Processing Of Numerical Information Research Articles

Floating-Point Embedding: Enhancing the Mathematical Comprehension of Large Language Models

The processing and comprehension of numerical information in natural language represent pivotal focal points of scholarly inquiry. Across diverse applications spanning text analysis to information retrieval, the adept management and understanding of the numerical content within natural language are indispensable in achieving task success. Specialized encoding and embedding techniques tailored to numerical data offer an avenue toward improved performance in tasks involving masked prediction and numerical reasoning, inherently characterized by numerical values. Consequently, treating numbers in text merely as words is inadequate; their numerical semantics must be underscored. Recent years have witnessed the emergence of a range of specific encoding methodologies designed explicitly for numerical content, demonstrating promising outcomes. We observe similarities between the Transformer architecture and CPU architecture, with symmetry playing a crucial role. In light of this observation and drawing inspiration from computer system theory, we introduce a floating-point representation and devise a corresponding embedding module. The numerical representations correspond one-to-one with their semantic vector values, rendering both symmetric regarding intermediate transformation methods. Our proposed methodology facilitates the more comprehensive encoding and embedding of numerical information within a predefined precision range, thereby ensuring a distinctive encoding representation for each numerical entity. Rigorous testing on multiple encoder-only models and datasets yielded results that stand out in terms of competitiveness. In comparison to the default embedding methods employed by models, our approach achieved an improvement of approximately 3.8% in Top-1 accuracy and a reduction in perplexity of approximately 0.43. These outcomes affirm the efficacy of our proposed method. Furthermore, the enrichment of numerical semantics through a more comprehensive embedding contributes to the augmentation of the model’s capacity for semantic understanding.

Exploring Equation as a Better Intermediate Meaning Representation for Numerical Reasoning of Large Language Models

Numerical reasoning is a vital capability for natural language processing models to understand and process numerical information in real-world scenarios. Most current methods first generate the Intermediate Meaning Representations (IMRs) of questions and then generate answers. Current SOTA methods generate programs as IMRs with large language models (LLMs). Intuitively, equations have fewer restrictions and closer semantics to the question than programs, leading to higher generation accuracy. However, current LLMs generate equations worse than programs, where we assume that the equation data is rare in pre-training data compared to programs. So in this paper, we try to use equations as IMRs to solve the numerical reasoning task by addressing two problems: (1) Theoretically, how to prove that the equation is an IMR with higher generation accuracy than programs; (2) Empirically, how to improve the generation accuracy of equations with LLMs. For the first problem, we propose and prove a proposition to theoretically compare the generation accuracy of different IMRs. For the second problem, we present a method called Boosting Numerical ReasonIng by Decomposing the Generation of Equations Bridge, which can improve the accuracy of LLMs in generating equations as IMRs by reducing the tendency of generating constant expressions and programs. Our method improves the performance by 2.2%, 0.9%, and 1.7% on GSM8K, SVAMP, and Algebra datasets compared to the previous state-of-the-art methods under the single reasoning path setting. Our code and prompts are available at https://github.com/zirui-HIT/Bridge_for_Numerical_Reasoning}.

Can rounding up price discounts reduce sales?

AbstractSome retailers round up price discounts, such as displaying a 7.7% discount as an 8% discount. In such instances, lay beliefs would suggest that displaying an 8% discount (vs. a 7.7% discount) would increase purchase intentions. In this research report, however, we show that displaying a rounded‐up, higher‐value discount (8%) versus a more precise but lower‐value discount (7.7%) reduces purchase intentions. Specifically, we show that using a more precise discount framing increases perceptions that the discount duration is shorter, in turn increasing purchase intentions. This research report presents a relevant and counterintuitive effect, and we propose contributions to work on both behavioral pricing and numerical information processing. Furthermore, this work has implications for practice, showing how to optimally display price discounts.

Are they equivalent? An examination of task variants of approximate number comparison.

Nonverbal numerical ability supports individuals' numerical information processing in everyday life and is also correlated with their learning of mathematics. This ability is typically measured with an approximate number comparison paradigm, in which participants are presented with two sets of objects and instructed to choose the numerically larger set. This paradigm has multiple task variants, where the two sets are presented in different ways (e.g., two sets are presented either simultaneously or sequentially, or two sets are presented either intermixed or separately). Despite the fact that different task variants have often been used interchangeably, it remains unclear whether these variants measure the same aspects of nonverbal numerical ability. Using a latent variable modeling approach with 270 participants (Mage = 20.75 years, SDage = 2.03, 94 males), this study examined the degree to which three commonly used task variants tapped into the same construct. The results showed that a bi-factor model corresponding to the hypothesis that task variants had both commonalities and uniqueness was a better fit for the data than a single-factor model, corresponding to the hypothesis that task variants were construct equivalent. These findings suggested that task variants of approximate number comparison did not measure the same construct and cannot be used interchangeably. This study also quantified the extent to which general cognitive abilities were involved in both common and unique parts of these task variants.

