Children’s Mapping between Non-Symbolic and Symbolic Numerical Magnitudes and Its Association with Timed and Untimed Tests of Mathematics Achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.     

Impact of high mathematics education on the number sense

In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.     

A Two-Minute Paper-and-Pencil Test of Symbolic and Nonsymbolic Numerical Magnitude Processing Explains Variability in Primary School Children's Arithmetic Competence

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.     

False Approximations of the Approximate Number System?

Prior research suggests that the acuity of the approximate number system (ANS) predicts future mathematical abilities. Modelling the development of the ANS might therefore allow monitoring of children's mathematical skills and instigate educational intervention if necessary. A major problem however, is that our knowledge of the development of the ANS is acquired using fundamentally different paradigms, namely detection in infants versus discrimination in children and adults. Here, we question whether such a comparison is justified, by testing the adult ANS with both a discrimination and a detection task. We show that adults perform markedly better in the discrimination compared to the detection task. Moreover, performance on discrimination but not detection, correlated with performance on mathematics. With a second similar experiment, in which the detection task was replaced by a same-different task, we show that the results of experiment 1 cannot be attributed to differences in chance level. As only task instruction differed, the discrimination and the detection task most likely reflect differences at the decisional level. Future studies intending to model the development of the ANS should therefore rely on data derived from a single paradigm for different age groups. The same-different task appears a viable candidate, due to its applicability across age groups.     

Preschoolers' precision of the approximate number system predicts later school mathematics performance

The Approximate Number System (ANS) is a primitive mental system of nonverbal representations that supports an intuitive sense of number in human adults, children, infants, and other animal species. The numerical approximations produced by the ANS are characteristically imprecise and, in humans, this precision gradually improves from infancy to adulthood. Throughout development, wide ranging individual differences in ANS precision are evident within age groups. These individual differences have been linked to formal mathematics outcomes, based on concurrent, retrospective, or short-term longitudinal correlations observed during the school age years. However, it remains unknown whether this approximate number sense actually serves as a foundation for these school mathematics abilities. Here we show that ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicts performance on school mathematics at 6 years of age. In contrast, ANS precision does not predict non-numerical cognitive abilities. To our knowledge, these results provide the first evidence for early ANS precision, measured before the onset of formal education, predicting later mathematical abilities.     

In How Many Ways is the Approximate Number System Associated with Exact Calculation?

The approximate number system (ANS) has been consistently found to be associated with math achievement. However, little is known about the interactions between the different instantiations of the ANS and in how many ways they are related to exact calculation. In a cross-sectional design, we investigated the relationship between three measures of ANS acuity (non-symbolic comparison, non-symbolic estimation and non-symbolic addition), their cross-sectional trajectories and specific contributions to exact calculation. Children with mathematical difficulties (MD) and typically achieving (TA) controls attending the first six years of formal schooling participated in the study. The MD group exhibited impairments in multiple instantiations of the ANS compared to their TA peers. The ANS acuity measured by all three tasks positively correlated with age in TA children, while no correlation was found between non-symbolic comparison and age in the MD group. The measures of ANS acuity significantly correlated with each other, reflecting at least in part a common numerosity code. Crucially, we found that non-symbolic estimation partially and non-symbolic addition fully mediated the effects of non-symbolic comparison in exact calculation.     

The Effects of Heart Rate Variability (HRV) Biofeedback on HRV Reactivity and Recovery During and After Anger Recall Task for Patients with Coronary Artery Disease

Patients with coronary artery disease (CAD) often experience anger events before cardiovascular events. Anger is a psychological risk factor and causes underlying psychophysiological mechanisms to lose balance of the autonomic nervous system (ANS). The heart rate variability (HRV) was the common index for ANS regulation. It has been confirmed that heart rate variability biofeedback (HRV-BF) restored ANS balance in patients with CAD during the resting state. However, the effects of HRV-BF during and after the anger event remain unknown. This study aimed to examine the effects of HRV-BF on ANS reactivity and recovery during the anger recall task in patients with CAD. This study was a randomized control trial with a wait-list control group design, with forty patients in the HRV-BF group (for six sessions) and 44 patients in the control group. All patients received five stages of an anger recall task, including baseline, neutral recall task, neutral recovery, anger recall task, and anger recovery. HRV reactivity in the HRV-BF group at the post-test was lower than that in the control group. HRV recovery at the post-test in the HRV-BF group was higher than that in the control group. The HRV-BF reduced ANS reactivity during anger events and increased ANS recovery after anger events for CAD patients. The possible mechanisms of HRV-BF may increase total HRV, ANS regulation, and baroreflex activation at anger events for patients with CAD, and may be a suitable program for cardiac rehabilitation.

