Improper Fractions Research Articles

A cross-sequential study of theory of mind, IQ, and fair sharing framed socially and non-socially in young children

Both ability and motivation underlie uniquely human prosociality but the study of resource allocation in young children has focused on prosocial motivations rather than mathematical ability to fairly allocate rewards. We examined the development of fair sharing by testing 3–6YO children at two time points (N at Time 1 = 158, N at Time 2 =111) with ToM, IQ, fractional quantities (proper or improper fraction problems), and framing of the problem (social or non-social) as predictors of children’s performance. As expected, children performed better with time and age, and on proper versus improper fractions. However, in contrast to our predictions, performance was not consistently related to general IQ, ToM, or framing of the problem although ToM interacted with age and framing to predict performance. Our results suggest that it is important to consider limitations in numerical ability rather than assuming selfish motivations when young children fail to share fairly.

Why fractions are difficult? Modeling optimal and sub-optimal integration strategies of numerators and denominators by educated adults

Many children and educated adults experience difficulties in understanding and manipulating fractions. In this study, we argue that a major cause of this challenge is rooted in the need to integrate information from two separate informational sources (i.e., denominator and numerator) according to a normative arithmetic rule (i.e., division). We contend that in some tasks, the correct arithmetic rule is replaced by an inadequate (sub-optimal) operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by applying two rigorous models of information integration : (a) functional measurement (Experiments 1-3) and (b) conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare participants’ integration strategies with that of an ideal-observer model. Functional measurement analyses on data from the number-to-line task, revealed that participants could represent the global magnitude of proper and improper fractions quite accurately and combine the fractions’ components according to an ideal-observer model. However, conjoint measurement analyses on data from the comparative judgment task, showed that most participants combined these fractions’ components according to a sub-optimal (saturated) observer model, that is inconsistent with an ideal-observer (additive) model. These results support the view that educated adults are capable of extracting multiple types of representations of fractions depending on the task at-hand. These representations can be either accurate and conform with normative arithmetic or approximated and inconsistent with normative arithmetic. The latter may lead to the observed difficulties people experience with fractions.

ANALISIS POLA KESULITAN BELAJAR MATEMATIKA DALAM MENYELESAIKAN SOAL OPERASI HITUNG PECAHAN PADA SISWA SEKOLAH DASAR

The main problem to be studied in this research is analyzing the pattern of mathematics learning difficulties when solving fraction calculation problems for elementary school students. This research aims to gain an in-depth understanding of the pattern of difficulties students face when understanding and answering fraction calculation questions. This study aims to analyze the pattern of mathematics learning difficulties of elementary school students when completing fraction calculation operations. The sample includes 5 subjects consisting of 5th grade students at SD Negeri 4 Mayonglor Jepara. Data collection was carried out through observation, as well as interviews with students and teachers. The data will be analyzed and used to explain the problem being studied. The results of data analysis show that students experience various difficulties in understanding the basic concepts of fractions and the ability to calculate fractions. One of the biggest difficulties is understanding the relationship between the numerator and denominator of a fraction. Students also often have difficulty converting fractions into their simplest form, changing fractions from mixed fractions to improper fractions and vice versa. Apart from that, understanding fractions with different denominators is also a challenge for students. The conclusion of this research is that elementary school students face many different difficulties when solving fraction calculation problems. These difficulties include understanding the basic concepts of fractions, changing the form of fractions, calculating the results of multiplying or dividing fractions, simplifying fractions, and applying fractions in the context of everyday situations

Strategies and Procedural and Conceptual Knowledge of Addition of Unlike Denominator Fractions: The Case of Grade 8 Children in Two Districts of the Central Region of Ghana

This study investigated children’s knowledge of the addition of fractions with unlike denominators. It focussed on the strategies used by children to add fractions with unlike denominators. A cross-sectional survey of Grade 8 children in schools in two districts in Ghana was used. A test on the addition of fractions with unlike denominators was administered to 543 high-, average- and low-achieving children, and their responses were categorised based on the characterisation of correct strategies by Braithwaite et al. and Siegler and Pykes. Cluster bar charts and analysis of variance were used to analyse the data. The results showed that the children’s use of correct strategies for adding proper fractions with unlike denominators was higher than their use of correct strategies for adding improper fractions with unlike denominators. Thus, the children’s procedural knowledge of the addition of proper fractions with unlike denominators was higher than that of improper fractions with unlike denominators. Conceptual knowledge of the addition of fractions with unlike denominators was much lower than procedural knowledge of the same. The less advantaged (females, rural and low achievers) used fewer correct strategies and demonstrated less procedural and conceptual knowledge. Among other recommendations, curricula and classroom teaching should emphasise conceptual knowledge and its support for procedural knowledge of the addition of fractions with unlike denominators. Research interventions focussed on the less advantaged are advised.

