Can Mathematics Education and History of Mathematics Coexist?

Despite the wide interest in combining mathematics education and the history of mathematics, there are grave and fundamental problems in this effort. The main difficulty is that while one wants to see historical topics in the classroom or an historical approach in teaching, the commitment to teach the modern mathematics and modern mathematical techniques necessary in thepure and applied sciences forces one either to trivialize history or to distortit. In particular, this commitment forces one to adopt a Whiggish approach to the history of mathematics. Two possible resolutions of the difficulty are (1) radical separation – putting the history of mathematics on a separate track from the ordinary course of instruction, and (2) radical accommodation – turning the study of mathematics into the study of mathematical texts.     

Content and form in mathematics

The present article is based on the premise that mathematical activity exhibits an important duality. On the one hand pre-school children do mathematics spontaneously; the mathematics they do is often called informal arithmetic. On the other hand, the mathematics we teach is a cultural product developed by generations of mathematicians (Byers and Herscovics, 1977). Whatever premium school mathematics puts on individuality and original thought, it seems obvious that the mathematics done in school is neither spontaneous nor informal; the bulk of it is formal mathematics. We expect our pupils to do mathematics in a conventionally acceptable manner. In short, mathematics is both an individual and a social activity. We shall endeavour to show that duality in mathematical activity reflects a duality of mathematics as a discipline. The latter is a duality between content and form; in fact, it is the importance of form in this discipline that resulted in mathematics being classified as a formal science.     

“Did I mean what I said?” — A reply to McCormick

In this note, I argue that the distinction between what one says and what one means by it is no more problematic (and no less) than the equally important distinction between evidence and theory. Both, I suggest, take a particular form in literate discourse.     

Orson Scott Card’s Ender and Bean: The Exceptional Child as Hero

Orson Scott Cards school stories in outer space, Enders Game and Enders Shadow, purportedly occur at the same time and tell the same story, but from the perspectives of two different child protagonists. Scenes in Enders Shadow even reproduce text from Enders Game. Nevertheless, 14 years elapsed between the publications of the two books. This essay brings child studies and exceptionality research to bear on the two novels, analyzing the development of Cards ideology regarding his view of the exceptional child and his view of the nature of heroism in a post-modern world.     

Consciously Thinking About Consciousness

Merker hypothesized that because mobile creatures move around and must constantly readjust their map of the world and because the demands are so great for continually processing information for a map of the world, evolution has created a space in the brain where such preprocessing has been eliminated. This space he calls consciousness with the preprocessing in the unconscious. Using Merker's view, this paper explores the process of changing paradigms, giving advice and the implications for learning.     

Education and the Contested Meanings of ‘Global Citizenship’

This article examines the oftentaken-for-granted educational policy and curricular discourses of globalization and global citizenship within their larger cultural, political, and economically uneven histories and unequal consequences. Drawing upon evidence from recent scholarship on the implications for classroom pedagogy embodied inefforts to internationalize curricula in the North American university contexts, as well as specific related efforts in one Canadian context, the University of British Columbia,this article unpacks the contested meanings of global citizenship from above and below.The author analyzes three dominant curricular discourses of global citizenship. Far from weakening the Canadian state and building instances of transnational democratic educational communities, these dominant discourses may actually reinforce notions of Canadian gendered and racialized nation-building and nationalism. The article raises questions about what an alternative curricular and educational policy discourse from below premised upon efforts to decolonize curricula might look like. By way of provisional conclusion, the author discusses some promising examples of such an alternative,showing how they depend on seeing the relations between the local and global neither as fixed abstractions nor as a slogan system to be applied in absolutist racial, geographical,national, or culturally essentialist terms.     

A multivariate approach to the assessment of quality

Most quality assessment studies utilize a univariate approach to the assessment of quality. While the univariate approach has some merit, this study presents an alternative—the multivariate approach. Factor analysis identified three clusters of variables potentially linked to quality. These clusters, labeled faculty research, input, and student, were later used as bases for measuring departmental excellence. Seven indicators based on these clusters were developed. The indicators that combined the first and second clusters and the indicators that combined all three clusters appeared to have produced the best estimates of departmental excellence.     

Higher educational impact on student objectives: Longitudinal change in Clark-Trow “educational philosophies”

This study explores the impact of higher education on 3,942 arts and sciences students as measured by change in their freshman and senior ratings of four Clark-Trow educational philosophies: vocational, academic, collegiate, and nonconformist (Clark and Trow, 1966). A repeated measures analysis of variance was applied to each of the four philosophies, controlling for sex, entering year, major, parents' educational background, scholastic aptitude, and academic motivation. As expected from Clark-Trow theory, students showed significant increases in academic and nonconformist philosophies, and decreases in vocational and collegiate philosophies. Relationships between independent variables and freshman-senior change suggested post hoc reinterpretation of the dimensions underlying the Clark-Trow phenotypes: i.e., from identification with the college and involvement with ideas to social interests and academic interests (the latter suggested by Terenzini and Pascarella, 1977).     

