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Background: The present study delves into the debate surrounding the effectiveness of homework, focusing specifically on the use of online homework in teaching decimal numbers to 5th and 6th grade primary school students. It highlights the challenges students face in representing and identifying decimal numbers, a concept that is often difficult for 5th and 6th graders to grasp. Objectives: The aim to evaluate the effectiveness of online homework in enhancing understanding of decimal numbers and compare the impact of instructional methodologies based on correct and incorrect examples. Methods: The research employs an experimental design, comparing the effects of online homework with correct and incorrect examples, assessing how these methodologies influence students' ability to understand and use decimal numbers, especially on a number line. Results: The results support the overall effectiveness of online homework in improving students' performance in decimal numbers, regardless of their initial level of understanding. Specifically, the study reveals that the type of examples used in homework plays a key role in the effectiveness of learning. Incorrect examples are more beneficial for students with higher prior knowledge, while correct examples are more appropriate for students with lower prior levels. Conclusions: These outcomes highlight the potential value of online homework in primary education and underline the importance of tailoring educational activities to the individual needs of pupils. The study contributes to the broader conversation about the role of homework in education, suggesting practical ways to optimise online learning experiences for young learners. Lay Description: What is currently known about this topic?: The effectiveness of online homework, especially at primary level, is the subject of debate due to the various impacts on pupils' understanding and performance. What does this paper add?: This study introduces misconception‐based tasks into online homework, showing differential impacts on decimal number learning by prior knowledge. Implications for practice/or policy: Suggests customising online homework based on student knowledge and common misconceptions, impacting educational methods and digital learning platform development. [ABSTRACT FROM AUTHOR]

Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta‐analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number (r = 0.33) and fractions number (r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta‐analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced (β = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students' fractions learning. Research Highlights: Two meta‐analyses examined the link between the number line and mathematics performance.The first revealed significant relations across domains (e.g., algebra and computation).The second controlled for general cognitive ability and resulted in reduced but still significant relations.The relation between number line and fractions performance was stronger than relations to other domains. [ABSTRACT FROM AUTHOR]

Number line estimation has been found to be strongly related to mathematical reasoning concurrently and longitudinally. However, the relationship between number line estimation and mathematical reasoning might differ according to children's level of performance. This study investigates whether findings from previous studies that show number line estimation significantly predicting mathematical reasoning replicate, and whether this relationship holds across several points of the mathematical reasoning distribution. Participants include 324 Singaporean children (162 girls, Mage = 74.1 months, SDage = 4.0) in their second year of kindergarten who were assessed on the number line estimation task (0–10 and 0–100) and mathematical reasoning skills. The results replicate previous findings showing that higher accuracy on the number line estimation task is predictive of higher mathematical reasoning, for both the 0–10 and 0–100 number line. Quantile regressions show that performance on the number line task similarly predict mathematical reasoning across the performance distribution. Possible differences between the 0–10 and 0–100 number line's predictive capacity are discussed. [ABSTRACT FROM AUTHOR]

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left‐to‐right‐oriented estimate patterns indicate task constraints or children's intuitive number–space mapping. Three‐ to six‐year‐old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non‐symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left‐to‐right‐oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space. [ABSTRACT FROM AUTHOR]

Number lines are acknowledged as an effective model for understanding fractions, yet students often face challenges in interpreting fractions on number lines. This study contributes to the field by investigating students’ performance on fraction number line tasks that require the coordination of their fraction and number line knowledge. To explore this, test items were developed, and 122 fourth-grade students participated in the assessment. Students’ written responses underwent analysis in three phases: a descriptive overview of overall student performance, latent profile analysis to identify subgroups with different competencies, and a qualitative analysis of each latent profile. The findings indicated lower performance among students across the tasks, revealing three distinct latent profiles with different competency characteristics: intuitive, emergent, and advanced understanding of fractions on the number line. From these findings, instructional implications were extrapolated for using number lines as a model for fractions. [ABSTRACT FROM AUTHOR]

