Number Of Lines Research Articles

Cognitive support for political partisans' understanding of policy data.

Political partisanship might lead educated adults-even the highly numerate-to reason selectively about numbers that are relevant to and support their ideology ("motivated numeracy"). In this pre-registered study, we sought to examine the replicability of motivated numeracy, and investigate whether cognitive support (number lines) that improves the reasoning of children might also improve the reasoning of political partisans. To test this, we asked 1000 adults about their political ideology and asked them to interpret fictional data, in a table or number-line format, about ideology relevant (i.e., the effect of gun control on crime) or irrelevant (i.e., the effect of skin cream on rash) issues. We failed to replicate motivated numeracy when political identity was used but observed motivated numeracy when prior attitude was used. Moreover, data presented on number lines elicited 75 percent greater accuracy than data presented in tables, regardless of whether the information was ideology-relevant, or whether data supported, was neutral to, or contradicted participants' political outlooks. Findings imply that political partisans require cognitive support to be more objective about policy data.

The Rank Length Effect

Rank lists vary in the number of items ranked on the list (e.g., top 5 vs. top 20 movies on IMDb), that is, the rank length. Across ten studies, including both field and laboratory experiments, the authors examine the influence of rank length on evaluations, willingness to pay, and choice. They document a novel rank length effect: The same ranked items elicit more positive judgments when the rank length is longer (vs. shorter), although the differences in judgments between the ranked items are smaller. This effect is driven both by consumers’ tendency to narrowly focus on the rank list and by the manner in which they map the rank list onto their mental number line. The rank length effect extends to willingness to pay, and choice. The authors explore three different kinds of choice contexts, discuss implications, and offer suggestions for future research.

Beading plot: a novel graphics for ranking interventions in network evidence

BackgroundNetwork meta-analysis is developed to compare all available treatments; therefore it enriches evidence for clinical decision-making, offering insights into treatment effectiveness and safety when faced with multiple options. However, the complexity and numerous treatment comparisons in network meta-analysis can challenge healthcare providers and patients. The purpose of this study aimed to introduce a graphic design to present complex rankings of multiple interventions comprehensively.MethodsOur team members developed a “beading plot” to summary probability of achieving the best treatment (P-best) and global metrics including surface under the cumulative ranking curve (SUCRA) and P-score. Implemented via the “rankinma” R package, this tool summarizes rankings across diverse outcomes in network meta-analyses, and the package received an official release on the Comprehensive R Archive Network (CRAN). It includes the `PlotBead()` function for generating beading plots, which represent treatment rankings among various outcomes.ResultsBeading plot has been designed based on number line plot, which effectively displays collective metrics for each treatment across various outcomes. Order on the -axis is derived from ranking metrics like P-best, SUCRA, and P-score. Continuous lines represent outcomes, and color-coded beads signify treatments.ConclusionThe beading plot is a valuable graphic that intuitively displays treatment rankings across diverse outcomes, enhancing reader-friendliness and aiding decision-making in complex network evidence scenarios. While empowering clinicians and patients to identify optimal treatments, it should be used cautiously, alongside an assessment of the overall evidence certainty.

Educational assessment without numbers.

Psychometrics conceptualizes a person's proficiency (or ability, or competence), in a cognitive or educational domain, as a latent numerical quantity. Yet both conceptual and empirical studies have shown that the assumption of quantitative structure for such phenomena is unlikely to be tenable. A reason why most applications of psychometrics nevertheless continue to treat them as if they were numerical quantities may be that quantification is thought to be necessary to enable measurement. This is indeed true if one regards the task of measurement as the location of a measurand at a point on the real number line (the viewpoint adopted by, for example, the representational theory of measurement, the realist theory of measurement as the discovery of ratios, and Rasch measurement theory). But this is not the only philosophically respectable way of defining the notion of measurement. This paper suggests that van Fraassen's more expansive view of measurement as, in general, location in a logical space (which could be the real continuum, as in metrological applications in the physical sciences, but could be a different mathematical structure), provides a more appropriate conceptual framework for psychometrics. Taking educational measurement as a case study, it explores what that could look like in practice, drawing on fuzzy logic and mathematical order theory. It suggests that applying this approach to the assessment of intersubjectively constructed phenomena, such as a learner's proficiency in an inherently fuzzily-defined subject area, entails recognizing the theory-dependent nature of valid representations of such phenomena, which need not be conceived of structurally as values of quantities. Finally, some connections are made between this "qualitative mathematical" theorization of educational assessment, and the application of techniques from machine learning and artificial intelligence in this area.