Intuitive mapping between nonsymbolic quantity and observed action across development

Adults’ concurrent processing of numerical and action information yields bidirectional interference effects consistent with a cognitive link between these two systems of representation. This link is in place early in life: infants create expectations of congruency across numerical and action-related stimuli (i.e., a small [large] hand aperture associated with a smaller [larger] numerosity). Although these studies point to a developmental continuity of this mapping, little is known about the later development and thus how experience shapes such relationships. We explored how number–action intuitions develop across early and later childhood using the same methodology as in adults. We asked 3-, 6-, and 8-year-old children, as well as adults, to relate the magnitude of an observed action (a static hand shape, open vs. closed, in Experiment 1; a dynamic hand movement, opening vs. closing, in Experiment 2) to either a small or large nonsymbolic quantity (numerosity in Experiment 1 and numerosity and/or object size in Experiment 2). From 6 years of age, children started performing in a systematic congruent way in some conditions, but only 8-year-olds (added in Experiment 2) and adults performed reliably above chance in this task. We provide initial evidence that early intuitions guiding infants’ mapping between magnitude across nonsymbolic number and observed action are used in an explicit way only from late childhood, with a mapping between action and size possibly being the most intuitive. An initial coarse mapping between number and action is likely modulated with extensive experience with grasping and related actions directed to both arrays and individual objects.

The Domain‐Specificity of Domain‐Generality: Attention, Executive Function, and Academic Skills

AbstractAttention and executive functions (EFs) play a critical role in academic skill development, including literacy and numeracy. Deficits in attention and EFs often accompany learning disorders, such as dyslexia and dyscalculia. Despite their well‐established link, we lack a nuanced understanding of the specific neurobiological mechanisms that integrate the higher order cognitive processes of EFs (e.g., cognitive control) with the control of lower‐level cognition related to domain‐specific skills (e.g., processing numerical information). While attention and EFs have been identified as key factors in domain‐specific cognition and learning, the focus on its domain‐generality may hinder exploration of its domain‐specific roots. Using the example of number processing, I detail a series of neuroimaging studies exploring how domain‐specific mechanisms interact with domain‐general processing. They suggest that a better understanding of the hierarchical nature of higher order control of lower‐level processing is necessary to explain the relation between attention, EFs, and academic skills.

Quantity discrimination in newly hatched zebrafish suggests hardwired numerical abilities

An intriguing hypothesis to explain the ubiquity of numerical abilities is that all vertebrates are born with hardwired neuronal networks for processing numbers. To date, only studies on human foetuses have clearly supported this hypothesis. Zebrafish hatch 48–72 h after fertilisation with an embryonic nervous system, providing a unique opportunity for investigating this hypothesis. Here, we demonstrated that zebrafish larvae exposed to vertical bars at birth acquired an attraction for bar stimuli and we developed a numerical discrimination task based on this preference. When tested with a series of discriminations of increasing difficulty (1vs.4, 1vs.3, 1vs.2, and 2vs.4 bars), zebrafish larvae reliably selected the greater numerosity. The preference was significant when stimuli were matched for surface area, luminance, density, and convex hull, thereby suggesting a true capacity to process numerical information. Converging results from two phylogenetically distant species suggests that numerical abilities might be a hallmark feature of vertebrates’ brains.

Implementation of interdisciplinary connections of informatics and mathematics in the study of the theme "Processing of numerical information in spreadsheets"

The article describes the possibilities of implementing interdisciplinary connections of informatics and mathematics at school when studying the theme "Processing numerical information in spreadsheets". It is substantiated that interdisciplinary connections in academic disciplines are established through interdisciplinary concepts. The key concepts in the educational theme "Processing numerical information in spreadsheets" of the school informatics course are highlighted. Based on these concepts, examples of solving problems from a school mathematics course in spreadsheets are presented: building a table of squares of two-digit numbers, plotting of functions, solving a system of equations graphically. The examples demonstrate the possibility of implementing interdisciplinary connections of informatics and mathematics in the context of studying the processing of numerical information using spreadsheets: on the one hand, the use of spreadsheets allows you to automate routine tabular calculations in mathematics, on the other hand, solving such problems allows students to better understand the organization of calculations in spreadsheets.