Early numerical skills are important not only for later mathematical achievement but for overall achievement and are associated with later income, health, and quality of life. Socioeconomic disparities in numerical skills are visible before children begin school, and widen throughout schooling. It is, therefore, important to support the development of early numerical skills in children from lower socioeconomic backgrounds. Previous studies have highlighted the effectiveness of linear number board games for improving early numerical skills, and the beneficial effect of counting backward as well as forward. We designed a number board game that required children to place number cards in order on a line, either forward-only or forward and backward in small groups in the classroom. The game's effectiveness was evaluated in 4- to 5-year-old children from schools located in socioeconomically disadvantaged areas. Children were randomly allocated to one of three conditions: playing the number game forward-only (n = 82), playing the number game forward and backward (i.e., bidirectional condition; n = 82), or an alphabet game (active control, n = 85). After eight gameplay sessions across 5 weeks, children's numerical and letter-sound knowledge skills improved, but there was no significant effect of the intervention condition. Neither forward nor bidirectional number line games (nor the alphabet game) added benefits beyond the learning already happening in the classroom. Short linear board game interventions may be effective compared to control groups who received minimal or absent numerical education, but they fail to provide an additional advantage when children are already learning mathematics in school. Educational Impact and Implications Statement: The study investigated the effectiveness of a brief number board game intervention aimed at enhancing early numerical skills in young children. Low attrition, high engagement, and enjoyment were observed among the children during the game sessions, demonstrating the game's potential in terms of usability. However, the game did not provide additional benefits beyond regular classroom learning. Short game-based interventions may not always improve numerical skills beyond the learning already happening in the classroom. [ABSTRACT FROM AUTHOR]

Sixty (35 girls and 25 boys) 9th-grade students’ conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students’ number line descriptions and the explanations coalesced around six features: sequential ordering (i.e., numbers are sequentially represented), positivity-negativity of numbers (i.e., the number line contains positive and negative numbers), non-centrality (i.e., zero does not have to be in the center), infinity, increment flexibility (i.e., number line increments can vary), and continuity (i.e., numbers can be placed anywhere between minus infinity and plus infinity without breaks). The students’ explanations show that these six features emerge in five successive stages in the conceptual understanding of the number line. These stages are (1) no knowledge, (2) sequential ordering and positivity-negativity, (3) infinity and non-centrality, (4) incremental flexibility, and (5) continuity. The last two stages were not found in most descriptions. The results suggest that students’ understanding of the number line is incomplete and may be overestimated by commonly used quantitative assessments of number line knowledge.

Research suggests that children with developmental dyscalculia (DD) have deficits in basic numerical skills. However, there is conflicting evidence on whether basic numerical skills in children with DD are qualitatively different from those in typically developing children (TD) or whether basic numerical skills development in children with DD is simply delayed. In addition, there are also competing hypotheses about deficits in basic numerical skills, assuming (1) a general deficit in representing numerosities (Approximate Number System, ANS), (2) specific deficits in an object-based attentional system (Object Tracking System, OTS), or (3) deficits in accessing numerosities from symbols (Access Deficit, AD). Hence, the purpose of this study was to investigate whether deficits in basic numerical skills in children with DD are more indicative of a developmental delay or a dyscalculia-specific qualitative deviation and whether these deficits result from (selective) impairment of core cognitive systems involved in numerical processing. To address this, we tested 480 children (68 DD and 412 TD) in the 2nd, 3rd, and 4th grades with different paradigms for basic numerical skills (subitizing, counting, magnitude comparison tasks, number sets, and number line estimation tasks). The results revealed that DD children's impairments did not indicate qualitatively different basic numerical skills but instead pointed to a specific developmental delay, with the exception of dot enumeration. This result was corroborated when comparing mathematical profiles of DD children in 4th grade and TD children in 2nd grade, suggesting that DD children were developmentally delayed and not qualitatively different. In addition, specific deficits in core markers of numeracy in children with DD supported the ANS deficit rather than the AD and OTS deficit hypothesis. [ABSTRACT FROM AUTHOR]

