A Fallacy in Geometry Reasoning

1930 ◽  
Vol 23 (1) ◽  
pp. 19-22
Author(s):  
H. C. Christofferson
Keyword(s):  
Mathematics Teacher ◽  
Isosceles Triangle ◽  
Angle Bisector ◽  
Usual Order

In the mathematics teacher for December, 1928, I attempted to expose a weakness in the usual proofs of beginning geometry and advocated a new and slightly different beginning which at the same time bas the virtue of being more simple. The usual order of proofs is (1) congruence by side-angle-side; (2) congruence by angle-sideangle; (3) the theorem that the angles opposite the equal sides of an isosceles triangle are equal; (4) congruence by three sides; and (5) sometime later, the construction of an angle bisector and of an angle equal to a given angle. We are concerned here with only (3), (4), and (5).

Proving the Equkaity of the Base Angles of an Isosceles Triangle

1929 ◽  
Vol 22 (6) ◽  
pp. 318-319
Author(s):  
Joseph A. Nyberg
Keyword(s):  
Mathematics Teacher ◽  
Isosceles Triangle ◽  
Plane Geometry ◽  
Angle Bisector ◽  
Significant Step

In a recent paper (A Different Beginning for Plane Geometry, by H. C. Christofferson, MATHEMATICS TEACHER, Dec., 1928) much emphasis is placed on the fact that the existence of an angle bisector must be assumed in proving that tho base angles of an isosceles triangle are equal. Because of this assumption it is argued that, we could better begin by assuming triangles congruent if the corresponding sides are equal. The following is a proof of the equality of the base angles of an isosceles triangle without the use of tho angle bisector. Since the quqestion seems important I present the proof with some detail. Step 4 is the significant step.

MATHEMATICS TEACHER EDUCATION AND THE FLEXIBLE USE OF TECHNOLOGIES IN DIGITAL AGE

2019 ◽  
Vol 8 (2) ◽  
Author(s):  
Hans-Georg Weigand
Keyword(s):  
Teacher Education ◽  
Mathematics Teacher ◽  
Industrial Revolution ◽  
Empirical Studies ◽  
Mathematics Teacher Education ◽  
Digital Technologies ◽  
Digital Age ◽  
Competence Model ◽  
Advantages And Disadvantages ◽  
Concept Representations

Advantages and disadvantages of the use of digital technologies (DT) in mathematics lessons are worldwidedissussed controversially. Many empirical studies show the benefitof the use of DT in classrooms. However, despite of inspiringresults, classroom suggestions, lesson plans and research reports,the use of DT has not succeeded, as many had expected during thelast decades. One reason is or might be that we have not been ableto convince teachers and lecturers at universities of the benefit ofDT in the classrooms in a sufficient way. However, to show thisbenefit has to be a crucial goal in teacher education because it willbe a condition for preparing teachers for industrial revolution 4.0.In the following we suggest a competence model, which classifies– for a special content (like function, equation or derivative) –the relation between levels of understanding (of the concept),representations of DT and different kind of classroom activities.The flesxible use of digital technologies will be seen in relationto this competence model, results of empirical investigations willbe intergrated and examples of the use of technologies in the upcoming digital age will be given.

THE SYSTEM OF INFORMATICS COMPETENCIES FOR THE MATHEMATICS TEACHER

2020 ◽  
pp. 57-72
Author(s):  
Nadiia Ponomareva
Keyword(s):  
Information Technology ◽  
Secondary Education ◽  
Mathematics Teachers ◽  
Mathematics Teacher ◽  
Systems Analysis ◽  
Web Technologies ◽  
Key Competencies ◽  
Teachers Training ◽  
Technology Specialists ◽  
Further Development

As a result of the analysis of domestic and foreign standards of key competencies, basic and complete secondary education, teachers’ training (in particular, mathematics teachers’ training) and information technology specialists training, the system of informatics competencies of a mathematics teacher, developed by Yu. S. Ramskyi, was updated in terms of the structure, content and indicators of competencies formation. It has been found out that the formation of the informatics competencies of a mathematics teacher begins with basic informatics competencies, the further development of which occurs primarily in the competencies in system administration, web technologies, programming and systems analysis.

