Two-digit Numbers Research Articles

Insights Chinese Primary Mathematics Teachers Gained into their Students’ Learning from Using Classroom Assessment Techniques

In this study, we explored the insights that Chinese primary mathematics teachers gained into their students’ mathematical understanding from using classroom assessment techniques (CATs). CATs are short teacher-initiated targeted assessment activities proximate to the textbook, which teachers can use in their daily practice to make informed instructional decisions. Twenty-five third-grade teachers participated in a two-week program of implementing eight CATs focusing on the multiplication of two-digit numbers, and filled in feedback forms after using the CATs. When their responses described specific information about their students, emphasized the novelty of the gained information, or referred to a fitting instructional adaptation, and these reactions went together with references to the mathematics content of the CATs, the teachers’ responses were considered as evidence of gained insights into their students’ mathematics understanding. This was the case for three-quarters of the teachers, but the number of gained insights differed. Five teachers gained insights from five or more CATs, while 14 teachers did so only from three or fewer CATs, and six teachers showed no clear evidence of new insights at all. Despite the differences in levels of gained insights, all the teachers paid more attention to descriptions of students’ performance than to possible instructional adaptations.

Direct evidence for linguistic influences in two-digit number processing.

Language-specific differences in number words influence number processing even in nonverbal numerical tasks. For instance, the unit-decade compatibility effect in two-digit number magnitude comparison (compatible number pairs [42_57: 4 < 5 and 2 < 7] are responded to faster than incompatible pairs [47_62: 4 < 6 but 7 > 2]) was shown to be influenced by the inversion of number words (e.g., in German the number word for 42 is zweiundvierzig [literally: two-and-forty]). In two studies, we used articulatory suppression to investigate whether previously observed cross-linguistic differences in two-digit number processing are indeed driven by differences in number word formation. In a two-digit number comparison task, German- and English-speaking participants had to identify the larger of two numbers presented in Arabic digits. In Study 1, participants performed the same task twice, with and without articulatory suppression. In Study 2, the percentage of within-decade filler items (36_39) was manipulated additionally. As expected, in both studies between-groups differences in the compatibility effect disappeared under articulatory suppression irrespective of the percentage of fillers included. Furthermore, paralleling results of previous studies including 33% or less filler items, we found that the compatibility effect was larger in German compared with English speakers in the 20% filler condition. However, this pattern was reversed in the 50% filler condition in both studies. Thus, results provide first direct evidence for influences of verbal number word formation on symbolic number processing. Moreover, these new findings suggest that linguistic influences and those of cognitive control processes associated with characteristics of the stimulus set interact in symbolic number processing. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Neuroplasticity, bilingualism, and mental mathematics: A behavior-MEG study

Bilingual experience alters brain structure and enhances certain cognitive functions. Bilingualism can also affect mathematical processing. Reduced accuracy is commonly reported when arithmetic problems are presented in bilinguals’ second (L2) vs. first (L1) language. We used MEG brain imaging during mental addition to characterize spatiotemporal dynamics during mental addition in bilingual adults. Numbers were presented auditorally and sequentially in bilinguals’ L1 and L2, and brain and behavioral data were collected simultaneously. Behaviorally, bilinguals showed lower accuracy for two-digit addition in L2 compared to L1. Brain data showed stronger response magnitude in L2 versus L1 prior to calculation, especially when two-digit numbers were involved. Brain and behavioral data were significantly correlated. Taken together, our results suggest that differences between languages emerge prior to mathematical calculation, with implications for the role of language in mathematics.

Sex Differences in Number Magnitude Processing Strategies Are Mediated by Spatial Navigation Strategies: Evidence From the Unit-Decade Compatibility Effect.

