EDUCATIONAL MATTERS /ВЪПРОСИ НА ПРЕПОДАВАНЕТО
Errors Related to Topics in Geometry, Data Representation and Analysis Madeby Fifth Grade Students in the Republic of Мacedonia /
Metodi Glavche, Risto Malčeski, Katerina Anevska – 567
Mathematical Skills are Required in the Changing World /
Desislava Georgieva – 582
EDUCATIONAL RESEARCH /НАУЧНО–МЕТОДИЧЕСКИ СТАТИИ
Исторически предпоставки за смятането с пръсти и обучението поматематика на ученици със специални образователни потребности
[Historical Prerequisites for Finger Math and the Mathematics Education ofStudents with Special Educational Needs] /
Милен Замфиров / Milen Zamfirov – 593
One more Proof for the Distance Between the Incentre and the Orthocentre ofthe Triangle /
Šefket Arslanagić – 604
Computer Discovered Mathematics: Pedal Corner Products /
Sava Grozdev, Deko Dekov – 609
Eдно обобщение на ортоцентричния триъгълник [A Generalization of theOrthocentric Triangle] /
Хаим Хаимов – 616
Теоремата на Фонтене по отношение на описани централни коничнисечения
[Fontené Theorem with Respect to Surcumscribed Central Conics] /
Сава Гроздев, Веселин Ненков / Sava Grozdev, Veselin Nenkov – 621
CONTEST PROBLEMS /КОНКУРСНИ ЗАДАЧИ
Конкурсни задачи на броя [Contest Problems of the Issue] – 639
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Решения на конкурсните задачи от брой 1, 2015
[Solutions of the ContestProblems from Issue 1, 2015] – 640
MATHEMATICAL EDUCATION WORLDWIDE/МАТЕМАТИЧЕСКОТО ОБРАЗОВАНИЕ ПО СВЕТА
Гений с български произход [Genius with Bulgarian Origin] /
Сава Гроздев / Sava Grozdev – 643
ANNUAL CONTENTS / ГОДИШНО СЪДЪРЖАНИЕ – 652
READ IN THE LATEST ISSUES OF „AZ BUKI“ JOURNALS /
В НОВИТЕБРОЕВЕ НА СПИСАНИЯТА НА ИЗДАТЕЛСТВО „АЗ БУКИ“ ЧЕТЕТЕ– 657
GUIDE FOR AUTHORS / УКАЗАНИЕ ЗА АВТОРИТЕ – 659
ERRORS RELATED TO TOPICS IN GEOMETRY, DATA REPRESENTATION AND ANALYSIS MADEBY FIFTH GRADE STUDENTS IN THE REPUBLICOF MACEDONIA
1Metodi Glavche, 2Risto Malčeski, 3Katerina Anevska
1Ss. Cyril and Methodius University – Skopje
2,3FON University – Skopje
Abstract. In the learning process it is natural for omissions to occur and if they arenot identifi ed and eliminated in a timely manner, they are multiplied further on in thecourse of education. Therefore, one of the tasks of the mathematics instruction is todetect the errors made by the students and to eliminate them on time. Errors that havemade students from fi fth grade in arithmetic and algebra are processed in (Glavche,Malčeski & Anevska, 2015). In this paper the results of the research are presentedabout some of the most prominent errors that fi fth grade students in the Republic ofMacedonia make. Additionally, the paper offers a systematic overview of the identifi ederrors and the reasons for their occurrence. It also offers procedures for elimination ofthe identifi ed errors.Keywords: data, error, fi fth grade, geometry, student
Dr. Metodi Glavche, Assoc. Prof
Faculty of Pedagogy
Ss. Cyril and Methodius University
Skopje, Macedonia
Prof. Risto Malčeski, DSc
Faculty of Informatics
FON University
Skopje, Macedonia
Dr. Katerina Anevska, Assoc. Prof.
Faculty of Informatics
FON University
Skopje, Macedonia
MATHEMATICAL SKILLS ARE REQUIREDIN THE CHANGING WORLD
Desislava Georgieva
University of Ruse “Angel Kanchev”
“In mathematics, know-how is more important than mere possession of information.”
George Pólya
Abstract. Different treatments of the notions “skill” and “mathematical skill” arepresented in the paper with regard to their relation to the notion of competence. Old andnew educational standards of different countries are collected and analysed, thus outliningthe framework of the mathematical skills which are necessary to face the needs ofthe 21st century education.
Keywords: mathematical skill, competence, iceberg model, KSAVE model.
Mrs. Desislava Georgieva, PhD student
Department of Mathematics
University of Ruse “Angel Kanchev”
8, Studentska Str.
7017 Ruse, Bulgaria
ИСТОРИЧЕСКИ ПРЕДПОСТАВКИ ЗАСМЯТАНЕТО С ПРЪСТИИ ОБУЧЕНИЕТО ПО МАТЕМАТИКА НАУЧЕНИЦИСЪС СПЕЦИАЛНИ ОБРАЗОВАТЕЛНИПОТРЕБНОСТИ
Milen Zamfirov
Sofia University
Absract. Статията разглежда историческото развитие на смятането с пръсти,като акцентира върху възможностите за приложение в обучението на ученици съсспециални образователни потребности, особено на ученици с увреден слух. Взетиса под внимание и други ученици с нарушения като дискалкулия и аутизъм, а такасъщо и ученици без нарушения.
