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Concept: Arithmetic

261

Human infants in the first year of life possess an intuitive sense of number. This preverbal number sense may serve as a developmental building block for the uniquely human capacity for mathematics. In support of this idea, several studies have demonstrated that nonverbal number sense is correlated with mathematical abilities in children and adults. However, there has been no direct evidence that infant numerical abilities are related to mathematical abilities later in childhood. Here, we provide evidence that preverbal number sense in infancy predicts mathematical abilities in preschool-aged children. Numerical preference scores at 6 months of age correlated with both standardized math test scores and nonsymbolic number comparison scores at 3.5 years of age, suggesting that preverbal number sense facilitates the acquisition of numerical symbols and mathematical abilities. This relationship held even after controlling for general intelligence, indicating that preverbal number sense imparts a unique contribution to mathematical ability. These results validate the many prior studies purporting to show number sense in infancy and support the hypothesis that mathematics is built upon an intuitive sense of number that predates language.

Concepts: Infant, Mathematics, Number, Real number, Set theory, Natural number, Infinity, Arithmetic

65

One in five adults in the United States is functionally innumerate; they do not possess the mathematical competencies needed for many modern jobs. We administered functional numeracy measures used in studies of young adults' employability and wages to 180 thirteen-year-olds. The adolescents began the study in kindergarten and participated in multiple assessments of intelligence, working memory, mathematical cognition, achievement, and in-class attentive behavior. Their number system knowledge at the beginning of first grade was defined by measures that assessed knowledge of the systematic relations among Arabic numerals and skill at using this knowledge to solve arithmetic problems. Early number system knowledge predicted functional numeracy more than six years later (ß = 0.195, p = .0014) controlling for intelligence, working memory, in-class attentive behavior, mathematical achievement, demographic and other factors, but skill at using counting procedures to solve arithmetic problems did not. In all, we identified specific beginning of schooling numerical knowledge that contributes to individual differences in adolescents' functional numeracy and demonstrated that performance on mathematical achievement tests underestimates the importance of this early knowledge.

Concepts: Mathematics, United States, Learning, Number, Arithmetic, Problem solving, Numeral system, Mathematics education

31

Machines that simultaneously process and store multistate data at one and the same location can provide a new class of fast, powerful and efficient general-purpose computers. We demonstrate the central element of an all-optical calculator, a photonic abacus, which provides multistate compute-and-store operation by integrating functional phase-change materials with nanophotonic chips. With picosecond optical pulses we perform the fundamental arithmetic operations of addition, subtraction, multiplication, and division, including a carryover into multiple cells. This basic processing unit is embedded into a scalable phase-change photonic network and addressed optically through a two-pulse random access scheme. Our framework provides first steps towards light-based non-von Neumann arithmetic.

Concepts: Mathematics, Optics, Addition, Computer, Arithmetic, Division, Multiplication, Elementary arithmetic

31

Many fields face an increasing prevalence of multi-authorship, and this poses challenges in assessing citation metrics. Here, we explore multiple citation indicators that address total impact (number of citations, Hirsch H index [H]), co-authorship adjustment (Schreiber Hm index [Hm]), and author order (total citations to papers as single; single or first; or single, first, or last author). We demonstrate the correlation patterns between these indicators across 84,116 scientists (those among the top 30,000 for impact in a single year [2013] in at least one of these indicators) and separately across 12 scientific fields. Correlation patterns vary across these 12 fields. In physics, total citations are highly negatively correlated with indicators of co-authorship adjustment and of author order, while in other sciences the negative correlation is seen only for total citation impact and citations to papers as single author. We propose a composite score that sums standardized values of these six log-transformed indicators. Of the 1,000 top-ranked scientists with the composite score, only 322 are in the top 1,000 based on total citations. Many Nobel laureates and other extremely influential scientists rank among the top-1,000 with the composite indicator, but would rank much lower based on total citations. Conversely, many of the top 1,000 authors on total citations have had no single/first/last-authored cited paper. More Nobel laureates of 2011-2015 are among the top authors when authors are ranked by the composite score than by total citations, H index, or Hm index; 40/47 of these laureates are among the top 30,000 by at least one of the six indicators. We also explore the sensitivity of indicators to self-citation and alphabetic ordering of authors in papers across different scientific fields. Multiple indicators and their composite may give a more comprehensive picture of impact, although no citation indicator, single or composite, can be expected to select all the best scientists.