Pengembangan E-modul Berbasis Smartphone Pada Materi Integral Tak Tentu Berorientasi Keterampilan Abad Ke-21

2020 ◽  
Vol 3 (2) ◽  
pp. 222-233
Author(s):  
Veronika Fany Monica Yuniarti ◽  
Nurul Anriani ◽  
Cecep Anwar H. F. Santosa
Keyword(s):  
Research And Development ◽  
21St Century ◽  
Mathematics Teacher ◽  
21St Century Skills ◽  
Research Method ◽  
Effective Medium ◽  
Developmental Research ◽  
High Category ◽  
Research Show

Abstrak: Penelitian ini bertujuan untuk menghasilkan aplikasi e-modul berbasis smartphone pada materi integral tak tentu berorientasi keterampilan abad ke-21 yang valid, praktis, dan efektif. Penelitian menggunakan metode Research and Development (R&D) dari Borg dan Gall yang dimodifikasi menjadi enam langkah. Hasil penelitian menyatakan bahwa hasil skor validasi ahli materi sebesar 4.2 termasuk kategori valid, ahli media diperoleh 3.8 termasuk kategori valid dan guru matematika diperoleh 3.7 termasuk kategori valid. Sedangkan hasil efektivitas diperoleh dari persentase ketuntasan 70% dengan kategori tinggi, dan n-gain sebesar 0,7682 termasuk kategori interpretasi tinggi. Lebih jauh, hasil praktis diperoleh 44.55 jumlah rata-rata skor pernyataan yang termasuk kategori sangat praktis dan 39.80 jumlah rata-rata skor pernyataan yang termasuk kategori praktis. Berdasarkan hasil tersebut maka dapat disimpulkan bahwa pengembangan aplikasi e-modul berbasis smartphone merupakan media yang valid, praktis, dan efektif. Abstract: The research aims to produce a valid, practical, and effective smartphone-based e-module applications 21st-century skills-oriented integral material. The Research method was developmental research (R&D) according to Borg and Gall which was modified into six steps. The results of the research show that the validity of material experts score is 4.2 (valid category), media experts obtained is 3.8 (valid category) and the mathematics teacher score is 3.7 (valid category). While the effectiveness results obtained from the percentage of completeness of 70% with a high category and n-gain of 0.7682 including the high interpretation category. As well as the practical results obtained 44.55 the average number of statements in the very practical category and 39.80 the average number of statements in the practical category. Based on these results it can be concluded that the development of smartphone-based e-module applications is a valid, practical, and effective medium.

scholarly journals Morality and Tribalism

Jus Cogens ◽  
2021 ◽  
Author(s):  
Claudio Corradetti
Keyword(s):  
Evolutionary Theory ◽  
Intergroup Relations ◽  
Moral Reason ◽  
Usual Order

AbstractThis contribution has two main goals which might be labelled for convenience as a pars construens and pars denstruens reversing the usual order of these terms. The first aim is to offer an overview of the main tenets of the book, while the second aim is to raise some critical concerns while remaining sympathetic to the author’s overall project. With regard to the first point, I present the context of intellectual debate where Buchanan’s contribution fits comfortably: Darwin’s evolutionary theory, anthropology, psychology, moral analysis etc. The target here is to show the internal complexity and different layers of analysis of the book. These initial reconstructions are, next, used to formulate some thoughts on what I consider possible problematic points in need of clarification. In particular, first, I hold that Buchanan presents too narrow oppositional views between intergroup relations whereupon the notion of “tribalism” is constructed. Such strong identitarian conception does not seem to depict adequately the sociological dynamics of intergroup relations. Second, I consider the terms in which it can be addressed the notion of the rise of the moral mind in evolutionary terms. The suggestion is to consider in a milder form the author’s key concept of a “Great Uncoupling” for the moral reason.

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