Applying Butterfly Method in the Learning of Addition and Subtraction of Fractions

Several researches attempt to improve the performance in addition and subtraction of fraction by focusing on students’ understanding, such as applying fraction tiles manipulatives and number line in the teaching, or using visual aids such as area model and drawings or virtual diagrams. Similarly, few studies focus on adopting simple procedures that will facilitate students in attempting the topic. Butterfly Method is an approach that not only simplifies the pedagogical approach of learning addition and subtraction of fraction, but also indirectly promotes students’ conceptual understanding in the topic. Yet this approach is not commonly practiced in Brunei. This action research employed quantitative approach involving fourty-one Year 9 IGCSE level students from a government secondary school in Brunei Darussalam. The study aimed to investigate the effect on applying Butterfly Method in the learning of addition and subtraction of fraction through pre-test and post-test. The study revealed a significant improvement in students’ performance through Wilcoxon Signed rank test (Z = 4.2332, p-value < 0.05), with large effect size (r = 0.6611). This is also supported with the improvement of the overall mark distribution of students from pre-test to post-test, with Hake’s normalised gain values indicated majority of the participants achieved high (56.1%), medium (7.32%) and low (2.44%) improvements. Item analysis also indicated overall improvement of correct attempts in questions related to addition and subtraction of fraction, most apparently on fraction problems involving denominators not multiple of each other, improper fractions and mixed numbers. Thus, applying Butterfly Method significantly improved the overall students’ performance in the learning of addition and subtraction of fractions.

Learning Improper Fractions with the Number Line and the Area Model

ABSTRACT Number lines and area models are both used pervasively in teaching fractions. Prior studies found that second and third graders demonstrated better magnitude knowledge of proper fractions after a 15-minute training using the number line as compared to using the area model. The current study aimed to extend these findings to improper fractions. We randomly assigned fourth and fifth graders to a number line training, an area model training, or a non-numerical control condition. The number line and area model trainings involved both proper and improper fractions and were closely modeled on the training procedures in prior studies. Fraction training with the area model produced improvements in children’s area model estimation of proper and improper fractions. However, contrary to our expectations, training with the number line did not improve number line estimation, and neither training led to improvements in transfer tasks assessing fraction magnitude knowledge. These findings suggest that children can develop the skill to represent improper fractions on area models with brief training. Nevertheless, it is unclear whether this skill enhances a comprehensive understanding of fraction magnitudes.

덧셈을 이용한 가분수와 대분수의 도입 방안 탐색

Whereas proper fractions are natural for young learners from the perspectives of histo-genetics or learning psychology, improper fractions or mixed numbers imply some learning difficulty due to their property of second concepts as they are extended from proper fractions. This study aims to propose an alternative method for teaching improper fractions and mixed numbers meaningfully. Henceforth, critical review was implemented about the teaching period, their definition, and relation to the addition of proper fractions through longitudinal and latitudinal analyses of elementary mathematics textbooks in Korea and some other countries. The results showed that the context of introducing fractions equal to or larger than 1 is especially unsatisfactory. Based on the results, an alternative method for introducing or defining improper fractions and mixed numbers using the addition of proper fractions or the addition of a natural number and a proper fraction has been proposed. To secure the validity of the method, several didactical discussions about the teaching period, method of definition, integration of fractions and their additions, and effective instructional representations have been included in the study.

Understanding primary school children’s learning on addition of fractions

Fractions are universally known to be difficult to learn. Learning basic fractions may not be sustainable if one is to apply it to more advanced mathematical content knowledge as it is interlinked with other mathematics topics. In the effort to understand how children grasp the learning of fractions additions, a study was carried out with a random sampling of Years 4, 5 and 6 children in Brunei Darussalam. The research participants consisted of 33 children who are currently studying at government and private primary schools in Brunei. A pen-and-paper self-designed test was used as an instrument in collecting the data. The quantitative data was analysed by using a descriptive analysis method. The overall reliability of the self-designed tests indicated that the items have relatively high internal consistency with a Cronbach alpha of .859. The findings revealed that the children’s understanding is stronger on questions with answers regarding proper fractions rather than improper fractions. This could be further explored through future studies employing a qualitative approach.

Design Research to Support Fourth Grader Learn Addition of Mixed Numbers in RME Learning

Mixed numbers andfraction in general have been a problem for students not only in understanding the concept of itself, but also in understanding the operation of it, such as addition.Thus, supporting students in learning addition of mixed numbers is needed. the aim of this study is to develop a local instructional theory and materials related to supporting students in learning addition mixed numbers. In order to develop the theory, this studyused design research method. The is collected by generatingthe video recording of the learning process, student’s written work, students’ result from pretest and posttest, the result of interview, and also the fields note from the observation. The participants of this study are fourth grade students in one of private elementary school in Palembang. From the study,it can be seen that students have built the reasoning in addition mixed numbers. Some of them are able to use the distribution principle and use equivalence fractions when they adding the mixed numbers, and some of them are directly used the algorithm of changing mixed number to improper fractions. However, it is insufficient to say that the fractions number line has a big role in the students’ finding.