Research has shown the importance of helping students, especially those with mild-to-moderate learning disabilities, to offload information during problem-solving. When students can get their thoughts onto paper, number line strategies can help them develop a firm foundation in mathematical problem-solving while understanding the relationships between mathematical operations. These strategies are helpful for the development of addition, subtraction, multiplication, division, and later, fractional mathematics. In this article, we describe the progression of number lines as a supportive strategy for elementary students and those with developmental delays in mathematics to improve mathematical understanding. This strategy is based on students being able to show their work and think about what they have written on paper or how they have used manipulatives. [ABSTRACT FROM AUTHOR]

There is ongoing debate regarding what performance on the number line estimation task represents and its role in mathematics learning. The patterns followed by children’s estimates on the number line task could provide insight into this. This study investigates children’s estimation patterns on the number line task and assesses whether mathematics achievement is associated with these estimation patterns. Singaporean children (n = 324, Age M = 6.2 years, Age SD = 0.3 years) in their second year of kindergarten were assessed on the number line task (0-100) and their mathematical performance (Numerical Operations and Mathematical Reasoning subtests from WIAT II). The results show that most children’s number line estimation patterns can be explained by at least one mathematical model (i.e., linear, logarithmic, unbounded power model, one-cycle power model, two-cycle power model). But the findings also highlight the high percentage of participants for which more than one model shows similar support. Children’s mathematical achievement differed based on the models that best explained children’s estimation patterns. Children whose estimation patterns corresponded to a more advanced model tended to show higher mathematical achievement. Limitations of drawing conclusions regarding what performance on the number line task represents based on models that best explain the estimation patterns are discussed.

Health risks, when presented as ratios (e.g., two out of seven people), are challenging to understand, but visual displays can foster accurate understanding. We conducted three experiments to test how characteristics of numbers (Experiment 1), icon arrays (Experiments 1, 2, and 3), and number lines (Experiments 1 and 3) influenced people's ability to accurately estimate the risk of experiencing side effects. Participants in each experiment saw smaller‐ (e.g., 2 out of 7) and larger‐component (e.g., 264 out of 924) equivalent ratios in one of three conditions: with number lines only, with icon arrays only, or in the form of Arabic numerals with no accompanying visual. We found that risk estimates were more accurate when presented in 10 × 10 icon arrays, long horizontal 1 × 99 arrays, or number lines. We theorize that hypothetical risks can be estimated more accurately when the display affords easy translation to a percentage. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Bayburt Education Faculty / Bayburt Eğitim Fakültesi Dergisi is the property of Bayburt University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article discusses the problems of definitions of concepts of school mathematics in textbooks. Excerpts from textbooks show how interpretations of a fraction having no necessary properties of a mathematical definition have changed over the decades. The alternative of a concept of a fraction based on the number line is discussed.

Zero is a magic number. It represents emptiness, nothing -- and yet it is considered one of the greatest cultural achievements of mankind. For a brain that has evolved to process sensory stimuli ('something'), conceiving of empty sets ('nothing') as a meaningful category demands high-level abstraction. Recent studies of cognitive neuroscience now provide an insight into how an abstract concept like zero can emerge. In both behavioral and neural processing, the emergence of zero passes through four stages. In the first stage, the absence of a stimulus, 'nothing', corresponds to a (mental/neural) resting state lacking a specific signature. In the second stage, the absence of stimulus is grasped as a meaningful behavioral category, but is still devoid of quantitative relevance. In the third stage, 'nothing' acquires a quantitative meaning and is represented as an empty set at the low end of a numerical continuum or number line. Finally, the empty-set representation is extended to become the number zero, thus becoming part of a combinatorial number of the symbols system used for calculation and mathematics. The concept of zero shows how our minds and brains originally evolved to represent stimuli and detach from empirical properties to achieve the ultimate in abstract thinking. Because of this, the story of zero tells us a great deal about the mind, leaving empirical grounds behind and raising it to new intellectual heights. [ABSTRACT FROM AUTHOR]

