Approaching Cauchy’s Theorem

Abstract

We hope to initiate a discussion about various methods for introducing Cauchy’s Theorem. Although Cauchy’s Theorem is the fundamental theorem upon which complex analysis is based, there is no “standard approach.” The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include Green’s Theorem, Goursat’s Lemma, Leibniz’ Rule, and homotopy theory, each of which has its positives and negatives.     

Educational reform and the public: Two case studies of Poland and Saskatchewan (Canada)

The involvement of the public in educational reform processes in modern democratic societies primarily serves the purpose of politically legitimizing the reform agenda. This study examines the rationales implicitly or explicitly submitted to the public to explain why educational reforms in the two countries should be endorsed. Although differences in the political culture caution against a hasty comparison of the two case studies, a number of politico-economic similarities allow for a valid juxtaposition. In Poland the context of socio-political and economic renewal prompted the reformers to emphasize the human-capital model which heightened public awareness and participation in the debate surrounding the reform. Public involvement in Saskatchewan was negatively affected for mainly two reasons. First, the government evidently manipulated public input by various means and thereby appears to have predetermined the outcome. Second, the rationale for the reform, based on a free-market model, tightened the linkage between the needs of the labour market and the mandate of the schools. As a result, public interest and participation was greatly diminished.

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Zusammenfassung Die Beteiligung der Öffentlichkeit an Reformen auf dem Gebiet der Erziehung in den modernen demokratischen Gesellschaften dient hauptsächlich der politischen Legitimierung der Reformpläne. In dieser Studie werden die der Bevölkerung offen oder unterschwellig vermittelten Gründe untersucht, warum den erzieherischen Reformen in beiden Ländern zugestimmt werden sollte. Obwohl die Unterschiede in der politischen Kultur Anlaß zur Vorsicht vor einem voreiligen Vergleich der beiden Fallstudien geben, so sprechen einige wirtschaftspolitische Ähnlichkeiten für eine sinnvolle Nebeneinanderstellung. In Polen führte der Zusammenhang sozialpolitischer und wirtschaftlicher Erneuerung dazu, daß die Reformer das Modell des menschlichen Kapitals, das das öffentliche Bewußtsein und die Teilnahme an der Debatte über die Reformen verstärkte, besonders betonten. Die öffentliche Beteiligung in Saskatchewan bekam aus zwei Gründen einen negativen Beigeschmack. Erstens manipulierte die Regierung offensichtlich in verschiedener Hinsicht die Beiträge der Öffentlichkeit, und sie scheint damit das Ergebnis vorbestimmt zu haben. Zweitens stärkten die Gründe für die auf ein Modell der freien Marktwirtschaft gegründeten Reformen die Verbindungen zwischen dem Bedarf des Arbeitsmarktes und dem Mandat der Schulen. Als Folge davon reduzierte sich das öffentliche Interesse und die Beteiligung erheblich.

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Résumé La participation du public aux processus de réforme éducative dans les sociétés démocratiques modernes sert essentiellement à légitimation politique du calendrier de réforme. Cette étude examine la raison d'être soumise implicitement ou explicitement au public et analyse pourquoi les réformes éducatives doivent être adoptées dans les deux pays. Bien que les divergences de cultures politiques mettent en garde contre une comparaison hâtive des deux études de cas, certaines correspondances politico-économiques permettent une juxtaposition valide. En Pologne, le contexte de la rénovation socio-politique et économique a pressé les réformateurs à mettre en valeur le modèle de capital humain qui a rehaussé l'information et la participation du public dans le débat autour de la réforme. La participation du public dans le Saskatchewan a été entravée pour deux raisons. Premièrement, il semblerait que le gouvernement ait manipulé les investissements publics de différentes manières et qu'il ait par cela prédéterminé le résultat obtenu. Deuxièmement, la légitimité de la réforme, orientée vers un modèle de marché libre, a renforcé les liens entre les besoins du marché de l'emploi et le mandat des écoles. Par conséquent, l'intérêt et la participation du public ont été considérablement affaiblis.

