The concept of chance in everyday teaching: Aspects of a social epistemology of mathematical knowledge

The paper analyzes the relationship between the epistemological nature of mathematical knowledge and its socially constituted meaning in classroom interaction. Epistemological investigation of basic concepts of elementary probability reveals the theoretical nature of mathematical concepts: The meaning of concepts cannot be deduced from more basic concepts; meaning depends in a self-referent manner on the concept itself. The self-referent nature of mathematical knowledge is in conflict with the linear procedures of teaching. The micro-analysis of a short teaching episode on the concept of chance illustrates this conflict. The interaction between teacher and students in everyday teaching produces a school-specific understanding of the epistemological status of mathematical concepts: the concept of chance is conceived of as a concrete generalization, which takes chance as a fixed and universalised pattern of explanation instead of unfolding potential and variable conceptual relations of chance or randomness and developing the theoretical nature of this concept in an appropriate way for students' comprehension.     

Student Self-Assessment in HOCS Science Examinations: Is There a Problem?

A specially-designed self-assessment questionnaire (SAQHOCS), containing higher-order cognitive skills (HOCS)-type questions, was administered to 71 biology majors, enrolled in a four-year college program. The gap between students' self-assessment marking, and that of their HOCS-biased teachers (the authors), is accounted for by the prevailing LOCS-orientation and the testing culture—a total separation between testing and learning—in contemporary science teaching. The majority of the students in the study evaluated themselves as capable of self-assessment, and felt reasonably confident in doing so. They were quite reserved as far as the applicability of the self-assessment method to nonalgorithmic (correct/incorrect) questions is concerned. The results of this study suggest that the potential for student self-assessment within college science teaching and learning exists. However, still a great purposed effort in HOCS-oriented teaching and learning is required in order for the student self-assessment practice to become a routine integral component of HOCS science examinations.     

Cognitive models and problem spaces in “purely random” situations

As part of a study on the natural interpretations of probability, experiments about elementary purely random situations (with dice or poker chips) were carried out using students of various backgrounds in the theory of probability. A prior study on cognitive models which analyzed the individual data of more than 600 subjects has shown that the most frequent model used is based on the following incorrect argument: the results to compare are equiprobable because it's a matter of chance; thus, random events are thought to be equiprobable by nature. The present paper is divided into two parts. In the first, the findings of a series of experiments are summarized. In the second, the following two hypotheses are tested: (1) Despite their incorrect model, subjects are able to find the correct response. (2) They are more likely to do so when the chance aspect of the situation has been masked. An experiment testing 87 students showed, as expected, that there is a way to induce the utilization of an appropriate cognitive model. However, the transfer of this model to a classical random situation is not as frequent as one might expect.     

The impetus theory: Between history of physics and science education

Through a physical, historical and epistemological analysis it is shown how much is wrong with the idea that relates impetus theory to a non-grown-up physical and epistemological conception. Indeed, it yields that impetus theory of Buridan and Oresme can be formalised and can furnish us a natural, non-violent interpretation of (classical) mechanics as well as a more general, physical hermeneutics of the world. Then, the possible relevance of impetus theory for science education is strongly pointed out.     

Whitehead and a new look at teaching elementary science

If Whitehead is right, science teachers who try to increase student interest by making the science they teach more pure and by covering more material are going about their work in just the wrong way. Science, for purposes of precision in measurement, translates the dynamic world of feeling and force, of causal efficacy (for example, the San Francisco earthquake), into a static representation spatialized and given presentational immediacy (for example, the Richter scale). But notice that the Richter scale isn't very interesting (even as abstract art) apart from its connection, via symbolic reference, to the earthquake. Such reference is essential to give both a sense of reality and a feeling of interes to the subject, but it makes the science less pure, and it takes more time to cover the material. An example of teaching pure and impure formal logic is given as a case study.     

Much ado about nothing: Science and hermeneutics

Insofar as the basic concepts (in particular, meaning, interpretation, text) and theses of Martin Eger's hermeneutics of the sciences are intelligible they are not distinctive, and insofar as they are distinctive they are unintelligible and/or lack backup arguments. Ambiguity characterises much of the argument. James Cushing's elaboration of hermeneutics, using the history of quantum mechanics as an illustration of its supposed strengths, does nothing to rescue the position against the criticisms here levelled at Martin Eger's formulations of it.     