Recent work reveals a left digit effect in number line estimation such that adults' and children's estimates for three-digit numbers with different hundreds-place digits but nearly identical magnitudes are systematically different (e.g., 398 is placed too far to the left of 401 on a 0-1000 line, despite their almost indistinguishable magnitudes; Lai et al., 2018, https://doi.org/10.1111/desc.12657). In two preregistered studies (N = 218), we investigate the scope and malleability of the left digit effect. Experiment 1 used a typical forwardoriented 0-1000 number line estimation task and an atypical reverse-oriented 1000-0 number line estimation task. Experiment 2 used the same forward-oriented typical 0-1000 number line estimation task from Experiment 1, but with trial-by-trial corrective feedback. We observed a large left digit effect, regardless of the orientation of the line in Experiment 1 or the presence of corrective feedback in Experiment 2. Further, analyses using combined data showed that the pattern was present across most stimuli and participants. These findings demonstrate a left digit effect that is robust and widely observed, and that cannot be easily corrected with simple feedback. We discuss the implications of the findings for understanding sources of the effect and efforts to reduce it. [ABSTRACT FROM AUTHOR]

This study recruited 428 Singaporean children at risk of math learning difficulties (MLD; Mage = 83.9 months, SDage = 4.35 months; 41% female). Using a factor mixture model that considered both quantitative and qualitative differences in math ability, two qualitatively different groups were identified: one with generalized difficulties across different math skills and the other with more focal difficulties in arithmetic fluency. Reading, working memory capacity, and numeracy (number line estimation skills and numerical discrimination) uniquely explained group membership. Children within each group differed in the extent of difficulties they exhibited, with numeracy variables differentially contributing to math ability in each group. Findings speak against a dimensional view of MLD and underscore the conceptual limitations of using basic numeracy performance to profile learning difficulties. [ABSTRACT FROM AUTHOR]

Copyright of Lietuvos Matematikos Rinkinys is the property of Vilnius University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Background: Number line accuracy (NL accuracy) shows improvement over the course of a school education. However, there are practically no cross‐country longitudinal studies of NL accuracy over the whole course of elementary school. Aims: This study investigated the developmental trajectories of NL accuracy and its types across the elementary school years in two countries—Russia and Kyrgyzstan. Sample(s): The analyses were carried out on the data collected from the sample of 508 schoolchildren at Grades 1, 2, 3 and 4 (aged 6.4–11.9 years) from Russia and Kyrgyzstan, who were surveyed as part of the 'Cross‐cultural Longitudinal Analysis of Student Success' project. Methods: The participants were administered the 'Number Line' computerized test task and a paper‐and‐pencil 'Standard Progressive Matrices' test at the end of each academic year. Results: During the course of the elementary school education, NL accuracy increases nonlinearly in both samples from Grade 1 to Grade 4, with a pronounced increase in the rate of improvement from the first to the second year. Cross‐country differences in NL accuracy were observed during each year of schooling as well as in the growth of NL accuracy. The development of NL accuracy is described by a model with two developmental types: (1) 'high start and growth' (93% of the pooled sample) and (2) 'low start and no growth' (7%). Conclusions: Both NL accuracy and the rate of its growth during elementary school depend on educational conditions. Cross‐country differences in the distribution of schoolchildren by these two developmental types were statistically insignificant. [ABSTRACT FROM AUTHOR]

An accumulating body of literature points to a link between spatial reasoning and mathematics learning. The present study contributes to this line of research by investigating sex differences both in spatial representations of magnitude and in the use of arithmetic strategies, as well as the relation between the two. To test the hypothesis that sex differences in spatial–numerical magnitude knowledge mediate sex differences in the use of advanced strategies (retrieval and decomposition), two studies were conducted. Study 1 included 96 US first graders (53% girls); Study 2 included 210 Russian first graders (49% girls). All participants completed a number line estimation task (a spatially based measure of numerical magnitude knowledge) and an arithmetic strategy task (a measure of strategy choice). The studies showed parallel results: boys produced more accurate numerical magnitude estimates on the number line estimation task and used advanced strategies more frequently on the arithmetic task. Critically, both studies provide support for the mediation hypothesis (although there were some differences in the pattern obtained for the two strategies). The results are discussed in the context of broader research about the relation between spatial and mathematical skills. [ABSTRACT FROM AUTHOR]