     

Autonomic nervous system responses as performance indicators among volleyball players

Complex motor skills require planning and programming before execution. The autonomic nervous system (ANS) is thought to transcribe these central operations at the peripheral level: a motor act is thought to be simultaneously programmed by central and autonomic nervous structures. The aim of this study was to verify that autonomic responses reflect the quality of central motor programming leading to successful or failed performance when subjects are required to perform a complex motor skill. The specificity of the ANS response has already been demonstrated through direct recording from sympathetic fibres. It has also been demonstrated through several mental tasks and closed motor skills such as shooting: ANS responses have been shown to be capable of distinguishing success from failure. The aim of this experiment was to test whether ANS responses are capable of distinguishing two levels of achievement during the performance of a skill involving uncertainty (open skill). The subjects had to intercept a ball on a volleyball court, using the forearm receive and pass technique, in order to pass it on to a moving human target. The results were displayed in terms of accuracy: accurate passes were successful and inaccurate passes missed the target. Six autonomic variables were recorded simultaneously during the task: skin resistance and potential, skin blood flow and temperature, instantaneous heart rate and respiratory frequency. Results showed that autonomic variables were capable of distinguishing success from failure in 22 subjects out of 24. This made it possible to build up autonomic patterns characterising subjects' performances, and to confirm that autonomic functioning may reveal information processing in the central nervous system. Thus, the study of autonomic responses may constitute an inferential model of central nervous system functioning. Such a method could be used as an index for the control of mental preparation.     

Mental representation of number in different numerical forms

How are numerical operations implemented within the human brain? It has been suggested that there are at least three different codes for representing number: a verbal code that is used to manipulate number words and perform mental numerical operations (e.g., multiplication), a visual code that is used to decode frequently used visual number forms (e.g., Arabic digits), and an abstract analog code that may be used to represent numerical quantities. Furthermore, each of these codes is associated with a different neural substrate. We extend these studies using dense-sensor event-related EEG recording techniques to investigate the temporal pattern of notation-specific effects observed in a parity judgement (odd versus even) task in which single numbers were presented in one of four different numerical notations. Contrasts between different notations demonstrated clear modulations in the visual evoked potentials (VEP) recorded. We observed increased amplitudes for the P1 and N1 components of the VEP that were specific to Arabic numerals and to dot configurations but differed for random and recognizable (die-face) dot configurations. These results demonstrate clear, notation-specific differences in the time course of numerical information processing and provide electrophysiological support for the triple-code model of numerical representation.     

Preschoolers' Dot Enumeration Abilities Are Markers of Their Arithmetic Competence

The abilities to enumerate small sets of items (e.g., dots) and to compare magnitudes are claimed to be indexes of core numerical competences that scaffold early math development. Insofar as this is correct, these abilities may be diagnostic markers of math competence in preschoolers. However, unlike magnitude comparison abilities, little research has examined preschoolers'' ability to enumerate small sets, or its significance for emerging math abilities; which is surprising since dot enumeration is a marker of school-aged children''s math competence. It is nevertheless possible that general cognitive functions (working memory, response inhibition in particular) are associated with preschoolers'' math abilities and underlie nascent dot enumeration abilities. We investigated whether preschoolers'' dot enumeration abilities predict their non-verbal arithmetic ability, over and above the influence of working memory and response inhibition. Two measures of dot enumeration ability were examined—inverse efficiency and paradigm specific (response time profiles) measures—to determine which has the better diagnostic utility as a marker of math competence. Seventy-eight 42-to-57 month-olds completed dot enumeration, working memory, response inhibition, and non-verbal addition and subtraction tasks. Dot enumeration efficiency predicted arithmetic ability over and above the influence of general cognitive functions. While dot enumeration efficiency was a better predictor of arithmetic ability than paradigm specific response time profiles; the response time profile displaying the smallest subitizing range and steepest subitizing slope, also displayed poor addition abilities, suggesting a weak subitizing profile may have diagnostic significance in preschoolers. Overall, the findings support the claim that dot enumeration abilities and general cognitive functions are markers of preschoolers'' math ability.     