Research on the Markov-Chain State Interval Division Based on Predicted Data Correction

By 2022, the total length of roads in Tibet Autonomous Region reached 121,447 kilometers. Due to the unique geological conditions in Tibet, various natural disasters such as earthquakes, mudslides, landslides, avalanches, and strong winds frequently occur. Along the Sichuan-Tibet Highway alone, over 300 disasters happen each year, significantly impacting the region’s economic development. This study focuses on the complexity and randomness of natural disaster mechanisms and combines Markov chain theory to improve the accuracy of prediction data for mudslides, landslides, and earth subsidence etc. The main method is to modify the state interval of the prediction model parameter-Markov chain based on the distribution of discrete points on the number axis. The following state interval division methods are proposed: (1) If the relative error of the predicted value exceeds 50%, adjust the prediction model. (2) Obtain the lower bound of state E1 by taking the floor value downward. (3) The width of each interval does not need to be uniform. (4) Arrange continuous, dense, and close points on the number line in the same side in batches, and represent a state continuously, dividing it into one suitable interval or batches. Using this method, an improved RMSE of 0.28mm and MAPE of 0.87% were obtained for engineering examples, outperforming other models such as GM(1,1), Verhulst, DGM(2,1) with corresponding RMSE values of 0.86 mm, 0.69 mm, and 1.38 mm, and MAPE values of 2.75%, 2.53%, and 5.99%. The combined prediction results for five sets of data yielded an RMSE of 0.14 mm and MAPE of 0.56%, which are quite close to the results obtained using Markov selection correction with an RMSE of 0.37 mm and MAPE of 1.01%. Furthermore, comparing the four sets of case, the average reduction in RMSE and MAPE is 3.56mm and 1.72%, respectively, demonstrating that this method can further improve the performance of Markov chain prediction.

From Sensations to Contrast, Opposition and Numbers

Language can be deceptively simple in its use yet frustratingly complex in its analysis. Its apparent clarity often masks an inherent complexity, rendering the true meaning of words simultaneously obvious and elusive. This inquiry investigates the multifaceted nature of meaning, encompassing not only language but also forms of expression that do not rely on spoken or written words. Central to this exploration is uncovering the body's role in creating and shaping meaning. The article presents a summary of the sensory schema theory, introduced in the author's previous publications, and examines its application through selected examples. These examples include the notions of scales, opposition, integers, and the number line derived from the sensory schema.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

"It's Different for Girls!" The Role of Anxiety, Physiological Arousal, and Subject Preferences in Primary School Children's Math and Mental Rotation Performance.

(1) Background: This study examines the role of subjective anxiety (mathematics and spatial anxiety), along with physiological responses, in mathematics or math and mental rotation performance in 131 German primary school students (65 girls, 66 boys; Mean age = 8.73 years). (2) Method: Students' preference for math vs. German and their subjective anxiety were assessed using standardized questionnaires. Emotional reactivity was measured using the Galvanic Skin Response (GSR). Math performance was evaluated via percentage scored and completion times on number line estimation, word problems, and missing terms tasks. Spatial skills were assessed using a novel mental rotation task (nMRT) incorporating gender-congruent and -neutral stimuli. (3) Results: Girls outperformed boys on percentage scored on the math task but took longer to complete this. No gender differences were found in performance on the nMRT. Girls demonstrated higher math anxiety and were less likely to prefer math over German. Math anxiety predicted math scores and accuracy on the nMRT while gender predicted math performance and mental rotation response time. Subject preference was associated with longer completion times and emotional reactivity with longer response times. Girls' preference for math and lower emotional reactivity was linked to shorter completion times, while lower math anxiety predicted higher scores. In contrast, these factors did not affect boys' math performance. Additionally, subjective anxiety, emotional reactivity, or subject preference did not impact spatial performance for either gender. (4) Conclusions: Supporting mathematical self-efficacy and emotional regulation, especially in girls, is crucial for enhancing STEM outcomes in primary education. Gender-fair assessment in mental rotation reveals equitable spatial performance and reduces the impact of anxiety.