There is ongoing debate regarding what performance on the number line estimation task represents and its role in mathematics learning. The patterns followed by children’s estimates on the number line task could provide insight into this. This study investigates children’s estimation patterns on the number line task and assesses whether mathematics achievement is associated with these estimation patterns. Singaporean children (n = 324, Age M = 6.2 years, Age SD = 0.3 years) in their second year of kindergarten were assessed on the number line task (0-100) and their mathematical performance (Numerical Operations and Mathematical Reasoning subtests from WIAT II). The results show that most children’s number line estimation patterns can be explained by at least one mathematical model (i.e., linear, logarithmic, unbounded power model, one-cycle power model, two-cycle power model). But the findings also highlight the high percentage of participants for which more than one model shows similar support. Children’s mathematical achievement differed based on the models that best explained children’s estimation patterns. Children whose estimation patterns corresponded to a more advanced model tended to show higher mathematical achievement. Limitations of drawing conclusions regarding what performance on the number line task represents based on models that best explain the estimation patterns are discussed. [ABSTRACT FROM AUTHOR]

Consumer payments elicited on slider scales can be systematically different from those elicited through text boxes because of the end point assimilation effect. When people use text boxes to make payments, they evaluate monetary values relative to the starting point of the response range. In contrast, when people use slider scales, they evaluate monetary values relative to the starting point as well as the end point of the response range. Consequently, payments elicited on slider scales tend to be assimilated toward the end point of the response range. This slider scale end point assimilation effect varies for ascending and descending payment formats. For ascending payment formats (e.g. eBay bids), slider scales elicit higher payments than text boxes. But for descending payment formats (e.g. Priceline bids), slider scales elicit lower payments than text boxes. This research not only documents how slider scales alter consumer payments, but also explains how the mental number line affects financial decisions. [ABSTRACT FROM AUTHOR]

The present study aimed to examine the opinions of in-and pre-service middle school mathematics teachers about teaching four operations with integers through "number line" and/or "counters". Participants were 42 middle school mathematics teachers and 51 pre-service mathematics teachers. The data were collected through open-ended interview questions. In data analysis, a content analysis technique was used. The results suggested that the number line was easier and more understandable and that the 'zero pair' contributed to conceptual understanding. It was also found that the use of these models in different operations with integers was due to different reasons. The recommendation of using traditional methods (eg memorization of rules) took its place in the study as another interesting result. [ABSTRACT FROM AUTHOR]

A strong foundation in early number concepts is crucial for students' future success in statistics. Despite its importance in statistics, many first-year students struggle to comprehend the normal distribution due to a lack of basic number sense. Students get confused about the order and magnitude of negative z-scores on a standard normal curve or when problems about normally distributed random variables are presented in word questions which involve phrases that indicate inequalities. As a result, students shade wrong areas on the bell-shaped curve when they have to calculate probabilities for normally distributed variables. Visual representations such as the number line can support students' development of quantitative literacy or number sense by helping them create a mental representation of the order and magnitude of numbers as well as inequalities. Based on a comprehensive investigation of evidence demonstrating this weakness, this experienced-based perspective proposes a framework that demonstrates how the number line can be used as a powerful teaching tool to promote students' conceptual understanding about the normal distribution. The framework illustrates with authentic examples how the number line relates to the horizontal axis of the normal and standard normal curve and how it can be used to address erroneous quantitative reasoning when students are required to calculate probabilities for normally distributed variables. To determine if the number line teaching intervention affects student performance, the researcher adopted a non-equivalent pretest-posttest design with two intact classes. The results of the analysis show that students who received the intervention performed significantly better in the posttest. [ABSTRACT FROM AUTHOR]

