Intuitive rules in science and mathematics: the case of ‘more of A ‐‐ more of B’

In the last twenty years researchers have studied students’ mathematical and scientific conceptions and reasoning. Most of this research is content‐specific. It has been found that students often hold ideas that are not in line with accepted scientific notions. In our joint work in mathematics and science education it became apparent that many of these alternative conceptions hail from the same intuitive rules. We have so far identified two such rules: ‘The more of A, the more of B’ and, ‘Everything can be divided by two’. The first rule is reflected in students’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.), in all tasks related to intensive quantities (density, temperature, concentration, etc.), and in tasks related to infinite quantities. The second rule is observed in responses related to successive division of material and geometrical objects, and in successive dilution tasks. In this paper we describe and discuss the first rule and its relevance to science and mathematics education. In a second paper (Tirosh and Stavy, in press) we shall describe and discuss the second rule.     

Intuitive rules in science and mathematics: the case of ‘Everything can be divided by two’

In the last twenty years, researchers have studied students’ mathematical and scientific conceptions and reasoning. Most of this research is content‐specific. It has been found that students often hold ideas that are not in line with accepted scientific notions. In our joint work in mathematics and science education, it became apparent that many of these alternative conceptions hail from a small number of intuitive rules. We have so far identified two such rules: ‘The more of A, the more of B’, and, ‘Everything can be divided by two’. The first rule is reflected in students’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.), all tasks related to intensive quantities (density, temperature, concentration, etc.), and tasks related to infinite quantities. The second rule is observed in responses related to successive division of material and geometrical objects, and in seriation tasks. In this paper we describe and discuss the second rule and its relevance to science and mathematics education. In a previous paper (Stavy and Tirosh 1995, in press) we described and discussed the first rule.     

The Best of Times,the Worst of Times: the ‘post’ in ‘postmodern’

As teacher educators we are in perhaps one of the most demanding of professions in terms of constant upgrading of qualifications and updating of skills and knowledge, keeping abreast of new developments in education. As we question the relevance of what has become traditional discourse to inform our practice (such as Marxism on the Left of the political spectrum and various forms of discourses of the Right) we raise serious questions as well in reference to the ideological basis for academic boundaries that have become established, that have in effect structured the way in which we have been taught, the way we have engaged in learning, and in turn, the way we engage in our own practices of teaching and learning in our dealings with subject disciples and knowledge formations. It is now that we are seriously criticizing such discourses as having ignored the socially constructed nature of what is taught, how it is taught, and not only the underpinnings of the way they are evaluated, but how they are actually valued. This paper points to some of the implications of postmodernism for practioners.     

The implementation of ‘Education for Citizenship’ in schools in Scotland: a research report

Changes in society and government have increased concerns of low participation and involvement by people, especially the young, in the political process and decision-making. As a result, citizenship has become a focus of recent curriculum developments in many countries. In Scotland, ‘Values and Citizenship’ has been made one of the Scottish Executive's national priorities for education and is linked to the major national initiative, ‘Education for Citizenship’. This is seen as encouraging pupils to make informed choices and decisions and to take action, individually and as part of the community. More recently, A curriculum for excellence 3–18 (SEED 2004 Shucksmith, J. and Spratt, J. 2001. Education for citizenship in Scotland: An analysis of responses to the consultation paper, Aberdeen: University of Aberdeen.  [Google Scholar]) placed promoting responsible citizenship at the heart of the curriculum. This article reports on the extent to which a sample of Scottish schools was making progress towards developing education for citizenship, the strategies they developed and the barriers encountered in this implementation. It compares progress and developments in Scotland to the model devised in a longitudinal study for citizenship in England (Ireland et al. 2004 Ireland, E. and Kerr, D. 2004. Making citizenship education real: Learning from current approaches in schools. Teaching Citizenship, 9: 20–25.  [Google Scholar], 2006 Ireland, E., Kerr, D., Lopes, J., Nelson, J. and Cleaver, E. 2006. Active citizenship and young people: Opportunities, experiences and challenges in and beyond school, London: DfES.  [Google Scholar]).     