Integrating Order Splitting and Acceptance with Batch Delivery in Parallel Machine Scheduling

Multiple production lines can work together to efficiently manufacture certain products. Thus, when capacity is insufficient, it is necessary to decide whether to develop new production lines to ensure the timely completion of all orders. For example, running a new production line for a small number of orders is not cost-effective. Therefore, decision-making involves choosing between paying tardiness costs for a few orders, abandoning some orders, or developing new production lines to maximize efficiency. Additionally, the timely transportation of completed orders is crucial and depends on vehicle usage efficiency. From a transportation perspective, fully loading vehicles is the most efficient, but this may impact the timeliness of orders, leading to potential tardiness costs. By comprehensively considering these aspects, a multi-machine production model is constructed that incorporates transportation batch sequences and uses heuristic algorithms to solve the problem. Finally, designed case examples validate the effectiveness of the model and algorithm.

Using diffusion models for symbolic numeracy tasks to examine aging effects.

We present a model-based analysis of aging effects in three symbolic numeracy tasks using three groups of subjects (young adults, 60- to 69-year-olds, and 70- to 90-year-olds). The tasks are number discrimination (is this number greater or less than 50), number memory (was this number in the list of numbers just displayed), and number line (point to where this number is on this number line). The first two tasks were fit by the standard two-choice diffusion model and the last one by the spatially continuous diffusion model (Ratcliff, 2018). Results showed good fits of the models to accuracy (choices) and response time distributions. In the tasks, nondecision time (the time to encode a stimulus and make a response) increased with age, but the amount of evidence needed for a decision (boundary settings) increased in the number discrimination and number memory tasks, but not the number line task. The number discrimination task produced conflicting accuracy and response time results as a function of age, but the model-based analyses resolved these differences. In the number memory task, drift rates (evidence used to drive the decision process) were lower for the older adults than for young adults, but for the other two (easier) tasks, there was no change in drift rate with age. The analyses extracted differences among individuals in model components, some of which were systematic across tasks. In particular, drift rates were correlated across tasks, which shows consistent individual differences across tasks, results that could not have been obtained without model-based analyses. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Pre-service teachers’ use of the jump strategy on the empty number line for mental computation

Background: South African teachers’ mathematical knowledge for teaching and number sense are markedly low, evidenced in national and international research. There is a call for supporting teachers in developing their knowledge and confidence in teaching mathematics.Aim: The aim of this article is to share how pre-service teachers (PSTs) used the empty number line (ENL) to demonstrate the jump strategy (JS) for efficient mental computation, related to the national Mental Starters Assessment Project (MSAP).Setting: This study was conducted at a tertiary institution in the Eastern Cape province of South Africa, with 40 first-year intermediate phase PSTs, during their mathematics didactics course.Methods: The data comprised 40 micro-teaching videos, analysed thematically, following an analytical model for video-recorded data. Videos were categorised according to the way they faithfully followed or departed from the MSAP teacher resources.Results: Findings suggest that more than 70% of the PSTs faithfully followed the teacher resources to teach the JS on the ENL, with regards to use of at least three of the four key elements, namely: ENL, jumps, gesturing, and key phrases and terminology.Conclusion: We suggest PSTs receive explicit instruction on the above-mentioned key elements for teaching mental calculation strategies with the ENL and opportunities to practise teaching these to their peers.Contribution: Most PSTs successfully used the ENL to demonstrate the JS for efficient mental computation, enabled by the opportunity to engage in developing (and reflecting on their) micro-teaching videos.

Effects of female-specific selection for reproductive investment on male fertility traits.

Despite sharing an autosomal genome, the often divergent reproductive strategies of males and females cause the selection to act in a sex-specific manner. Selection acting on one sex can have negative, positive, or neutral fitness consequences on the opposite sex. Here, we test how female-limited selection on reproductive investment in Japanese quail (Coturnix japonica) affects male fertility-related traits. Despite there being no difference in the size of males' testes from lines selected for high female reproductive investment (H-line) or low female reproductive investment (L-line), in both lines, the left testis had a greater volume of sperm-producing tissue. Since H-line females have a larger left-side restricted oviduct, this suggests a positive genetic correlation between male and female gonad function and that internal testis structure is a target of sexual selection. However, despite H-line males having previously been found to have greater fertilization success in a competitive scenario, we found little evidence of a difference between the lines in sperm number, motility, velocity, length, or the number of sperm that reached the ova. Precopulatory cues and/or the role of seminal fluid in sperm motility may thus be more likely to contribute to the H-line male fertilization advantage in this species.

