Multi-digit Numbers Research Articles

Reversing the Manual Digit Bias in Two-Digit Number Comparison.

Though recent work in numerical cognition has supported a strong tie between numerical and spatial representations (e.g., a mental number line), less is known about such ties in multi-digit number representations. Along this line, Bloechle, Huber, and Moeller (2015) found that pointing positions in two-digit number comparison were biased leftward toward the decade digit. Moreover, this bias was reduced in unit-decade incompatible pairs. In the present study, we tracked computer mouse movements as participants compared two-digit numbers to a fixed standard (55). Similar to Bloechle et al. (2015) , we found that trajectories exhibited a leftward bias that was reduced for unit-decade incompatible comparisons. However, when positions of response labels were reversed, the biases reversed. That is, we found a rightward bias for compatible pairs that was reduced for incompatible pairs. This result calls into question a purely embodied representation of place value structure and instead supports a competition model of two-digit number representation.

Josephson Fourier spectrometer based on HTSC: Construction and the problem of physical implementation of a quantum computer

The possibility of developing the broadband Fourier spectrometer based on the unsteady Josephson effect in HTSC, with a frequency range including the terahertz region, is considered. A simple design of a precisely tuned point Josephson junction is developed, which allows “fitting” of its parameters immediately under cryogenic conditions. A modification of such a spectrometer for solving the problems of factorizing multidigit numbers is proposed, which is currently one of the best known motivations of works on the quantum computer development.

Is place-value processing in four-digit numbers fully automatic? Yes, but not always.

Knowing the place-value of digits in multi-digit numbers allows us to identify, understand and distinguish between numbers with the same digits (e.g., 1492 vs. 1942). Research using the size congruency task has shown that the place-value in a string of three zeros and a non-zero digit (e.g., 0090) is processed automatically. In the present study, we explored whether place-value is also automatically activated when more complex numbers (e.g., 2795) are presented. Twenty-five participants were exposed to pairs of four-digit numbers that differed regarding the position of some digits and their physical size. Participants had to decide which of the two numbers was presented in a larger font size. In the congruent condition, the number shown in a bigger font size was numerically larger. In the incongruent condition, the number shown in a smaller font size was numerically larger. Two types of numbers were employed: numbers composed of three zeros and one non-zero digit (e.g., 0040-0400) and numbers composed of four non-zero digits (e.g., 2795-2759). Results showed larger congruency effects in more distant pairs in both type of numbers. Interestingly, this effect was considerably stronger in the strings composed of zeros. These results indicate that place-value coding is partially automatic, as it depends on the perceptual and numerical properties of the numbers to be processed.

P.6.e.005 - Internet use and its relationship with loneliness and social phobia in a sample of southern Italian high school students from a rural area

Children's early knowledge of the base-10 structure of multi-digit numbers has been hypothesized to play a critical role in subsequent learning of mathematics, in particular arithmetic operations. The present study investigated the relation between base-10/place value understanding and arithmetic accuracy in early elementary school. Children were assessed in kindergarten (N = 90) and then a subgroup of participants was assessed again two years later in second grade (N = 21). Mediation analyses indicated that, in kindergarten, base-10 knowledge had a direct effect on arithmetic accuracy as well as an indirect effect through the use of a decomposition strategy. Furthermore, kindergarten base-10 knowledge had a direct effect on arithmetic accuracy in second grade and an indirect effect through second grade place-value notation understanding. Implications for understanding early mathematical development are discussed.