Numerical Capacities as Domain-Specific Predictors beyond Early Mathematics Learning: A Longitudinal Study

The first aim of the present study was to investigate whether numerical effects (Numerical Distance Effect, Counting Effect and Subitizing Effect) are domain-specific predictors of mathematics development at the end of elementary school by exploring whether they explain additional variance of later mathematics fluency after controlling for the effects of general cognitive skills, focused on nonnumerical aspects. The second aim was to address the same issues but applied to achievement in mathematics curriculum that requires solutions to fluency in calculation. These analyses assess whether the relationship found for fluency are generalized to mathematics content beyond fluency in calculation. As a third aim, the domain specificity of the numerical effects was examined by analyzing whether they contribute to the development of reading skills, such as decoding fluency and reading comprehension, after controlling for general cognitive skills and phonological processing. Basic numerical capacities were evaluated in children of 3^rd and 4^th grades (n=49). Mathematics and reading achievements were assessed in these children one year later. Results showed that the size of the Subitizing Effect was a significant domain-specific predictor of fluency in calculation and also in curricular mathematics achievement, but not in reading skills, assessed at the end of elementary school. Furthermore, the size of the Counting Effect also predicted fluency in calculation, although this association only approached significance. These findings contrast with proposals that the core numerical competencies measured by enumeration will bear little relationship to mathematics achievement. We conclude that basic numerical capacities constitute domain-specific predictors and that they are not exclusively “start-up” tools for the acquisition of Mathematics; but they continue modulating this learning at the end of elementary school.     

Interaction of the fluorescent probe, 1-anilino-8-napthalene sulfonate, with the sulfate transport system of Ehrlich ascites tumor cells.

The addition of the fluorescent dye, ANS, to intact ascites tumor cells results in an enhancement of fluorescence intensity. The increase in fluorescence intensity as a function of time is biphasic which suggests that at least two processes occur. The first associated with the rapid initial rise in fluorescence represents binding to the cell surface while the second or slower phase reflects entrance of ANS into the intracellular phase. The relationship between bound and free ANS in 0.50 mM sulfate medium was used to calculate the apparent dissociation constant of ANS-membrane complex (Kd = 6.53 times 10(-5) M) and the total number of ANS binding sites (4.49 nmoles/mg dry weight). Kinetic analysis of steady state sulfate transport in the presence and absence of ANS suggests that (1) sulfate exchange can be described by Michaelis Menten type kinetics (Km = 2.05 times 10(-3) M), (2) a small fraction of surface associated ANS competitively inhibits sulfate exchange (Ki = 4.28 times 10(-6) M) and (3) the transport system has a higher affinity for ANS than for sulfate. These data are consistent with the hypothesis that inhibition of sulfate exchange is related to the direct, reversible interaction of the negatively charged sulfonate group of ANS with superficial positively charged membrane sites.     

Intracellular fluorescence decay time of anilinonaphthalene sulfonate

This paper presents the intracellular fluorescence decay time of the probe anilinonaphthalene sulfonic acid (ANS) and compares the results to certain ANS complexes in vitro. There is relatively constant decay time for intracellular ANS over a range of concentrations in the incubating medium, compared with marked variation in results with the complex of ANS-bovine serum albumin in vitro when concentration of the probe is varied. Calculation of the apparent rotational relaxation time from the Perrin equation, using ANS intracellular decay time and polarization data gave a tentative value of circa 66 ns. By comparison with the results of ANS complexes with cell fractions and with certain lipids these data support the concept that intracellularly the compound may be largely membrane located with a portion of the molecule in the lipid phase. Cells damaged by heating or alcohol show longer decay time than those which have taken up ANS in the living state. Suggestions for refinement of technique are included in the discussion.     

Gluconeogenesis and the peroxisome

The interaction between hydroperoxides, cytochrome P450 and 8-anilino-1-naphthalenesulfonic acid (ANS) has been investigated. The addition of ANS to the cytochrome P450 solution did not effect the P450 Soret absorption peak or the reduced CO difference spectrum, suggesting that ANS may not bind to P450 heme directly. H2O2 or CuOOH alone did not effect ANS fluorescence and absorption spectra indicating that no detectable reaction occurs between hydroperoxide and ANS in the absence of P450. The reconstituted system of cytochrome P450, P450 reductase, lipid and NADPH did not mediate ANS metabolism. In the presence of P450, the addition of either H2O2 or CuOOH, however, leads to a decrease in ANS absorption around 258 nm and 350 nm indicating possible destruction of ANS. ANS destruction was confirmed with the disappearance of the ANS elution peak in the reverse phase HPLC profiles and with the changes in P450-bound ANS fluorescence intensity and the shift of max of ANS. Moreover , a very sensitive method to detect trace fluorescent products of ANS by thin layer chromatography has been developed based on the fact that ANS fluorescence is enhanced more than 1000-fold by the organic solvent butanol. A UV-sensitive fluorescent product was detected on thin layer chromatography profiles of the reaction mixtures. P450 was also observed to be modified by a fluorescent derivative of ANS, when the fluorescence was enhanced by butanol. These results also show that an organic compound which can not be metabolized by the reconstituted system of cytochrome P450 and NADPH-P450 reductase is metabolized by the reconstituted system of P450 and hydroperoxide, suggesting the activities of these two systems may not be completely comparable. (Mol Cell Biochem 167: 159-168, 1997)     

