Number Of Lines Research Articles

An insect brain organizes numbers on a left-to-right mental number line

The "mental number line" (MNL) is a form of spatial numeric representation that associates small and large numbers with the left and right spaces, respectively. This spatio-numeric organization can be found in adult humans and has been related to cultural factors such as writing and reading habits. Yet, both human newborns and birds order numbers consistently with an MNL, thus raising the question of whether culture is a main explanation for MNL. Here, we explored the numeric sense of honey bees and show that after being trained to associate numbers with a sucrose reward, they order numbers not previously experienced from left to right according to their magnitude. Importantly, the location of a number on that scale varies with the reference number previously trained and does not depend on low-level cues present on numeric stimuli. We provide a series of neural explanations for this effect based on the extensive knowledge accumulated on the neural underpinnings of visual processing in honey bees and conclude that the MNL is a form of numeric representation that is evolutionarily conserved across nervous systems endowed with a sense of number, irrespective of their neural complexity.

Modeling the left digit effect in adult number line estimation

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0–100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

A near-isogenic line for spikelet number in rice with a genetic background of IR64 under various fertilizer conditions

ABSTRACT The total spikelet number (TSN) per panicle is an essential factor involved in grain yield among yield components in rice. IR64-NIL12 carrying qTSN12.2-YP4, which was detected on chromosome 12 for TSN per panicle, has been developed in a previous study. It is important to characterize the environment of gene function to increase yields. Thus, we attempted to evaluate IR64-NIL12 grown under no, low, and high fertilizer conditions in 2019 and 2020. Grain yields of IR64-NIL12 were significantly higher than that of IR64 in all conditions. The average grain yield of each condition for IR64 and IR64-NIL12 ranged from 387 to 616 g m−2. Therefore, qTSN12.2-YP4 would have an increasing effect on grain yield within this range. The ratio of fertility improved in IR64-NIL12. This rice line exhibited modified traits, such as tiller number and plant height (during the vegetative stage), as well as panicle number (at maturity), which may confer an advantage in terms of light-intercepting characteristics. We also focused on the number of unproductive tillers and dry weight per tiller at the heading stage to investigate the mechanism of the increasing effect of qTSN12.2-YP4 on TSN per panicle. There is a significant difference in dry weight per tiller at heading between IR64 and IR64-NIL12, but not in the percentage of unproductive tillers. Therefore, this result indicated that qTSN12.2-YP4 contributes to increasing dry weight per tiller until the heading stage and, consequently, TSN per panicle.

From integers to fractions: The role of analogy in developing a coherent understanding of proportional magnitude.

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions would improve understanding of fractions, but little is known about how the representations develop and interact with one another. To address this, we asked 177 third, fourth, and fifth graders (M = 9.85 years, 87 girls, 69% White, 19% low income) to estimate the position of proportionally equivalent integers and fractions on number lines (e.g., 3 on a 0-8 number line vs. 3/8 on a 0-1 number line, Study 1). With increasing age, children's estimates of implicit and explicit proportions became more coherent, such that a child's estimates of fractions on a 0-1 number-line was a linear function of the same child's estimates of equivalent integers. To further investigate whether preexisting integer knowledge can facilitate fraction learning through analogy, we assigned 100 third to fifth graders (M = 10.04 years, 55 girls, 76% White) to an Alignment condition, where children estimated fractions and integers on aligned number lines, or to a No Alignment condition (Study 2). Results showed that aligning integers and fractions on number lines facilitated a better understanding of fractional magnitudes, and increased children's fraction estimation accuracy to the level of college students'. Together, findings suggest that analogies can play an important role in building a coherent understanding of proportions. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Summing up: A functional role of eye movements along the mental number line for arithmetic

In Western cultures, small-left and large-right spatial-numerical associations are constantly found in various simple number processing tasks. It has recently been suggested that spatial associations are also involved in more complex number processing, for example that individuals make rightward or upward “mental” movements along the number line during addition, and leftward or downward movements during subtraction. In line with this, it has been shown that participants' spontaneous eye movements on a blank screen during upward and downward counting follow such associations. The present research investigated whether eye movements along the number line are simply an epiphenomenon of the recruitment of a spatial reference frame, or whether they play a functional role for the arithmetic computation. This question was addressed by using multi-step problems (e.g., 59 + 5 + 4 + 3) that show a larger proportion of computation (vs. retrieval) when compared to single-step problems (e.g., 59 + 5), as confirmed in Pretest 1. Moreover, the question was addressed only for addition problems and vertical eye movements, because spatial-arithmetic associations were not found in the other conditions (subtraction, horizontal eye movements) in Pretest 2. In the main experiment, participants (n = 29) solved addition problems while following a moving dot with their eyes (smooth pursuit) either in a congruent (upward) or incongruent (downward) direction, or while keeping their eyes fixated on to the center of the screen, or while moving their eyes freely on a blank screen. Arithmetic performance was faster in the congruent condition (upward eye movements) when compared to the other conditions (downward eye movements, central fixation, free viewing). These results suggest that vertical shifts in spatial attention along the mental number line are functionally involved in addition. The results support the view of shared mechanisms for directing spatial attention in external (visual) and representational (number space). Implications for embodied views of number processing are discussed.