The hybrid model of number magnitude processing suggests that multi-digit numbers are simultaneously processed holistically (whole number magnitudes) and in a decomposed manner (digit magnitudes). Thus, individual tendencies and situational factors may affect which type of processing becomes dominant in a certain individual in a given situation. The unit-decade compatibility effect has been described as indicative of stronger decomposed number processing. This effect occurs during the comparison of two-digit numbers. Compatible items in which the larger number contains the larger unit digit are easier to solve than incompatible items in which the larger number contains the smaller unit digit. We have previously described women show a larger compatibility effect than men. Furthermore, the compatibility effect is modulated by situational factors like the vertical spacing of the presented numbers. However, it has not been addressed whether situational factors and sex affect the unit-decade compatibility effect interactively. We have also demonstrated that the unit-decade compatibility effects relates to global-local processing, which in turn also affects spatial processing strategies. However, a link between spatial processing strategies and the unit-decade compatibility effect has not yet been established. In the present study we investigate, whether sex differences in the unit-decade compatibility effect (i) depend on the vertical spacing between numbers, (ii) are mediated via sex hormone levels of participants, and (iii) relate to sex differences in spatial processing strategies. 42 men and 41 women completed a two-digit number comparison task as well as a spatial navigation task. The number comparison task modulates compatibility and vertical spacing in a 2 × 2 design. The results confirm a larger compatibility effect in women compared to men and with dense compared to sparse spacing. However, no interactive effect was observed, suggesting that these factors modulate number magnitude processing independently. The progesterone/testosterone ratio was related to the compatibility effect, but did not mediate the sex difference in the compatibility effect. Furthermore, spatial processing strategies were related to the compatibility effect and did mediate the sex difference in the compatibility effect. Participants with a stronger focus on landmarks in the spatial navigation task showed a larger compatibility effect.

English and Chinese Children's Performance on Numerical Tasks.

East Asian pupils have consistently outperformed Western pupils in international comparisons of mathematical performance at both primary and secondary school level. It has sometimes been suggested that a contributory factor is the transparent counting systems of East Asian languages, which may facilitate number representation. The present study compared 35 7-year-old second-year primary school children in Oxford, England and 40 children of similar age in Hong Kong, China on a standardized arithmetic test; on a two-digit number comparison test, including easy, misleading and reversible comparisons; and on a number line task, involving placing numbers in the appropriate position on four number lines: 1–10, 1–20, 1–100, and 1–1000. The Chinese children performed significantly better than the English children on the standardized arithmetic test. They were faster but not significantly more accurate on the Number Comparison and Number Line tasks. There were no interactions between language group and comparison type on the number comparison task, though the performance of both groups was faster on easy pairs than those where there was conflict between the relative magnitudes of the tens and the units. Similarly, there were no interactions between group and number line range, though the performance of both groups was influenced by the range of the number line. The study supports the view that counting systems affect aspects of numerical abilities, but cannot be the full explanation for international differences in mathematics performance.

A Novel Design of a Multiplier Using Reversible Ternary Gates

ABSTRACT Reversible ternary circuit design is now being taken into consideration for its application within the quantum computer and other technologies associated with the nanotechnology domain, in addition to the advantages of the ternary logic application as opposed to binary logic. Multiplier circuit is one of the most important computing circuit in the ALU. Therefore, the circuit optimization for the multiplication will lead to efficient processor. In this paper, the reversible circuit for multiplication of two unsigned two-digit ternary numbers has been proposed. To construct reversible ternary multiplier, a new component termed TPPG with the quantum cost of 13 for partial product generation was proposed. In addition, we have introduced four new blocks of reversible ternary adder with the quantum cost of 11, 10, 7, 5 and one constant input, which can be used in the proposed multiplier circuit. We attempted to optimize the design of the circuits in terms of quantum cost, a number of primitive gates, constant inputs, and garbage outputs as far as possible. To realize all proposed circuits, one-qutrit shift gates and two-qutrit Muthukrishnan-Stroud gates were used.