Concepts: Science, Arithmetic, Impact factor, Order theory, Nobel Prize, Bibliometrics, H-index, Citation impact

30

To interpret visual-motion events, the underlying computation must involve internal reference to the motion status of the observer’s head. We show here that layer 6 (L6) principal neurons in mouse primary visual cortex (V1) receive a diffuse, vestibular-mediated synaptic input that signals the angular velocity of horizontal rotation. Behavioral and theoretical experiments indicate that these inputs, distributed over a network of 100 L6 neurons, provide both a reliable estimate and, therefore, physiological separation of head-velocity signals. During head rotation in the presence of visual stimuli, L6 neurons exhibit postsynaptic responses that approximate the arithmetic sum of the vestibular and visual-motion response. Functional input mapping reveals that these internal motion signals arrive into L6 via a direct projection from the retrosplenial cortex. We therefore propose that visual-motion processing in V1 L6 is multisensory and contextually dependent on the motion status of the animal’s head.

Concepts: Brain, Angular momentum, Summation, Visual perception, Arithmetic, Lateral geniculate nucleus, Retinotopy, Motion perception

27

A photonic approach to instantaneously identify frequency components of microwave signals with multiple tones is conceived and practically demonstrated. A mathematical model was first developed to predict the behavior of the system. Then the system operation was tested in practice. The system employs a double mixing technique that enables high-frequency measurement without the need for any high-frequency RF component or broadband photodetector. The system operation was demonstrated over a frequency range of 0.1-40 GHz. Frequency measurement of two simultaneous RF tones is demonstrated; however, the system has the potential to be expanded to measure a larger number of simultaneous RF tones. It also has the potential to operate over a wider frequency range.

Concepts: Mathematics, Optics, Measurement, Electromagnetic spectrum, Frequency, Arithmetic, Identification

27

OBJECTIVE: The relatively low success rate of the treatment maneuver for horizontal semicircular canal (HSC) benign paroxysmal positional vertigo (BPPV) may be caused by the difficulty determining the affected side. We developed a 180-degree supine roll test (SRT) by modifying the 90-degree SRT to increase diagnostic accuracy and evaluated its significance. STUDY DESIGN: A prospective study. SETTING: Tertiary referral center. PATIENTS: A total of 122 patients with HSC-BPPV performed both the 90- and 180-degree SRTs. INTERVENTIONS: The affected side was determined by the 90- and 180-degree SRTs. The bow and lean (BL) test was also performed in cases with ambiguous or opposite results on both SRTs. MAIN OUTCOME MEASURE: A comparison of the difference in slow phase velocity (SPV) of nystagmus among the 90- and 180-degree SRTs and BL test. RESULTS: The maximum SPV of nystagmus during the 180-degree SRT was significantly greater than that during the 90-degree SRT. The SPV difference was less in the 180-degree SRT than that in the 90-degree SRT. Although the 180-degree SRT showed fewer meaningful results (n = 65) than the 90-degree SRT (n = 71), the affected side was determined by the 180-degree SRT in 15 cases with ambiguous results on the 90-degree SRT. Among 10 cases showing opposite results, 7 were identified by the BL test. Five (71.4%) of 7 cases had consistent affected sides with the 180-degree SRT. CONCLUSION: The 180-degree SRT can be an additional method when it is difficult to determine the affected side from the 90-degree SRT.

