the answers by gender revealed that the male pre-service teachers were more likely to give false answers compared to the female pre-service teachers.

There is evidence of a relation between the approximate number system (ANS) and later mathematics achievement. Researchers have proposed various mediators of this relation, including executive functioning (EF), numeral knowledge, and mathematical language. The goal of the present study was to determine which factors mediate the relation between preschoolers' ANS and the change in their mathematics achievement over a five-month period. We collected data from 125 preschoolers ("mean age" = 4.2 years) in the fall and spring. We tested mediation models using path analysis models, controlling for children's fall mathematics achievement, age, sex, and parent education. EF was not a statistically significant predictor of mathematics achievement when controls were included. Numeral knowledge was not a significant mediator, but mathematical language was a significant mediator in both individual mediation and multiple mediation models. Although children's ANS predicts their later mathematics achievement, the relation is fully mediated by children's mathematical language.

Most any students can explain the meaning of "a[superscript b]", for "a" ? R and for "b" ? Z. And some students may be able to explain the meaning of "(a + bi)[superscript c]," for "a, b" ? R and for "c" ? Z. However, few may be able to explain "a[superscript di]" or "a[superscript (c+di)]" let alone "(a + bi)[superscript (c+di)]" for "a, b, c, d" ? R. Through both mathematical explanation and reader investigations via a dynamic graphing applet, we will investigate the meaning of "(a + bi)[superscript (c+di)]". Additionally, we provide some course appropriate investigations which can be used by instructors in various grades 11-16 courses.

This paper reports on an experiment involving 108 sixth-grade French students in which the history of Chinese numeration was used in mathematics education. The aim of this study is to evaluate the relevance of using ancient number systems in mathematics education, which is common in French textbooks. In the experiment a pre-test with usual mathematics exercises and an activity with a post-test comprising history-based ones have been proposed. On this data, a factor analysis shows that the tasks based on the history of mathematics combined well with specific mathematical skills. Then, an analysis of the difficulty levels (1PL model) of the regular items and the history-based items helps identifying some key points for which the history of mathematics is relevant.

We performed a meta-analysis of approximate number system (ANS) training studies to investigate the strength of the causal effects of practicing ANS related tasks on symbolic math performance. Across 33 effect sizes from 11 studies involving 754 participants, for which neither the treatment nor control group received symbolic training, we found a small nonsignificant and sensitive effect of ANS training on symbolic math task performance (g = 0.11, 95% confidence interval, CI [-0.01, 0.22]; precision-effect estimate with standard errors (PEESE) adjusted g = -0.04, 95% CI [-0.58, 0.50]). Some heterogeneity was accounted for by participant age, with larger estimates for adults than for children. Estimates did not vary significantly by ANS training type, training duration, and control group type. An exploratory analysis on the transfer effects of ANS training on untrained nonsymbolic tasks suggests weak support for the key auxiliary assumption that ANS training has substantial effects on a general ANS, indicating that the training literature may not adequately represent theories of how ANS influences symbolic number performance.

In this study, it was aimed to determine the errors and misunderstandings of 5th and 6th grade middle school students in fractions and operations with fractions. For this purpose, the case study model, which is a qualitative research design, was used in the research. In the study, maximum diversity sampling, which is a purposeful sampling method, was used. For this reason, this study was conducted in a state and a private secondary school on a voluntary basis with 105 5th graders and 84 6th graders, with a total of 189 secondary school students from different levels of achievement. In order to determine students' errors and misconceptions about fractions, two tests were prepared by researchers, ten open-ended questions for 5th graders and twelve open-ended questions for 6th graders. Before these tests were formed, the related field was scanned and the existing misunderstandings were determined. Content analysis method was used on the analyzing of the data. The answer papers of the students were coded and students' solving were examined in three categories as correct, wrong and blank. In addition to this the solving belongs to wrong category was examined in detail and mistakes of the students were coded and the reason of these mistakes were dwelled on. It is seen from the result of the study that students have misunderstanding and mistakes about fractions. It is seen that students are not using the modellings on the operations with fractions because of that most of the mistake and misunderstandings are about operations with fractions.

