Third Graders' Interpretations of Equality and the Equal Symbol

This paper analyzes the interpretations of equality and the equal symbol of the third-grade children who participated in a year long whole-class socio constructivist teaching experiment. These children initially interpreted the equal symbol as a command to perform an arithmetical operation; it was less natural to them to interpret it as a relational symbol to compare two quantities. By the end of the school year, children were able to conceptualize the quantitative sameness of two numerical expressions and describe it by using the phrase is the same as, the words equal or equals, or the symbols = or =s. These children expanded their conceptualizations of equality due to their active role in class discussions, the arithmetical tasks that took into account children's difficulties, and the teacher's intellectual sensitivity to strike a delicate balance between the force of teaching and the freedom of learning (Freudenthal, 1991, p. 55).     

Scale economies in academic excellence: An exploratory analysis of the United Kingdom's 1992 research selectivity exercise

Taken in aggregate, bigger university departments did disproportionately well in the 1992 U.K. Higher Education Research Selectivity Exercise (RSE). A number of reasons are reviewed whereby such an economies of scale effect might apply both in general and with respect to the RSE. A methodology is developed whereby the RSE performance of the UK's universities across academic units can be attributed to size and non size components, the relative importance of which are then calculated for each of the old universities, paying attention to their independent designations as research and teaching institutions. Possible implications for academic planning by universities are drawn out.     

On culture,geometrical thinking and mathematics education

This article confronts a widespread prejudice about mathematical knowledge, that mathematics is culture-free, by demonstrating alternative constructions of euclidean geometrical ideas developed from the traditional culture of Mozambique. As well as establishing the educational power of these constructions, the article illustrates the methodology of cultural conscientialization in the context of teacher training.     

Analyzing children's errors on written mathematical tasks

Methods developed by Newman and Casey for analyzing errors made by children attempting verbal arithmetic problems are described, with particular emphasis being given to Newman's hierarchy of error causes. Data obtained by Newman, Casey, and Clements are presented. These show that a large proportion of errors made by children in grades 5–7 in Victoria on verbal arithmetic problems are in the Newman categories Comprehension, Transformation, Process Skills, and Carelessness.     

Impact and adjustment: foreign students in comparative perspective

Foreign study is a multifaceted phenomenon - its impact is felt on academic institutions in both the host and sending countries, on the economies of nations, and of course on the individuals involved. With more than one million students studying abroad, foreign study has assumed considerable importance in higher education planning. This article considers the many aspects of foreign study and discusses the interrelationships of these elements. The policies of the host nations, for example, have an impact on higher education planning in the sending countries. The non-return of foreign students, traditionally referred to as the brain drain, is considerably more complex than was once thought since Third World graduates settled in the industrialized nations often retain contacts with their home countries and increasingly return after a period abroad. This article also considers the various push and pull factors which determine the constantly changing flow of foreign students.This study was supported by the Organisation for Economic Co-operation and Development, Centre for Educational Research and Innovation. It was presented at the International Seminar on Higher Education and the Flow of Foreign Students, convened by the Hochschul-Informations-System, with support from the Federal Ministry of Education and Science and the Lower Saxony Ministry of Science and Arts in Hannover, Germany on 26th–28th April 1990.     

The development of semantic categories for addition and subtraction

Research conducted in several countries has shown consistent patterns of performance on change, combine and compare word problems involving addition and subtraction. This paper interprets these findings within a theoretical framework that emphasizes the development of empirical, logical and mathematical knowledge.     

Career Development and the Needs of Young College-Educated Females in Taiwan

This interpretive study examined the lived experiences of young college-educated Taiwanese women employed as clerical workers. Using hermeneutic phenomenology as the research methodology, the study explored the meanings of the participants lived experiences. Systematic text analysis of the data revealed various experiences and expectations. Experience themes identified in the study were: (a) bored and busy; and (b) disrespected and mistrusted. Major expectation themes revealed are: (a) career dreams and aspirations; and (b) desire for meaningful work. The study emphasised the importance of career development education in young Taiwanese women. The experiences of Taiwanese female clerical workers as recorded in the study reflected a need for career development, which might provide insightful information for career development scholars and practitioners.     