Quantifying cardiac sympathetic and parasympathetic nervous activities using principal dynamic modes analysis of heart rate variability

The ratio between low-frequency (LF) and high-frequency (HF) spectral power of heart rate has been used as an approximate index for determining the autonomic nervous system (ANS) balance. An accurate assessment of the ANS balance can only be achieved if clear separation of the dynamics of the sympathetic and parasympathetic nervous activities can be obtained, which is a daunting task because they are nonlinear and have overlapping dynamics. In this study, a promising nonlinear method, termed the principal dynamic mode (PDM) method, is used to separate dynamic components of the sympathetic and parasympathetic nervous activities on the basis of ECG signal, and the results are compared with the power spectral approach to assessing the ANS balance. The PDM analysis based on the 28 subjects consistently resulted in a clear separation of the two nervous systems, which have similar frequency characteristics for parasympathetic and sympathetic activities as those reported in the literature. With the application of atropine, in 13 of 15 supine subjects there was an increase in the sympathetic-to-parasympathetic ratio (SPR) due to a greater decrease of parasympathetic than sympathetic activity (P=0.003), and all 13 subjects in the upright position had a decrease in SPR due to a greater decrease of sympathetic than parasympathetic activity (P<0.001) with the application of propranolol. The LF-to-HF ratio calculated by the power spectral density is less accurate than the PDM because it is not able to separate the dynamics of the parasympathetic and sympathetic nervous systems. The culprit is equivalent decreases in both the sympathetic and parasympathetic activities irrespective of the pharmacological blockades. These findings suggest that the PDM shows promise as a noninvasive and quantitative marker of ANS imbalance, which has been shown to be a factor in many cardiac and stress-related diseases.     

Distracting the mind improves performance: an ERP Study

Background

When a second target (T2) is presented in close succession of a first target (T1), people often fail to identify T2, a phenomenon known as the attentional blink (AB). However, the AB can be reduced substantially when participants are distracted during the task, for instance by a concurrent task, without a cost for T1 performance. The goal of the current study was to investigate the electrophysiological correlates of this paradoxical effect.

Methodology/Principal Findings

Participants successively performed three tasks, while EEG was recorded. The first task (standard AB) consisted of identifying two target letters in a sequential stream of distractor digits. The second task (grey dots task) was similar to the first task with the addition of an irrelevant grey dot moving in the periphery, concurrent with the central stimulus stream. The third task (red dot task) was similar to the second task, except that detection of an occasional brief color change in the moving grey dot was required. AB magnitude in the latter task was significantly smaller, whereas behavioral performance in the standard and grey dots tasks did not differ. Using mixed effects models, electrophysiological activity was compared during trials in the grey dots and red dot tasks that differed in task instruction but not in perceptual input. In the red dot task, both target-related parietal brain activity associated with working memory updating (P3) as well as distractor-related occipital activity was significantly reduced.

Conclusions/Significance

The results support the idea that the AB might (at least partly) arise from an overinvestment of attentional resources or an overexertion of attentional control, which is reduced when a distracting secondary task is carried out. The present findings bring us a step closer in understanding why and how an AB occurs, and how these temporal restrictions in selective attention can be overcome.     