Learning fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of children with MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind their TA peers in fraction understanding and fall further behind during middle school. In contrast, Chinese children, who like the MD children in the United States score in the bottom one third of the distribution in their country, possess reasonably good fraction understanding. We interpret these findings within the framework of the integrated theory of numerical development. By emphasizing the importance of fraction magnitude knowledge for numerical understanding in general, the theory proved useful for understanding differences in fraction knowledge between MD and TA children and for understanding how knowledge can be improved. Several interventions demonstrated the possibility of improving fraction magnitude knowledge and producing benefits that generalize to fraction arithmetic learning among children with MD. The reasonably good fraction understanding of Chinese children with MD and several successful interventions with U.S. students provide hope for the improvement of fraction knowledge among American children with MD.

In recent times, the endangerment of Yorùbá has highly been speculated among Yorùbá intellectuals, indigenes and enthusiasts alike. In an effort to promote the learning and use of Yorùbá numeral system in carrying out day-to-day activities and transactions, the development of a Yorùbá arithmetic learning system will help bridge the gap between lost Yorùbá traditions of the transaction and modern day business activities. The design of this project was conceptualized using Unified modelling language (UML) tools. The resulting design and software specification was implemented using Python programming language due to its cross-platform compatibility and PyQt graphical user interface (GUI) module for the GUI. System evaluation was done using questionnaire to determine how the system is perceived by potential users. It was observed that 90% were unable to evaluate the addition expressions; 55% were unable to evaluate subtraction expressions, 65% were unable to evaluate multiplication expressions and 90% were unable to evaluate division expressions. It can be concluded from both research and the system developed that in order to make the Yorùbá language more relevant in the world today, attempts such as this project must be encouraged. The issue of the almost non-existent use of Yorùbá numeral must be addressed by producing learning systems or aids to help the teaching and learning of the Yorùbá numerals.

This study investigated the impact of a chatbot-based micro-learning system on students' learning motivation and performance. A quasi-experiment was conducted with 99 first-year students taking part in a basic computer course on number system conversion. The students were assigned to a traditional learning group or a chatbot-based micro-learning group. After the experiment, both groups achieved a comparable performance, suggesting that students are sufficiently competent to learn independently in the chatbot-based learning environment without the need for continuous face-to-face delivery. Moreover, students in the chatbot learning group attained significantly higher intrinsic motivation than the traditional learning group with perceived choice and perceived value as core predictors of intrinsic motivation. Further analysis with the Johnson-Neyman procedure revealed differences on interaction between the perceived choice and the learning environments. For students with a high initial perceived choice (>=5.1), chatbot-based learning further enhances their post choice motivation whereas for students with a low initial perceived choice (<=3.0), the traditional classroom is more suitable to enhance their post choice motivation. The implications of the findings can help instructors to incorporate chatbot-based learning in the classroom.

Research over the past 20 years has suggested that our intuitive sense of number--the Approximate Number System (ANS)--is associated with individual differences in symbolic math performance. The mechanism supporting this relationship, however, remains unknown. Here, we test whether the ANS contributes to how well adult observers judge the "direction" and "magnitude" of symbolic math equation errors. We developed a novel task in which participants view symbolic equations with incorrect answers (e.g., 47 + 21 = 102), and indicate whether the provided answer was too high or too low. By varying the ratio between the correct and the provided answers, we measured individual differences in how well participants detect the "magnitude" and "direction" of symbolic equation errors. We find that individual differences in equation error detection were uniquely predicted by ANS acuity--that is, the precision of each participant's intuitive number representations--even when controlling for differences in surface area perception, working memory span, and operational span. This suggests that the ANS can act as a unique source of error detection variability for formal mathematics, providing a plausible mechanism for how our universally shared number sense might link with human-specific symbolic math abilities.