“Without Contraries is no Progression”: Dust as an All-Inclusive, Multifunctional Metaphor in Philip Pullman's “His Dark Materials”

Drawing on John Milton's Paradise Lost and on motifs found within Gnostic mythology and the poetry of William Blake, this article explores Philip Pullman's reworking of the Judeo-Christian myth of the Fall. At the centre of this investigation is Dust: a conventional metaphor for human physicality inspired by God's judgment on humanity. This article suggests that Dust is re-presented in the trilogy in a more positive manner through the development of Milton's metaphor of the dark materials into a substance in which good and evil, and spirit and matter—conceptual opposites that form the basis of religious dualism—coexist.     

The cultural grid: A framework for multicultural counseling

The Cultural Grid suggests a structure for interpreting a person's behavior in the context of culturally learned expectations. When two persons' expectations are similar, but their behaviors are dissimilar, cross cultural misunderstanding is likely to result. When their expectations are dissimilar, the result is likely to be a personal misunderstanding even though their behaviors are similar. This article introduces a framework for differentiating personal and cultural differences in the interview which is essential for appropriate multicultural counseling. The alternatives are to allow apparent and obvious cultural differences to mask underlying personal differences or to incorrectly assume that two persons who appear culturally similar are having a personal disagreement.     

Growth in mathematical understanding: How can we characterise it and how can we represent it?

There has been a variety of approaches to the study of mathematical understanding, and some of these are reviewed before outlining the background to the model we are proposing for the growth of such understanding. The model is explained in detail and illustrated with reference to the concept of fractions. Key features of the model include don't need boundaries, folding back, and the complementarities of acting and expressing that occur at each level of understanding. The theory is illustrated by examples of pupils' work from a variety of topics and stages. Finally one of the practical applications of the theory, mapping, is explained in some detail.     

The Pseudo-Conceptual and the Pseudo-Analytical Thought Processes in Mathematics Learning

This paper suggests a theoretical framework to deal with some well known phenomena in mathematical behavior. Assuming that the notions conceptual and analytical are clear enough in the domain of mathematical thinking, the notions pseudo-conceptual and pseudo-analytical are proposed and explained. Examples from mathematics classrooms, mathematics exams, and homework assignments are analyzed and discussed within the proposed theoretical framework.The notions pseudo-conceptual and pseudo-analytical proposed in this paper, actually narrow the extension of the notion cognitive by restricting it to the domain of meaningful contexts. Analysis of meaningless behaviors, it is claimed, requires a different theoretical framework. The attempt to analyze meaningless behaviors in the same way as meaningful behaviors is called here the cognitive approach fallacy.     

Education for work: Reflections towards a theory of vocational education

By integrating aspects of a philosophy of work and a sociology of work, this article draws conclusions relevant to framing a theory of vocational education. Education for work is accepted in this paper as the common central meaning of the term vocational education. The two concepts, education and work, are first examined separately: work is seen to include recreational and occupational work, while occupational work is seen to fall on a continuum ranging from constrained to unconstrained occupational work. Education and training are distinguished. After a discussion of the conceptual links which might be drawn between education and work, recommendations are made about the aims, the place, and the curriculum of vocational education within the general institution of education.

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Zusammenfassung Durch die Integration von Aspekten einer Philosophie der Arbeit und einer Soziologie der Arbeit kommt der Verfasser dieses Artikels zu Schlußfolgerungen, die eine Theorie der Berufsbildung entwickeln. Bildung für die Arbeit wird in diesem Artikel als die allgemeine, zentrale Bedeutung des Begriffes Berufsbildung verstanden. Die beiden Begriffe Erziehung, und Arbeit werden zunächst einzeln untersucht: Arbeit umfaßt hier Freizeit- und Berufsarbeit; Berufsarbeit ist ein Kontinuum von mehr oder weniger vorgegebener Tätigkeit. Es wird zwischen Bildung und Ausbildung unterschieden. Im Anschluß an eine Diskussion über begriffliche Gemeinsamkeiten zwischen Erziehung und Arbeit werden Vorschläge über Ziele, Einordnung und Curriculum der Berufsbildung innerhalb des allgemeinen Systems der Erziehung unterbreitet.