A qualitative study into the experience of living with acalculia after stroke and other forms of acquired brain injury

ABSTRACT Acalculia, an acquired disability following a brain injury, involves difficulty processing numerical information and/or calculations. Acalculia is not routinely screened for, and as a result there is a lack of understanding about the nature and prevalence and the impact of the condition. This qualitative study was initiated by stroke survivors with a strong interest in acalculia. Sixteen stroke/brain injury survivors with acalculia and seven carers were interviewed using semi-structured online interviews. Participants ranged in age, gender, time post-onset, country of residence and numeracy level prior to brain injury. Data were analysed using thematic analysis. Three main themes were identified: Awareness and Diagnosis; Emotional and Practical Impact (independence); Support, Coping Strategies and Self-training. Participants and carers repeatedly referred to the lack of awareness and treatment for acalculia and the impact acalculia has had on their lives and independence. Practical impacts included managing money, making appointments, using timetables, organizing social activities and employment, and managing medication. Our results highlight the urgent need to develop suitable assessments and interventions for acalculia and the scope for this to be Patient, Carer and Public involvement (PCPI)-led. The data also reveal useful strategies and suggestions regarding effective timing, targets and approaches for intervention.

Individual differences in numerical representations of risk in health decision making: A fuzzy-trace theory approach.

Fuzzy-trace theory predicts that decisionmakers process numerical information about risk at multiple levels in parallel: the simplest level, nominal (categorical some-none) gist, and at more fine-grained levels, involving relative comparison (ordinal less-more gist) and exact quantities (verbatim representations). However, little is known about how individual differences in these numerical representations relate to judgments and decisions, especially involving health tradeoffs and relative risks. To investigate these differences, we administered measures of categorical and ordinal gist representations of number, objective numeracy, and intelligence in two studies (Ns=978 and 956). In both studies, categorical and ordinal gist representations of number predicted risk judgments and decisions beyond objective numeracy and intelligence. Participants with higher scores in categorical gist were more likely to choose options to avoid cancer recurrence risks; those who were higher in ordinal gist of numbers were more likely to discriminate relative risk of skin cancer; and those with higher scores in objective numeracy were more likely to choose options that were numerically superior overall in terms of relative risk of skin cancer and of genetic risks of breast cancer (e.g., lower numerical probability of cancer). Results support parallel-processing models that assume multiple representations of numerical information about risk, which vary in precision, and illustrate how individual differences in numerical representations are relevant to tradeoffs and risk comparisons in health decisions. These representations cannot be reduced to one another and explain psychological variations in risk processing that go beyond low versus high levels of objective numeracy.

The effect of proportion manipulation on the size-congruency and distance effects in the numerical Stroop task.

In the numerical Stroop task, participants are presented with two digits that differ in their numerical and physical size and are requested to respond to which digit is numerically larger. Commonly, slower responses are observed when the numerical distance between the digits is small (the distance effect) and when the numerical and physical size are incongruent (the size-congruency effect). The current study will use proportion manipulation, which consists of two experimental lists with high versus low frequency of trials belonging to different conditions, as a tool to reduce these effects. Specifically, it will be used to examine how these two interference effects depend on each other, and how a reduction of one effect will affect the other. In Experiment 1, the size-congruency proportions were manipulated; in Experiment 2, the distance proportions were manipulated. The results show that manipulating size-congruency proportions modulates the size-congruency effect but not the distance effect, while manipulating the distance proportions modulates the distance effect but not the size-congruency effect. These results demonstrate for the first time that the distance effect can be modulated by the distance proportions. Furthermore, these results indicate that proportion manipulation is specific and only modulates the variable being manipulated. Together, these results shed new light on the specificity of proportion manipulation in the context of numerical information processing. These results are further discussed in the context of various numerical models that suggest a different relationship between these effects and demonstrate how proportion manipulation can aid to investigate numerical processes.