The fault is in ourselves: looking at ‘failures in implementation’

While implementers are often blamed, reformers themselves are usually responsible for sub‐optimal implementation of educational reform projects. This article shows that the conditions for successful implementation, which are known and accessible, are frequently ignored. The case study concerns a university that is engaged in in‐service teacher education reform in Pakistan, focusing on introducing a ‘learner‐centred’ approach in schools. The research identified severe shortcomings related to implementation of the new approach, mostly relating to a failure by the teacher‐educators to understand and adapt to the actual contexts in which the teachers work. Finally, we suggest reasons why there are so many instances of failed reforms in developing countries.     

‘Being in the World’: The event of learning

This paper employs an eclectic mix of paradigms in order to discuss constituting characteristics of young children's learning experiences. Drawing upon a phenomenological perspective it examines learning as a form of ‘Being’ and as the result of learners’ engagement with the world in their own, unique, intentional manners. The learners’ intentions towards their world are expressed in everyday activity and participation. A social constructivist perspective is thus employed to present learning as situated in meaningful socio‐cultural contexts of the everyday, lived world and as a form of participation in those settings. These characteristics of learning are brought together into a holistic, synthesised model, a Gestalt of learning. The proposed synthesis has relevance for and is applicable to educational contexts as a means of making sense of children's learning experiences and of promoting and facilitating them.     

‘It's more flexible’: persistence of women engineers in the academy

The under-representation of women in engineering is an issue of concern for policy-makers. While much of the existing literature has focussed on understanding reasons for women's under-representation in engineering and related technical fields, there is a paucity of research that examines why women who are already in engineering persist. This study aims to answer the question, why do women engineers in academe persist in a discipline that is generally perceived to be inhospitable to women. Three main challenges, namely gender discrimination, lack of institutional and disciplinary support, and the rigid nature of the workplace, emerged as key impediments to women engineers in the private sector. In contrast, women engineers in the academy identified the flexibility offered by academic environments, the presence of personal support networks, and a passion for teaching and students as critical factors that influenced their decision to remain in academia.     

‘The 5 cm rule’: biopower,sexuality and schooling

This paper explores ‘the 5 cm rule’, a regulation around student contact discovered during an investigation of the sexual culture of schooling with 16–19-year-olds in New Zealand. Implemented to stem ‘inappropriate and unwanted’ touching, it stipulates that students must maintain a physical distance of 5 cm at all times. It is argued this rule represents a contemporary type of biopower which forms part of the sexual culture of schooling. As a technique of corporeal regulation it is characterised by a ‘loose’ exercise of power, that allows for student resistance while producing subjects’ ‘docility-utility’ (Foucault, 1980). The paper contends that the rule contributes negatively to ‘the sexual culture of schooling’ by constituting student sexuality as ‘unruly’ and ‘problematic’. This stipulation also prescribes a set of gender relations that are inhibitive of mutually negotiated and pleasurable corporeal experience.     

A Good Example of Hemingway’s Iceberg Principle:The Killers

Hemingway established a well-known literary theory - iceberg principle.He said he always attempted to create his literary works according to this principle.In his opinion,if a prose writer has a pretty clear idea of what he writes about,then he can omit the things he knows,and the readers will strongly feel that what the writer has omitted seems to have been written as long as what he writes is authentic.Hemingway himself once said,\"The dignity of movement of an iceberg is due to only one eighth of it being above water.\"Here,one eighth refers to the information given by the author,and the rest refers to the unwrit ten information that can be acquired in readers’imagination on the basis of the one eighth.The Killers is a good example of Hemingway’s iceberg principle.This essay will analyse the characters in it to show how the principle was used in this short story.With no doubt the iceberg principle leaves a large space for the readers to imagine and better understand Hemingway’s works.     