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Résumé En intégrant des aspects d'une philosophie et d'une sociologie du travail, cet article tire des conclusions pertinentes pour l'élaboration d'une théorie de l'éducation professionnelle. L'éducation pour le travail est prise dans cet article au sens commun central du terme éducation professionnelle. Tout d'abord, on examine séparément les deux concepts éducation et travail: on considère que le travail inclut les occupations récréatives et professionnelles alors que le travail professionnel est vu en tant qu'un continuum allant du travail professionnel contraint au travail non contraint. On distingue l'éducation de la formation. Après une discussion sur les liens conceptuels qui pourraient être établis entre l'éducation et le travail, des recommandations sont faites sur les objectifs, la place et le contenu de l'éducation professionnelle au sein de l'institution générale de l'éducation.

     

Innovations in adult education: The changing perspective of the post-literacy curriculum in Tanzania

Conclusion The post-literacy Curriculum was developed, on the one hand, to empower the people through the development of appropriate knowledge, skills, and attitudes to improve their living conditions. On the other hand, it was developed to create conditions and facilities for lifelong education. One of the main factors which influenced the reforms was the unsuitability of the post-literacy content: one could argue that it was developed in a top-down manner.Clearly, curriculum design is not merely a technical issue, but rather a cooperative activity among the agents involved. Any innovation in the post-literacy curriculum, therefore, should take into account the views of the target population. This would not only make the curriculum feasible but would also reduce uncertainty about its mission.     

Styles of scientific thinking

It is a main contention of this paper that the history of science is not so much a story of the progressive advance in our understanding and discovery of the facts of nature, but rather, an account of different ways of seeing things; where the things thus seen are to a considerable extent themselves the result of realizational processes operating in terms of some theory or other. But further, such theories are in turn controlled by some respective methodology which has its history: with the latter itself a record of different views about those elements believed to be essential for any adequate constructionof scientific theories. The paper then distinguishes between three views, the rationalist, the empiricist, and the systemic processing of scientific facts; the last-named view operating under the guidance of certain leading maxims and principles. Finally, the paper formulates a triadic type of methodology whose three components mirror the three views just mentioned: the probative, the explicative and the systemic components; which in turn are then shown to generate three corresponding ontologies.     

Research and scholarship: Perceptions of senior academic administrators

This paper reports selected findings from the first stage of a study on the research role within academic work in Australian universities. These findings come from the interview component of the study and discuss the perceptions that senior academic administrators hold on research and scholarship. The analysis of the interviews indicates that research covers a wide and varied range of activities across the disciplines found in a university and therefore needs to be defined broadly. However, research has three major attributes: new knowledge, enquiry and publication of results and views. Scholarship was perceived to be part of the research process, providing the context for good research by adding the element of breadth to the depth of research. In addition, scholarship describes the manner of pursuing a serious, sustained line of enquiry as well as the dissemination process.     

The indigenous and the imported: The New Zealand school system

New Zealand (Aotearoa) was colonized from Britain and the colonizers imposed on the indigenous Maori people a foreign view of education. From then on tradition has vied with local adaptations to produce a school system with substantial traces of the Old Country but with many local features. The curriculum for boys continued to dominate, with that for girls struggling to make itself felt. There has been constant debate about basics and frills though these terms have not been clearly defined. More recently there has been more serious consideration of the curriculum but this has been overtaken by a market forces view of schooling. A new administration system comes into operation on 1st October 1989. The future is unclear but it is reasonable to hope that there will continue a dialectic which may one day produce a genuine synthesis suited to the multicultural nature of Aotearoa.