PENGEMBANGAN MEDIA ARISAN ANTIK PADA PEMBELAJARAN BILANGAN BULAT DI KELAS INKLUSI SD SWASTA AL-AZHAR BUKITTINGGI

Making this innovative work aims to increase motivation and student learning outcomes in learning negative and positive integers by using Antique Arisan media (Manual and Electric Number Lines). This innovative work is the development of a number line media made using a board with an attractive appearance using green led lights on positive integers, red led lights on negative integers, each number is embossed for students with special needs, namely the blind and low vision. How to operate this media in two ways. (1) Use of this media by pressing the switch for each number to be addressed to complete simple number operations such as addition and subtraction of negative and positive integers. (2) Using a rail equipped with a toy that moves forward to show the addition of numbers, moves backwards for integer subtraction operations, this rail is installed under the number line which functions as a substitute for carrying out negative and positive integer operations using the number line, if the use of lights cannot be used because there is no power source. This use helps regular students and especially students with special needs in adding and subtracting negative and positive integers, so that students with special needs are served according to their needs and can solve problems encountered related to adding and subtracting integers, so that learning outcomes can increase.

Symbolic and non-symbolic representations of numerical zero in the human brain

Representing the quantity zero as a symbolic concept is considered a unique achievement of abstract human thought.1,2 To conceptualize zero, one must abstract away from the (absence of) sensory evidence to construct a representation of numerical absence: creating "something" out of "nothing."2,3,4 Previous investigations of the neural representation of natural numbers reveal distinct numerosity-selective neural populations that overlap in their tuning curves with adjacent numerosities.5,6 Importantly, a component of this neural code is thought to be invariant across non-symbolic and symbolic numerical formats.7,8,9,10,11 Although behavioral evidence indicates that zero occupies a place at the beginning of this mental number line,12,13,14 in humans zero is also associated with unique behavioral and developmental profiles compared to natural numbers,4,15,16,17 suggestive of a distinct neural basis for zero. We characterized the neural representation of zero in the human brain by employing two qualitatively different numerical tasks18,19 in concert with magnetoencephalography (MEG) recordings. We assay both neural representations of non-symbolic numerosities (dot patterns), including zero (empty sets), and symbolic numerals, including symbolic zero. Our results reveal that neural representations of zero are situated along a graded neural number line shared with other natural numbers. Notably, symbolic representations of zero generalized to predict non-symbolic empty sets. We go on to localize abstract representations of numerical zero to posterior association cortex, extending the purview of parietal cortex in human numerical cognition to encompass representations of zero.10,20.

How Numerical Cognition Explains Ambiguity Aversion

Abstract Consumers generally prefer precise probabilities or outcomes over imprecise ranges with the same expected value, a bias known as “ambiguity aversion.” We argue that two elementary principles of numerical cognition explain great heterogeneity in this bias, affecting consumer choices in many domains where options are characterized by varying levels of uncertainty (e.g., lotteries, discounts, investment products, vaccines, etc.). The first principle, the “compression effect,” stipulates that consumers’ mental number lines are increasingly compressed at greater number magnitudes. This alone suffices to predict ambiguity aversion as it causes a midpoint (e.g., $40) to be perceived as closer to the upper bound of a range (e.g., $60) compared to its lower bound (e.g., $20). Furthermore, as the compression effect distorts the mental number line especially at lower numbers, it follows that ambiguity aversion should decrease around greater numbers. The second principle, the “left-digit effect” causes a range’s relative attractiveness to decrease (increase) disproportionately with every left-digit transition in its lower (upper) bound, thus increasing (decreasing) ambiguity aversion. Due to the overall compression effect, the impact of the left-digit effect increases at greater numbers. We present 34 experiments (N = 10,634) to support the theory’s predictions and wide applicability.