Difficulties of Hong Kong teachers’ understanding and implementation of ‘play’ in the curriculum

Play is a characteristic behaviour of young children. It is a vital way through which children reconcile their inner lives with the outside world. It is a central activity in kindergarten and nurseries. ‘Learning through play’ is recommended as the mode of learning and teaching for young children. In Hong Kong, it was documented in an official education report in 1986 and is the key guideline for the implementation of the aims of early childhood education for the next millennium. A case study of two kindergarten teachers in Hong Kong was conducted to investigate the understanding and implementation of play in the kindergarten curriculum. The findings showed immense difficulties in understanding the concept ‘learning through play’ and a gap between the practitioners’ espoused theories and practice. The study reflected the complex nature of learning and teaching and highlighted the core issue for reform, namely, the need for bridging the gap between theory and practice.     

‘Success’ and ‘Failure’ in the Education Marketplace: An Example from New Zealand

This paper details the impact of market policies in New Zealand on one low socio-economic school, raising questions about the underlyingassumptions of the programmes designed to turn around so called failing schools. The impact of macro policies on disadvantaged schoolsin the market place is documented, as is recent evidence on the impactof schools entering a spiral of decline on their examination outcomes.It is argued that school success and failure cannot be understood inisolation but needs to be placed in the wider context of stateeducational policy. Consequently, change strategies which ignore thewider policy settings risk failure.     

Secondary teachers’ knowledge of elementary number theory proofs: the case of general-cover proofs

In light of recent reform recommendations, teachers are expected to turn proofs and proving into an ongoing component of their classroom practice. Two questions emerging from this requirement are: Is the mathematical knowledge of high school teachers sufficient to prove various kinds of statements? Does teachers’ knowledge allow them to determine the validity of an argument made by their students? The results of this study, in which 50 secondary school teachers participated, point to a positive answer to the first question in the framework of elementary number theory (ENT). However, the picture is more complex with respect to the second one. Results indicated that some teachers may over-value the generality of symbolic mode of representation and under-value the generality of verbal ones. Possibly, the verbal representation of an argument is less transparent and more difficult to understand.     

Bartleby the Example and Eros the Idea of the Work: Some considerations on Giorgio Agamben’s ‘The idea of study’

Abstract

The present article investigates the rhythm of study as described by Giorgio Agamben in ‘The idea of study’, present in Idea of prose. In this short treatise, Agamben presents Melville’s scrivener Bartleby as the exemplary embodiment of study. Bartleby’s paradigmatic status, according to Agamben’s interpretation, does, however, exclude him from belonging to the ‘class of study’. Bartleby’s exclusion leads to the discovery of an unmentioned member of the ‘class of study’: Eros. The surprising absence of Eros dissolves, however, once he is recuperated as the image of ‘The idea of the work’. Bartleby’s exclusion and Eros’s absent inclusion and presence as the ‘The idea of the work’ not only demonstrate the close relation between study and work, but also explain Agamben’s considerations on the connection between study and the nature of his own work. The article concludes with a call for the remaining importance of Agamben’s festina lente, the rhythm of study, within the broader context of our contemporary university institutions. Once the incessant shuttling between passion and action has been called to a halt—in favour of one of the two extremities—study becomes impossible.     

The ‘nature of science’ in the school curriculum: the great survivor

This paper explores the ways in which the ‘nature of science’ (NoS) has been interpreted, accommodated and justified within school curricula since science was first schooled in the mid-nineteenth century. It explores how different interpretations of ‘the NoS’ have been invoked by those seeking to reform school science education in response to wider political, economic or social concerns such as the demand to ‘humanise’ school science teaching, to increase the supply of qualified scientists or to promote scientific literacy. It offers some comments upon the implications of these interpretations for current attempts to promote the ‘NoS’ in school science education. The focus of attention is England and, to a lesser extent, the USA but the issues raised are of contemporary relevance to many other parts of the world.     

