The representation of fraction magnitudes and the whole number bias reconsidered

Two experiments investigated the whole number bias in the representation of fraction magnitudes with adults. A fraction magnitude comparison task was used where half of the comparisons were consistent with whole number ordering and the other half were not. Distance effects were found in Experiment 1 indicating that participants were comparing the magnitude of the whole fraction rather than just the parts. However, accuracy and response time also depended on the comparisons' consistency with whole number ordering. Experiment 2 manipulated the distance between the fraction pairs and showed that the whole number effect was strongest when the distance between the fraction pairs was very small. The results suggest that even skilled adults do not always have direct access to a fraction's magnitude on the number line. When the magnitudes are especially close together, adults may rely on alternative implicit or explicit strategies, such as examining the whole number parts, to evaluate the comparison.     

Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents

Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The integrated theory of numerical development posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of the magnitudes to which they refer, and this magnitude understanding is central to general mathematical competence. We investigated relations among fraction magnitude understanding, arithmetic and general mathematical abilities in countries differing in educational practices: U.S., China and Belgium. Despite country-specific differences in absolute level of fraction knowledge, 6th and 8th graders' fraction magnitude understanding was positively related to their general mathematical achievement in all countries, and this relation remained significant after controlling for fraction arithmetic knowledge in almost all combinations of country and age group. These findings suggest that instructional interventions should target learners' interpretation of fractions as magnitudes, e.g., by practicing translating fractions into positions on number lines.     

Fractions: Could they really be the gatekeeper’s doorman?

The National Mathematics Advisory Panel (NMAP, 2008) asserts that a foundational knowledge of fractions is crucial for students’ success in algebra; however, empirical evidence for this claim is relatively nonexistent. In the present study, we examine the impact of middle school students’ fraction and whole number magnitude knowledge on various components of their algebra readiness. Results suggest that fraction knowledge is related to algebra readiness, more so than number magnitude knowledge in general; students’ magnitude knowledge of unit fractions (i.e., those with a numerator of 1) appears particularly important. Findings confirm the intuition of the NMAP (2008) and support the recommendation of the Common Core Standards (National Governors Association Center for Best Practices (NGA Center), 2010) that students’ fraction knowledge should be cultivated using number lines.     

Analogue equivalents in number processing of simple arithmetic sums

This study had two aims. The first was to test the postulate of analogical equivalents in number processing using a stimulus set based on the differences between pairs of numbers, and second, to look for IQ-dependent differences in this processing. Participants were asked to make judgments concerning the differences between pairs of numbers—each number pair being defined according to overall numerical size and level of difference—and to draw the magnitude of the differences using a free-hand line. In agreement with previous findings, results indicated that the magnitude of participants' responses was dependent on the sizes and levels of the differences within the number pairs. In particular, participants' responses to a difference of nine units at the highest level were based on unit lengths smaller in magnitude compared with other number pairs. In contrast, participants' responses to a second test requiring them to estimate the length of a line using an independent number scale showed remarkable accuracy across all lengths. The results of the two tests did not, however, indicate any difference in responses based on IQ. The results gave support to the idea that transformation of numerical quantities to their analogical equivalents occurs during simple arithmetic sums, and that this transformation was flawed as suggested by the number size effect.     

Children’s mental representation when comparing fractions with common numerators

Researchers debate whether one represents the magnitude of a fraction according to its real numerical value or just the discrete numerosity of its numerator or denominator. The present study examined three effects based on the notion that people possess a mental number line to explore how children represent fractions when they compare fractions with common numerators. Specifically, the effect of the spatial numerical association of response codes (SNARC), the distance effect and the size effect in representing fractions were examined in a sample of 72 sixth graders, who successfully solved the fraction comparison task with a real number (.2) or a fraction (1/5) as the reference. Results showed that in the fraction-reference group (1/5 as the reference), there was a significant reverse SNARC effect and a distance effect between the denominators of the target fractions and the reference fraction; in the real number-reference group, the three effects were also observed. These results revealed that both groups used the mental number line to represent fractions and did not represent their real numerical values but rather the discrete numerosities of denominators when comparing fractions with common numerators. It seems that the way people represent fractions may depend on their strategy choices.     