• We examine new aspects of children's ordinal understanding of a number line. • We measure if children place numbers correctly relative to each other and benchmarks. • Our new measures may reflect unique aspects of children's ordinal understanding. • Children with fewer rather than more ordinal errors scored higher on math ability. • Children with fewer vs. more ordinal errors made larger estimation gains (lower PAE). Children's performance on the number line estimation task, often measured by the percentage of absolute error, predicts their later mathematics achievement. This task may also reveal (a) children's ordinal understanding of the target numbers in relation to each other and the benchmarks (e.g., endpoints, midpoint) and (b) the ordinal skills that are a necessary precursor to children's ability to understand the interval nature of a number line as measured by percentage of absolute error. Using data from 104 U.S. kindergartners, we measured whether children's estimates were correctly sequenced across trials and correctly positioned relative to given benchmarks within trials at two time points. For both time points, we found that each ordinal error measure revealed a distinct pattern of data distribution, providing opportunities to tap into different aspects of children's ordinal understanding. Furthermore, children who made fewer ordinal errors scored higher on the Test of Early Mathematics Ability and showed greater improvement on their interval understanding of numbers as reflected by a larger reduction of percentage of absolute error from Time 1 to Time 2. The findings suggest that our number line measures reveal individual differences in children's ordinal understanding of numbers, and that such understanding may be a precursor to their interval understanding and later mathematics performance. [ABSTRACT FROM AUTHOR]

An accumulating body of literature points to a link between spatial reasoning and mathematics learning. The present study contributes to this line of research by investigating sex differences both in spatial representations of magnitude and in the use of arithmetic strategies, as well as the relation between the two. To test the hypothesis that sex differences in spatial–numerical magnitude knowledge mediate sex differences in the use of advanced strategies (retrieval and decomposition), two studies were conducted. Study 1 included 96 US first graders (53% girls); Study 2 included 210 Russian first graders (49% girls). All participants completed a number line estimation task (a spatially based measure of numerical magnitude knowledge) and an arithmetic strategy task (a measure of strategy choice). The studies showed parallel results: boys produced more accurate numerical magnitude estimates on the number line estimation task and used advanced strategies more frequently on the arithmetic task. Critically, both studies provide support for the mediation hypothesis (although there were some differences in the pattern obtained for the two strategies). The results are discussed in the context of broader research about the relation between spatial and mathematical skills.

• Number line accuracy and estimation patterns were studied in children at risk of MLD. • Number line accuracy predicted growth in numerical operations and math reasoning. • Number line accuracy predicted intercept but not growth (slope) in math fluency. • Children best fit by the linear outperformed those best fit by the logarithmic model. • Math skills did not differ based on models from the proportion judgment perspective. Children with mathematics learning difficulties (MLD) show poorer performance on the number line task, but how performance on this task relates to other mathematical skills is unclear. This study examined the association between performance on the number line task and mathematical skills during the first 2 years of school for children at risk of MLD. Children (N = 100; M age = 83.63 months) were assessed on four occasions on the number line task and other mathematical skills (math fluency, numerical operations, and mathematical reasoning). Estimation patterns were analyzed based on the representational shift and proportional judgment accounts separately. More consistent longitudinal trends and stronger evidence for differences in mathematical skills based on estimation patterns were found within the representational shift account. Latent growth curve models showed accuracy on the number line task as a predictor of growth in some mathematical skills assessed. We discuss impacts of methodological limitations on the study of estimation patterns. [ABSTRACT FROM AUTHOR]

The notion of negative numbers is one of the most fundamental in mathematics. For many years, they have caused confusion and they have been an object of controversy among great researchers, until the development of symbolic algebra led to their acceptance in the form they have today. When students are first introduced to the concept of negative numbers and the operations between them, various difficulties occur and misconceptions arise, as expected. Their comprehension is one of the most challenging tasks of teaching mathematics due to their complexity and abstract nature. For this purpose, various models have been devised and employed. In the present study we expose some of the epistemological obstacles commonly observed in understanding negative numbers and then exhibit and compare two different models frequently used for introducing the four basic operations between them.

