Unraveling the gap between natural and rational numbers

The foundations for more advanced mathematics involve a good sense of rational numbers. However, research in cognitive psychology and mathematics education has repeatedly shown that children and even adults struggle with understanding different aspects of rational numbers. One frequently raised explanation for these difficulties relates to the natural number bias, i.e., the tendency to inappropriately apply natural number properties to rational number tasks. This contribution reviews the four main areas where systematic errors due to the natural number bias can be found, i.e., their size, operations, representations and density. Next, we discuss the major theoretical frameworks from which rational number understanding is currently investigated. Finally, an overview of the various papers is provided.     

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.     

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.     

Irrational Numbers: The Gap between Formal and Intuitive Knowledge

This report focuses on prospective secondary mathematics teachers’ understanding of irrational numbers. Various dimensions of participants’ knowledge regarding the relation between the two sets, rational and irrational, are examined. Three issues are addressed: richness and density of numbers, the fitting of rational and irrational numbers on the real number line, and operations amongst the elements of the two sets. The results indicate that there are inconsistencies between participants’ intuitions and their formal and algorithmic knowledge. Explanations used by the vast majority of participants relied primarily on considering the infinite non-repeating decimal representations of irrationals, which provided a limited access to issues mentioned above.     

Designs of concept maps and their impacts on readers' performance in memory and reasoning while reading

From the perspective of the Fuzzy Trace Theory, this study investigated the impacts of concept maps with two strategic orientations (comprehensive and thematic representations) on readers' performance of cognitive operations (such as perception, verbatim memory, gist reasoning and syntheses) while the readers were reading two history articles that argue from different perspectives about a historical incident that had a profound impact on Taiwan. The results show that the design and focus of the concept maps may influence the formation of mental representations, and that this may be facilitative or constraining in regard to the readers' memory formation and reasoning about the reading materials. Based on these findings, the meaning of constraining effects of concept maps is discussed, and an instructional method involving the progressive elaboration of concept map systems is recommended.     

Students' Early Mathematical Representation Knowledge: The Effects of Emphasizing Single or Multiple Perspectives of the Rational Number Domain in Problem Solving

This study examined changes in 26 fourth-grade students' early conceptions of rational number representations as a function of receiving one of two curricular interventions. The first group of 12 students received a curriculum that emphasized constructing knowledge through extended problem solving with a single perspective of the rational number domain based on part-whole relations. A second group of 14 students received a curriculum that emphasized a more conceptually diverse multiple perspective view of the domain through problem solving with operator and ratio relations. Analyses of the students' rational number knowledge before and after the interventions indicated that students in the single perspective group produced organizations of knowledge that more frequently diverged from a formal domain analysis than those produced by students in the multiple perspective group. Further, students in the single perspective group increased their focus on superficial surface features. Alternatively, students in the multiple perspective group demonstrated an increased focus on operations that more frequently reflected the underlying mathematical relation conveyed by the representation. The findings indicate that an early exposure to more diverse perspectives of rational numbers assists students in developing more interconnected and viable representation knowledge for rational numbers.     

What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density

It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number based prior knowledge, certain intermediate states of understanding emerge. This paper presents a study of Greek and Flemish 9th grade students who solved a test about the infinity of numbers in an interval. The Flemish students outperformed the Greek ones. More importantly, the intermediate levels of understanding—where the type of the interval endpoints (i.e., natural numbers, decimals, or fractions) affects students’ judgments—were very similar in both groups. These results point to specific conceptual difficulties involved in the shift from natural to rational numbers and raise some questions regarding instruction in both countries.     

Historical Objections Against the Number Line

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations.     