We determined the relative importance of the so-called approximate number system (ANS), symbolic number comparison (SNC) and verbal and spatial short-term and working memory (WM) capacity for mathematics achievement in 1,254 Grade 2, 4 and 6 children. The large sample size assured high power and low false report probability and allowed us to determine effect sizes precisely. We used reading decoding as a control outcome measure to test whether findings were specific to mathematics. Bayesian analysis allowed us to provide support for both null and alternative hypotheses. We found very weak zero-order correlations between ANS measures and math achievement. These correlations were not specific to mathematics, became non-significant once intelligence was considered and ANS measures were not selected as predictors of math by regression models. In contrast, overall SNC accuracy and spatial WM measures were reliable and mostly specific predictors of math achievement. Verbal short-term and WM and SNC reaction time were predictors of both reading and math achievement. We conclude that ANS tasks are not suitable as measures of math development in school-age populations. In contrast, all other cognitive functions we studied are promising markers of mathematics development.

We assessed a range of theoretically critical predictors (numerosity discrimination, number knowledge, counting, language, executive function and finger gnosis) of early arithmetic development in a large unselected sample of 569 children at school entry. Assessments were repeated 12 months later. Although all predictors (except finger gnosis) were moderate to strong correlates of arithmetic, a latent variable path model showed that only number knowledge and numerosity discrimination were unique predictors of arithmetic development over the 1st year of formal education. These findings are consistent with the Triple Code Model (Dehaene & Cohen, 1995) suggesting that both nonsymbolic representations of numerosity and visual symbolic representations of number are critical for the development of arithmetic. Our results suggest that the assessment of numerosity judgments and Arabic number knowledge at school entry may be useful for identifying children at risk of difficulties in acquiring arithmetic skills.

In this paper, dialogue among a group of adult women with basic adult education is examined. Women read books of mathematics in the context of a "dialogic mathematics gathering" (DMG). There is limited evidence in research on how adults develop their 'numerical understanding'. The objective of this study is to analyze how adults engage in critical thinking on numeracy, expanding their skills when becoming "critics" as readers. Women participating in the DMG create a context in which they exchange reports of their personal experiences in making meaning for the mathematical objects appearing in the readings. Three different episodes are discussed, involving units of measurement, number system, and the concept of base. Using a methodological framework grounded on Freire's work, evidence shows how those women unpack the mathematical meaning embedded in the readings, examining critically the objects discussed. DMGs result in a successful educational action to empower those women with low mathematics education experience.

Some children struggle with mathematics. Among these children, some of them may be learners with mathematical difficulties. While research has revealed multiple deficits as candidate causes for mathematical difficulties and probable subtypes, the cognitive profiles of these subtypes are not fully understood. Moreover, we have yet to discover whether children may move in or out of these subtypes over time. This study set out to identify the subtypes of mathematical difficulties as well as their stability. Using cluster analyses, we discovered 5 distinctive subtypes among children with mathematical difficulties over the first 2 years of elementary school: the number sense deficit subtype, the numerosity coding deficit subtype, the symbolic deficit subtype, the working memory subtype, and a mild difficulty group. These subtypes showed moderate stability. While some subtypes appeared to be somewhat stable over time, some developed into another profile of deficits and still some emerged only at later points in the development. Understanding the cognitive profiles of different subtypes of mathematical difficulties can better help researchers and educators to devise intervention strategies that build on what the children are able to do and improve what they are still struggling.

In this article, we review research literature concerning number sense from several related fields. Whereas other authors have pointed to difficulty defining number sense or to some degree of inconsistency in the literature, we argue instead that this is a case of polysemy: three constructs given the same name. Our purpose is to clarify the research literature concerning number sense by naming and defining these three constructs, identifying similarities and differences among them, categorizing the research traditions associated with each construct, and summarizing the methods used and findings reported. We compare and contrast themes in each body of literature by drawing on a sample of 141 research articles that were focused on number sense. We find evidence that there is confusion of number sense constructs across research traditions. We emphasize the need for clarity in order for research in each of these traditions to progress.

