Number Of Lines Research Articles

Supplemental Material for Number Lines Can Be More Effective at Facilitating Adults’ Performance on Health-Related Ratio Problems Than Risk Ladders and Icon Arrays

Supplemental Material for Number Lines Can Be More Effective at Facilitating Adults’ Performance on Health-Related Ratio Problems Than Risk Ladders and Icon Arrays

Mimicking tracing actions enhances young children's mathematical learning

AbstractLearning mathematical concepts and procedures typically requires extended cognitive effort, presenting a challenge for many children. People can make tracing actions with the index finger, as well as mimic another's movements, with little or no conscious effort. From the perspective of cognitive load theory, such biologically primary actions may facilitate learning biologically secondary concepts and skills requiring extensive cognitive effort, such as mathematics. The present study investigated effects on learning processes and outcomes of students mimicking a teacher's tracing actions from the perspective of an evolutionarily informed cognitive load theory. One hundred and thirteen Grade Two children learned about number lines, either observing a teacher tracing out elements of worked examples with her index finger, or mimicking the teacher's tracing actions with their own index finger. In accord with hypotheses, results indicated enhanced performance due to mimicking on cognitive load, motivation, and post‐test performance. Directions for future research are discussed.

Spatial-numerical associations without a motor response? Grip force says ‘Yes’

In numerical processing, the functional role of Spatial-Numerical Associations (SNAs, such as the association of smaller numbers with left space and larger numbers with right space, the Mental Number Line hypothesis) is debated. Most studies demonstrate SNAs with lateralized responses, and there is little evidence that SNAs appear when no response is required. We recorded passive holding grip forces in no-go trials during number processing. In Experiment 1, participants performed a surface numerical decision task (“Is it a number or a letter?”). In Experiment 2, we used a deeper semantic task (“Is this number larger or smaller than five?”). Despite instruction to keep their grip force constant, participants' spontaneous grip force changed in both experiments: Smaller numbers led to larger force increase in the left than in the right hand in the numerical decision task (500–700 ms after stimulus onset). In the semantic task, smaller numbers again led to larger force increase in the left hand, and larger numbers increased the right-hand holding force. This effect appeared earlier (180 ms) and lasted longer (until 580 ms after stimulus onset). This is the first demonstration of SNAs with passive holding force. Our result suggests that (1) explicit motor response is not a prerequisite for SNAs to appear, and (2) the timing and strength of SNAs are task-dependent. (216 words).

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children.

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3-5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1-9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

A Virtual Number Line Intervention Package to Support Addition and Subtraction for Students With Intellectual Disability

Researchers established virtual manipulatives as an evidence-based practice for students with intellectual and developmental disabilities (IDD). However, much of the existing research for this population targets middle school students with IDD. This study examined a virtual number line paired with teacher modeling and the system of least prompts to support high school students with IDD in acquiring, maintaining, and generalizing addition and subtraction problems. Through a single case research study with three high school students with IDD, the researchers found a functional relation between the intervention package and student accuracy. Students also became more independent but struggled to maintain their skills when prompting was not provided. Further, the students were unable to generalize the skills of addition and subtraction to real-life scenarios targeting time and money. These findings and their implications for the field of IDD are examined.

Number estimation in Down syndrome: Cognition or experience?

BackgroundThe ability to place numbers on a visual “number line” is a hallmark of the understanding of numerical magnitude and it is a strong predictor of mathematical achievement. AimWe examined whether the performance in the number line estimation task is more driven by mental age or experience with numbers in a sample of Italian children with Down syndrome (DS). Method and procedureSixty-three children with DS (Mmonths = 128.62, SD = 30.73) and sixty-three typically developing children (Mmonths = 54.98, SD = 6.34) matched one to one for mental age completed number line estimation tasks and other tests to assess their numerical knowledge. Outcomes and resultsNo significant differences emerged between the two groups in terms of accuracy of positioning numbers on the 1–10 and 1–20 interval. In addition, the accuracy on the 1–10 interval was related to the ability to recognize numbers, while the accuracy on the 1–20 line was related to the ability to compare magnitudes. Conclusion and ImplicationResults suggest that in individuals with DS the linear mapping of numbers is driven by mental age, but the accuracy of positioning numbers is also shaped by the experience with symbolic numbers. Therefore, the improvement of numerical estimation abilities should be a target of intervention programs.

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.