The link between number and action in human infants

Humans' inborn ability to represent and manipulate numerical quantities is supported by the parietal cortex, which is also involved in a variety of spatial and motor abilities. While the behavioral links between numerical and spatial information have been extensively studied, little is known about the connection between number and action. Some studies in adults have shown a series of interference effects when simultaneously processing numerical and action information. We investigated the origins of this link by testing forty infants (7- to 9-month-old) in one of two experimental conditions: one group was habituated to congruent number-hand pairings, where the larger the number, the more open the hand-shape associated; the second group was habituated to incongruent number-hand pairings, where the larger the number, the more close the hand-shape associated. In test trials, both groups of infants were presented with congruent and incongruent pairings. We found that only infants habituated to congruency showed a significantly higher looking time to the test trial depicting incongruent pairings. These findings show for the first time that infants spontaneously associate magnitude-related changes across the dimensions of number and action-related information, thus offering support to the existence of an early, preverbal number-action link in the human mind.

An undeniable interplay: Both numerosity and visual features affect estimation of non-symbolic stimuli

Converging lines of evidence suggest that the numerical abilities in Humans are rooted in the approximate number system (ANS): an innate, non-verbal mechanism that enables to estimate the numerosity of a set of items with little effort. Nevertheless, the high correlation between visual features and numerosity in the natural environment always constituted a relevant methodological problem that gathered growing concern throughout the years. This issue led some researchers to cast doubts on the existence of a system able to process numerical information independently from the influence of visual features. In the present study, we sought to shed light on the interplay between numerosity and visual features. To this aim, we implemented a non-symbolic estimation task which included a calibration phase. After performing a pre-calibration block, participants were presented with the calibration image for 20 s, and they were divided in three groups, according to the calibration stimulus they attended to: the three calibration stimuli contained the same number of items (30), but were characterized by a different amount of visual features. Results showed that performance was affected by numerosity and visual features in both phases of the experiment. However, calibration increased the weight of numerosity on performance while decreasing the weight of visual features. These results are hard to be reconciled with theories that attempt to explain human performance in non-symbolic number processing without taking into account both numerical and non-numerical aspects of the stimuli.

Clinical and psychophysiological manifestations of dyscalculia in children

The article presents an overview of scientific publications devoted to the study of dyscalculia. Present-day data on the prevalence of this pathology, the features of the clinical picture, and theories of neurocognitive deficit underlying the disorder in the processing of numerical information are presented. The results of neurophysiological and neuroimaging studies devoted to the study of the features of the functional activity of various brain structures in dyscalculia are considered.

ИНТЕЛЛЕКТУАЛЬНЫЙ ПРОГРАММНЫЙ КОМПЛЕКС ДЛЯ МОНИТОРИНГА И УПРАВЛЕНИЯ ГОРЯЧИМ ВОДОСНАБЖЕНИЕМ (ГВС) НА ОСНОВЕ РЕГУЛЯРИЗИРУЮЩЕГО БАЙЕСОВСКОГО ПОДХОДА

Digitalization and intellectualization of housing and communal services is one of the most important and complex tasks of the development of the modern regional economy. The difficult operating conditions of housing and communal services complexes, the lack of the necessary information base, uncertainty, incompleteness, unstructured information form increased requirements for the information systems being created and the methodology of their development. The article offers an example of the implementation of an intelligent software package for monitoring and supporting management decisionmaking for hot water systems (DHW). Based on the RBP methodology, a working sample of the complex for a section of the DHW network has been developed. The unique properties of the intellectual complex are the possibilities of integrating different types of information, intellectual processing of numerical and linguistic information in conditions of considerable uncertainty, metrological justification of the solutions obtained, implementation of the principles of traceability, transparency, explainability of solutions. The complex can be used for training specialists of the DHW system. It can also serve as an example for the development of intelligent monitoring and control systems for complex distributed energy networks, oil and gas pipelines.

Increased structural covariance in brain regions for number processing and memory in children with developmental dyscalculia.

Developmental dyscalculia (DD) is a developmental learning disability associated with deficits in processing numerical and mathematical information. Several studies demonstrated functional network alterations in DD. Yet, there are no studies, which examined the structural network integrity in DD. We compared whole‐brain maps of volume based structural covariance between 19 (4 males) children with DD and 18 (4 males) typically developing children. We found elevated structural covariance in the DD group between the anterior intraparietal sulcus to the middle temporal and frontal gyrus (p < 0.05, corrected). A hippocampus subfield analysis showed higher structural covariance in the DD group for area CA3 to the parahippocampal and calcarine sulcus, angular gyrus and anterior part of the intraparietal sulcus as well as to the lingual gyrus. Lower structural covariance in this group was seen for the subiculum to orbitofrontal gyrus, anterior insula and middle frontal gyrus. In contrast, the primary motor cortex (control region) revealed no difference in structural covariance between groups. Our results extend functional magnetic resonance studies by revealing abnormal gray matter integrity in children with DD. These findings thus indicate that the pathophysiology of DD is mediated by both structural and functional abnormalities in a network involved in number processing and memory function.

