Rational Number Concepts Research Articles

Teaching Rational Number Concepts to Fifth-Grade Students Who Struggle With Mathematics

This study’s purpose was to examine the effects of the concrete-representational-abstract integrated (CRA-I) sequence on the performance of students who struggled with rational number concepts. Three students in southeastern U.S. fifth grade class participated in the study. The CRA-I intervention was grounded in the principles of explicit instruction and showed rational number concepts related to fractions and decimals using fraction blocks, number lines, base 10 blocks, and coins. Students learned about unit fractions, fraction magnitude, fraction equivalence, addition of fractions with unlike denominators, equivalent decimals, and notation of fractions as decimals. The researchers used a multiple probe across behaviors design and demonstrated a functional relation between CRA-I and three behaviors: decreased fraction estimation error, accuracy in adding fractions with unlike denominators, and accuracy writing fractions as decimals.

What mathematics do Grade 7 learners take to high school in the context of the COVID-19 pandemic?

Background: Poor mathematics performance in South Africa is well known. The COVID-19 pandemic was expected to exacerbate the situation.Aim: To investigate Grade 7 learners’ mathematical knowledge at the end of primary school and to compare mathematical performance of Grade 7 and 8 learners in the context of the pandemic.Setting: Data were collected in term four of 2020 at 11 primary schools and five secondary schools. All schools drew learners from poor communities in Gauteng.Methods: A multiple-choice test covering mathematical content from Grades 4–7 was designed and piloted. Learner performance was measured through number of correct responses. Qualitative error analyses were conducted on learners’ choices of distractors.Results: The difference in performance of the two grade groups was not statistically significant. There were similar response patterns in learners’ choices of distractors with strong evidence of cue-based reasoning and evidence of additive reasoning in items requiring multiplicative reasoning.Conclusion: Grade 8 learners made very small gains, likely due to reduced learning time. Learner errors show many similarities with the international literature and show that Grade 7 learners are not yet ready for algebra.Contribution: The findings provide starting points for addressing the most common errors and highlight the need for: greater attention to whole and rational number concepts in Grade 8; strategies for teacher support in teaching primary maths content; and innovative teaching strategies to fast-track learning of this content.

Teaching Fraction-to-Decimal Translation Using the Number Line

Rational number proficiency is predictive of later mathematics achievement, especially for rigorous mathematics courses, such as Algebra I. Students with learning disabilities (LD) in mathematics struggle to make adequate progress on rational number concepts and skills that lay the foundation for successfully completing secondary mathematics courses. However, recent research has demonstrated that utilizing effective instructional design, such as the use of a number line with systematic and explicit instruction, can result in academic progress for students with LD in mathematics. This article presents teachers with a four-step teaching sequence utilizing the number line and effective strategies for teaching students with LD in mathematics to introduce fraction-to-decimal relationships in upper elementary, a prerequisite for middle school mathematics.

Let’s be rational: worked examples supplemented textbooks improve conceptual and fraction knowledge

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students’ fraction number line acuity, rational number concepts and procedures. Finally, the study assessed whether individual differences in prior knowledge of rational numbers moderate the effect of the textbooks. The experimental textbooks resulted in significantly greater improvements overall in fraction number line estimation skills than control textbooks. Further, example-supplemented textbooks were more effective than control textbooks at improving understanding of rational number concepts for those with low prior conceptual knowledge. Both textbooks were equally effective at improving procedural skills.

General Understanding Regarding to the Fraction and Rational Number Concepts

The aim of the current research is to examine the studies carried out to determine the understanding of the concepts of fraction and rational number of students, teachers and prospective teachers and to reveal the general understanding by synthesizing the findings of these studies. For this purpose, meta-synthesis research method, in which qualitative studies are again interpreted and synthesized with a qualitative understanding, was used in the study. Within the scope of the study, a total of 14 studies conducted in Turkey within the framework of the qualitative research paradigm were examined in order to reveal the understanding regarding to the concepts of fraction and rational number. Content analysis was used in the analysis of the obtained data. As a result of the research; It has been determined that students, teachers and prospective teachers have various understandings regarding to the concepts of fraction and rational number. One of the remarkable results is that the prospective teachers have an understanding that the numerator and denominator of the fraction are real numbers, while the teachers have an understanding that the numerator and denominator can be integer, natural number or positive integer. Also, it was found that students associate the rational number with negativity while associating the fraction with positivity.