“Can any fraction be turned into a decimal?” A case study of a mathematical group discussion

This case study examines two days of teacher-led large group discussion in a fifth grade about a mathematical question intended to support student exploration of relationships among fraction and decimal representations and rational numbers. The purpose of the analysis is to illuminate the teacher’s work in supporting student thinking through the use of a mathematical question embedded in a position-driven discussion. The focus is an examination of the ways that the emergence of mathematical ideas is partially shaped by complex interactions among the mathematical contents of the question, the inherent properties of the discourse format and participant structure, and the available computational methods. The teacher’s work is conceptualized in terms of actions and practices that coordinate these diverse tools, in constant response to students’ concurrent use of them. This revised version was published online in July 2006 with corrections to the Cover Date.     

Students' mental load,stress, and performance when working with symbolic or symbolic–textual molecular representations

In science education, representations are necessary inter alia for the understanding of relationships between structures and systems. However, several studies have identified difficulties of students when working with representations. In the present study, we investigated students' responses (regarding their preference, test performance, mental load (ML), and stress) toward two kinds of representations: symbolic representations, which only use abstract symbols, versus combined symbolic–textual representations, which additionally comprise textual elements. Therefore, students were randomly assigned to one of two treatment groups: one group worked on test tasks accompanied by symbolic representations, and the others worked on the same tasks, but with symbolic–textual representations. Thereafter, the students' test performance and ML were assessed. The level of perceived stress and the salivary cortisol concentration were measured before and after the test and again a few minutes later. Additionally, heart rate variability parameters were assessed continuously. We found a strong preference of the test version with symbolic representations. Additionally, the students showed better test performance and lower ML when they worked with symbolic representations. However, the level of perceived stress was comparable between both groups and there was no strong physiological stress response: The cortisol concentration decreased in both groups and the heart rate was relatively similar. However, during the second half of the test, we observed a significantly higher ratio between low and high heartbeat frequencies in the group with symbolic–textual representations and we found an indirect influence of the kind of representation on test performance through its effect on ML. The poorer test performance and higher ML in connection with symbolic–textual representations confirm previous studies, which found that symbolic–textual representations pose major problems for students. Thus, teachers should enable students to understand symbolic–textual representations and consider carefully whether they can use symbolic representations instead, especially when they teach complex content.     

Disrupting the 'boys don't read' discourse: Primary school boys who love reading fiction

This article disrupts dominant discourses around boys and reading that often homogenise young males as reluctant, disengaged and, at times, adversarial readers. Rather than essentialising boys, we argue there is a need for a more sophisticated knowledge base about the influences, constraints and diverse experiences of boys as readers in society today. Drawing on interviews (n = 30) with Year 4 (8 to 9-year-old) boys at six schools, we consider their personal recounts of their enjoyment in reading, their preferred reading choices and narratives related to their experiences as readers at school. Analysis highlights boys' emerging reading interests, sophisticated and specific reading preferences, and changes in reading identities over time. Boys' preferences for particular fiction authors, novel series and genres dispute the common assumption in educational contexts that boys prefer to engage with non-fiction books. This finding is significant, as negative gendered stereotypes can impact on boys' reading self-concepts. It is also critical given Jerrim and Moss's recent research highlighting the importance of fiction in the development of reading skills. We consider implications for pedagogical practices that broaden reading experiences for the diversity of emerging masculine reading identities in nations such as Australia, where there is an absence of reading for pleasure in education policies.     

Distinguishing adaptive from routine expertise with rational number arithmetic

Adaptive expertise is a valued, but under-examined, feature of students' mathematical development (e.g. Hatano & Oura, 2012). The present study investigates the nature of adaptive expertise with rational number arithmetic. We therefore examined 394 7th and 8th graders’ rational number knowledge using both variable-centered and person-centered approaches. Performance on a measure of adaptive expertise with rational number arithmetic, the arithmetic sentence production task, appeared to be distinct from more routine features of performance. Even among the top 45% of students, all of whom had strong routine procedural and conceptual knowledge, students varied greatly in their performance the arithmetic sentence production task. Strong performance on this measure also predicted later algebra knowledge. The findings suggest that it is possible to distinguish adaptive expertise from routine expertise with rational numbers and that this distinction is important to consider in research on mathematical development.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

