Number Line Representation Research Articles

Conceptualizations and representations of fraction division in online open access GeoGebra resources

Dividing fractions is often difficult for students to understand. Many GeoGebra applets have been developed to support students’ understanding of fraction division, but it is less clear how fraction division is conceptualized and represented in these tool-based resources. This study aimed to examine the quality of the GeoGebra applets for fraction division by attending to the conceptualizations, representations, and cognitive actions prevalent in these digital resources. The results reveal that a majority of the existing applets conceptualize fraction division as measurement while other conceptualizations of fraction division are underrepresented. Among the various representations used, the length model in the form of fraction bar and area model are the predominant choices, with the number line representation being notably less prominent. The results from this study also show that although most applets attempt to visualize the process of fraction division, there are limited opportunities in most applets for users to enact mental actions associated with fraction division, such as partitioning, unitizing, iterating, and disembedding. These results not only increase our understanding of the affordances and limitations of existing applets for fraction division so that we can become more intentional in our choice of them but also inform the design of new applets that support students’ development of a rich and robust understanding of fraction division.

Investigating the influence of body movements on children's mental arithmetic performance

Several lines of research have demonstrated spatial-numerical associations in both adults and children, which are thought to be based on a spatial representation of numerical information in the form of a mental number line. The acquisition of increasingly precise mental number line representations is assumed to support arithmetic learning in children. It is further suggested that sensorimotor experiences shape the development of number concepts and arithmetic learning, and that mental arithmetic can be characterized as “motion along a path” and might constitute shifts in attention along the mental number line. The present study investigated whether movements in physical space influence mental arithmetic in primary school children, and whether the expected effect depends on concurrency of body movements and mental arithmetic. After turning their body towards the left or right, 48 children aged 8 to 10 years solved simple subtraction and addition problems. Meanwhile, they either walked or stood still and looked towards the respective direction. We report a congruency effect between body orientation and operation type, i.e., higher performance for the combinations leftward orientation and subtraction and rightward orientation and addition. We found no significant difference between walking and looking conditions. The present results suggest that mental arithmetic in children is influenced by preceding sensorimotor cues and not necessarily by concurrent body movements.

Adaptive variability in children’s conceptual models of division

This study examined whether different types of commonly used mathematical tasks affect how children think about whole number division problems. Prior research suggested that children tend to rely on the partitive model to understand whole number division, which is likely problematic when students transition to learning about fraction division. We assessed variability in correct whole number division problem-solving strategies among 63 elementary school children (41.5% female, 58.5% male, 0% nonbinary/gender expansive; 69.2% White, 10.7% multiracial, 6.1% Black, 4.6% Latino, 3.3% other/unidentified, 6.1% preferred not to answer). Each participant was asked to demonstrate four whole number division problems in each of three contexts (within participants): objects, story problems, and number lines. Most children displayed understanding of multiple conceptual models of division, but strategies varied by context. Story problems elicited partitive models, number lines elicited quotative models, and objects elicited both. Finally, elementary school children used strategies adaptively. Number line representations may afford conceptual connections between earlier-learned whole number concepts and analogous later-learned fraction concepts, supporting the integration of children’s whole number and fraction knowledge.