The Effect of Using Multiple Mathematical Representations of Rational number concepts in Basic Grades Students in Jordan

This study, aimed at the effect of using multiple mathematical representations of rational number concept in basic grades students in jordan. The current study employed the content analysis approach to investigate the multiple mathematical representations and transitions among them in 8th Grade Mathematics Textbook. An observation method was used to analyses the teacher practices (n=35 observations), and record the representations and transitions. The results showed that there was an existence of symbol and verbal representations in the textbook and teachers' implementation. Meanwhile, the other three representations (pictures and figures, models and Cutters, and life situations) . And This study explored the nature of difficulties of eighth-grade students who struggled to build their conceptual understanding of early fraction ideas. interviews with Pre and post of students were conducted for a sufficient identification of the nature of the students’ difficulties. The study revealed The students also minimal use of informal ordering strategies that involve more conceptual than a procedural understanding of the concept of initial fraction ideas.

Rational number understanding: The big picture, not the essence

Background: The learning of rational numbers is a complex and difficult process that begins in the early grades. This teaching often focuses on the mastery of essential knowledge, including particular skills (e.g. using fractions to describe part–whole diagrams) and interpretations (e.g. sharing), which often results in an incomplete and inflexible understanding of these numbers.Aim: This article proposes a holistic and relational perspective on rational number knowing and sense-making.Setting: This possibility emerged through research into the learning of rational number concepts by Foundation Phase and Grade 4 children.Methods: This research forms part of an ongoing, in-depth, exploratory research programme into the processes of learning rational numbers. Clinical interviews and classroom observations were the primary methods of data collection and an in-depth, constant comparative method of analysis was performed on the data.Results: Thinking relevant to rational numbers was identified within four different perspectives through which children make sense of their interactions with the world, namely, social, instrumental, personal and symbolic sense-making.Conclusion: The learning of rational numbers may be usefully seen as arising from the interrelation of multiple aspects of knowing and doing that develop as children balance these different ways of sense-making.

The role of rational number density knowledge in mathematical development

Many students still have not developed a robust understanding of rational number concepts at the end of primary school, despite several years of instruction on the topic. The present study aims to examine the patterns, predictors, and outcomes of the development of rational number knowledge in lower secondary school. Latent transition analysis revealed that rational number development from primary to lower secondary school (N = 362) appears to follow similar patterns as in younger students. In particular, a majority of students had poor knowledge of the density of the rational number set. Whole number magnitude knowledge appeared to be an important predictor of the development of rational number size knowledge, but not density knowledge. Finally, fraction density knowledge appeared to be related to concurrent algebra knowledge. Together these results point to an important role for density knowledge in mathematical development.

Developing Namibian Grade 8 Learners’ Conceptions of Fractions Using Visual Models

Learning rational number concepts is acknowledged as an important task but many learners find it difficult to make sense of them. This paper reports on a case study of the learning in a short (nine-lesson) learning programme for Grade 8 learners in a Namibian school, which sought to use visual models (circle area, bar area and number line) to deepen learners’ understanding of fractions as a means to represent rational quantities. The initial benchmark test indicated a number of ways of working with fraction representations, many of which were inappropriate to the rational quantity presented. Although most learners were able to use a fraction to appropriately describe a part–whole area diagram with a single whole, few were able to appropriately label a similar diagram with multiple wholes, or a quantity greater than 1 on the number line. In the learning programme learners worked with visual models that incorporated multiple reference wholes, to explicitly identify the reference whole, to quantify the size of appropriately subdivided units using unit fraction names and to use these units in a measurement process to quantify the quantities these models indicated. A final test showed a sound conceptualisation of the use of fractions to represent rational quantities less than and greater than 1, in such models.