Thomas Kuhn (1962/2012) introduced the term "paradigm shift" to the scientific literature to describe how knowledge in science develops. The aims of this article are to identify paradigm shifts, or revolutions, that have occurred in mathematics, and to discuss their relevance to teaching mathematics in schools. The authors argue that Kuhn's ideas could be used to promote awareness that mathematics adapts to different cultural and historical contexts. In this article, they give a brief outline of the key ideas in Kuhn (1962/2012). Then, they consider its relevance to the development of mathematics. Have there been revolutions in mathematics? They go on to explore links between Kuhn's ideas and Piaget's theory of constructivism. They propose five topics from the "Australian Curriculum: Mathematics" that could be regarded as revolutionary in the sense of Kuhn: the Hindu-Arabic numeral system, algebra, irrational numbers, coordinate geometry, and infinity. They then suggest a sequence of lessons aimed at presenting one of these topics to middle years and senior secondary classes. Conclusions are presented in the final section.

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number three, whose proofs are explanatory in the sense of Steiner. Explanatory proofs transform what might initially seem mysterious or even magical into lucid mathematics. We propose possible ways to use these statements and others similar to them in a course in abstract algebra emphasizing proofs, or in an introductory course on proofs based on group theory.

Earlier institution-sponsored research revealed that about 20% of students in community college basic math and pre-algebra programs lacked a sense of part-whole relationships with whole numbers. Using the same tool with a group of 86 workforce students, about 75% placed five whole numbers on an empty number line in a way that indicated lack of part-whole thinking. This concept, needed to understand fraction and percent relationships, carries over as a grasp of the relationship between details and the main idea in factual prose, in critical thinking in job situations, and on the current high school equivalency tests.

The study was undertaken to establish the relationship between the roots of the perfect numbers and the "n" consecutive odd numbers. Odd numbers were arranged in such a way that their sums were either equal to the perfect square number or equal to a cube. The findings on the patterns and relationships of the numbers showed that there was a relationship between the roots of numbers and "n" consecutive odd numbers. From these relationships a general formula was derived which can be used to determine the number of consecutive odd numbers that can add up to get the given number and this relationship was used to define the other meaning of "n"th root of a given perfect number from consecutive "n" odd numbers point of view. If the square root of 4 is 2, then it means that there are 2 consecutive odd numbers that can add up to 4, if the cube root of 27 is 3, then there are 3 consecutive odd number that add up to 27, if the fifth root of 32 is 5, then it means that there are 5 consecutive odd numbers that add up to 32.

Magnitude understanding is critical for students to develop a deep understanding of fractions and more advanced mathematics curriculum. The research reports in this special issue underscore magnitude understanding for fractions and emphasize number lines as both an assessment and an instructional tool. In this commentary, we discuss how number lines broaden the concept of fractions for students who are tied to the more general part-whole representations of area models. We also discuss how number lines, compared to other representations, are a superior and more mathematically correct way to explain fraction concepts.

In this article, I present research on some traditional numeracy practices of the Gusilay people group in Senegal and make recommendations for developing a numeracy programme for women. Based on a strong foundation of traditional knowledge and practices, the programme will aim to meet felt needs of women who are faced with new numeracy related challenges due to changes in society. My research is placed in the framework of social practice theory, which emphasizes the fact that numeracy is not a set of skills that are learned and used in isolation, but rather practices that happen in context and vary with it. After a brief outline of social practice theory and the methodology I have chosen for my research, I analyze my findings from that perspective and suggest some practical implications for developing a numeracy programme for Gusilay women.

Prospective elementary school teachers (PTs) come to their mathematics courses fluent in using procedures for adding and subtracting multidigit whole numbers, but many are unaware of the essential features inherent in understanding the base-ten place-value system (i.e., grouping, place value, base). Understanding these features is crucial to understanding and teaching number and place value. The research aims of this paper are (1) to present a local instructional theory (LIT), designed to familiarize PTs with these features through comparison with historical number systems and (2) to present the effects of using the LIT in the PT classroom. A theory of learning (variation theory) is paired with a framework related to motivation (intellectual need) to illustrate the mutually supporting roles they may play in mathematical learning and task design. The LIT, a supporting task sequence, and the rationale for task design are shared. This theoretical contribution is then paired with evidence of PTs' changing growth in their conceptions of whole number before and after courses leveraging this task sequence.