     

Comparing State SAT Scores: Problems, Biases, and Corrections

Comparing SAT scores among states using regression analysis leads to biased results because states differ in the proportion of students taking the exam. When the proportion of students taking the exam is included in the regression equation, the results can be biased because of misspecifieation bias. A method intended to correct for selection bias is presented, and empirical results suggest that sample selection bias is present in SAT score regressions. Regression equations and state rankings are compared between the selection-corrected equation and equations for which the selection problem is not addressed. The proposed method is one of many available as possible solutions to the selection problem. Alternative methods may produce different results     

Didactising: Continuing the work of Leen Streefland

In this paper we present three cases of instructional design that illustrates both horizontal didactising, the activity of using already established principles to design instruction, and vertical didactising the activity of developing new tools and principles for instructional design. The first case illustrates horizontal didactising by elaborating how the constructs chains of signification and models were used to design an instructional sequence involving linear growth. The second and third cases illustrate vertical didactising by developing argumentation analyses and generative listening, respectively, as instructional design tools. In the second case, argumentation analyses emerge as a tool that other designers can use to anticipate the quality of conversations that can occur as students engage in tasks prior to implementing the instructional sequence. The third case develops the notion of generative listening as a conceptual tool within the context of designing differential equations instruction to gain insights into what are, for students, experientially-real starting points that are mathematical in nature and to provide inspirations for revisions to instructional sequences. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dropouts in Swiss vocational education and the effect of training companies’ trainee selection methods

In the Swiss vocational education system, which is often called a ‘Dual System’, trainees enter into an apprenticeship contract with a training company. On average, 25% of those contracts are terminated prematurely (PCT). This article examines the relationship between training companies’ selection methods and PCTs. The investigation is based on a quantitative study with a cross-sectional design and on qualitative interviews. Trainers and operations managers (N = 335) of Swiss training companies for cooks and painters were surveyed. Half of the companies have been affected by a PCT in the past and the other half have not. The findings indicate that a certain amount of the explained variance between companies with and without reported PCTs can be explained by the companies’ selection process. Companies that were affected by a PCT use a job interview and a company visit significantly less often in their selection process than companies without a PCT.     

Advanced Linear Algebra: A Call for the Early Introduction of Complex Numbers

Abstract

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by the early introduction of complex arithmetic. Most elementary properties of complex numbers have immediate matrix analogues and many important theorems can be deduced, or at least postulated, from the basics of complex arithmetic alone.     

An Empirical Assessment of Interdisciplinarity: Perspectives from Graduate Students and Program Administrators

Interdisciplinarity is heralded as a relatively new educational and research paradigm that can effectively address complex problems at disciplinary boundaries. Yet little is known about the extent to which interdisciplinarity has penetrated higher education, nor about how students and program administrators view its usefulness. Through a case study of one field, urban planning, we assessed the extent to which graduate students and program administrators understand and support interdisciplinarity in the departments. We also assessed the extent to which specific factors might affect levels of support. Our survey results suggest differing attitudes between students and administrators and diverse opinions as to whether interdisciplinarity provides a practicable approach to addressing societal challenges. We offer a model for fostering interdisciplinarity and developing empirical evaluations of programs.     

Optimal presentation of graphic organizers and text: A case for large bites?

In three experiments, we investigated the optimal presentation of graphic organizers (GOs) and text by having undergraduates read a chapter-length text in its entirety, a few pages at a time, or a few sentences at a time, before or after viewing seven GOs in their entirety (large bites), one at a time (small bites), or one cell at a time (animated). Large bites were better than small bites for learning concept relations and applying that knowledge, and for recalling macropropositions when the GOs were presented before text. Animated GOs offered no advantages over the large-bites GOs at first presentation and considerably extended total study time. Presenting several GOs first, followed by the entire text, appears to be optimal in directing students’ attention to across-chapter relations in text.