The clinical effect of enhanced recovery after surgery to tibial plateau fractures patients applied arthroscopic minimally invasive treatment

Objective To explore and analyze the clinical effect of enhanced recovery after surgery to tibial plateau fractures patients applied arthroscopic minimally invasive treatment. Methods A total of 60 tibial plateau fractures patients were selected in our orthopedics department from January 2016 to July 2017 and who applied arthroscopic minimally invasive treatment, according to the last two-digit number of patient ID, divided them into observation group (n=30) and control group (n=30) randomly. The control group used regular perioperative strategies. The observation group used multidisciplinary cooperation fast track surgery idea, through preoperative assessment and education, nutrition and fasting, advance pre-rehabilitation and preventive analgesia; intraoperative optimization of anesthesia, body fluid management and body temperature control; postoperative nutritional support, multimodal analgesia, early ambulation and rehabilitation exercises, implied standardized and professional perioperative overall optimization management. The differences of the condition of 6 h, 12 h, 24 h after surgery, VAS score on discharge, time of first ambulation, active knee flexion 120° days; self-care ability at discharge and AKSS score one month after surgery between 2 groups were compared. Results The VAS scores 6 h, 12 h, 24 h after surgery and at discharge were 4.48±1.18, 3.81±1.68, 3.05±1.63, 2.65±1.65 in the observation group, and were 5.45±1.15, 4.15±1.05, 3.71±1.15, 3.23±1.68 in the control group. The differences were statistically significant (t= 0.796~0.902, P<0.05). The time of first ambulation, active knee flexion 120°days, self-care ability at discharge and AKSS scores one month after surgery were (5.61±1.4) hours, (4.01±1.1) days, 80.22±3.6, 71.89±6.56 and 64.13±6.15 in the observation group, and (35.8±8.1) hours, (6.82±1.6) days, 64.25±3.8, 63.45±8.36 and 60.95±8.98 in the control group. The differences were statistically significant (t=2.789~10.200, P<0.05). Conclusion Enhanced recovery after surgery to tibial plateau fractures patients applied arthroscopic minimally invasive treatment is worthy of being popularized as it’s beneficial to tibial plateau fractures patients. It can also fasten recovery and improve quality of life for postoperative patients. Key words: Enhanced recovery after surgery; Arthroscopy; Fracture of tibial plateau; Perioperative management

Attention allows the SNARC effect to operate on multiple number lines

To investigate whether participants can activate only one spatially oriented number line at a time or multiple number lines simultaneously, they were asked to solve a unit magnitude comparison task (unit smaller/larger than 5) and a parity judgment task (even/odd) on two-digit numbers. In both these primary tasks, decades were irrelevant. After some of the primary task trials (randomly), participants were asked to additionally solve a secondary task based on the previously presented number. In Experiment 1, they had to decide whether the two-digit number presented for the primary task was larger or smaller than 50. Thus, for the secondary task decades were relevant. In contrast, in Experiment 2, the secondary task was a color judgment task, which means decades were irrelevant. In Experiment 1, decades’ and units’ magnitudes influenced the spatial association of numbers separately. In contrast, in Experiment 2, only the units were spatially associated with magnitude. It was concluded that multiple number lines (one for units and one for decades) can be activated if attention is focused on multiple, separate magnitude attributes.

Spatial grounding of symbolic arithmetic: an investigation with optokinetic stimulation

Growing evidence suggests that mental calculation might involve movements of attention along a spatial representation of numerical magnitude. Addition and subtraction on nonsymbolic numbers (numerosities) seem to induce a “momentum” effect, and have been linked to distinct patterns of neural activity in cortical regions subserving attention and eye movements. We investigated whether mental arithmetic on symbolic numbers, a cornerstone of abstract mathematical reasoning, can be affected by the manipulation of overt spatial attention induced by optokinetic stimulation (OKS). Participants performed additions or subtractions of auditory two-digit numbers during horizontal (experiment 1) or vertical OKS (experiment 2), and eye movements were concurrently recorded. In both experiments, the results of addition problems were underestimated, whereas results of subtractions were overestimated (a pattern that is opposite to the classic Operational Momentum effect). While this tendency was unaffected by OKS, vertical OKS modulated the occurrence of decade errors during subtractions (i.e., fewer during downward OKS and more frequent during upward OKS). Eye movements, on top of the classic effect induced by OKS, were affected by the type of operation during the calculation phase, with subtraction consistently leading to a downward shift of gaze position and addition leading to an upward shift. These results highlight the pervasive nature of spatial processing in mental arithmetic. Furthermore, the preeminent effect of vertical OKS is in line with the hypothesis that the vertical dimension of space–number associations is grounded in universal (physical) constraints and, thereby, more robust than situated and culture-dependent associations with the horizontal dimension.