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Zusammenfassung Neuseeland (Aotearoa) wurde von Großbritannien kolonisiert, und die Kolonialherren zwangen dem einheimischen Volk der Maori ein fremdes Erziehungssystem auf. Von da an konkurrierten Tradition und lokale Anpassungsformen. so daß ein Schulsystem mit wesentlichen Merkmalen des Alten Landes, aber trotzdem mit vielen lokalen Grundzügen aufgebaut wurde. Das Curriculum der Jungen dominiert weiterhin das Curriculum der Mädchen, welches nur mühsam zur Geltung kommt. Es wurde ständig über Grundzüge und Extras debattiert, obwohl diese Begriffe nicht klar definiert sind. In letzter Zeit machte man sich ernsthaftere Gedanken über das Curriculum, dies wurde aber von einer marktorientierten Erziehungsauffassung überholt. Am 1. Oktober 1989 wurde ein neues Verwaltungssystem eingeführt. Die Zukunft ist unklar, aber es besteht begründete Hoffnung, daß eine Dialektik fortgeführt wird, die eines Tages eine echte Synthese hervorbringt, die der multikulturellen Kultur Aotearoas gerecht wird.

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Résumé La Nouvelle Zélande a été colonisée par la Grande Bretagne, et les colonisateurs ont imposé un système d'éducation étranger à la population indigène maori. Dès lors, la tradition et les adaptations locales rivalisèrent pour produire un système scolaire comportant des traces substantielles du vieux pays et de nombreuses caractéristiques locales. Le curriculum pour les garçons n'a pas cessé de dominer alors que celui pour les filles lutte toujours pour s'imposer. Il y a un débat permanent sur l'essentiel et le superflu, bien que ces termes n'aient pas été clairement définis. Récemment, on s'est occupé plus sérieusement du curriculum, mais cet intérêt a été dépassé par une vision de l'instruction déterminée par les forces du marché. Un nouveau système administratif entre en vigueur le l^er octobre 1989. Le futur est incertain mais il est raisonnable d'espérer que se poursuivra la dialectique qui pourrait produire un jour une véritable synthèse adaptéc à la nature multiculturelle d'Aotearoa.

     

Socialist mathematics education: Does it exist?

A recent book by Frank Swetz and others has postulated the existence of the phenomenon socialist mathematics education. In this paper we consider critically Swet's arguments and assertions and suggest an alternative methodology whereby one might attempt to describe socialist mathematics education.     

‘Discrete analysis’ — An approach to fundamental ideas of analysis

The paper investigates conceptual relationships between the Continuous and the Discrete. Differentiation, integration and the fundamental theorem of calculus are based on corresponding operators and a corresponding theorem with functions on finite domains (discrete functions). The final section discusses the possibilities for introducing elements of a discrete analysis into the mathematics curriculum.     

Understanding: ‘Knowledge’, ‘Belief’ and ‘Understanding’

The following paper is intended as an exercise in friendly criticism of one of Harvey Siegel's and Mike Smith's (Knowing, Believing and Understanding, this volume). I'm in substantial sympathy with the general thrust of their paper and my remarks merely provide some criticism of their discussion's conceptual coherence and clarity and a correspondingly slightly adjusted version of what they have to say. My focus is limited to the conceptions of knowledge, belief and understanding and their inter-relationships in terms of which they offer suggestions to science educators.     

Child-methods in secondary mathematics

An earlier research project, the Concepts in Secondary Mathematics and Science (Mathematics) project, identified both a hierarchy of levels of understanding in different areas of secondary mathematics, and a number of particular errors which were made by significant proportions of the children tested. Preliminary consideration of these errors and the strategies which appear to have given rise to them suggests that the use of informal naive methods which are limited in their applicability is widespread even at fourth-year level. The suggestion is made that there may be two systems of mathematics coexisting in the secondary school classroom: the formal taught system, and a system of child-methods which are based upon a counting, adding-on or building-up approach, and by which children attempt to solve mathematical problems within a human-sense framework. The difficulties which some children appear to experience in mathematics is suggested to be due in part to these children's non-initiation into the formal taught system. The implication of such a view for teaching and research are indicated.