Generally, people respond faster to small numbers with left-sided responses and large numbers with right-sided responses, a pattern known as the SNARC (spatial numerical association of response codes) effect. The SNARC effect is interpreted as evidence for amodal automatic access of magnitude and its spatial associations, because it occurs in settings where number is task-irrelevant and for different number formats. We report five studies designed to establish the degree to which activation of magnitude and its spatial associations is truly automatic and amodal. Based on the notion of autonomous automaticity, we hypothesized that the mere presence of a number form (to which participants made a color decision) would be sufficient to elicit the SNARC effect. However, we found no evidence of a SNARC effect for simple color decisions to Arabic digits (Experiment 1). There was a SNARC effect for color decision to digits when participants recognized the stimulus as a digit before responding (Experiment 2), participants viewed the digit for sufficient time before color onset (Experiments 3 and 5), or there was temporal uncertainty regarding color onset (Experiment 3). There was no SNARC effect for color decision to arrays of circles (Experiment 4), regardless of viewing time or temporal uncertainty. Overall, our results suggest that, while access to magnitude and its spatial associations is not automatic in an "all-or-none" sense, it is certainly at the strong end of automaticity, and that this automatic activation is modality dependent. Our findings are most supportive of conceptual coding accounts of the SNARC effect.

When Johann Bernoulli published his lectures on integrals in 1742, integral calculus had become very advanced since the time of their composition in 1692. Nevertheless, these lectures are of excellent clarity and simplicity even when the book deals with major problems of Mathematical Physics. Just to pique some interest, we offer a commented translation of the first lecture and some general information about the whole treatise. In his introductory lecture, Johann Bernoulli develops many examples of how the integration of polynomials could lead to the integration of radicals and then provides some relevant changes of variables. When studying the integration of [equation omitted] the integrable combinations appear for the first time; several developments will stem from them in the future.

Humans and other animals are able to make rough estimations of quantities using what has been termed the "approximate number system" (ANS). Much evidence suggests that sensitivity to numerosity correlates with symbolic math capacity, leading to the suggestion that the ANS may serve as a start-up tool to develop symbolic math. Many experiments have demonstrated that numerosity perception transcends the sensory modality of stimuli and their presentation format (sequential or simultaneous), but it remains an open question whether the relationship between numerosity and math generalizes over stimulus format and modality. Here we measured precision for estimating the numerosity of clouds of dots and sequences of flashes or clicks, as well as for paired comparisons of the numerosity of clouds of dots. Our results show that in children, formal math abilities correlate positively with sensitivity for estimation and paired-comparisons of the numerosity of visual arrays of dots. However, precision of numerosity estimation for sequences of flashes or sounds did not correlate with math, although sensitivities in all estimations tasks (for sequential or simultaneous stimuli) were strongly correlated with each other. In adults, we found no significant correlations between math scores and sensitivity to any of the psychophysical tasks. Taken together these results support the existence of a generalized number sense, and go on to demonstrate an intrinsic link between mathematics and perception of spatial, but not temporal numerosity.

There is currently considerable discussion about the relative influences of evolutionary and cultural factors in the development of early numerical skills. In particular, there has been substantial debate and study of the relationship between approximate, nonverbal (approximate magnitude system [AMS]) and exact, symbolic (symbolic number system [SNS]) representations of number. Here we examined several hypotheses concerning whether, in the earliest stages of formal education, AMS abilities predict growth in SNS abilities, or the other way around. In addition to tasks involving symbolic (Arabic numerals) and nonsymbolic (dot arrays) number comparisons, we also tested children's ability to translate between the 2 systems (i.e., mixed-format comparison). Our data included a sample of 539 kindergarten children (M = 5.17 years, SD = 0.29), with AMS, SNS, and mixed-comparison skills assessed at the beginning and end of the academic year. In this way, we provide, to the best of our knowledge, the most comprehensive test to date of the direction of influence between the AMS and SNS in early formal schooling. Results were more consistent with the view that SNS abilities at the beginning of kindergarten lay the foundation for improvement in both AMS abilities and the ability to translate between the 2 systems. It is important to note that we found no evidence to support the reverse. We conclude that, once one acquires a basic grasp of exact number symbols, it is this understanding of exact number (and perhaps repeated practice therewith) that facilitates growth in the AMS. Though the precise mechanism remains to be understood, these data challenge the widely held view that the AMS scaffolds the acquisition of the SNS.