INVESTIGATION OF THE SKILLS IN SOLVING AND POSING RATIONAL NUMBER PROBLEMS IN CONTEXT OF MULTIPLE REPRESENTATIONS

In this study, it is aimed to examine the problem solving and problem posing skills of seventh grade students with rational numbers in the context of multiple representations. When the literature is examined, although there is a study examining the relationship between multiple representation and rational numbers, there is no research examining the relationship between students' problem posing skills in addition to this relationship. It has been considered important to examine problem posing in the context of multiple representation, which helps students to look at the subject from a new perspective and allows them to understand it more deeply. The research was carried out with 25 students attending a public secondary school in Istanbul in the 2022-2023 academic year. The case study method, which is one of the interactive designs included in the qualitative research approach, was used in the research. The "Rational Numbers Multiple Representation Test" developed by the researchers was used as a data collection tool. In line with the data obtained, the representation preferences of the students and the interpretations of the success in these preferences were given using descriptive statistics. In addition, the problems posed by the students were evaluated using the problem posing assessment rubric introduced by Katrancı (2014). The reason for choosing this rubric is that it allows grading students' problem posing performance. According to the research findings, the most preferred representation types in problem posing questions were verbal (textual), table, number line, algebraic (numerical) and model representation types, respectively. The problems posed by the students were found to be 64 points out of 100 according to the problem posing evaluation rubric. When the transition between representations were evaluated, it was found that the transition from the verbal representation type to the table representation type was the most successful, and the transition from the number line to the model type was the least successful. The type of representation that the students liked to solve the most was the model type (40%), while the number line (36%) took the second place. The types of representations that the students liked least were expressed as algebraic and verbal. Considering the reasons for liking the model representation type more, the reasons such as being visual, comprehensible and easy to construct were stated. On the other hand, reasons such as being easy and practical are shown in the reasons for liking the number line representation type.

Teaching Fraction Magnitude Using the Number Line

The number line is a powerful tool for supporting students’ understanding of fraction magnitude. Fractions are a critical component of mathematics instruction in the elementary and intermediate grades. More specifically, understanding fraction magnitude is central to mathematical development. Yet, fractions are challenging for many students, particularly students with or at risk of learning disabilities (LD) in mathematics. This article shares (a) key recommendations when planning and implementing fraction number line instruction, (b) sample fraction number line activities for supporting students’ understanding of fraction magnitude and overall mathematics achievement, and (c) strategies for helping students grasp the abstract number line representation.

Using the Number Line to Develop Understanding of Whole Number Magnitude and Operations

This article illustrates how teachers can use number lines to support students with or at risk for learning disabilities (LD) in mathematics. Number lines can be strategically used to help students understand relations among numbers, approach number combinations (i.e., basic facts), as well as represent and solve addition and subtraction problems. The authors draw from the existing research base to describe practices for incorporating number line representations to (a) support students’ conceptual understanding of number and (b) teach addition and subtraction computational strategies. Specific examples are provided for teachers to integrate number line representations with other mathematical models to support the development of whole number understanding for students with LD in mathematics.

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.

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.

Development of the Mental Number Line Representation of Numbers 0-10 and Its Relationship to Mental Arithmetic.

The internal representation of numbers on the mental number line (MNL) was demonstrated by performing the computerized version of the number-to-position (CNP) task on a touchscreen while restricting response time. We found that the estimation pattern is best fit by a sigmoid function, further denoted as the “sigmoidal model”. Two developmental leaps occurring during elementary school were recognized: (1) the division of the number line into two segments and (2) consistent use of different anchor points on the number line—the left endpoint in first grade, the right endpoint in second grade, and finally the midpoint in third grade. Additionally, when examining the differences between the breakpoints, we found that first graders demonstrated a breakpoint close to 6, which linearly decreased over the years until stabilizing close to 5. The relation between the ability to place individual numbers on a number line and performance of mental arithmetic showed that the consistent use of anchor points correlated significantly with faster responses in mental arithmetic.

Efficacy of a Computer-Based Learning Program in Children With Developmental Dyscalculia. What Influences Individual Responsiveness?

This study presents the evaluation of a computer-based learning program for children with developmental dyscalculia and focuses on factors affecting individual responsiveness. The adaptive training program Calcularis 2.0 has been developed according to current neuro-cognitive theory of numerical cognition. It aims to automatize number representations, supports the formation and access to the mental number line and trains arithmetic operations as well as arithmetic fact knowledge in expanding number ranges. Sixty-seven children with developmental dyscalculia from second to fifth grade (mean age 8.96 years) were randomly assigned to one of two groups (Calcularis group, waiting control group). Training duration comprised a minimum of 42 training sessions à 20 min within a maximum period of 13 weeks. Compared to the waiting control group, children of the Calcularis group demonstrated a higher benefit in arithmetic operations and number line estimation. These improvements were shown to be stable after a 3-months post training interval. In addition, this study examines which predictors accounted for training improvements. Results indicate that this self-directed training was especially beneficial for children with low math anxiety scores and without an additional reading and/or spelling disorder. In conclusion, Calcularis 2.0 supports children with developmental dyscalculia to improve their arithmetical abilities and their mental number line representation. However, it is relevant to further adapt the setting to the individual circumstances.