La Croissance des depenses publiques d'education en Allemagne: 1829–1989

Résumé Le retournement de conjoncture du début des années 1970 s'est traduit par un renouveau des travaux relatifs aux cycles longs de type Kondratieff. Face à la fluctuation atypique de certaines variables (prix, production, etc ...), les acquis théoriques antérieurs semblaient avoir épuisé leur capacité explicative. Il devenait nécessaire d'énoncer et de vérifier des hypothèses nouvelles pour expliquer les traits originaux de la crise de structure qui se dessinait. Les résultats des travaux développés au sein du groupe de recherche en Histoire Quantitative et Théories du Long Terme de l'Université de Montpellier I répondent à cette nécessité. Ils introduisent une relation entre les dépenses engagées pour le développement des hommes (salaires, éducation, santé etc ...) et celles consacrées à la croissance matérielle.

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The downturn in the economic situation in the early 1970s gave rise to new work on long-term Kondratieff cycles. Given the atypical fluctuation in some variables (prices, production, etc.), previous theories appeared to have exhausted their ability to explain events. To explain the original features of the structural crisis that was developing, it has become necessary to put forward and test new hypotheses. The results of work carried out by the research group on Quantitative History and Long-Term Theories at the University of Montpellier I meet this requirement. They introduce a relationship between expenditure on human development (wages, education, health, etc.) and that devoted to material growth.

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Zusammenfassung Die Verschlechterung der wirtschaftlichen Situation in den frühen 70ern gab der Arbeit an langfristigen Kontratieff — Zyklen Aufschwung. In Anbetracht der atypischen Fluktuation einiger Variablen (Preise, Produktion, usw.) schienen vorangegangene Theorien die Ereignisse nicht mehr erklären zu können. Um die Ursachen der sich entwickelnden strukturellen Krise erklären zu können wurde es notwendig, neue Hypothesen aufzustellen und auszuprobieren. Die Ergebnisse der von der Forschungsgruppe über Quantitative Geschichte und Langzeittheorien an der Universität von Montpellier I durchgeführten Studien erfüllen diese Anforderungen. Sie stellen Ausgaben für die menschliche Entwicklung (Lohn, Bildung, Gesundheit usw.) und Ausgaben für materielles Wachstum in Relation.

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Resumen La degradación de la situación económica a comienzos de los a~nos setenta ha originado un nuevo trabajo sobre ciclos Kondratieff de largo plazo. Dada la fluctuación atípica de algunas variables (precios, producción, etc.), las teorías previas aparentemente habían agotado su capacidad de explicar los hechos. A efectos de explicar las características originales de la crisis estructural que se estaba desarrollando, surgió la necesidad de adelantar y comprobar nuevas hipótesis. Los resultados del trabajo realizado por el grupo de investigación de Historia Cuantitativa y Teorías de Largo Plazo de la Universidad de Montpellier I satisfacen estos requisitos. Ellos introducen una relación entre los gastos y esfuerzos destinados al desarrollo humano (salarios, educación, sanidad, etc.) y aquellos dedicados al crecimiento material.

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A Textbook Case of the Nature of Science: Laws and Theories in the Science of Biology

The two concepts law and theory are among the most important elements of the nature of science. They represent both the tools and products of science itself. Unfortunately, the variable meanings and use of these terms in general discourse and in other school disciplines results in much confusion with respect to their proper application in a science context. The project included the design of a six-part model definition for law and theory based on a review of the literature of the philosophy of science with special reference to biology. These model definitions were then compared with those provided in a range of U.S. secondary school biology textbooks. The majority of all current major U.S. secondary school biology texts were reviewed and analyzed with respect to how the concepts of law and theory were defined and applied, in an attempt to determine whether students and teachers using such texts would gain an accurate impression of these terms and the distinction between them. This study focuses on biology instruction since a life science course is completed as a graduation requirement by virtually all U.S. high school students and as such serves as a widely shared educational experience across the nation. The term law is rarely defined in any text but various laws such as those found in genetics are frequently included as examples. The term theory is frequently defined but with a wide range of completeness of the definitions. Only rarely are theories in biology included as examples.     