INTERCULTURAL FEATURES AND THE THEME OF TRAVELLING IN BILINGUAL MATHEMATICS LESSONS

The present qualitative research is focused on bilingual mathematics education. The research presents findings of a case study of one bilingual Slovak and English mathematics 40-minute lesson within an after school elective bilingual mathematics course running weekly since October, 2015. The lesson took place in March, 2016, and was attended by nine learners aged 12-13, eight boys and one girl. The learners are cases of successive school additive bilingual education. The elective course as a whole is a case of immerse bilingual educational programme. In terms of sociolinguistic settings, the course lessons are cases of bilingual education with external second language. The researcher designed and realized the course lessons in terms of CLIL approach, i.e. Content and Language Integrated Learning. The main aim of the case study was to examine if bilingual mathematics instruction does or does not prevent learners from solving math word problems. Secondly, the analysis of transcription of the lesson audio-record served for identification of intercultural features which might hinder the learning process. The analysis of the transcribed audio-record indicates that the bilingual context did not prevent students from solving math word problems, although each of the students worked at their individual rate. On the other hand, some students were confused by the comma as a thousands-separator in multi-digit numbers, and this actually hindered their learning and problem solving process. This fact has been identified as an intercultural difference which had to be explicitly explained to the students. In order to lessen the possible negative influences of bilingual context on mathematics education, teachers need to predict students’ responses to various intercultural differences which students are unfamiliar with. Key words: bilingual mathematics education, CLIL, thousands-separator, land area, per cents.

Kindergartners' base-10 knowledge predicts arithmetic accuracy concurrently and longitudinally

Children's early knowledge of the base-10 structure of multi-digit numbers has been hypothesized to play a critical role in subsequent learning of mathematics, in particular arithmetic operations. The present study investigated the relation between base-10/place value understanding and arithmetic accuracy in early elementary school. Children were assessed in kindergarten (N=90) and then a subgroup of participants was assessed again two years later in second grade (N=21). Mediation analyses indicated that, in kindergarten, base-10 knowledge had a direct effect on arithmetic accuracy as well as an indirect effect through the use of a decomposition strategy. Furthermore, kindergarten base-10 knowledge had a direct effect on arithmetic accuracy in second grade and an indirect effect through second grade place-value notation understanding. Implications for understanding early mathematical development are discussed.

The way of effective use of incremental with multiple thinning of test values for Fermat's factoring method

Among the modern methods of cryptographic information protection asymmetric algorithms is most widely used. A special place among them is occupied by RSA (Rivest-Shamir-Adleman) encryption algorithm, which recommended the use a number of international standards and recommendations. RSA cryptographic resistance is based on the difficulty of the task execution of multi-digit numbers factorization and is not an effective problem of compromising its software and hardware implementations. Currently the “fastest” ways of decomposition the big numbers into factors are methods of the general number field sieve (GNFS), the quadratic sieve (QS) algorithm and the elliptic curve factorization method (ECM). It is known that the basis of these methods are based on a number of fundamental relations of Fermat’s classical algorithm, proceeding from which it can be argued that the improvement of Fermat’s method can have an impact on reducing the computational complexity of modern factorization methods listed above. One of the ways to increase the efficiency of the improved Fermat's factoring method is a modification of existing or developing new algorithms of execution the algebraic operations with large numbers. Among these can be the operation of the modular division in procedures for advance sieving of test values X. A modified method of thinning test values in Fermat’s factoring algorithm is proposed, the main advantage of which is the refusal to perform complex arithmetic operations of modular division of large sequences and replacing them with the procedure of the modular division of small numbers.

Spacing and Presentation Modes Affect the Unit-Decade Compatibility Effect During Number Comparison.

. The unit-decade compatibility effect has challenged the model of holistic number magnitude processing, suggesting decomposed processing of multi-digit numbers. Recent evidence confirms that decomposed processing of decade and unit magnitudes occurs in parallel. However, the mode of presentation of multi-digit numbers may affect the processing mode (holistic vs. decomposed, parallel vs. sequential). We therefore investigated in two studies, whether presentation mode (vertical, horizontal, or consecutive) or the distance between two vertically presented numbers affects the unit-decade compatibility effect during number comparison. We found that the compatibility effect did not differ significantly between vertical and horizontal presentation, adding to previous results on perceptual generality, but was nonsignificant with consecutive presentation. However, the compatibility effect was significantly smaller, if numbers are spaced further apart. Thus, stimulus size and distance between numbers affect the processing of multi-digit numbers and should be reported in future studies.

Place-value understanding in number line estimation predicts future arithmetic performance.