Deconstructing South African Grade 1 learners’ awareness of number in terms of cardinality, ordinality and relational understandings

The cardinal and ordinal aspects of number have been widely written about as key constructs that need to be brought together in children’s understanding in order for them to appreciate the idea of numerosity. In this paper, we discuss similarities and differences in the ways in which understandings not only of ordinality, cardinality but also additive and multiplicative relations have been theorized. We examine how the connections between these can be considered through a focus on number line representations and children positioning and comparing numbers. The responses of a cohort of South African Grade 1 learners’ (6- and 7-year-olds) to a numerical magnitude estimation task and to a numerical comparison task are analysed and the findings compared to those in the international literature, some of which argue that children’s early, informal, understandings of cardinality and ordinality are underpinned by an intuitive logarithmic model relating number order and size. A main finding presented here is that the responses from learners in this study exhibited a better fit with an exponential model of the relationship between cardinality and ordinality. These findings raise questions about whether some of the findings in previous research are as universal as sometimes claimed.

Accuracy of Area Model and Number Line Representations of Fractions for Students with Learning Disabilities

AbstractAssessment results are used to investigate relations between performance on a fraction number line estimation task and a circular area model estimation task for students with LD in Grades 6–8. Results indicate that students’ abilities to represent fractions on number lines and on circular area models are distinct skills. In addition, accurate fraction magnitude estimation using number lines was more strongly related to other fractions skills (e.g., fraction magnitude comparison) than accurate fraction magnitude estimation using circular area models. Our findings call for greater integration of the number line into early fractions instruction, and highlight the importance of explicitly teaching students to make connections between different visual representations of fractions. Implications for research and practice are presented.

The Importance of Ordinal Information in Interpreting Number/Letter Line Data.

The degree to which the ability to mark the location of numbers on a number-to-position (NP) task reflects a mental number line (MNL) representation, or a representation that supports ordered lists more generally, is yet to be resolved. Some argue that findings from linear equation modeling, often used to characterize NP task judgments, support the MNL hypothesis. Others claim that NP task judgments reflect strategic processes; while others suggest the MNL proposition could be extended to include ordered list processing more generally. Insofar as the latter two claims are supported, it would suggest a more nuanced account of the MNL hypothesis is required. To investigate these claims, 84 participants completed a NP and an alphabet-to-position task in which they marked the position of numbers/letters on a horizontal line. Of interest was whether: (1) similar judgment deviations from linearity occurred for number/letter stimuli; (2) left-to-right or right-to-left lines similarly, affected number/letter judgments; and (3) response times (RTs) differed as a function of number/letter stimuli and/or reverse/standard lines. While RTs were slower marking letter stimuli compared to number stimuli, they did not differ in the standard compared to the reverse number/letter lines. Furthermore, similar patterns of non-linear RTs were found marking stimuli on the number/letter lines, suggesting that similar strategic processes were at play. These findings suggest that a general mental representation may underlie ordered list processing and that a linear mental representation is not a unique feature of number per se. This is consistent with the hypothesis that number is supported by a representation that lends itself to processing ordered sequences in general.

Electrophysiological markers of poor versus superior math abilities in healthy individuals.