Profiles of rational number knowledge in Finnish and Flemish students – A multigroup latent class analysis

Students have a great deal of difficulties learning about rational number concepts, as they are confounded by misapplying reasoning about natural numbers to fractions and decimals, referred to as a natural number bias. For example, students often think that the number of digits of a decimal, or the size of the component numbers of fractions, is enough information to determine the magnitude of rational numbers. As well, students have trouble understanding that there is an infinite number of numbers between any two fractions or decimals. Using multigroup latent class analysis, the present study examines the structure of 611 Finnish and Flemish students' rational number knowledge in order to determine the similarities and differences between these two sub-samples. Results reveal that, while the Flemish students performed somewhat better, there were only relatively minor differences in the structure of the two sub-samples' rational number knowledge. In general, it appears that the natural number bias affects these Finnish and Flemish students' knowledge of the size and density of fractions and decimals in similar ways.

Future Mathematics Teachers' Knowledge of Rational and Irrational Number Concepts

Bu arastirmada matematik ogretmen adaylarinin rasyonel ve irrasyonel sayi kavramlarina iliskin bilgilerinin incelenmesi amaclanmistir. Nitel arastirma yontemlerinin kullanildigi bu calisma, durum calismasi seklinde tasarlanmistir. Arastirmanin katilicilarini 40 ortaogretim matematik ogretmen adayi olusturmaktadir. Arastirmacilar tarafindan olusturulan acik uclu soru formu ve yari yapilandirilmis gorusmeler araciligiyla toplanan verilerin analizi sonucu katilimcilarin bircogunun rasyonel ve irrasyonel sayilar konusundaki kavram bilgilerinin eksik ya da hatali oldugu belirlenmistir. Katilimcilarin bu konuda bazi kavram yanilgilarina sahip olduklari, rasyonel ve irrasyonel sayilari birbirinden ayirt etmede yeterince basarili olamadiklari, bir sayinin farkli temsillerinin esitliginin bilincinde olmadiklari ve rasyonel ve irrasyonel sayi kumelerinin kardinalitelerinin belirleme ve birbiri ile mukayese etme konusunda yetersiz kaldiklari belirlenmistir.

Fraction Multiplication and Division Models: A Practitioner Reference Paper

It is well documented in literature that rational number is an important area of understanding in mathematics. Therefore, it follows that teachers and students need to have an understanding of rational number and related concepts such as fraction multiplication and division. This practitioner reference paper examines models that are important to elementary and middle school teachers and students in the learning and understanding of fraction multiplication and division.

Measuring Middle School Students’ Algebra Readiness

Students’ understanding and proficiency with rational number concepts and operations is considered a key foundational skill for future success in algebra. As middle school students work with these concepts, teachers need timely data to determine whether students are making adequate progress. The purpose of this article is to document the content specifications and technical adequacy data of three experimental measures for Grades 6 to 8 that are intended to provide teachers with ongoing information about students’ development of algebra-readiness concepts and skills. Three experimental measures were administered to a total of 575 students in Grades 6 to 8 in two states. Results from a systematic content review indicate that the content assesses key algebra-readiness concepts and skills. Pilot study results suggest that the measures may provide teachers with technically adequate data.

Second-Graders’ Mathematical Practices for Solving Fraction Tasks

Recently, over 40 states in the United States adopted the Common Core State Standards for Mathematics (CCSSM) which include standards for content and eight standards for mathematical practices. The purpose of this study was to better understand the nature of young children’s mathematical practices through an exploratory examination of the practices of a group of second-grade students engaged in several mathematical tasks focused on rational number concepts. Twenty-five second-grade students completed three fraction tasks in structured clinical interviews. The interviews and student work were analyzed using an interpretational analysis to examine the data for constructs, themes, and patterns that were useful in explaining children’s mathematical practices. The results reveal that children used a variety of mathematical practices during the interviews to respond to the mathematical problems presented. Children’s mathematical practices were both a product that they used to solve the mathematical situations, and a process that was developing during the interactions of the interview. The findings lead to new insights about how mathematical practices develop and what promotes their development.