In multi-digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place-value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place-value understanding in first grade. We hypothesized that estimating the position of two-digit numbers requires place-value understanding. Therefore, we fitted a linear function to children's estimates of two-digit numbers and considered the resulting slope as a measure of children's place-value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place-value understanding. Additionally, the slope for two-digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first-graders' place-value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place-value understanding) may drive the association between number line estimation and arithmetic performance.

Development of children’s early understanding of numeric structure

Understanding of the base-10 structure of multi-digit numbers is one of the critical aspects in early mathematics learning. It has been documented that children from different countries vary in their use of base-10 representations. Questions concerning potential sources of this variability have been debated for decades. One commonly posited explanation is that some languages provide explicit cues about the structure of multi-digit numbers, facilitating the development of base-10 representations. In the present study, we tested this view against an alternative view, positing that variability in children’s learning of numeric structure may reflect differences in their experiences with numbers. The study examined kindergartners and first-graders from four countries: Taiwan, South Korea, the USA, and Russia. Results showed that the use of base-10 representations by American first-graders increased dramatically over the last decades, following changes in curricular guidelines. First-graders across the four countries showed some differences in performance (however, not consistent with the language account), whereas kindergartners performed comparably despite the differences in their languages. The results suggest that the nature of early math instruction may be critical for children’s developing understanding of numeric structure.

Handwriting recognition of digits, signs, and numerical strings in Persian

This paper presents an important step towards the standardization of research works on Optical Character Recognition in Persian language. It describes the formations of a standard handwritten database, including isolated digits, isolated signs, multi-digit numbers, numerical strings, courtesy amounts, and postal codes. In this regard, binary images of 72,180 samples were extracted from the designed forms. These forms were filled by 180 writers selected from different ages, genders, and jobs. Then these forms were scanned at 300 dpi with a high-speed scanner. Finally, forms are segmented into samples and are stored in bitmap format. This database is named PHOND, Persian Handwritten Optical Numbers & Digits, and it is available to the research community. Comparisons with the previous related databases illustrate the advantages of PHOND against other databases. Different experiments are done using PHOND database and the results are compared with other research works in handwritten recognition.

Processing multi-digit numbers: a translingual eye-tracking study.

The present study aimed at investigating the underlying cognitive processes and language specificities of three-digit number processing. More specifically, it was intended to clarify whether the single digits of three-digit numbers are processed in parallel and/or sequentially and whether processing strategies are influenced by the inversion of number words with respect to the Arabic digits [e.g., 43: dreiundvierzig ("three and forty")] and/or by differences in reading behavior of the respective first language. Therefore, English- and German-speaking adults had to complete a three-digit number comparison task while their eye-fixation behavior was recorded. Replicating previous results, reliable hundred-decade-compatibility effects (e.g., 742_896: hundred-decade compatible because 7 < 8 and 4 < 9; 362_517: hundred-decade incompatible because 3 < 5 but 6 > 1) for English- as well as hundred-unit-compatibility effects for English- and German-speaking participants were observed, indicating parallel processing strategies. While no indices of partial sequential processing were found for the English-speaking group, about half of the German-speaking participants showed an inverse hundred-decade-compatibility effect accompanied by longer inspection time on the hundred digit indicating additional sequential processes. Thereby, the present data revealed that in transition from two- to higher multi-digit numbers, the homogeneity of underlying processing strategies varies between language groups. The regular German orthography (allowing for letter-by-letter reading) and its associated more sequential reading behavior may have promoted sequential processing strategies in multi-digit number processing. Furthermore, these results indicated that the inversion of number words alone is not sufficient to explain all observed language differences in three-digit number processing.

On the limits of language influences on numerical cognition - no inversion effects in three-digit number magnitude processing in adults.