Interindividual differences in the numerical ability of healthy adults have been previously demonstrated, mainly with tasks involving mental number line or size representation. However, electrophysiological correlates of superior versus poor arithmetic ability (in the healthy population) have been scarcely investigated. We correlated electric potentials with math performance in 13 skilled and 13 poor calculators selected from a sample of 41 graduate students on the basis of their poor or superior math abilities assessed through a timed test. EEG was recorded from 128 channels while participants solved 352 arithmetical operations (additions, subtractions, multiplications, divisions) and decided whether the provided solution was correct or incorrect. Overall skilled individuals correctly solved a higher number of operations than poor calculators and had faster response times. Consistently, the latency of fronto-central P300 component of event-related potentials (ERPs) peaked earlier in the skilled than poor group. The P300 was larger in amplitude to correct than incorrect solutions, but just in the skilled group, with a tendency found in poor calculators. Spearman's ρ correlation coefficient analyses showed that the larger P300 response was to correct arithmetic solutions, the better the performance; conversely, the larger the P300 amplitude was to incorrect solutions, the worse the performance. The results suggest that poor calculators had a less clear representation of arithmetic solutions and difficulty in quickly accessing it. This study provides a standard method for directly investigating math abilities throughout ERP recordings that could be useful for assessing acalculia/dyscalculia in the clinical population (children, elderly, brain-damaged patients).

Symmetry and Measuring: Ways to Teach the Foundations of Mathematics Inspired by Yupiaq Elders

Evident in human prehistory and across immense cultural variation in human activities, symmetry has been perceived and utilized as an integrative and guiding principle. In our long-term collaborative work with Indigenous Knowledge holders, particularly Yupiaq Eskimos of Alaska and Carolinian Islanders in Micronesia, we were struck by the centrality of symmetry and measuring as a comparison-of-quantities, and the practical and conceptual role of qukaq [center] and ayagneq [a place to begin]. They applied fundamental mathematical principles associated with symmetry and measuring in their everyday activities and in making artifacts. Inspired by their example, this paper explores the question: Could symmetry and measuring provide a systematic and integrative way to teach the foundations of mathematical thinking? We illustrate how the fundamental structures of symmetry, measuring, and comparison-of-quantities, starting with the embodied orthogonal axes, form a basis for properties of equality, aspects of numbers and operations (including place value), geometry and number line representations, functions, algebraic reasoning, and measurement. We conclude by embedding the earlier geometric constructions of triangles and squares within the unit circle and making explicit connections to trigonometric functions.

Spatial Presentations, but Not Response Formats Influence Spatial-Numerical Associations in Adults.

According to theories of embodied numerosity, processing of numerical magnitude is anchored in bodily experiences. In particular, spatial representations of number interact with movement in physical space, but it is still unclear whether the extent of the movement is relevant for this interaction. In this study, we compared spatial-numerical associations over response movements of differing spatial expansion. We expected spatial-numerical effects to increase with the extent of physical response movements. In addition, we hypothesized that these effects should be influenced by whether or not a spatial representation of numbers was presented. Adult participants performed two tasks: a magnitude classification (comparing numbers to the fixed standard 5), from which we calculated the Spatial Numerical Association of Response Codes (SNARC) effect; and a magnitude comparison task (comparing two numbers against each other), from which we calculated a relative numerical congruity effect (NCE), which describes that when two relatively small numbers are compared, responses to the smaller number are faster than responses to the larger number; and vice versa for large numbers. A SNARC effect was observed across all conditions and was not influenced by response movement extent but increased when a number line was presented. In contrast, an NCE was only observed when no number line was presented. This suggests that the SNARC effect and the NCE reflect two different processes. The SNARC effect seems to represent a highly automated classification of numbers as large or small, which is further emphasized by the presentation of a number line. In contrast, the NCE likely results from participants not only classifying numbers as small or large, but also processing their relative size within the relevant section of their mental number line representation. An additional external presentation of a number line might interfere with this process, resulting in overall slower responses. This study follows up on previous spatial-numerical training studies and has implications for future spatial-numerical trainings. Specifically, similar studies with children showed contrasting results, in that response format but not number line presentation influenced spatial-numerical associations. Accordingly, during development, the relative relevance of physical experiences and presentation format for spatial-numerical associations might change.