Concepts: Measurement, Semicircular canal, Arithmetic, Ear, Determinacy, Pathologic nystagmus, Benign paroxysmal positional vertigo, Horizontal semicircular canal

16

Studies in the literature have provided conflicting evidence about the effects of background noise or music on concurrent cognitive tasks. Some studies have shown a detrimental effect, while others have shown a beneficial effect of background auditory stimuli. The aim of this study was to investigate the influence of agitating, happy or touching music, as opposed to environmental sounds or silence, on the ability of non-musician subjects to perform arithmetic operations. Fifty university students (25 women and 25 men, 25 introverts and 25 extroverts) volunteered for the study. The participants were administered 180 easy or difficult arithmetic operations (division, multiplication, subtraction and addition) while listening to heavy rain sounds, silence or classical music. Silence was detrimental when participants were faced with difficult arithmetic operations, as it was associated with significantly worse accuracy and slower RTs than music or rain sound conditions. This finding suggests that the benefit of background stimulation was not music-specific but possibly due to an enhanced cerebral alertness level induced by the auditory stimulation. Introverts were always faster than extroverts in solving mathematical problems, except when the latter performed calculations accompanied by the sound of heavy rain, a condition that made them as fast as introverts. While the background auditory stimuli had no effect on the arithmetic ability of either group in the easy condition, it strongly affected extroverts in the difficult condition, with RTs being faster during agitating or joyful music as well as rain sounds, compared to the silent condition. For introverts, agitating music was associated with faster response times than the silent condition. This group difference may be explained on the basis of the notion that introverts have a generally higher arousal level compared to extroverts and would therefore benefit less from the background auditory stimuli.

Concepts: Mathematics, Addition, Arithmetic, Sound, Music, Noise, Elementary arithmetic, Subtraction

16

We present the results of a gamified mobile device arithmetic application which allowed us to collect vast amount of data in simple arithmetic operations. Our results confirm and replicate, on a large sample, six of the main principles derived in a long tradition of investigation: size effect, tie effect, size-tie interaction effect, five-effect, RTs and error rates correlation effect, and most common error effect. Our dataset allowed us to perform a robust analysis of order effects for each individual problem, for which there is controversy both in experimental findings and in the predictions of theoretical models. For addition problems, the order effect was dominated by a max-then-min structure (i.e 7+4 is easier than 4+7). This result is predicted by models in which additions are performed as a translation starting from the first addend, with a distance given by the second addend. In multiplication, we observed a dominance of two effects: (1) a max-then-min pattern that can be accounted by the fact that it is easier to perform fewer additions of the largest number (i.e. 8x3 is easier to compute as 8+8+8 than as 3+3+…+3) and (2) a phonological effect by which problems for which there is a rhyme (i.e. “seis por cuatro es veinticuatro”) are performed faster. Above and beyond these results, our study bares an important practical conclusion, as proof of concept, that participants can be motivated to perform substantial arithmetic training simply by presenting it in a gamified format.

Concepts: Scientific method, Regression analysis, Mathematics, Addition, Experiment, Arithmetic, Multiplication, Elementary arithmetic

14

In this article, we tested, using a 1-year longitudinal design, whether symbolic numerical magnitude processing or children’s numerical representation of Arabic digits, is as important to arithmetic as phonological awareness is to reading. Children completed measures of symbolic comparison, phonological awareness, arithmetic, reading at the start of third grade and the latter two were retested at the start of fourth grade. Cross-sectional and longitudinal correlations indicated that symbolic comparison was a powerful domain-specific predictor of arithmetic and that phonological awareness was a unique predictor of reading. Crucially, the strength of these independent associations was not significantly different. This indicates that symbolic numerical magnitude processing is as important to arithmetic development as phonological awareness is to reading and suggests that symbolic numerical magnitude processing is a good candidate for screening children at risk for developing mathematical difficulties.

Concepts: Mathematics, Cross-sectional study, Number, Arithmetic, Numerical digit