Recent years have witnessed an increase in research on how numeral ordering skills relate to children's and adults' mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers. [ABSTRACT FROM AUTHOR]

Mapping number to space is natural and spontaneous but often nonveridical, showing a clear compressive nonlinearity that is thought to reflect intrinsic logarithmic encoding of numerical values. We asked 78 adult participants to map dot arrays onto a number line across nine trials. Combining participant data, we confirmed that on the first trial, mapping was heavily compressed along the number line, but it became more linear across trials. Responses were well described by logarithmic compression but also by a parameter-free Bayesian model of central tendency, which quantitatively predicted the relationship between nonlinearity and number acuity. To experimentally test the Bayesian hypothesis, we asked 90 new participants to complete a color-line task in which they mapped noise-perturbed color patches to a "color line." When there was more noise at the high end of the color line, the mapping was logarithmic, but it became exponential with noise at the low end. We conclude that the nonlinearity of both number and color mapping reflects contextual Bayesian inference processes rather than intrinsic logarithmic encoding. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Inquiry Based Activities (JIBA) / Araştırma Temelli Etkinlik Dergisi (ATED) is the property of Journal of Inquiry Based Activities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Number line estimation (NLE) performance is usually believed to depend on the magnitudes of presented numerals, rather than on the particular digits instantiating those magnitudes. Recent research, however, shows that NLE placements differ considerably for target numerals with nearly identical magnitudes, but instantiated with different leftmost digits. Here we investigate whether this left digit effect may be due, in part, to the ordering of digits in number words. In English, the leftmost digit of an Arabic numeral is spoken first (“forty-one”), but Dutch number words are characterized by the inversion property: the rightmost digit of a two-digit number word is spoken first (“eenenveertig” – one and forty in Dutch). Participants (N = 40 Dutch-English bilinguals and N = 20 English-speaking monolinguals) completed a standard 0-100 NLE task. Target numerals were read aloud by an experimenter in either English or Dutch. Preregistered analyses revealed a strong left digit effect in monolingual English speakers’ estimates: e.g., 41 was placed more than two units to the right of 39. No left digit effect was observed among Dutch-English bilingual participants tested in either language. These findings are consistent with the idea that the order in which digits are spoken might influence multi-digit number processing, and suggests linguistic influences on numerical estimation performance.

This research aims to describe the utilization of number line media through peer teaching methods in student learning with hearing impairments. This study used an action research design with research participants of class III SLB B Karnnamanohara Yogyakarta. The data collection techniques used in this study were participative observation, field notes, interviews, and documentation. Data analysis used data triangulation techniques. The results showed that (1) Teachers and students could be assisted in the learning process using number line media through the peer teaching method. (2) The peer teaching method combined with number line media helps to establish interactions between teachers and students in the learning process. (3) Student learning outcomes in Mathematics can achieve KKM scores after using number line media through the peer teaching method.

In modern mathematics curricula in primary and secondary education, number line is an important supervisory tool for understanding many concepts, such as different types of numbers, equations, and more. The use of the number line is supported by a large number of researchers, but there are also studies showing that students find it difficult to use. Although the concept of number line is important for teaching and there is a great deal of debate about its use, as far as we know, there are very few systematic studies that examine the epistemological development of some components regarding the concept of number line throughout history and correlate this development by learning this concept from the students. However, there are no studies that examine the concept of historical development of the number line as a whole or relate it to student behavior. In this paper, therefore, the first attempt has been made to examine the overall development of the concept of number line in the history of mathematics. We have therefore studied the historical evolution of the concept of number line and divided it into periods, according to the characteristics of this evolution. It seems that based on the slow mathematical integration of the concept of number line at the end of the 19th century, but also on some other critical points in the four historical periods that we have analyzed, some of the difficulties that students encounter when using it are likely to be epistemological obstacles.