Training,stability and control

This paper presents a system-theoretic approach to the analysis of the problem of training formally relating it to the control of an abstract dynamic system, the adaption automaton of the trainee. The utility of this formulation and the possibility of basing real training strategies upon it are discussed, and it is argued that further constraints upon the automaton are both necessary, and available, in so far as the theory corresponds to practical reality. The minimal constraints generate an extended theory in which training is related to the stability of the adaption automaton. More practical constraints lead to theoretical foundations for strategies of feedback or adaptive training. Corresponding to each set of constraints a training theorem is proved which demonstrates that the constraint is adequate to lead to a simple universal training strategy.Although this paper is highly theoretical it is argued that the formal concepts introduced correspond to intuitive models of the phenomena of learning and training which are implicit in the design of many training systems. It is hoped that the formal analysis will throw new light on these implicit assumptions and help to clarify discussion of practical approaches to training, including the possibility of computer-aided instruction given on our present level of knowledge of human cognitive skills or individual students.     

Preservice elementary teachers' explicit beliefs about multiplication and division

This study, conducted in the United States, was designed to assess the extent to which the beliefs, multiplication always makes bigger and division always makes smaller, are explicitly held by preservice elementary teachers. Paper and pencil instruments were administered to 136 preservice elementary teachers in a large university in the southeastern United States. Responses on the instruments and in interviews provide insight into the sources of preservice teachers' beliefs and into the relationship among preservice teachers' beliefs, computational skills, and performance on word problems.     

Can we make sense of learning theory?

This article discusses a range of theoretical models of the learning process that have been published in recent years, and examines whether it is possible to relate them systematically to one another. Problems in such an enterprise include those of terminology; globe trotting; the development of separate schools of research that have proceeded largely independently, each with their own central concerns, perspectives, and methodology; and the problem of whether the various different activities popularly described by the word learning do in fact have anything much in common. Despite these difficulties, however, the article argues that the various major models each contain certain key concepts, not necessarily identical in each model, from which it is possible to produce a meta-model - a framework or cognitive map within which the individual models can be located and therefore better understood. The article ends by proposing one such meta-model, and suggesting some ways in which it might be used.     

State Testing and Inquiry-Based Science: Are They Complementary or Competing Reforms?

The effect of district strategies for improving high-stakes test scores on science teachers practice is explored in case studies of six middle schools in six Massachusetts districts. At each school, science teachers, curriculum coordinators, principals, and superintendents shared their strategies for raising scores, their attitudes towards the test, the changes that they were implementing in their curriculum and pedagogical approaches, and the effects that the test was having on staff and on students. Results from these case studies suggest that districts chose markedly different strategies for raising scores on high stakes tests, and that the approaches taken by districts influenced the nature of pedagogical and curriculum changes in the classroom. District strategies for raising scores that were complementary to the districts prior vision of science reform tended to cause less teacher resentment towards the test than strategies that departed from previously adopted goals. Differing effects on teachers in socio-economically advantaged, middle, and challenged districts are discussed.     

Paranoia: A few final remarks

Paranoid is given a tolerably clear meaning. It is suggested that conceptual understanding must be the first step before seeing how far unconscious or other pressures distract teachers and others from retaining a firm grasp of concepts. Concepts are not relative.     

The non-education of Kurds: A Kurdish perspective

Educational provision in Kurdistan (embracing parts of Iran, Iraq, Syria and Turkey) violates most of the language-in-education requirements of international law. The same is true for Kurds in diaspora in most parts of the world. Linguistic and cultural genocide is attempted, with the tacit complicity of the West. The future of Kurdish education depends to a large extent on the political situation in the Middle East. Political solutions are needed before educational problems can be tackled. Within international law, a new interpretation by the UN Human Rights Committee of Article 27 in the International Convenant on Civil and Political Rights (UN 1986) might give grounds for hope for the future.

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Zusammenfassung Bildungsmöglichkeiten in Kurdistan (einschließlich Teilen Irans, Iraks, Syriens und der Türkei) entsprechen nicht den internationalen Forderungen zur Sprachpolitik in der Bildung. Das gleiche gilt für Diaspora Kurden in den meisten Teilen der Welt. Mit stillschweigender Übereinkunft des Westen wird der linguistische und kulturelle Massenmord angestrebt. Die Zukunft der kurdischen Bildung hängt größtenteils von der politischen Situation im mittleren Osten ab. Bevor jedoch das Bildungsproblem gelöst werden kann, braucht man politische Lösungen. Innerhalb des internationalen Gesetzes könnte eine neue Auslegung des Artikel 27 im Internationalen Convent über zivile und politische Rechte (UN 1968) durch das Menschenrechtskommittee der Vereinten Nationen Hoffnung für die Zukunft bringen.