The inversion of number words influences numerical cognition even in seemingly non-verbal tasks, such as Arabic number comparison. However, it is an open question whether inversion of decades and units also influences number processing beyond the two-digit number range. The current study addresses this question by investigating compatibility effects in both German- (a language with inverted) and English-speaking (a language with non-inverted number words) university students (mean age 22 years) in a three-digit number comparison task. We observed reliable hundred-decade as well as hundred-unit compatibility effects for three-digit number comparison. This indicates that, comparable two-digit numbers, three-digit numbers are processed in a parallel decomposed fashion. However, in contrast to previous results on two-digit numbers as well as on children’s processing of three-digit numbers, no reliable modulation of these compatibility effects through language was observed in adults. The present data indicate that inversion-related differences in multi-digit number processing are limited. They seem to be restricted to the number range involving those digits being inverted (i.e., tens and units in two-digit numbers) but do not generalize to neighboring digits. Possible reasons for this lack of generalization are discussed.

Testing a model of componential processing of multi-symbol numbers-evidence from measurement units.

Research on numerical cognition has addressed the processing of nonsymbolic quantities and symbolic digits extensively. However, magnitude processing of measurement units is still a neglected topic in numerical cognition research. Hence, we investigated the processing of measurement units to evaluate whether typical effects of multi-digit number processing such as the compatibility effect, the string length congruity effect, and the distance effect are also present for measurement units. In three experiments, participants had to single out the larger one of two physical quantities (e.g., lengths). In Experiment 1, the compatibility of number and measurement unit (compatible: 3 mm_6 cm with 3 < 6 and mm < cm; incompatible: 3 cm_6 mm with 3 < 6 but cm > mm) as well as string length congruity (congruent: 1 m_2 km with m < km and 2 < 3 characters; incongruent: 2 mm_1 m with mm < m, but 3 > 2 characters) were manipulated. We observed reliable compatibility effects with prolonged reaction times (RT) for incompatible trials. Moreover, a string length congruity effect was present in RT with longer RT for incongruent trials. Experiments 2 and 3 served as control experiments showing that compatibility effects persist when controlling for holistic distance and that a distance effect for measurement units exists. Our findings indicate that numbers and measurement units are processed in a componential manner and thus highlight that processing characteristics of multi-digit numbers generalize to measurement units. Thereby, our data lend further support to the recently proposed generalized model of componential multi-symbol number processing.

Toward a model framework of generalized parallel componential processing of multi-symbol numbers.

In this article, we propose and evaluate a new model framework of parallel componential multi-symbol number processing, generalizing the idea of parallel componential processing of multi-digit numbers to the case of negative numbers by considering the polarity signs similar to single digits. In a first step, we evaluated this account by defining and investigating a sign-decade compatibility effect for the comparison of positive and negative numbers, which extends the unit-decade compatibility effect in 2-digit number processing. Then, we evaluated whether the model is capable of accounting for previous findings in negative number processing. In a magnitude comparison task, in which participants had to single out the larger of 2 integers, we observed a reliable sign-decade compatibility effect with prolonged reaction times for incompatible (e.g., -97 vs. +53; in which the number with the larger decade digit has the smaller, i.e., negative polarity sign) as compared with sign-decade compatible number pairs (e.g., -53 vs. +97). Moreover, an analysis of participants' eye fixation behavior corroborated our model of parallel componential processing of multi-symbol numbers. These results are discussed in light of concurrent theoretical notions about negative number processing. On the basis of the present results, we propose a generalized integrated model framework of parallel componential multi-symbol processing.

Reexamining the language account of cross-national differences in base-10 number representations

East Asian students consistently outperform students from other nations in mathematics. One explanation for this advantage is a language account; East Asian languages, unlike most Western languages, provide cues about the base-10 structure of multi-digit numbers, facilitating the development of base-10 number representations. To test this view, the current study examined how kindergartners represented two-digit numbers using single unit-blocks and ten-blocks. The participants (N=272) were from four language groups (Korean, Mandarin, English, and Russian) that vary in the extent of “transparency” of the base-10 structure. In contrast to previous findings with older children, kindergartners showed no cross-language variability in the frequency of producing base-10 representations. Furthermore, they showed a pattern of within-language variability that was not consistent with the language account and was likely attributable to experiential factors. These findings suggest that language might not play as critical a role in the development of base-10 representations as suggested in earlier research.