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems--the approximate number system (ANS) and the object tracking system (OTS)--but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single-mechanism (ANS-only and OTS-only) and two dual-mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single-mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.

Previous studies in both typically and atypically developing children have shown that approximate number system (ANS) abilities predict formal mathematical knowledge later on in life. The current study investigated whether playing specially designed training games that targets the ANS system using nonsymbolic stimuli only would improve preschool children's ANS abilities. Thirty-eight preschool children were randomly allocated to either the training or control group. For 5 weeks, 20 preschoolers (9 girls) in the training group played daily games for 10 min that included guessing and comparing numerosities, whereas 18 control children (6 girls) were involved in interactive picture book reading sessions. Children's ANS abilities were assessed using a computerized task before and after the training program. An analysis of covariance with posttraining ANS scores as dependent variable and pretraining scores as a covariate showed that the children in the training group had higher ANS abilities after the training, in contrast to children in the control group (p = 0.012, ?[superscript 2][subscript p] = 0.171). This study provides evidence that ANS abilities can be improved in preschool children through a daily training program that targets the ANS specifically. These findings provide support for further training programs for preschool children who show mathematical difficulties early on in life.

While conceptual understanding of properties, operations, and the base-ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three elementary schools in a school district implements mandatory automaticity drills for 10 minutes each day while the remaining two elementary schools, with the same curriculum and very similar demographics, do not. This study looks at (a) the impact that schoolwide implementation of automaticity drills has on schoolwide computational math skills as measured by the ITBS and (b) the relationship between automaticity and conceptual understanding as measured by statewide standardized testing. The results suggest that while there may be an association between automaticity and higher performance on standardized tests, caution should be taken before assuming there are benefits to promoting automaticity drills. These results are consistent with those that support a process-driven approach to automaticity based on familiarity with properties and strategies associated with the base-ten number system; they are not consistent with those that support an answer-driven approach to automaticity based on memorization of answers.

A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non-symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5- and 6-year-old children using comparison tasks of non-symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non-numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non-symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non-numerical magnitude.

The natural numbers may be our simplest, most useful, and best-studied abstract concepts, but their origins are debated. I consider this debate in the context of the proposal, by Gallistel and Gelman, that natural number system is a product of cognitive evolution and the proposal, by Carey, that it is a product of human cultural history. I offer a third proposal that builds on aspects of these views but rejects one tenet that they share: the thesis that counting is central to number. I suggest that children discover the natural numbers when they learn a natural language: especially nouns, number words, and the rules that compose quantified noun phrases. This learning, in turn, depends both on cognitive systems that are innate and shared by other animals, and on our species-specific language faculty. Thus, natural number concepts are unique to humans and culturally universal, yet they are learned.

In this commentary, I examine some of the distinctive, foundational difficulties in learning fractions and other types of rational numbers encountered by students with a mathematical learning disability and how these differ from the struggles experienced by students classified as low achieving in math. I discuss evidence indicating that students with math disabilities exhibit a significant delay or deficit in the numerical transcoding of decimal fractions, and I further maintain that they may face unique challenges in developing the ability to effectively translate between different types of fractions and other rational number notational formats--what I call "conceptual transcoding." I also argue that characterizing this level of comprehensive understanding of rational numbers as "rational number" sense is irrational, as it misrepresents this flexible and adaptive collection of skills as a biologically based percept rather than a convergence of higher-order competencies that require intensive, formal instruction.