Task instructions modulate unit-decade binding in two-digit number representation.

Previous studies have found decomposed processes, as well as holistic processes, in the representation of two-digit numbers. The present study investigated the influence of task instruction on such processes. Participants completed both magnitude and parity tasks in one of three instructional conditions, where they were asked to either consider two-digit numbers as a whole or to focus on one specific digit. In two experiments, we found that when participants were asked to consider the two digits as an integrated number, they always exhibited a unit-decade compatibility effect, indicating a failure of selective attention on the digit relevant to the given task. However, the mere presence of the neighboring digit is not a sufficient condition for the compatibility effect: when participants were explicitly asked to process a specific digit, their success/failure to selectively ignore the irrelevant digit depended on task requirements. Further, computer mouse tracking indicated that the locus of the compatibility effect was related to late response-related processing. The results signify the deep involvement of top-down processes in unit-decade binding for two-digit number representation.

A Mental Odd-Even Continuum Account: Some Numbers May Be "More Odd" Than Others and Some Numbers May Be "More Even" Than Others.

Numerical categories such as parity, i.e., being odd or even, have frequently been shown to influence how particular numbers are processed. Mathematically, number parity is defined categorically. So far, cognitive, and psychological accounts have followed the mathematical definition and defined parity as a categorical psychological representation as well. In this manuscript, we wish to test the alternative account that cognitively, parity is represented in a more gradual manner such that some numbers are represented as “more odd” or “more even” than other odd or even numbers, respectively. Specifically, parity processing might be influenced by more specific properties such as whether a number is a prime, a square number, a power of 2, part of a multiplication table, divisible by 4 or by 5, and many others. We suggest that these properties can influence the psychologically represented parity of a number, making it more or less prototypical for odd- or evenness. In the present study, we tested the influence of these numerical properties in a bimanual parity judgment task with auditorily presented two-digit numbers. Additionally, we further investigated the interaction of these numerical properties with linguistic factors in three language groups (English, German, and Polish). Results show significant effects on reaction times of the congruity of parity status between decade and unit digits, even if numerical magnitude and word frequency are controlled. We also observed other effects of the above specific numerical properties, such as multiplication attributes, which facilitated or interfered with the speed of parity judgment. Based on these effects of specific numerical properties we proposed and elaborated a parity continuum account. However, our cross-lingual study also suggests that parity representation and/or access seem to depend on the linguistic properties of the respective language or education and culture. Overall, the results suggest that the “perceived” parity is not the same as objective parity, and some numbers are more prototypical exemplars of their categories.

Sex differences and functional hemispheric asymmetries during number comparison

BackgroundGlobal-local stimuli are hierarchical structures consisting of a larger global structure which is composed of smaller local stimuli. Numbers are also constructed hierarchically, with multi-digit numbers being made up from single digits. During two-digit number comparison, compatible items (larger number contains larger unit digit, e.g., 53 vs. 68) are processed faster and more accurately than incompatible items (smaller number contains larger unit digit, e.g., 58 vs. 63). This so-called unit-decade-compatibility effect has challenged the holistic model of number processing and suggests that the processing of number magnitudes occurs at least in part, decomposed, i.e., separately for each digit. Thus, the compatibility effect is indicative of how decomposed numbers are processed, thereby sharing similarities with traditional global-local processing of hierarchical stimuli.The goal of this study was to investigate whether factors that have been shown to reliably influence global-local processing also affect the compatibility effect during number comparison. Those include visual hemifield, sex, and menstrual cycle phase in women.MethodOne hundred sixty participants, 77 naturally cycling women and 83 men, completed a two-digit number comparison task twice, with test-sessions time-locked to the early follicular or mid-luteal cycle phase in women. Number comparison stimuli were presented to the right or left hemifield, respectively.ResultsWe observed a stronger compatibility effect in the right visual hemifield compared to left visual hemifield and in women compared to men, but no evidence for an influence of menstrual cycle phase in women could be found.ConclusionHemispheric asymmetries in holistic versus decomposed number processing could be demonstrated for the first time, suggesting a similar hemispheric modulation for number magnitude processing as for global-local processing.