Semantic Domains of Rational Numbers and the Acquisition of Fraction Equivalence

This study examined how the semantic meanings of rational numbers embodied in graphical representations pose a constraint on children's construction of the concept of fraction equivalence. A test consisting of graphical representations representing the part–whole and measuring semantic meaning was given to 205 fifth-graders and 208 sixth-graders from China. Results showed that the fifth-graders' overall performance on equivalent fraction items was poor across the semantic domains of rational numbers. The sixth-graders showed significantly improved performance on the equivalent fraction items representing the part–whole relation but not on those representing the measure aspect of rational number. The results provide evidence that semantic meanings of fractional numbers in different relevant contexts also constitute a source of difficulty in children's constructing the concept of fraction equivalence, in addition to the multiplicative nature of the concept.     

Developmental changes in the comparison of decimal fractions

Learning about decimal fractions is difficult because it requires an extension of the number concept built on natural numbers. The aim of the present study was to investigate developmental changes in children's misconceptions in decimal fraction processing. A large sample of children from Grades 3 to 6 performed a numerical comparison task on different categories of pairs of decimal fractions. The success rate and the type of error they made varied with age and categories. We distinguished the impact of the value of the digits from the impact of the length of the fractional part on children's pattern of responses. Although both kinds of impact affected the success rate, the digit values had a stronger impact and were mastered later than the length. Our results also showed that a zero just after the decimal point was understood better and earlier than a zero at the end of the fractional part of a number. Cluster analysis was conducted to determine groups of children who answered similarly regarding the response type across the various categories of decimal fractions. To interpret the data the conceptual change framework was used.     

Teaching prospective teachers about fractions: historical and pedagogical perspectives

Research shows that students, and sometimes teachers, have trouble with fractions, especially conceiving of fractions as numbers that extend the whole number system. This paper explores how fractions are addressed in undergraduate mathematics courses for prospective elementary teachers (PSTs). In particular, we explore how, and whether, the instructors of these courses address fractions as an extension of the whole number system and fractions as numbers in their classrooms. Using a framework consisting of four approaches to the development of fractions found in history, we analyze fraction lessons videotaped in six mathematics classes for PSTs. Historically, the first two approaches—part–whole and measurement—focus on fractions as parts of wholes rather than numbers, and the last two approaches—division and set theory—formalize fractions as numbers. Our results show that the instructors only implicitly addressed fraction-as-number and the extension of fractions from whole numbers, although most of them mentioned or emphasized these aspects of fractions during interviews.     

Which Type of Rational Numbers Should Students Learn First?

Many children and adults have difficulty gaining a comprehensive understanding of rational numbers. Although fractions are taught before decimals and percentages in many countries, including the USA, a number of researchers have argued that decimals are easier to learn than fractions and therefore teaching them first might mitigate children’s difficulty with rational numbers in general. We evaluate this proposal by discussing evidence regarding whether decimals are in fact easier to understand than fractions and whether teaching decimals before fractions leads to superior learning. Our review indicates that decimals are not generally easier to understand than fractions, though they are easier on some tasks. Learners have similar difficulty in understanding fraction and decimal magnitudes, arithmetic, and density, as well as with converting from either notation to the other. There was too little research on knowledge of percentages to include them in the comparisons or to establish the ideal order of instruction of the three types of rational numbers. Although existing research is insufficient to determine the best sequence for teaching the three rational number formats, we recommend several types of research that could help in addressing the issue in the future.     

Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.

     

Young Children'S Notations For Fractions

This paper focuses on the kinds of notations young children make for fractional numbers. The extant literature in the area of fractional numbers acknowledges children's difficulties in conceptualizing fractional numbers. Some of the research suggests possibly delaying an introduction to conventional notations for algorithms and fractions until children have developed a better understanding of fractional numbers (e.g., Hunting, 1987; Sáenz-Ludlow, 1994, 1995). Other research (e.g., Empson, 1999), however, acknowledges the interaction between conceptual understandings and the representations for those understandings. Following the latter line of thinking, this paper argues that children's notational competencies and conceptual understandings are intertwined, addressing the following research questions: (a) What kinds of notations do five and six-year-old children make for fractional numbers?; and (b) What can be learned about young children's learning of fractional numbers by analyzing the notations they make? This paper presents data from interviews carried out with twenty-four children in Kindergarten and first grade (five and six-year-old children), exploring the nuances of their notations and the meanings that they attach to fractions.     

Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.     