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Resumen La educación prestada en Kurdistán (que comprende partes de Irán, Iraq, Siria y Turquía) viola la mayor parte de las exigencias que el derecho internacional establece en cuanto a la educación en la lengua correspondiente del individuo. Esto también rige para los kurdos, que se encuentran en la diáspora, en muchas partes del mundo. Con ello se intenta un genocidio cultural, con la complicidad tácita del mundo occidental. El futuro de la educación kurda depende en gran medida de la situación política de Oriente Medio. Se requieren soluciones políticas que permitan abordar los problemas de la educación, Dentro del derecho internacional, puede dar lugar a esperanzas una nueva interpretación que el Comité de Derechos Ilumanos de las Naciones Unidas hizo del artículo 27 en el Convenio Internacional de Derechos Civiles y Políticos (UN 1986).

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Résumé Les formes d'enseignement dispensé au Kurdistan (partagé entre l'Iran, l'Iraq, la Syrie et al Turquie) violent la plupart des exigences envers les langues dans l'enseignement établies par le droit international. La même constatation est valable pour la diaspora kurde disséminée sur la quasi-totalité de la planète. Il ya a en fait tentative de génocide linguistique et culturel, avec la complicité tacite de l'Occident. L'avenir de l'éducation kurde dépendant dans une large mesure de la situation politique au Moyen-Orient, les problèmes pédagogiques ne peuvent être abordés que si la situation politique est résolue. Dans le adre du droit international, une nouvelle interprétation de l'article 27 de la Convention internationale des droits civils et politiques par le Comité des droits de l'hommes (ONU 1986) pourrait donner un nouvel espoir pour l'avenir.

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The applicability of democratic principles and the mission of the university

The university's mission of serving as a fortress for freedom of speech, inquiry, and criticism of established truths is inextricably connected with democracy. Such a mission can be pursued only in a climate of genuine political and academic freedom. Three other aspects of democracy in the university are discussed: equality of access, university governance, and the public utility of the university. It is argued that equal representation or one man one vote cannot be applied in university governance because it comes into conflict with the notion of competence—a conflict that cannot be resolved by taking votes. Interest groups inside and outside of the university cannot be allowed to interfere with the free pursuit of knowledge. The internal audit, that is, peer review (by, ultimately, the international community of scholars) has to take precedence over vertical review by government bureaucracy.     

Wherever I look,I find

The best source of infant learning toys is the trash can. A noted authority on infancy made this statement...me. Many commercially made toys and learning materials are available. Some of them were developed out of a firm knowledge of developmental tasks of infancy and with aesthetic sensitivity. Unfortunately they cost, and many of us who have our own infants or work in day care do not have resources for a lot of commercial materials.Mr. Altham taught infants and toddlers while studying at Bank Street College. After teaching in a Head Start program for four years, he moved to his current job as teacher/director of a college campus child care center.     

Different aspects in designing mathematics education: Three examples from the Freudenthal Institute

Three teaching designs are offered. They were developed for 12 to 16 year old pupils taking into account their personal differences in terms of gender (in Making Up the Cube), culture (in Mecca), knowledge, and use of calculators in The integrating calculator. The designs concern the representation of three-dimensional objects in the plane, directions and great circles on the sphere, and exploration of the integrating calculator.     

Three-dimensional symmetry

The errors made when high-school students locate reflections in horizontal, vertical or slanting mirrors were examined. A questionnaire consisting of thirty-six perspective drawings, each portraying an upright transparent cube containing a two-sided mirror, was given to 394 Australian students from grades 7, 9 and 11. On a corner, an edge, or a face of each cube was drawn either a point or a straight-line segment. The task was to draw its image (reflection). Analysis of errors revealed a strong tendency towards horizontal or vertical sliding, suggesting that the edges of the cube act in much the same way as a horizontal-vertical background grid does in the two-dimensional case.     