“Anything goes”⋯ connecting children to literature: A personal view

In a recent article in the weekly Times Educational Supplement,Roger Knight attacked some of the various kits and packages which are currently on offer to support teachers and children in preparing for examinations in English literature. Compilers are accused of trivializing the study of literature and distracting teachers and children away from a proper concern with the language of the novel or poem. Some aspects of the compilations are seen by Knight as being akin to party games from a Christmas cracker, and in becoming involved, the teacher for whom literature matters as literature is likely to feel superannuated, disregarded and excluded from the party. Such compilations numb the texts and numb the understanding, Knight continues, and the mentality they so fully express derives partly from the erroneous and fashionable belief that any response to a work of literature is valuable. In this paper, Michael O'Hara offers a personal view on literature teaching in British schools and takes issue with Knight's view that certain approaches to the teaching of literature can be ruled out a priori. He suggests that in the right hands all kinds of approaches are possible. In short, in the teaching of literature, and as long as imagination and judgment prevail, anything goes!Michael O'Hara is a Lecturer in Education at the University of Ulster, Northern Ireland. He has taught in a College of Education in the Republic of Ireland, in a Belfast secondary school, and briefly in Nigeria.     

Critical thinking and the Spock Fallacy

Conventional instruction in critical thinking technique tends to conflate logical thinking with rational or good thinking. But in doing so, it ignores the creative and intuitive functions of rationality, thereby encouraging a mechanically rote approach to textual analysis, problem solving and problem construction. This overemphasis upon straightforwardly logical calculus of justification functions, and its concomitant deemphasis of intuitive pattern of discovery ones, constitute the Spock Fallacy. Its epistemological weakness and pedagogical hazards are explored.The author of two books and numerous articles, one of his specialties is philosophy of education and critical thinking.     

School effectiveness and effectiveness indicators: Parents', students', teachers', and principals' perspectives

The purpose of this study was to examine perceptions of school effectiveness among parents, students, teachers and principals, and differences in their perceptions across school levels (primary vs. secondary) and types (religious state vs. non-religious state). Finally, the results were compared with school effectiveness indicators in the literature. A sample of all categories of subject in eight Israeli schools were interviewed (N=64). Analysis reveals that parents stressed school outputs, teachers stressed their skills and teaching processes, students emphasized both inputs and outcomes, and principals chiefly inputs. Religious schools gave more weight than others to values. These findings are broadly in line with the literature, but any comprehensive analysis of school effectiveness demands reference to all aspects of the concept.

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Zusammenfassung Ziel dieser Studie war eine Untersuchung darüber, für wie effektiv Eltern, Schüler, Lehrer und Direktoren die Schule beurteilen und wie unterschiedlich diese Wahrnehmungen auf den verschiedenen Schulebenen und -arten sind (Primarstufe und Sekundarstufe, religiös und nicht-religiös). Danach wurden die Ergebnisse mit Indikatoren über Schuleffektivität in der Literatur verglichen. Man führte Interviews mit den Zielgruppen in acht israelischen Schulen durch (N=64). Die Analysen zeigten, daß Eltern am meisten Wert auf Schulergebnisse legten, Lehrer auf ihre Fähigkeiten und Lehrprozesse, Schüler empfanden Lernen und Ergebnisse als gleich wichtig und Direktoren interessierte hauptsächlich das Lernen. Religiöse Schulen legten mehr Gewicht auf Wertvorstellungen als andere. Diese Ergebnisse stehen im allgemeinen mit der Literatur in Einklang, aber jede umfassende Analyse der Schuleffektivität erfordert die Berücksichtigung aller Aspekte des Konzepts.

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Resumen El propósito de este estudio residía en examinar las percepciones referentes a la eficacia de la escuela entre padres, estudiantes, docentes y directores, y las diferencias en sus percepciones a través de los diferentes niveles (primario vs. secundario) y tipos de escuelas (religiosas vs. laicas). Finalmente, los resultados se compararon con los indicadores de eficacia escolar establecidos en la literatura. Se ha interrogado a un grupo representativo de todas las categorias de sujetos en ocho escuelas israelíes (N=64). Los análisis revelan que los padres ponían énfasis en los rendimientos escolares, los maestros enfatizaban sus habilidades y procedimientos de enseñanza, los estudiantes enfatizaban tanto los esfuerzos como los resultados, y los directores, principalmente, los esfuerzos. Las escuelas religiosas, más que otras, adjudicaban una mayor importancia a los valores. Estos resultados coinciden de forma global con los datos que indica la literatura, pero todo análisis integrado de eficacia escolar exige la referencia a todos los aspectos del concepto.

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Résumé Le but de cette étude était d'appréhender comment parents, élèves, enseignants et directeurs d'école perçoivent l'efficacité scolaire, et si cette appréciation est différente selon le niveau scolaire (primaire ou secondaire) et le type d'école (statut religieux ou laïque). En dernier lieu, les résultats ont été comparés avec les indicateurs de l'efficacité scolaire relevés dans la littérature. Un échantillon de chaque groupe de sujets de huit écoles israéliennes a été soumis à des entretiens. L'analyse révèle que les parents font valoir le rendement scolaire, les enseignants s'attachent à leurs compétences et leurs méthodes pédagogiques, les élèves soulignent à la fois le contenu des programmes et les résultats scolaires, et les directeurs d'établissement se concentrent en premier lieu sur les contenus pédagogiques. Les écoles religieuses quant à elles donnent davantage de poids aux valeurs. Ces résultats correspondent d'une façon générale à ceux de la littérature, mais toute analyse complète sur l'efficacité scolaire doit couvrir tous les aspects du concept.