Individual differences influence two-digit number processing, but not their analog magnitude processing: a large-scale online study.

Symbolic magnitude comparison is one of the most well-studied cognitive processes in research on numerical cognition. However, while the cognitive mechanisms of symbolic magnitude processing have been intensively studied, previous studies have paid less attention to individual differences influencing symbolic magnitude comparison. Employing a two-digit number comparison task in an online setting, we replicated previous effects, including the distance effect, the unit-decade compatibility effect, and the effect of cognitive control on the adaptation to filler items, in a large-scale study in 452 adults. Additionally, we observed that the most influential individual differences were participants' first language, time spent playing computer games and gender, followed by reported alcohol consumption, age and mathematical ability. Participants who used a first language with a left-to-right reading/writing direction were faster than those who read and wrote in the right-to-left direction. Reported playing time for computer games was correlated with faster reaction times. Female participants showed slower reaction times and a larger unit-decade compatibility effect than male participants. Participants who reported never consuming alcohol showed overall slower response times than others. Older participants were slower, but more accurate. Finally, higher grades in mathematics were associated with faster reaction times. We conclude that typical experiments on numerical cognition that employ a keyboard as an input device can also be run in an online setting. Moreover, while individual differences have no influence on domain-specific magnitude processing-apart from age, which increases the decade distance effect-they generally influence performance on a two-digit number comparison task.

ChilDiBu – A Mobile Application for Bulgarian Children with Special Educational Needs

Designing applications adapted to the capabilities of each user is an important part of modern software engineering. Children with Autism spectrum disorder and/or intellectual disabilities are a special group of users where standard application design is not a good approach. Some of them have speech disorders; others have disorders of the musculoskeletal system and in their cognitive development. In the literature, these children are referred to as children with special educational needs. Only a few researchers are involved in the development of various computer technologies, including mobile applications appropriate for these children. Some specialized mobile applications for these users can be found in the different mobile stores. However, most of these apps are rather expensive or with inappropriate user interfaces. Another disadvantage in all the apps is the lack of support in the Bulgarian language. It makes them unusable for the youngest users with special needs in Bulgaria. A conceptual model based on the investigation of some existing applications, the well-established theoretical practices, and on the opinion of children without disorders is proposed in this paper. An approach to designing and developing the interactive mobile application ChilDiBu for children with special needs, suitable for the Windows Phone mobile operating system, is presented. ChilDiBu is a picture-based app that combines graphic images, texts and audio files which can be used to learn the Bulgarian alphabet, the numbers up to 20, some basic colors and the daily activities that children perform. The application is focused on the development of the fine motor skills, observation and decision-making skills which is completed with a puzzle for arranging the two-digit numbers. At the end of the paper the app interface, the way it is used and its app architecture are shown.

Is there variation across individuals in processing? Bayesian analysis for systems factorial technology

Systems factorial technology (Townsend and Nozawa, 1995) is a leading methodology for assessing the processing of multiple-feature items. By using certain experimental designs and analyses, researchers can assess whether features are processed in serial, in parallel, or coactively. Current practice is to categorize each individual as displaying one of these three architectures. We argue this approach implicitly assumes heterogeneity of processing strategies across participants. A more scientifically meaningful approach may be to first ask whether all people are serial or parallel or coactive before assuming heterogeneity. We develop a series of Bayesian hierarchical models that captures both situations where everyone follows a common architecture and, alternatively, where there is heterogeneity in architecture. These models use g-prior structures that make computation of Bayes factors convenient. We report an application to investigate Miller’s (1956) notion of chunking. We asked participants to compare objects that are composed of separable features simultaneously, a perception task, and sequentially, a memory task. We assessed whether processing changed across the perception and memory tasks with the notion that participants might have to chunk features to store them, and that this chunking might make processing more efficient. The answer is “no.” We find a serial architecture for processing for highly separable features (size of circle and the orientation of its diameter) in both the perception and memory tasks. We also find parallel processing for less separable features (first and second digits in a two-digit number) in both perception and memory tasks. Taken together, while processing may depend on the separability of features, it does not vary across perception and memory. As importantly, we find that all people had the same processing strategy; that is models that stated no heterogeneity outperformed those with heterogeneity. This result indicates that architecture may be universal in this setting and not under strategic control.