초등 예비교사의 수직선을 활용한 분수 과제 변형 사례 연구

With the importance and students’ difficulties with the number line in learning fractions, this study aims to investigate prospective teachers’ textbook fraction task modification using the number line and identify their strengths and weaknesses. The analysis of 58 prospective teachers’ modification tasks about fraction concept and addition and subtraction of fraction indicated that the prospective teachers chose a wide range of topics in the textbooks and modified the external contexts and types of the number lines. Moreover, the prospective teachers showed strengths in making efforts to help students understand the number line such as designing additional number line activities and anticipating students’ strategies about the modified number line tasks. However, some limitations and mathematical errors were found in modifying the tasks in a conceptual way and establishing specific lesson plan for the modified tasks.

Using mathematics game-based intervention on children with special educational needs: Preliminary findings

Many video games incorporate positive learning principles, stimulating students' cognitive functioning and promoting problem-solving and spatial abilities. The high levels of engagement and involvement that some students can achieve with video games are notable. Hence, video games for children with a long history of school failure, such as children with special educational needs (SEN). Moreover, it may give educators immediate and ongoing assessment of students' progress. When complemented with human tutoring, video games as a game-based intervention may improve mathematics performance since instruction is more effective when adapted to students' learning needs and pace.
 As part of the research project "GBl4deaf – Game-based Learning for Deaf Students (PTDC/COM-CSS/32022/2017), the video game "Space adventure: Defend the planet!" was designed to stimulate arithmetic competencies in deaf and hearing children. The player must use elementary arithmetical and spatial concepts to rebuild an abandoned space station. Each challenge has three difficulty levels. In the game challenge used in this study, the player must add or remove particles to collect resources. The current research focuses on two questions: The study follows two research questions: Q1: Did the students make any progress in mathematics achievement after playing the video game?; Q2: Is the gameplay of Space adventure: Defend the planet! an engaging experience for players? A pre-and post-game mathematical test was applied to measure mathematics achievement and an observational grid to gather information about arithmetical procedures. Ten fourth- to ninth-graders participated in the study - four girls and six boys, aged between 9 and 16, three deaf and seven hearing students with different special needs (dyscalculia, cognitive deficits, autism spectrum disorder, deafness and Asperger syndrome). Due to the COVID-19 pandemic lockdown, children played the video game using Zoom video conference software in 8-12 sessions (50 minutes, two a three times/week). The results show 4% to 19% of mathematics progression after children played the video game and indicate that they maintain the use of counting-based procedures throughout the game sessions. For instance, they kept counting both addends starting from 1 or counting by 1, 2, 5 or 10 using a number line. The current data suggest that the videogame "Space adventure: Defend the planet!" allows educators to gather immediate information about students' difficulties and progression.

The role of math anxiety in the relationship between approximate number system and math performance in young children

AbstractAlthough interest in mathematics anxiety in young children (6–11 years) has increased recently, the results of previous studies on the role of mathematics anxiety in the relationship between number sense and math performance are inconsistent. The study's main objective was to establish whether math anxiety is a mediator or a moderator in the relationship between number line estimation (NLE) and mathematical outcomes. A cross‐sectional, correlational design was used. A total of 234 Turkish public primary school children (third and fourth graders) participated in the study. The following hypotheses were tested: (1) math anxiety mediates the relationship between the approximate number system (ANS) (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency); (2) math anxiety moderates the relationship between ANS (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency). The results mostly suggest no mediation effect of math anxiety between NLE and math performance in young children. The results also suggest that increases in difficulty level in NLE (increases of errors in NLE) is related to more anxiety regarding mathematics. Furthermore, moderation analyses indicated that the acuity of mental representation of numbers in high math‐anxious children is a key factor contributing to math performance. Developmental aspects of the obtained results are also discussed in the paper.