중학생들의 분수 지식과 대수식 표현 사이의 관계 분석

The aim of this study is to explore various ways in which students’ algebraic representations relate to their fraction knowledge. From clinical interviews with seven middle school students, three students each of whom represented distinct ways of reasoning were chosen as the focus of the present paper. The in-depth analysis of the data suggests that students’ quantitative reasoning related to their iterative fraction schemes was involved in their ways of representing situations into algebraic equations. More specifically, students who did not construct iterative fraction schemes, only used whole numbers as multipliers for unknowns. Students who weakly constructed iterative fraction schemes, conceived whole numbers, unit fractions, and proper fractions as multipliers for unknowns. Lastly, students who constructed iterative fraction schemes, conceived whole numbers, unit fractions, proper fractions, and improper fractions as multipliers for unknowns, and could represent given situations in various forms of algebraic representation. We finalize our paper by discussing how the study complements past research on students’ knowledge of fractions and its relationship to algebraic knowledge and by providing implications for further research.

Preservice Teachers’ Exploration of Model Breaking Points

This study is an investigation of elementary preservice teachers’ (PSTs’) exploration of model breaking points in fractions. Eighty-three PSTs at two universities in the Midwestern and Southwestern regions of the USA were asked to explore the area model of fraction addition through a series of tasks and explain their reasoning in determining its affordances and constraints. The PSTs presented their critical mathematical ideas on modeling fraction addition and area models’ breaking points when used to add two fractions, as well as suggestions for alternative models to overcome model breaking points. An inductive content analysis showed that most PSTs represented fraction addition well with simple fractions but had difficulty representing fraction addition with improper fractions or fractions with unlike and relatively large denominators and tended to use algorithm-based thinking. Also, the area models drawn by several PSTs revealed various misconceptions, suggesting the impact of their content knowledge on their understanding of area models. Based on the findings from this study, we provide some implications for teacher educators.

An Investigation of SRSD to Teach Fractions to Students With Disabilities

In the current study we examined the effect of a self-regulated strategy development intervention on the fraction calculations of students with or at risk for disabilities using a multiple-baseline-across-sites, single-case design. Specifically, the intervention package addressed the following skills: adding and subtracting fractions with unlike denominators, simplifying fractions, and converting improper fractions to mixed numbers. The intervention was implemented with high levels of treatment fidelity and social validity across three separate intervention agents. Results of the study showed the intervention to be effective, with 15 of the 16 participants making marked gains on fraction probes. We discuss the results of the study with respect to the research questions, provide limitations to the study, and propose areas for future research.

Promoting sophisticated fraction constructs through instructional changes in a mathematics course for PreK-8 prospective teachers

In a previous study, we validated the fractions schemes and operations trajectory (Norton and Wilkins 2012; Steffe in J Math Behav 20(3): 267–307, 2002; Steffe and Olive Children’s fractional knowledge. Springer, New York, 2010; Wilkins and Norton 2011) with PreK-8 prospective teachers and explored PreK-8 prospective teachers’ understanding of fractions based on fraction schemes and operations (Lovin et al. 2016). In that study, we found that most of the prospective teachers had constructed the lower-level fraction schemes and operations, but less than half had constructed the more sophisticated ones. In the current study, we explored ways of evoking the more sophisticated fraction schemes and operations, in particular, by implementing simple instructional changes to a mathematics course for PreK-8 prospective teachers. One change focused on fraction language, specifically encouraging the use of iterative fraction language (e.g., 3/5 is three equal-size pieces, each of which is 1/5 of the whole). Another change focused on intentional number choice, specifically emphasizing improper fractions. After focused fraction instruction that included these instructional changes, we found significant increases in PreK-8 prospective teachers’ interiorization of coordinating three levels of units as well as in PreK-8 prospective teachers’ construction of the reversible partitive fraction scheme and the iterative fraction scheme. We conclude with implications for practice and research in mathematics courses intended for PreK-8 prospective teachers.

Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade.