Sources of Individual Differences in Children's Understanding of Fractions

Longitudinal associations of domain‐general and numerical competencies with individual differences in children's understanding of fractions were investigated. Children (n = 163) were assessed at 6 years of age on domain‐general (nonverbal reasoning, language, attentive behavior, executive control, visual‐spatial memory) and numerical (number knowledge) competencies; at 7 years on whole‐number arithmetic computations and number line estimation; and at 10 years on fraction concepts. Mediation analyses controlling for general mathematics ability and general academic ability revealed that numerical and mathematical competencies were direct predictors of fraction concepts, whereas domain‐general competencies supported the acquisition of fraction concepts via whole‐number arithmetic computations or number line estimation. Results indicate multiple pathways to fraction competence.     

Transactional Presence as a Critical Predictor of Success in Distance Learning

This paper argues that, apart from interactive activities, the perceptions of psychological presence that distance education students hold of their teachers, peer students, and the institution can be significant predictors of their learning. The "perception of presence" in this paper is defined as the degree to which a distance education student senses the availability of, and connectedness with, each party. This form of presence is designated here as "Transactional Presence" (TP). In this study, distance education student learning was assessed in the light of students' perceived learning achievement, satisfaction, and intent-to-persist. An analysis of student survey data indicates that a distance student's sense of institutional TP predicts all the selected measures to do with success in distance learning. While a sense of peer student TP is significantly related to satisfaction and intent-to-persist, the effect of teacher TP is found to relate only to student-perceived learning achievement. Implications of the TP construct are discussed with respect to the theory, research, and practice of distance education, along with recommendations for future research.     

Student Journals and Student-constructed Questions in a Primary Mathematics Classroom

Eleven Grade 6 students from a class of 26 Canadian children (comprising both Grade 5 and Grade 6 students) were the focus of a 15-week study on students' mathematical learning through writing, where Student Journals (SJ) and Student-Constructed Questions (SCQ) were used by the class teacher as an integral part of her lessons on common and decimal fractions. Student interviews, classroom observations and a teacher interview complemented the analysis of SJ and SCQ. Both writing tasks evidenced students' mathematical learning, but while the SCQ indicated students' fraction knowledge more clearly, the SJ did not communicate students' understanding of fractions as well as student's verbal discussion and explanations did.     

Keller's Personalized System of Instruction: the search for a basic distance learning paradigm

This paper presents research results and arguments supporting the possibility of using Keller's Personalized System of Instruction as a basic paradigm for distance learning. A study connected with PSI at Athabasca University is reported and discussed. A list of advantages and disadvantages of PSI as a basic paradigm for distance learning is also presented and discussed. The report concludes by suggesting that a paradigm for distance learning would facilitate communication on distance learning activity and that PSI would provide a basic paradigm for distance learning while still allowing for educational variations within distance learning to occur and develop.     

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

Assessment 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 symbolic distance effect in monkeys (Cebus apella)

In making comparative judgments about pairs of stimuli that are linearly ordered, human subjects usually respond faster the greater the separation between-the-items of a test pair—the symbolic distance effect. A similar result has been obtained for associatively related items, such as the alphabet. We report evidence for a distance effect in monkeys tested with pairs of items drawn from a five-item series with which they had considerable previous experience-in a serial learning setting. This finding provides independent evidence that in learning a serial list of items, monkeys acquire knowledge about the ordinal positions of the items. Analysis of the positive results obtained in Experiment 2 and of the failure to find a distance effect in Experiment 1 suggested that in learning a serial list, monkeys construct both an associative chain representation of the series and a spatial representation, with the latter supplying the spatial markers that convey positional information. This dual coding of sequential events, which may be rather general among mammals, probably supports a variety of cognitive competencies.     

Going beyond Kirkpatrick's Training Evaluation Model: The role of workplace factors in distance learning transfer

Abstract

This article emanates from a longitudinal study of the impact of a distance education programme for teacher training on graduates' job performance, in which the authors built on the findings of a previous pilot study. After using Kirkpatrick's Training Evaluation Model in a previous study, one of the authors found there to be a strong relationship between graduates' completion of the programme and their performance at school. However, the model does not probe factors that impede on transfer of learning. Quite a number of the graduate participants indicated that they were faced with this problem. In order to further probe this phenomenon, the authors fused Baldwin's Transfer of Training Model with the second level of Kirkpatrick's model by using a mixed-methods enquiry. It became clear that the organizational climate of schools has a strong influence on the transfer of learning in the workplace. Suggestions are presented on how educators and school managers can work together effectively.