Some Conceptual Metaphors for Rational Numbers as Fractions in Swedish Mathematics Textbooks for Elementary Education

ABSTRACT This study examines the presence of three conceptual metaphors for fractions – The measuring stick metaphor, Arithmetic is motion along a path, and Arithmetic is object construction – in four common and popular Swedish mathematics textbook series for years 1–3. I analyse the introduction of fractions and the kinds of tasks students are given in these books. The results show an abundance of labelling exercises related to Arithmetic is object construction, with representations of geometric shapes, in three of the book series. One book series reversely introduces fractions with The measuring stick metaphor and Arithmetic is motion along a path, with a focus on number line representations. The consequences of these variations in fraction introduction and treatment are discussed in relation to previous research and to process-object theories.

Contributions of the psychology of mathematical cognition in early childhood education using apps.

Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and promote the use of these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57–79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group (N = 49), high-performance group (N = 21), and control group (N = 123). The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance.

Whole Number Bias of Students in Fraction Number Line Tasks

Whole Number Bias of Students in Fraction Number Line Tasks

The Teaching and Learning of Integer Operations: The Case of Number Rule and Conventional Method

This study looked at how to employ the Number Rule to help JHS students perform better when learning about integer addition and subtraction. To ascertain the impact of the Number Rule as an instructional aid in the teaching of addition and subtraction of integers on the performance of Junior High School students in mathematics, a mixed-method approach (mostly quasi-experimental and use of interviews) was used. It was simple to choose two schools. Both classes remained unaltered, with one serving as the control group (34 students) and the other (37 students) as the experimental group. The control group received instruction using the conventional approach, while the experimental group received instruction utilizing the Number Rule. With the help of a teacher-made achievement test with fifteen (15) essay-style questions, both groups were pre- and post-tested. According to the study's findings, pupils who were taught using the Number Rule approach did better on the post-test than those who were taught using the Conventional method. Additionally, the Number Rule method makes the lessons more time-efficient, simple to learn, engaging, and practical. As a result, it was suggested that teachers employ the number rule to teach mathematical concepts, particularly the addition and subtraction of integers. The study is significant to teachers and teacher trainees as the result depicts the effectiveness to the use of manipulative (number rule) in teaching and learning of integer additions and subtraction. Keywords: Addition of Integers, Convention Method, Integers, Integer Operations, Manipulatives, Number Line, Number Rule, Operations on Integers, Subtraction of Integers, Whole Numbers DOI: 10.7176/JEP/13-27-01 Publication date: September 30 th 2022

The differential relations between sub-domains of spatial abilities and mathematical performance in children

Spatial abilities are closely connected to mathematical performance. However, previous work do not determine whether sub-domains of spatial abilities are equally important for different mathematics tasks. This study addressed this gap by adopting Uttal et al.’s (2013) typology of spatial abilities along the intrinsic/extrinsic and static/dynamic dimensions. Three hundred twenty-four Chinese first graders were tested on a battery of spatial and mathematics tasks, including hidden figures, mental rotation, spatial scaling, perspective taking, mental number line representation, place-value understanding, calculation, word problems, geometry, measurement, and algebra. Hierarchical regression models showed that after controlling for age and gender, spatial abilities explained 12.6% to 25.7% of the variance across seven measures of mathematical performance. Sub-domains of spatial abilities were found to have varying roles in explaining mathematical performance. Among them, hidden figures and perspective taking were significant predictors of all mathematics measures. The findings offer new insights into developing spatial training to support children’s mathematical learning.

The role of cognitive control in the SNARC effect: A review.

The spatial-numerical association of response codes (SNARC) effect, in which people respond to small numbers faster with the left hand and to large numbers faster with the right hand, is a popular topic in cognitive psychology. Some well-known theoretical accounts explaining this effect include the mental number line model, polarity correspondence principle, dual-route model, and working memory account. However, these fail to explain the finding that the size of the SNARC effect is modulated by cognitive control. Here, we propose a new account-a cognitive control-based view of the SNARC effect. This view argues that the SNARC effect is fundamentally determined by cognitive control in resolving conflicts during stimulus-response mapping. Several subcomponents of cognitive control, such as working memory, mental or task set shifting, inhibition control, and conflict adaptation, can easily modulate the SNARC effect. The cognitive control-based view can account for the flexible SNARC effect observed in diverse task situations while providing new insight into its mechanism.

The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6484 Luxembourgish ninth-graders

The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.

The SNARC effect: a preregistered study on the interaction of horizontal, vertical, and sagittal spatial–numerical associations

Small numbers are processed faster through left-sided than right-sided responses, whereas large numbers are processed faster through right-sided than left-sided responses [i.e., the Spatial–Numerical Association of Response Codes (SNARC) effect]. This effect suggests that small numbers are mentally represented on the left side of space, whereas large numbers are mentally represented on the right side of space, along a mental number line. The SNARC effect has been widely investigated along the horizontal Cartesian axis (i.e., left–right). Aleotti et al. (Cognition 195:104111, 2020), however, have shown that the SNARC effect could also be observed along the vertical (i.e., small numbers-down side vs. large numbers-up side) and the sagittal axis (i.e., small numbers-near side vs. large numbers-far side). Here, we investigated whether the three Cartesian axes could interact to elicit the SNARC effect. Participants were asked to decide whether a centrally presented Arabic digit was odd or even. Responses were collected through an ad hoc-made response box on which the SNARC effect could be compatible for one, two, or three Cartesian axes. The results showed that the higher the number of SNARC-compatible Cartesian axes, the stronger the SNARC effect. We suggest that numbers are represented in a three-dimensional number space defined by interacting Cartesian axes.