Development of fraction number line estimation was assessed longitudinally over 5 time points between 4th and 6th grades. Although students showed positive linear growth overall, latent class growth analyses revealed 3 distinct growth trajectory classes: Students who were highly accurate from the start and became even more accurate (n = 154); students who started inaccurate but showed steep growth (n = 121); and students who started inaccurate and showed minimal growth (n = 197). Younger and minimal growth students typically estimated both proper and improper fractions as being less than 1, failing to base estimates on the relation between the numerator and denominator. Class membership was highly predictive of performance on a statewide-standardized mathematics test as well as on a general fraction knowledge measure at the end of 6th grade, even after controlling for mathematic-specific abilities, domain-general cognitive abilities, and demographic variables. Multiplication fluency, classroom attention, and whole number line estimation acuity at the start of the study predicted class membership. The findings reveal that fraction magnitude understanding is central to mathematical development. (PsycINFO Database Record

Abordagem exploratória com representações múltiplas na aprendizagem dos números racionais: Um estudo de desenvolvimento curricular

This paper presents the preparation and implementation of a teaching unit for grade 5 pupils on the concept and representation of rational numbers, the comparison and sorting of rational numbers and equivalence of fractions, based on an exploratory approach with multiple representations, aligned with current curriculum guidelines. The way the unit unfolded and the results regarding pupils’ learning support, overall, the general teaching and learning conjecture. It was noted the need to continue working with improper fractions, and the density of the rational numbers. Some possible alternatives to investigate in the future are suggested, regarding the representation on number line, the approach to the equivalence of fractions, and work in the ration meaning of rational numbers.

Spin-up and spin-down electron transport through a single-atom imperfect metallic wire: An analytical picture

Within a new perspective which includes the consideration of spin-up and spin-down electrons, a quantum-box approach is used to find a closed mathematical expression for the quantized electrical conductance of an imperfect point–metal contact. From this expression, both proper and improper fractions of the fundamental conductance quantum are obtained and discussed in the light of both resonant and off-resonant conduction states. Issues concerning Fermi energy and electrochemical potential are discussed. In addition, essential aspects related to the atom–lead coupling are examined; in particular, a tensor–dyadic formalism is introduced. Our results are found to be in excellent agreement with previous theoretical results and with experimental observations.

A quantitative analysis of children's splitting operations and fraction schemes

Teaching experiments with pairs of children have generated several hypotheses about students’ construction of fractions. For example, Steffe (2004) hypothesized that robust conceptions of improper fractions depends on the development of a splitting operation. Results from teaching experiments that rely on scheme theory and Steffe's hierarchy of fraction schemes imply additional hypotheses, such as the idea that the schemes do indeed form a hierarchy. Our study constitutes the first attempt to test these hypotheses and substantiate Steffe's claims using quantitative methods. We analyze data from 84 students’ performances on written tests, in order to measure students’ development of the splitting operation and construction of fraction schemes. Our findings align with many of the hypotheses implied by teaching experiments and, additionally, suggest that students’ construction of a partitive fraction scheme facilitates the development of splitting.

Mixed Numbers Made Easy: Building and Converting Mixed Numbers and Improper Fractions

After several years of observing and interacting with children, I have noticed their struggles with converting mixed numbers and improper fractions. I have tried several types of manipulatives, computer programs, pictures, and algorithm-based activity sheets. Even after being exposed to all these teaching approaches, many students still struggle with these representations. The concept of a mixed number—a whole number paired with a fraction—really boggles students' minds. More so, the idea of an improper fraction—a fraction with a numerator that is greater than or equal to the denominator—is virtually impossible for some fifth graders to grasp. They cannot connect their written algorithms to the concept or explain how they derived their answers. The reason for rewriting their answer in a different form was too abstract for them. Through direct work with my fifth-grade classes over the years, I have developed a new approach that produces great results and a deeper understanding of mixed numbers, improper fractions, and their relationship to each other.

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

This article communicates findings from a year-long constructivist teaching experiment about the relationship between four sixth-grade students’ multiplicative structures and their construction of improper fractions. Students’ multiplicative structures are the units coordinations that they can take as given prior to activity—i.e., the units coordinations that they have interiorized. This research indicates that the construction of improper fractions requires having interiorized three levels of units. Students who have interiorized only two levels of units may operate with fractions greater than one, but they don’t produce improper fractions. These findings call for a revision in Steffe's hypothesis (Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior, 20, 267–307) that upon the construction of the splitting operation, students’ fractional schemes can be regarded as essentially including improper fractions. While the splitting operation seems crucial in the construction of improper fractions, it is not necessarily accompanied by the interiorization of three levels of units.

A Game Involving Fraction Squares

Fractions are a major area of concept and skill study in elementary school mathematics. The game in this article helps students practice many of these concepts and skills in a motivational and informal setting. The major concepts covered by this game include identifying fractions, equivalent fractions, and improper fractions; performing operations with fractions; and reading and writing numerals for fractions. This game also helps develop number sense and uses models to explore operation sense with fractions. The emphasis is on developing and understanding fraction concepts and operations.