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Exploring the Concept of ‘Habitus’ with Preservice Teachers Through the Use of Popular School Films

The thesis of this paper is that it is a productive venture to introduce preservice students to social theories that are typically not a part of teacher preparation programs. Examples of these theories are those associated with terms such as habitus, the carnivalesque, power / knowledge, seduction, detournement, the spectacle,la perruque, (the wig), and the art of making do,as well as others. In this paper, I describe a project that involved exploring the theory of habitus to problematize the tendency of preservice students to not think in terms of relations between what goes on in the classroom and what goes on in society. I introduced habitus by having preservice students read selected print materials and by having them view, analyze, and respond in writing to popular school films. Students also analyzed their experiences in classrooms in terms of habitus, making a connection between theory and practice.     

Early childhood family education

If they came with instructions, then you wouldn't need us reads the poster on the classroom door. It is a pleasant morning in early September and my twelve-month-old daughter, Laura, and I are going to our first day of school. Laura and I enter the infant and toddler room with anticipation.Barb Rose is a free lance writer living in St. Paul, MN, and the mother of a 2-year-old named Laura.     

Reducing Abstraction: The Case of School Mathematics

There is a growing interest in the mathematics education community in the notion of abstraction and its significance in the learning of mathematics. Reducing abstraction is a theoretical framework that examines learners behavior in terms of coping with abstraction level. It refers to situations in which learners are unable to manipulate concepts presented in a given problem; therefore, they unconsciously reduce the level of abstraction of the concepts involved to make these concepts mentally accessible. This framework has been used for explaining students conception in different areas of undergraduate mathematics and computer science. This article extends the applicability scope of this framework from undergraduate mathematics to school mathematics. We draw on recently published research articles and exemplify how students behavior can be described in terms of various interpretations of reducing abstraction level.     

Normative Support for Improving Undergraduate Education in Teaching-Oriented Colleges

A number of recommendations to improve undergraduate education have been put forth in the higher education literature. However, we maintain that the implementation of these recommendations at our colleges and universities is unlikely if norms or group standards of appropriate and inappropriate behavior do not support these recommendations. Consequently, this study investigated the normative support for 6 selected recommendations to improve undergraduate education at teaching-oriented colleges. Results indicate support for 3 of the 6 recommendations: systematic program of advisement, providing students with prompt, formative feedback on assignments, and fostering an egalitarian classroom climate. These are the same recommendations supported by norms at higher education institutions that are more focused toward research and graduate education. The implications suggest that the normative structure that supports teaching improvement efforts may be more similar among various institutions than previous research might have indicated.     

Transitions to school-college partnership: a phenomenological inquiry

This study took place during the first 2 years of a partnership between three cooperating elementary schools and a college of education. The purpose of the interpretive phenomenological study was to learn from cooperating teachers, student-teachers and college instructors about the transitions that accompanied the change in the culture of schooling they experienced through the newly established partnership, and to attempt to understand their experiences through their explanations of the transitions they identified. On the basis of the data analysis, we identified seven transitions experienced by the participants: From discrete processes to parallel, open processes. From individual work to collaboration. From seeing the other as an object to seeing the other as a critical friend. From leading to responding. From the conception of knowledge as an object to the conception of knowledge as a process. From viewing learning as a linear process to viewing it as a spiral process. From viewing their work environment as a closed system to viewing it as an open system. In this paper, we describe and provide examples of each of these transitions and show how they are interrelated and how each one is both a cause and an effect of the others. These findings increased our knowledge about how the school and college altered each others consciousness.     

Connecting Teacher Beliefs to the Use of Children’s Literature in the Teaching of Mathematics

This article presents examples that illustrate how teachers use childrens literature in the teaching of mathematics. The examples are related to four curriculum ideologies that have influenced mathematics education in the USA for the last 75 years. It discusses why it is relevant to help teachers understand the ideological positions that influence their use of childrens literature during mathematics instruction, summarizes the four ideological positions, and presents results of a study of how teachers ideological positions relate to their use of childrens literature in the teaching of mathematics. The study examines two research questionsCan an instructional tool be developed that will highlight for teachers the different ways in which they and others use childrens literature to teach mathematics? and Can that instructional tool stimulate teacher discussion and reflection about their own beliefs and the ideological nature of the instructional environment in which they learned (as students) and teach (as teachers)? Study results indicate that both questions can be answered in the affirmative.