Partially folded conformations of bovine liver glutamate dehydrogenase induced by mild acidic conditions

The acid-induced unfolding of bovine liver glutamate dehydrogenase (GDH) was studied using various spectroscopic methods such as far- and near-UV circular dichroism (CD), intrinsic and 1-anilino naphthalene-8-sulphonate (ANS) extrinsic fluorescence spectroscopy, light scattering and fluorescence quenching in 20 mM mixed buffer at various pHs. CD spectra show that at pH 3.5, GDH retains its secondary structure substantially, whereas its tertiary structure content is reduced considerably. Intrinsic fluorescence of GDH and ANS binding suggest that, at pH 3.5, the hydrophobic surface of enzyme is more exposed in comparison to the native form. Acrylamide quenching indicates more exposure of tryptophan residues of enzyme at pH 3.5 in comparison to pH 7.5. Another partially unfolded intermediate was detected at pH 5.0, which with its ANS binding capacity lies between the pH 3.5 intermediate and the native form of the enzyme. Gel filtration results revealed that the enzyme at pH 3.5 is dissociated into trimeric species whereas it exists as hexamer at pH 7.5 and 5.0. All the data taken together suggest the existence of two partially unfolded states of GDH at moderate acidic pHs which may be considered as molten and pre-molten globule-like states.     

The interaction of the anionic fluorescence probe, 1-anilinonaphthalene-8-sulfonate, with hepatocytes and hepatoma tissue culture cells

The fluorescence probe 1-anilinonaphthalene-8-sulfonate (ANS) has been used to characterize the anion transport properties of normal hepatocytes and hepatoma tissue culture cells. Incubation of hepatocytes in the presence of ANS (20 micron) resulted in a 35-fold enhancement of fluorescence and a 50 nm blue shift. The time course of this process is biphasic. A rapid initial fluorescence enhancement suggests ANS binding to the plasma membrane, and a slower component reflects the uptake of ANS into intracellular compartments. Analysis of ANS uptake showed this latter process to be saturable, with a Km of 10 micron, to be temperature dependent and to occur only in viable cells. The above observations suggest a carrier-mediated anion transport mechanism. Incubation of hepatoma tissue culture cells with ANS (20 micron) gave a fluorescence emission spectrum similar to that obtained from purified plasma membranes. The kinetics of this interaction only exhibited a rapid initial binding of ANS. The second slow component was now absent, suggesting that ANS transport by the malignant cell system was greatly reduced. Transport of ANS could, however, be stimulated in the presence of the local anesthetic tetracaine. The observed transport was now saturable, temperature dependent, and as in normal hepatocytes, required viable cells, again indicating a carrier-mediated transport system. These studies suggest a significant alteration in membrane function in hepatoma tissue culture cells resulting in a major defect in anion transport.     

A study of carbachol-atropine interaction on intestinal smooth muscle vesicles, using a fluorescent probe.

Plasma membrane vesicles were prepared from guinea pig ileum longitudinal muscle. The vesicles were characterized by electron microscopy and analysis of lipid and protein content. They were shown to be free of gross contamination from actomyosin, sarcoplasmic reticulum, and mitochondria. 8-Anilino-1-naphthalene sulphonic acid (ANS) binding characteristics were similar to those found in other membranes. Both carbachol and atropine increased the fluorescence of ANS bound to this membrane, the maximum increase for atropine being greater than that for carbachol. Since neither drug effected the apparent affinity constant for the ANS-membrane interaction. It may be assumed that the increased fluorescence was due to an increase in the number of ANS binding sites. The carbachol-dependent increase in ANS fluorescence was blocked noncompetitively by atropine but not by tubocurarine or diphenhydramine. These latter two antagonists also increased ANS fluorescence but at much higher concentrations than either carbachol or atropine. Neither atropine nor carbachol increased ANS fluorescence on either erythrocyte ghosts or liposomes (prepared from a lipid extract of the muscle membrane).     

Beliefs about genetic influences on mathematics achievement: a cross-cultural comparison

The poor mathematics performance of children in the United States has become a topic of national concern. Numerous studies have shown that American children consistently perform worse than their counterparts in many parts of the world. In contrast, children in China, Japan, Taiwan, and other Asian countries consistently perform at or near the top in international comparisons. This paper examines possible causes of the poor performance of American children and the excellent performance of Asian children. Contrary to the beliefs of many Americans, the East Asian advantage in mathematics is probably not due to a genetically-based advantage in mathematics. Instead, differences in beliefs about the role of genetics may be partly responsible. Asians strongly believe that effort plays a key role in determining a child's level of achievement, whereas Americans believe that innate ability is most important. In addition, despite the relatively poor performance of their children, American parents are substantially more satisfied with their children's performance than Asian parents. The American emphasis on the role of innate ability may have several consequences for children's achievement. For example, it may lead children to fear making errors and to expend less effort on mathematics than their Asian counterparts. As research on genetic influences on behavior, traits, and abilities increases scientists should be careful to ensure that the public understands that genetics does not directly determine the exact level of a child's potential achievement. This revised version was published online in August 2006 with corrections to the Cover Date.