Ninth-grade students’ conceptual understanding of the number line

Ninth-grade students’ conceptual understanding of the number line

Rapid automatised naming is related to reading and arithmetic for different reasons in Chinese: Evidence from Hong Kong third graders

BackgroundRapid automatised naming (RAN) has been found to predict children's reading and arithmetic abilities. However, the underlying mechanisms for its involvement in the two abilities are not clear. This study examines how RAN shared variances with domain‐general and domain‐specific abilities in predicting reading and arithmetic in Chinese children.MethodsOne hundred and sixty‐four children (mean age = 8 years 0 months, SD = 4 months) were administered with RAN tasks, word reading and arithmetic tasks and measures of working memory, processing speed, morphological awareness, phonological awareness and number line estimation.ResultsRAN mainly shared variance with morphological awareness in predicting word reading, while it shared variance with processing speed and number line estimation in predicting arithmetic calculation.ConclusionsThe findings indicated that RAN was related to reading and arithmetic for different reasons. The RAN–reading relationship partly reflected the semantic facilitation of the orthography–phonology links for both RAN stimuli and Chinese characters, while the RAN–arithmetic relationship partly reflected the shared process of retrieving semantic information from long‐term memory embedded in the two tasks.

Impacts of number lines and circle visual displays on caregivers’ fraction understanding

Playful fraction picture books, together with math instructional content called “back matter,” may promote fraction learning, which is crucial because fractions are difficult and often disliked content. However, open questions remain regarding how different types of back matter may affect caregivers’ ability to use fraction picture books as a teaching tool. The current study offers a novel investigation into how back matter affects caregivers’ (N = 160) fraction understanding (i.e., equivalence and arithmetic) and subjective beliefs about math using a pretest/posttest design. We contrasted existing back matter text with research-informed back matter text crossed with either circle area or number line visual displays. Caregivers’ performance improved from pretest to posttest in the Researcher-Generated + Circles condition (fraction equivalence) and in the Existing + Circles, Researcher-Generated + Circles, and Researcher-Generated + Number Lines conditions (fraction arithmetic). In addition, caregivers were aware of their learning; they predicted improvements in their fraction arithmetic performance over time. These findings suggest that brief interventions, such as back matter in children’s picture books, may improve adults’ fraction understanding.

Effect of Using Number Line and Tale Assisted Instruction in the Learning of Redox Reaction (Electrochemistry) in Senior Secondary School

This research aimed to explore the impact of incorporating number line and tale assisted instruction on secondary school students in chemistry, specifically focusing on the topic of redox reactions. Additionally, the study investigated whether gender influenced student performance in learning environments utilizing these instructional methods. Employing a quasi-experimental design with a 3 x 2 factorial structure, the research involved 120 senior secondary school students (SSS II) drawn from three public secondary schools in Anambra State, Nigeria. Pre-test and post-test scores were analysed using Analysis of Covariance (ANCOVA). The results revealed that students exposed to either number line or combined number line and tale assisted instruction demonstrated significantly higher performance compared to those taught through conventional classroom instruction. Also, there was no notable disparity in performance between male and female students across the experimental groups. Drawing from these findings, recommendations were proposed to implement relevant tales assisted and number line methodologies into chemistry teaching practices within Nigerian secondary schools.

Evaluating the content structure of intelligent tutor systems—A psychological network analysis

The adoption of intelligent tutoring systems (ITSs) worldwide has led to a considerable accumulation of process data as students interact with different learning topics within these systems. Typically, these learning topics are structured within ITSs (e.g., the fraction topic includes subtopics such as a fraction number line subtopic). However, there is a lack of methods that offer quick, data-driven insights into the content structure of ITSs, particularly through easily accessible visualizations. Here, we applied psychological network analysis to process data (230,241 students; 5,365,932 problem sets) from an ITS for learning mathematics to explore performance interdependencies between 40 different subtopics. We argue that the visualization of these content interdependencies allows for a quick empirical evaluation of the validity of the existing structuring of the respective learning content. These insights allow for deriving recommendations concerning potential changes in the ITS structure and are thus highly valuable for ITS developers. Our results are also relevant for researchers as the interdependencies illustrated through psychological network analysis can contribute towards a better understanding of the interplay between mathematical skills. Together, our results indicate that psychological network analysis represents a valuable data-driven method to evaluate and optimize ITSs.