Recent Developments on Drawing of Multi-Digit Random Numbers from Single Table of Random Two-Digit Numbers and of Random Three-Digit Numbers

Recent Developments on Drawing of Multi-Digit Random Numbers from Single Table of Random Two-Digit Numbers and of Random Three-Digit Numbers

AN EXAMPLE OF PREPARED-PLANNED CREATIVE DRAMA IN SECOND GRADE MATHEMATICS EDUCATION

The aim of this research is teaching addition with natural numbers and the concept of large and small natural numbers in the second grade mathematics course, through creative drama method. The study has been applied to 31 elementary school second grade students studying at a public school in the province of Aydin. In this research, case study based on observation method, which is one of the qualitative research methods, has been employed. The data have been collected through drama lesson plans, observation, and interview methods. The activities involving the acquisitions of ''adding two-digit natural numbers'' and ''absorbing large-small concepts'' prepared by the researcher, have been applied in the classroom. The drawings of students, literary notes, and observation reports kept by researchers have been used in the evaluation of the activity. As a result of this application, students' views towards the questions covering the whole application have been gathered. The results of the study suggest that prepared-planned creative drama activities are an entertaining and enjoyable way for students to learn. The notes taken by the participative observers and findings obtained from the interviews indicate that drama activities increase sharing among students and that the students could associate what they have learned with some of the situations they encounter in their every day-life. It is observed that the students learn by having more fun while playing games, which involves a high amount of physical movements. Drawing the attention of students through instructions in the warm-up phase of the activity increased students' curiosity in learning. In the animation phase of the activity, it was observed that the students' interest in the mathematics increased, they had fun while making the mathematical operations, their participation to the lesson was high throughout the activity, the energy they received from the lesson was at the maximum level, they learned by doing and living, and they obtained an experience they could use in their every day-life. In the evaluation phase of the activity, the students were asked to describe how they felt by drawing a picture or writing a story as a summary of the creative drama activity they experiences by doing. According to the stories they have written and the pictures they have drawn, it is apparent that they were entertained and happy throughout the activity. The results of this study indicate that the students relieve their mental exhaustion and a relaxed classroom setting is achieved through the relaxation activities. Article visualizations:

Automatic Recognition of Speed Limits on Speed-Limit Signs by Using Machine Learning

This study describes a method for using a camera to automatically recognize the speed limits on speed-limit signs. This method consists of the following three processes: first (1) a method of detecting the speed-limit signs with a machine learning method utilizing the local binary pattern (LBP) feature quantities as information helpful for identification, then (2) an image processing method using Hue, Saturation and Value (HSV) color spaces for extracting the speed limit numbers on the identified speed-limit signs, and finally (3) a method for recognition of the extracted numbers using a neural network. The method of traffic sign recognition previously proposed by the author consisted of extracting geometric shapes from the sign and recognizing them based on their aspect ratios. This method cannot be used for the numbers on speed-limit signs because the numbers all have the same aspect ratios. In a study that proposed recognition of speed limit numbers using an Eigen space method, a method using only color information was used to detect speed-limit signs from images of scenery. Because this method used only color information for detection, precise color information settings and processing to exclude everything other than the signs are necessary in an environment where many colors similar to the speed-limit signs exist, and further study of the method for sign detection is needed. This study focuses on considering the following three points. (1) Make it possible to detect only the speed-limit sign in an image of scenery using a single process focusing on the local patterns of speed limit signs. (2) Make it possible to separate and extract the two-digit numbers on a speed-limit sign in cases when the two-digit numbers are incorrectly extracted as a single area due to the light environment. (3) Make it possible to identify the numbers using a neural network by focusing on three feature quantities. This study also used the proposed method with still images in order to validate it.

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.

Manual Method of Construction: Table of Random Two-Digit Numbers

Manual Method of Construction: Table of Random Two-Digit Numbers