Keywords: fi nger Math, mathematics education, disability, hearing loss, dyscalculia,autism
HISTORICAL PREREQUISITES FOR FINGER MATH AND THEMATHEMATICS EDUCATION OF STUDENTS WITH SPECIALEDUCATIONAL NEEDS
Abstract. This article overviews the historical development of fi nger Math accentingon the possibilities for its implication in the teaching process of students with specialeducational needs, namely students with hearing loss. Other students with disabilities likedyscalculia and autism are taken under consideration and also students without disabilities.
Dr. Milen Zamfi rov, Assoc. Prof.
Department of Special Education and Speech Therapy
Sofi a University
15, Tsar Osvoboditel Blvd.
1504 Sofi a, Bulgaria
ONE MORE PROOF FOR THE DISTANCEBETWEEN THE INCENTRE ANDTHE ORTHOCENTRE OF THE TRIANGLE
Šefket Arslanagić
University of Sarajevo
Abstract. One more proof is proposed in the present paper concerning the thedistance between the incentre (I) and the orthocentre (H) of the triangle.Keywords: distance, incentre, orthocentre, sine law, cosine law, centroid of atriangle.
Prof. Dr. Šefket Arslanagić
Faculty of Mathematics and Natural Sciences
University of Sarajevo
Sarajevo, Bosna and Herzegovina
COMPUTER DISCOVERED MATHEMATICS:PEDAL CORNER PRODUCTS
1Sava Grozdev, Deko Dekov
1University of Finance, Business and Entrepreneurship
Abstract. Theorems about pedal corner products obtained by the computer program“Discoverer” are presented in the paper. 2010 Mathematics Subject Classifi cation:Primary 51-04, 68T01, 68T99.
Keywords: pedal corner product, triangle geometry, remarkable point, computerdiscovered
mathematics, Euclidean geometry, Discoverer.
Prof. Sava Grozdev, DSc
University of Finance, Business and Entrepreneurship
1, Gusla Street
1618 Sofi a, Bulgaria
Dr. Deko Dekov, Assoc. Prof.
81, Zahari Knjazheski Street
6000 Stara Zagora, Bulgaria
EДНО ОБОБЩЕНИЕНА ОРТОЦЕНТРИЧНИЯ ТРИЪГЪЛНИК
Хаим Хаимов
Absract. Авторът предлага обобщение на ортоцентричния триъгълник. Обобщеният триъгълник е перспективен на триъгълника ABC и зависи от произволнаточка на Чева P . Наричаме го H − триъгълник на точката P . Изображението,което на точка на Чева P съпоставя перспектора на H − триъгълника на P итриъгълника ABC , наричаме H − изображение на P . Показваме, че H − изображението е композиция на две известни преобразувания на точки: допълнениеи изогонална спрегнатост. Намираме образите на някои забележителни точки приH − изображението. Резултатите в тази статия могат да бъдат използвани за тестване на компютърната програма „Откривател“.
Keywords: generalization of the orthic triangle, perspective triangle, perspector, notable points
A GENERALIZATIONOF THE ORTHOCENTRIC TRIANGLE
Abstract. The author proposes a generalization of the orthic triangle. Thegeneralized triangle is defi ned by an arbitrary point of Ceva P and it is in perspectivewith the reference triangle ABC . We call it the H − triangle of point P . We call theH − transform of point P the mapping which assigns to point P the perspector ofthe H − triangle of point P and triangle ABC . We show that the H − transform iscomposition of two known transforms: the complement and the isogonal conjugate. Weidentify the images of a few remarkable points under the H − transform. The results inthis paper could be used for testing of the computer program “Discoverer”.
Haim Haimov
16, Bratia Shkorpil Str.
9000 Varna, Bulgaria
ТЕОРЕМАТА НА ФОНТЕНЕ ПО ОТНОШЕНИЕНА ОПИСАНИ ЦЕНТРАЛНИ КОНИЧНИСЕЧЕНИЯ
1Сава Гроздев, 2Веселин Ненков
1VUZF University
2Технически колеж – Ловеч
Absract. В настоящата статия е описано едно обобщение на забележителнататеорема на Фонтене от геометрията на триъгълника.
Keywords: triangle, conic, pedal circle, pedal curve, Feuerbach confi guration, Eulercurve.
FONTENÉ THEOREM WITH RESPECTTO SURCUMSCRIBED CENTRAL CONICS
Sava Grozdev, Veselin Nenkov
Abstract. A generalization of the remarkable Fontené theorem from the geometry oftriangle is described in the present paper.
Prof. Sava Grozdev, DSc
University of Finance, Business and Entrepreneurship
1, Gusla Str.
1618 Sofi a, Bulgaria
Dr. Veselin Nenkov, Assoc. Prof.
Technical College Lovech
31, Sajko Saev Str.
Lovech, Bulgaria