The development of teachers' knowledge in a lesson study

PurposeThis paper seeks to understand how teachers' mathematical knowledge for teaching (MKT) is developed during a lesson study regarded as a professional development process that places particular emphasis on teachers' knowledge of tasks, representations and students' learning.Design/methodology/approachThis study is based on the networking of two theories, namely the Interconnected Model of Teacher Professional Growth (IMTPG) and MKT. The methodology is qualitative and interpretative. The participants are three primary school teachers and team members involved in a lesson study. Data collection was conducted by means of participant observation, with audio recordings of the sessions and a final individual interview.FindingsThe participant teachers developed their knowledge of tasks and representations, which are essential elements of content and teaching knowledge, as well as knowledge of the curriculum, content and students. Additionally, they developed specialized knowledge of mathematics, the meaning of fraction as a measure and its representation on the number line. This development arose from activities conducted in the Domain of Practice, in many cases prompted by the External Domain, and was particularly important when these activities gave rise to unforeseen consequences. Connections among the various domains were established through enactment and reflection processes.Originality/valueThe study shows how the networking of two different theories, in this case the IMTPG and MKT, may enhance further understanding of educational phenomena. This networking involved the coordination of these two theories, which were superimposed in the Group Domain.

A cross-national comparison of fourth and eighth grade students’ understanding of fraction magnitude

Without a conceptual understanding of fraction magnitude, students have difficulties understanding more advanced mathematics in high school and beyond. The purpose of this study is to highlight 4th and 8th grade students’ misconceptions in fraction magnitude using the Trends in International Mathematics and Science Study-2015 data. The study is informed by the theory of numerical development that focuses on the understanding of the magnitude of numbers and the recognition of the location of numbers on a number line. The results indicate that fraction magnitude understanding is a challenge at the 8th grade. Implications for higher education are elaborated.

Children's use of egocentric reference frames in spatial language is related to their numerical magnitude understanding.

Numerical magnitude information is assumed to be spatially represented in the form of a mental number line defined with respect to a body-centred, egocentric frame of reference. In this context, spatial language skills such as mastery of verbal descriptions of spatial position (e.g., in front of, behind, to the right/left) have been proposed to be relevant for grasping spatial relations between numerical magnitudes on the mental number line. We examined 4- to 5-year-old’s spatial language skills in tasks that allow responses in egocentric and allocentric frames of reference, as well as their relative understanding of numerical magnitude (assessed by a number word comparison task). In addition, we evaluated influences of children’s absolute understanding of numerical magnitude assessed by their number word comprehension (montring different numbers using their fingers) and of their knowledge on numerical sequences (determining predecessors and successors as well as identifying missing dice patterns of a series). Results indicated that when considering responses that corresponded to the egocentric perspective, children’s spatial language was associated significantly with their relative numerical magnitude understanding, even after controlling for covariates, such as children’s SES, mental rotation skills, and also absolute magnitude understanding or knowledge on numerical sequences. This suggests that the use of egocentric reference frames in spatial language may facilitate spatial representation of numbers along a mental number line and thus seem important for preschoolers’ relative understanding of numerical magnitude.

Learn Number Lines with Interactive Video Learning Media

This research is motivated by the difficulties of the teacher in implementing online learning, which makes students less enthusiastic and less able to understand the material in online learning, so the development of interactive video media is needed. This study aims to produce interactive video media for grade three elementary school number line materials tested for validity. The development model used in this study is the ADDIE model. The subjects in this study consisted of 1 teacher and four dose experts who were experts in their fields as media experts and material experts, and 12 students consisting of 3 people as individual tests and nine students as small group tests. The method of data collection in this study is the method of questionnaires and interviews. The data generated in this study are qualitative and quantitative. The data obtained were then analyzed using the mean formula to determine the average validation of interactive video media. The development of interactive video media was declared valid based on the assessments of media experts, design experts, material experts, individual test students, and small group students with an average range of 4.01 < X 5, with the qualification "very good.” Thus, it can be indicated that the interactive video product is declared valid and suitable for use in learning, especially on the number line material for third grade of elementary school.