Can a Single Representational Object Account for Different Number-Space Mappings?

Numbers are mapped onto space from birth on, as evidenced by a variety of interactions between the processing of numerical and spatial information. In particular, larger numbers are associated to larger spatial extents (number/spatial extent mapping) and to rightward spatial locations (number/location mapping), and smaller numbers are associated to smaller spatial extents and leftward spatial locations. These two main types of number/space mappings (number/spatial extent and number/location mappings) are usually assumed to reflect the fact that numbers are represented on an internal continuum: the mental number line. However, to date there is very little evidence that these two mappings actually reflect a single representational object. Across two experiments in adults, we investigated the interaction between number/location and number/spatial extent congruency effects, both when numbers were presented in a non-symbolic and in a symbolic format. We observed a significant interaction between the two mappings, but only in the context of an implicit numerical task. The results were unaffected by the format of presentation of numbers. We conclude that the number/location and the number/spatial extent mappings can stem from the activation of a single representational object, but only in specific experimental contexts.

Numbers (but not words) make math anxious individuals sweat: Physiological evidence

The study aimed to determine the specificity of math anxiety by measuring physiological arousal to the presentation of numerical and non-numerical stimuli. It also investigated whether math and trait anxieties share similar behavioral and physiological manifestations. Fifty-two female university students performed an experimental task including simple or complex arithmetical equations and math-related or neutral words. Participants’ electrodermal activity (skin conductance response) was monitored during the task. Math and trait anxieties were measured using common explicit questionnaires. Results showed math anxiety levels were significantly related to physiological arousal during the performance of complex numerical tasks. Importantly, math anxiety significantly mediated the links between trait anxiety and physiological arousal in complex numerical tasks. The findings support previous work finding relations between math and trait anxieties, but also show math anxiety is a unique phenomenon with specific behavioral and physiological manifestations, especially during the processing of complex numerical information.

The unexplored role of handshape similarity in processing numbers on the hands

With two simple experiments we investigate the overlooked influence of handshape similarity for processing numerical information conveyed on the hands. In most finger-counting sequences there is a tight relationship between the number of fingers raised and the numerical value represented. This creates a possible confound where numbers closer to each other are also represented by handshapes that are more similar. By using the American Sign Language (ASL) number signs we are able to dissociate between the two variables orthogonally. First, we test the effect of handshape similarity in a same/different judgment task in a group of hearing non-signers and then test the interference of handshape in a number judgment task in a group of native ASL signers. Our results show an effect of handshape similarity and its interaction with numerical value even in the group of native signers for whom these handshapes are linguistic symbols and not a learning tool for acquiring numerical concepts. Because prior studies have never considered handshape similarity, these results open new directions for understanding the relationship between finger-based counting, internal hand representations and numerical proficiency.

Perceived Motion and Operational Momentum: How Speed, Distance, and Time Influence Two-Digit Arithmetic

Operational momentum was originally defined as a bias toward underestimating outcomes of subtraction and overestimating outcomes of addition. It was suggested that these estimation biases are due to leftward attentional shift along the mental number-line (spatially organized internal representation of number) in subtraction and rightward shift in addition. This assumes the use of “recycled” mechanisms of spatial attention, including “representational momentum” – a tendency to overestimate future position of a moving object, which compensates for the moving object’s shift during preparation of a reaction. We tested a strong version of this assumption directly, priming two-digit addition and subtraction problems with leftward and rightward motion of varied velocity, as velocity of the tracked object was found to be a factor in determining representational momentum effect size. Operands were subsequently moving across the computer screen, and the participants’ task was to validate an outcome proposed at the end of the event, which was either too low, correct, or too high. We found improved accuracy in detecting too-high outcomes of addition, as well as complex patterns of interactions involving arithmetic operation, outcome option, speed, and direction of motion, in the analysis of reaction times. These results significantly extend previous evidence for the involvement of spatial attention in mental arithmetic, showing movement of the external attention focus as a factor directing internal attention in processing numerical information. As a whole, however, the results are incompatible with expectations derived from the strong analogy between operational and representational momenta. We suggest that the full model may be more complex than simply “moving attention along the mental number-line” as a direct counterpart of attention directed at a moving object.