Breaking down number syntax: Spared comprehension of multi-digit numbers in a patient with impaired digit-to-word conversion

Can the meaning of two-digit Arabic numbers be accessed independently of their verbal-phonological representations? To answer this question we explored the number processing of ZN, an aphasic patient with a syntactic deficit in digit-to-verbal transcoding, who could hardly read aloud two-digit numbers, but could read them as single digits (“four, two”). Neuropsychological examination showed that ZN's deficit was neither in the digit input nor in the phonological output processes, as he could copy and repeat two-digit numbers. His deficit thus lied in a central process that converts digits to abstract number words and sends this information to phonological retrieval processes. Crucially, in spite of this deficit in number transcoding, ZN's two-digit comprehension was spared in several ways: (1) he could calculate two-digit additions; (2) he showed good performance in a two-digit comparison task, and a continuous distance effect; and (3) his performance in a task of mapping numbers to positions on an unmarked number line showed a logarithmic (nonlinear) factor, indicating that he represented two-digit Arabic numbers as holistic two-digit quantities. Thus, at least these aspects of number comprehension can be performed without converting the two-digit number from digits to verbal representation.

A pilot study of the effects of RightStart instruction on early numeracy skills of children with specific language impairment

This pilot study investigated the effects of an early numeracy program, RightStart Mathematics (RS), on Finnish kindergartners with specific language impairment (SLI). The study applied a pre-test-instruction-post-test design. The children with SLI (n=9, Mage=82.11 months) received RS instruction two to three times a week for 40 min over seven months, which replaced their business-as-usual mathematics instruction. Mathematical skill development among children with SLI was examined at the individual and group levels, and compared to the performance of normal language-achieving age peers (n=32, Mage=74.16 months) who received business-as-usual kindergarten mathematics instruction. The children with SLI began kindergarten with significantly weaker early numeracy skills compared to their peers. Immediately after the instruction phase, there was no significant difference between the groups in counting skills. In Grade 1, the children with SLI performed similarly to their peers in addition and subtraction skills (accuracy) and multi-digit number comparison, but showed weaker skills in arithmetical reasoning and in matching spoken and printed multi-digit numbers. Our pilot study showed encouraging signs that the early numeracy skills of children with SLI can be improved successfully in a kindergarten small-classroom setting with systematic instruction emphasizing visualization.

Inequality of computational effort of Pollard p-method according to dispaly selection and initial estimate for small-scale factorizable amounts

For a range of information security tasks the cryptostrengthof algorithms applied is linked to solving acomputational problem of multi-digit numbers factorization.Algorithms of modern factorization methods canuse existing algorithms as integral part. That is why aresearch of existing methods and development of the wayto accelerate their work is considered to be a highly topicalproblem. For Pollard p factorization method generalevaluation of iterations number is known, but the resultsof initial approximation influence study have not beenpresented. To evaluate such influence it is suggested todefine the average number of iterations for Pollard factorization method in the context of numbersoptions of 2*107 not exceeding 231, of the N=p*q type,where p and q are prime factors. While defining the averageiterations number the total iterations count for alloptions of N under study was calculated and divided bythe number of these options. To provide factoring constantc in the polynomial was increased by one wheneveriteration process ran into a cyclic path. Studies are conductedto evaluate the average iterations number dependingon the choice of constant c in the polynomial, representingiteration process 21 ( )mod    k k x x c N , as well ason initial estimate. It has been defined that for the numberoptions under study the average iterations number islower than existing evaluations, and it can be decreased byone third, due to the initial estimate.

Impacting Early Childhood Teachers’ Understanding of the Complexities of Place Value

In order to help children gain a more robust understanding of place value, teachers must understand the connections and relationships among the related concepts as well as possess knowledge of how children learn early number concepts. Unfortunately, teachers’ familiarity with the base-ten number system and/or lack of an understanding of multidigit whole numbers leads to instruction that encourages a superficial understanding of the concepts of our base-ten system rather than instruction that pushes students to go beyond recitation of the value of the digit in the ten's place. This study highlights the attempts of three mathematics educators in reconceptualizing and impacting teachers’ understanding of place value concepts.