Sensing the Impact of Elementary School Science Reform: A Study of Stakeholder Perceptions of Implementation,Constructivist Strategies,and School–Home Collaboration

This case study of a midwestern school districts reform effort explored the treatment fidelity of the Science: Parents, Activities, and Literature (Science PALs) project. Data were collected from the perspectives of three stakeholders: students, school district, and parents. Students and school district supervisors perceptions indicated that teachers were implementing predicted curricular and instructional features promoted by the Science PALs project. Students perceptions of their current teachers classroom performance did not parallel the supervisors perceptions of the teachers implementation. Parents perceptions and comments indicated a high level of satisfaction with the implemented features. Collectively, the three perspectives suggest that the Science PALs project was successful in changing the classroom practices of K–6 teachers involved in the professional development. The results of this study appear to support the findings of other recent studies that indicate the need for extended professional development and support to fully implement changes in elementary school science instruction.     

Educational language planning and linguistic identity

There are cases in which a high form of a language is taught and used in formal situations, but linguistic variation is also caused by geography, ethnicity and socioeconomic class. Certain variants are regarded as inferior and restricted in expressive capacity, and are disadvantageous. The paper suggests that it is possible to map each person's linguistic identity in two dimensions: the number of languages spoken, and the situation-specific variants of each language. Further, it is argued that the distance between a low variant and a high standard form of a language may present to the low learner of a standardized mother tongue a barrier just as great as that posed by the learning of a related foreign language to a speaker of the high variant. It is proposed that greater tolerance be exercised in acceptance of variation and in recognition of linguistic identity, so that this can be built on in the necessary and desirable expansion of linguistic competence, rather than being devalued. The relevance of the communicative approach to language teaching is touched on.

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Zusammenfassung Es gibt Fälle, in denen eine hohe Form einer Sprache gelehrt und in formellen Situationen gebraucht wird, aber linguistische Veränderungen werden auch durch Geographie, ethnische Identität und sozialkonomische Klassen hervorgerufen. Gewsse Varianten gelten hinsichtlich ihres Ausdrucksvermögens als unterlegen und restriktiv und sind unvorteilhaft. In dem Artikel wird vorgeschlagen, die linguistische Identität jedes Einzelnen in zwei Dimensionen darzustellen: die Anzahl der gesprochenen Sprachen und die situationsspezifische Variante jeder Sprache. Weiterhin wird argumentiert, daß die Distanz zwischen einer niedrigen Variante und einer hohen Standardform einer Sprache für den niedrigen Lernenden einer standardisierten Muttersprache ebenso schwierig ist wie das Erlernen einer verwandten ausländischen Sprache für denjenigen, der die hohe Form spricht. Es wird vorgeschlagen, größere Toleranz durch Akzeptieren der Variation und Anerkennung der linguistischen Identität zu üben, so daß man darauf die notwendige und wünschenswerte Erweiterung der linguistischen Fähigkeit aufbauen kann anstatt sie zu entwerten. Die Relevanz des kommunikativen Ansatzes zum Unterrichten einer Sprache wird kurz behandelt.

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Résumé Il existe des cas où une haute forme d'une langue est enseignée et utilisée dans des situations formelles, mais la différentiation est causée également par la géographie, l'ethnicité et la classe socioéconomique. Certaines variantes sont considérées inférieures, d'une capacité expressive limitée, et sont désavantageuses. L'article suggère qu'il est possible de tracer l'identité linguistique de chacun dans deux dimensions: le nombre de langues parlées, et les variantes de chaque langue reliées à des situations spécifiques. En plus, on raisonne que la distance entre une variante basse d'une langue et la haute forme standard peut présenter à l'apprenant, ayant la forme basse d'une langue maternelle standardisée, une barrière aussi grande que celle qui se dresse devant celui qui parle la variante haute et qui apprend une langue étrangère apparentée. On propose donc qu'une tolérance plus large soit exercée en acceptant la variation et en reconnaissant l'identité linguistique, afin que cette dernière serve de fondation pour l'élargissement nécessaire et souhaitable des compétences linguistiques, plutôt que d'être dévalorisée. La relevance de l'approche communicative à l'enseignement des langues est mentionnée.

     

The decline of the modern paradigm in education

The education system is widely perceived to be functioning poorly, and this paper argues that there is in fact a deep crisis caused by the gap between the system and the changing world that surrounds it. Post-modern conditions stress the production of information rather than the productivity of land or capital, and a relativistic world-view rather than adherence to a fixed religion or ideology, which in turn gives rise to pluralism and shifting social frameworks. At the same time, the pursuit of Truth is no longer seen as the highest goal in life, and the possession of proof of education is no longer a guarantee of access to the Good Life. Educational institutions are dysfunctional because they remain wedded to outmoded parameters in their aims, activities, structures, methods and perceptions of their clientele. Through a wide-ranging review of recent conceptual debate and assessment of social trends, the paper explores the implications of radical contemporary changes in the parameters: the search for a new paradigm of education has only just begun.