‘The rules above the bed’: academic language and learning as hospitality

ABSTRACT

In times of widening participation, Australian universities trade on notions of diversity, framing themselves as hospitable places of access and inclusion. In this space, Academic Language and Learning (ALL) practitioners may be seen as extending the welcome of the university through practices aimed at addressing students’ diverse needs. These include identifying students ‘at-risk’ of failure and attrition, conducting one-to-one appointments, and embedding academic literacies by team-teaching within disciplines. In this paper, we reflect on these practices through the lens of Derrida’s notion of hospitality, drawing on key themes such as the constitutive power relationships of hospitality, the aporia between conditional and unconditional hospitality, and hospitality as an ethics of difference and openness to the ‘arrival of the new’. For each of the explored practices, we problematise the institutionalised framing of ALL practitioners as hosts in benevolent universities providing an unreserved welcome to the student ‘foreigner’. We examine the practice of identifying students-at-risk and question a conditional hospitality that risks closing the door on the unforeseeable that students may bring. Our reflection on the managerially devalued one-to-one appointments highlights hospitality as ethics, with each appointment presenting a crisis of choice in responsibly welcoming student difference. Team-teaching exposes the ambiguity of the ALL practitioner ‘being at-home’ in embedded contexts while presenting the possibility for disrupting established roles and practices. Engaging with Derrida’s hospitality thus allows us to uncover power dynamics shaping the role of ALL practitioners and offers the possibility of ethical responsiveness to student difference and a radical opening to the new.     

The origin of counting: A study of the early meaning of ‘one’, ‘two’ and ‘three’ among Basque- and Spanish-speaking children

This study examines the beginning of the conceptual understanding of the first number-words and what role language can play in developing the notion of numbers. To that end, 2?- and 3?-year-old Basque and Spanish monolingual children's (N = 131) basic numeracy skills are analysed by means of two different experimental procedures: Give-N (in which children are requested to gather sets of objects) and How-Many (based on the ability to count collections). The paper accounts for differences as regards the performance in the Give-N procedure between children with different linguistic backgrounds. In accordance with previous research, this finding can be related to the dissimilar ways by which languages (Basque and Spanish, in this case) express grammatical number, supporting the idea that language plays a definite role in the emergence of the earliest set-size meanings of ‘one,’ ‘two,’ and ‘three’ number-words. Eventually, the work attempts to contribute to the growing body of evidence that shows that the meaning of the count-list appears from mapping numerals onto numerical cognitive representations produced by early core systems of numbers.     

‘The fear of the lord is the beginning of wisdom’: A defence of religious education in county schools

ABSTRACT

It is argued here that understanding John Dewey's thought as that of a prodigal liberal or a fellow traveller does not capture the complexity of his work. It is also important to recognise the portion of his work that is historie morale. In the very best sense it is epic, encapsulating the hopes and dreams of a history of the American people in the early 1900s. It is a work that simultaneously pursues modernity and the past – for the sake of coping with the present. It also calls attention to an unnoticed element in his work ‘inexperience’.     

The performative shift: middle leadership ‘in the line of fire’

ABSTRACT

State school inspection in Norway is currently changing with targeted schools becoming subject to more complex methods of inquiry. Not only school principals but middle leaders are exposed to this shifting system, for the latter are in the frontline of their schools’ everyday practices. The article examines how state school inspection is used as means of controlling legal compliance, as well as evaluating the formative assessment routines and practices of schools, middle leaders, and individual teachers. Drawing on the concepts of accountability and performativity, field observations of inspectors interrogating department heads in primary education are analysed. The empirical study demonstrates how use of standardised rubrics steer the inspection process in schools, aiming more towards completing on task, rather than supporting middle leaders in their struggle to comply with legal standards. During such interrogation, the department heads comply with the system, and are at the same time open towards the inspectors’ questioning concerning the school’s lack of fully implemented routines.