A good education: girls' extracurricular pursuits and school choice

This paper draws on the case studies of six girls between the ages of 10 and 13 and considers their transition to secondary school and involvement in extracurricular sports. Within it, I explore how the girls' understandings of ‘a good education’ affects both their academic and extracurricular/sporting choices. Despite government educational targets which seem to value ‘performative’ academic results, I argue that ideas about a ‘well‐rounded student’ continue to hold particular resonance for middle‐class parents and students. Conversely, I suggest that models of excellence and achievement within education are increasingly echoed in students' sporting participation and that this has specific consequences for working‐class and middle‐class students. I draw particular attention to the girls' experiences of these systems with a view to their ability to convert physical capital accrued in physical activity into other forms of social or cultural capital.     

A Study of the Performance of 5th Graders in Number Sense and its Relationship to Achievement in Mathematics

In order to investigate the performance of number sense and its relationships with mathematics achievement of Taiwanese students who had just completed the 5th-grade mathematics curriculum, a computerized number sense scale has been developed. This number sense scale includes four factors which are recognizing relative number size, using multiple representations of numbers and operations, judging the reasonableness of estimates of computed results and recognizing the relative effect of operations on numbers. A total of 1,212 students in Taiwan participated in this study. The main findings of this study are summarized as follows. First, the students perform best on “recognizing relative number size” and perform worst on “judging the reasonableness of estimates of computed results”. This finding is consistent with previous studies. It shows that students in Taiwan seem quite poor on judging the reasonableness of estimates of computed results. Second, female students, on average, have higher scores on recognizing the relative number size than male students, even though only a small effect size is found. And, third, the achievements of the students in mathematics are significantly correlated with their number sense, as measured by the average grade for the academic year of 5th-grade students.     

Being ‘fun’ at work: emotional labour,class, gender and childcare

This paper reports on data drawn from an Economic and Social Research Council‐funded project investigating the experiences of UK‐based students training on level‐2 and level‐3 childcare courses. We focus on the concept of emotional labour in relation to learning to care for and educate young children and the ways in which the students’ experiences of emotional labour and the expectations placed upon their behaviour and attitudes are shaped by class and gender. We consider the ways in which students are encouraged to manage their own and the children's emotions and we identify a number of ‘feeling rules’ that demarcate the vocational habitus of care work with young children. We conclude by emphasising the importance of specific contexts of employment in order to understand workers’ emotional labour and argue for more recognition of the intense demands of emotional labour in early childhood education and care work.     

A Novel Approach to Investigate the Impact of Mindset and Physiology on the Choice to Invest Effort During an Arithmetic Task

Growth mindset, the belief that personal attributes such as intelligence are malleable, has previously been related to more effort investment. Here, we investigated how undergraduates' mindset (N = 114) relates to the choice to invest effort during an arithmetic task, indexed by whether they make low vs. high effort-related choices. Social cognitive theory suggests that past performance experiences (mastery vs. failure) and physiological state are important sources for competence self-evaluations. Therefore, in addition to mindset, we also investigated how effort-related choices are influenced more dynamically, by failures and physiological responses during the task. Growth mindset and physiological effort mobilization did not predict effort-related choices but making mistakes did predict lower effort choices in the subsequent round. This study further supports the importance of mastery experiences for effort investment and provides a novel approach for integrating different levels of influence on effort-related choices during an educationally-relevant task.     

Michel Foucault: liberation,freedom, education1

This article is a critical examination of practices and representations that constitute international education. While international education has provided substantial contributions and benefits for nation‐states and international students, we question the discourses and practices which inform the international education export industry. The ‘brand identities’ of receiving or host countries imply that they are welcoming, respectful of multiculturalism and have a well established intellectual history, in contrast to international students' embodied experiences. There is also a tendency to represent and regard international students as disembodied learners. We conclude that these disjunctures between disembodied representations and embodied experiences are undermining the kinds of cosmopolitan sensibilities that international education claims to provide.