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Zusammenfassung Das Bildungssystem wird weitgehend als unzulänglich betrachtet, und dieser Artikel argumentiert dahingehend, daß es in der Tat eine tiefgreifende Krise gibt, die durch die Kluft zwischen dem System und der sich ändernden Welt um es herum ist. Postmoderne Bedingungen legen eher Wert auf die Erstellung von Informationen als auf die Produktivität von Land oder Kapital. Man zieht eine relativistische Weltanschauung einer festgelegten Religion oder Ideologie vor, die wiederum Pluralismus und sich ändernde soziale Rahmenbedingungen fördert. Gleichzeitig wird die Suche nach der Wahrheit nicht länger als oberstes Ziel im Leben angesehen und der Besitz von Bildungsnachweisen ist nicht länger eine Garantie für ein gutes Leben. Bildungsinstitutionen funktionieren nicht mehr, weil sie in ihren Zielen, Aktivitäten, Strukturen, Methoden und Wahrnehmungen ihrer Zielen, Aktivitäten, Strukturen, Methoden und Wahrnehmungen ihrer Zielgruppen mit unzeitgemäßen Normen verknüpft sind. Mittels eines weitreichenden überblicks über jüngste Debatten über Konzepte und die Einschätzung sozialer Trends, untersucht der Artikel die Auswirkungen radikaler zeitgenössischer Anderungen der Normen; die Suche nach einem neuen Paradigma der Bildung hat gerade erst begonnen.

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Resumen Hay una percepción general de que el funcionamiento del sistema de educación es deficiente, y este trabajo sostiene que, efectivamente, se produce una profunda crisis causada por la brecha existente entre el sistema y el mundo cambiante que lo rodea. Las condiciones postmodernistas ponen énfasis en la producción de informaciones más que en la productividad de la tierra o del capital, y en una óptica del mundo relativista más que en la adhesión a una religión o ideología fija, lo que por su parte da origen al pluralismo y a sistemas sociales cambiantes. Al mismo tiempo, la busca de la Verdad ya no es considerada como la meta más importante en la vida, y la titularidad de pruebas de formación ya no garantiza el acceso a la Buena Vida. Las instituciones de la educación muestran una disfunción porque aún se atienen a parámetros obsoletos en cuanto a objetivos, actividades, estructuras, métodos y percepciones de su clientela. A través de una amplia reseña de debates conceptuales recientes y de la valoración de tendencias sociales, el trabajo explora las implicaciones que los cambios radicales contemporáneos tienen en los parámetros; la busca de un nuevo paradigma de educaión solamente acaba de comenzar.

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Résumé Le système éducatif a la réputation largement répandue de mal fonctionner, et l'article expose qu'il existe en effet une crise profonde due à l'écart entre le système d'éducation et le monde en évolution qui l'entoure. Les conditions de vie post-modernes font passer au premier plan la production de l'information à la place de la productivité de la terre et du capital, et une vision du monde relativiste remplace l'adhésion à une religion ou une idéologie fixe, qui à son tour engendre le pluralisme et la modification des structures sociales. Parallèlement, la recherche de la vérité n'est plus le but suprême dans la vie, et la détention d'une preuve d'éducation n'est plus une garantie d'accès à une vie réussie. Le dysfonctionnement des institutions éducatives réside donc dans leur attachement obstiné à des modèles périmés quant à leurs objectifs, leurs activités, leurs structures, leurs méthodes et leur vision de la clientèle. Sur la base d'une vaste étude sur le récent débat conceptuel et une évaluation des tendances sociales, l'article analyse les conséquences des changements radicaux contemporains sur ces modèles: la recherche d'un nouveau type d'éducation ne fait que commencer.

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Real-time simulation in computer-assisted instruction

Large scale computer-assisted instruction systems generally impose severe constraints upon the demands that individual users may make. Nevertheless, it is possible to overcome these problems and a program is described which teaches students how to fly a specific maneuver through real-time simulation of the flight of a student-controlled airplane. This is achieved in spite of the fact that the student is flying his plane through the use of a manually controlled analog input device. Both computational and educational implications are discussed.Contractual support for this work was provided in part by the Life Sciences Program Air Force Office of Scientific Research, Contract number F44620-70-C-0105. Dr. Charles Hutchinson was the scientific monitor of the contract.