Performance of Understanding Students’ Construction In The Naming Fraction of The Three Representation

The purpose of this study is to describe students’ thinking processes in constructing the names of the fraction in circular, rectangular and number line representations by using APOS Theory (Action, Process, Object, and Scheme). For that purpose, the researcher used qualitative research approach. The subject of this research is the students of class XI in Malang. The process of collecting data begins with giving the questions to the subject to be completed. Then the students are interviewed to reinforce the answers that have given. From the performance of the subject and results of the interview, the data are analyzed using indicators of APOS Theory is action, process, object, and scheme. The results of this study indicate that for the thinking process students with circle representation S1 reached the stage of the scheme, while for S2 and S3 only reached the stage of the process. With the rectangular representation, S1 and S2 reach the stage of the scheme, but different with S3, S3 that reached the stage of the process. Moreover, the thinking process by using the number line representation S1, S2 and S3 can reach the stage of the scheme. To construct the name of the fraction with a circle and rectangular representation the subject is using the whole-part concept and area model of the fraction. As for the representation that uses the number line the subject using the concept of measurement and associates it with the length model in constructing the name of the fraction.

The relation between spatial skills and mathematical abilities: The mediating role of mental number line representation

Spatial skills are positively related to mathematical abilities, yet the mechanism of such relation remains unclear. This study examined the mediating role of mental number line representation in the relations between spatial skills and performance in various mathematical tasks. One hundred and nine second-graders were tested on mental rotation, perspective taking, mental number line representation, visuospatial working memory, arithmetic fact retrieval, calculation and word problems. Using structural equation modelling with the bootstrap procedure, we found that after controlling for age, gender and visuospatial working memory, mental number line representation fully mediated the relation between spatial skills and calculation and the relation between spatial skills and word-problem solving. However, it was not a significant mediator between spatial skills and arithmetic fact retrieval. This study highlights the important role of mental number line representation in explaining the relation between spatial skills and mathematical abilities.

Use of feedback to improve mental number line representations in primary care clinics

BackgroundAs patients become more engaged in decisions regarding their medical care, they must weigh the potential benefits and harms of different treatments. Patients who are low in numeracy may be at a disadvantage when making these decisions, as low numeracy is correlated with less precise representations of numerical magnitude. The current study looks at the feasibility of improving number representations. The aim of this study was to evaluate whether providing a small amount of feedback to adult subjects could improve performance on a number line placement task and to determine characteristics of those individuals who respond best to this feedback.MethodsSubjects from two outpatient clinic waiting rooms participated in a three phase number line task. Participants were asked to place numbers on a computerized number line ranging from 0 to 1000 in pre-test, feedback, and post-test phases. Generalized estimating equations were used to model log-transformed scores and to test whether 1) performance improved after feedback, and 2) the degree of improvement was associated with age, education level or subjective numeracy.ResultsThere was an overall improvement in task performance following the feedback. The average percent absolute error was 7.32% (SD: 6.00) for the pre-test and 5.63% (SD: 3.71) for the post-test. There was a significant interaction between college education and post-test improvement. Only subjects without some college education improved with feedback.ConclusionsAdults who do not have higher levels of education improve significantly on a number line task when given feedback.

Dancing with the SNARC: Measuring spatial-numerical associations on a digital dance mat.

According to the concept of embodied numerosity, bodily experiences influence the way in which we process numerical magnitude. The development of this influence could be anchored in the spatial ordering of numbers along a mental number line representation, which is measured by effects of spatial-numerical associations. The aim of this study was to investigate whether horizontally oriented full-body movement and visual presentation of a number line both contribute to spatial-numerical associations in children. We presented fourth-graders with 2 magnitude comparison tasks that differed in the relevance of magnitude information. In both tasks, we varied the amount of bodily movement in different response conditions (responding verbally, with a foot tap, or by jumping) and the visual presentation (items were presented with or without a number line). From the data, we calculated 2 spatial-numerical effects and expected to find the strongest effects if a full-body response was combined with a number line presentation. The 2 effects were differentially influenced by response modalities, but not presentation. The SNARC (= Spatial Numerical Association of Response Codes) effect was present in all conditions and was not influenced by our manipulations. In contrast, a new relative numerical congruity effect was influenced by the variations in responses in accordance with our hypotheses. The relative numerical congruity effect results suggest that responses involving bodily movement increase activation of spatial-numerical associations compared to verbal responses. These results are the first to demonstrate such an influence in a full-body approach in elementary schoolchildren. (PsycINFO Database Record