Modeling the developmental trajectories of rational number concept(s)

The present study focuses on the development of two sub-concepts necessary for a complete mathematical understanding of rational numbers, a) representations of the magnitudes of rational numbers and b) the density of rational numbers. While difficulties with rational number concepts have been seen in students' of all ages, including educated adults, little is known about the developmental trajectories of the separate sub-concepts. We measured 10- to 12-year-old students' conceptual knowledge of rational numbers at three time points over a one-year period and estimated models of their conceptual knowledge using latent variable mixture models. Knowledge of magnitude representations is necessary, but not sufficient, for knowledge of density concepts. A Latent Transition Analysis indicated that few students displayed sustained understanding of rational numbers, particularly concepts of density. Results confirm difficulties with rational number conceptual change and suggest that latent variable mixture models can be useful in documenting these processes.

The effect of using a research-based curriculum on learning basic rational number concepts by Lebanese students

Abstract This research study investigates the effects of using a research-based curriculum, the Rational Number Project (RNP) Curriculum, on the acquisition of basic rational number concepts by 5th grade students in contrast with those using the traditional Lebanese curriculum. The study sample comprised five schools randomly selected from a population of 14 schools. Eighteen 5th grade classrooms were randomly assigned to experimental (RNP) and control (TRAD) groups. Two instruments were used for data collection: a RNP post-test and a school test. Both tests were administered to assess students’ achievement after curricular intervention. After a series of six-hour professional training sessions, RNP teachers were given a culturally relevant Arabic version of the RNP curriculum lesson plans and materials for implementation. Three techniques were employed for data analysis: reliability, fidelity and multi analysis of variance (MANOVA). Results showed that students instructed using the research-based curricu...

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.

Rules and Reasons: Decimal Instruction for Academically Low Achieving Students

A study that contrasted two instructional approaches for teaching decimals to remedial students and students with learning disabilities in junior high school is presented. Students in the experimental group were taught decimals in a way that emphasized conceptual understanding. Visual representations of decimal concepts were particularly important instructional materials for this group. Students in the comparison group learned decimals using a highly procedural approach to the topic. Materials for this group emphasized a mastery of basic operations (e.g., addition, division) as they are applied to decimal numbers. Although the results indicate significant effects on conceptual tasks for students in the experimental group, the overall findings suggest that academically low-achieving students require considerable time to learn rational number concepts like decimals. It appears that much more time than many special educators anticipate is required to teach decimals and other rational number concepts adequately.

Development in mastering of rational number concepts and rational number operations: the role of gender, cognitive style and intellectual skills

In this study we report data on the role played by cognitive factors in mathematics learning. A total of 54 Italian seventh‐grade students were tested to determine their cognitive characteristics, intellectual development and cognitive style. These variables, together with the students’ gender, were correlated to the score results in two multiple‐choice tests on rational number concepts (RNC) and rational number operations (RNO). While the cognitive factors were correlated both to RNC and RNO, gender seemed to have a limited significance on the RNO scores only. The possible implications of these results for mathematics teaching are discussed.

Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment

Fourth-grade students' understanding of the order and equivalence of rational numbers was investigated in 11 interviews with each of 12 children during an 18-week teaching experiment. Six children were instructed individually and as a group at each of two sites. The instruction relied heavily on the use of manipulative aids. Children's explanations of their responses to interview tasks were used to identify strategies for comparing fraction pairs of three types: same numerators, same denominators, and different numerators and denominators. After extensive instruction, most children were successful but some continued to demonstrate inadequate understanding. Previous knowledge relating to whole numbers sometimes interfered with learning about rational numbers. Rational number concepts are among the most complex and most important mathematical ideas that children encounter before they reach secondary school. The increased attention being given to research on children's acquisition of such concepts reflects their importance. Recent results from national assessments have shown that children have significant difficulty learning and applying concepts related to rational numbers. In a recent national assessment, 30% of the nation's 13-year-olds added the numerators and the denominators to find the sum of 1/2 and 1/3 (Post, 1981), even though a bit of reflection would have suggested that the sum of two positive quantities should not be less than one of them. Only 24% of the 13-year-olds were able to estimate 12/13 + 7/8 by selecting the correct response, 2, from [1, 2, 19, 21, I don't know]. The most recent Minnesota State Assessment in Mathematics identified fractions as the topic most in need of attention (Minnesota Department of Education & Minnesota Council of Teachers of Mathematics, 1976), a finding that is not unique to Minnesota. State and national assessments suggest that students have often failed to internalize a workable concept of rational number. Students often do not consider numerators and denominators in relation to one another but, rather,