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TEACHING OF ANALYTIC GEOMETRY AND VECTOR …

Acta Didactica Universitatis Comenianae Mathematics, Issue 5, 2005 TEACHING OF ANALYTIC GEOMETRY AND VECTOR calculus AND PROPOSALS OF PROBLEMS SOLUTIONS1 MICHAELA REGECOV Abstract. The aim of our work was to describe situation in TEACHING of ANALYTIC GEOMETRY and VECTOR calculus at secondary schools in the Slovak Republic, to specify possible stu-dents problems those are due to the application of methods of ANALYTIC GEOMETRY and vec-tor calculus in various geometric tasks and to propose their possible solutions. The aim was realized in three parts: theoretical, experimental and applications. Following the con-clusions of experimental part we proposed new strategies in ANALYTIC GEOMETRY and VECTOR algebra that can help to eliminate the students' problems in solving mathematical prob-lems not only from ANALYTIC GEOMETRY and VECTOR algebra .

lems not only from analytic geometry and vector algebra. ... teaching of analytic geometry and vector calculus at secondary schools is prac- ... formed our ideas to improve the teaching of analytic geometry and vector calcu-lus at secondary schools in the Slovak Republic.

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Transcription of TEACHING OF ANALYTIC GEOMETRY AND VECTOR …

1 Acta Didactica Universitatis Comenianae Mathematics, Issue 5, 2005 TEACHING OF ANALYTIC GEOMETRY AND VECTOR calculus AND PROPOSALS OF PROBLEMS SOLUTIONS1 MICHAELA REGECOV Abstract. The aim of our work was to describe situation in TEACHING of ANALYTIC GEOMETRY and VECTOR calculus at secondary schools in the Slovak Republic, to specify possible stu-dents problems those are due to the application of methods of ANALYTIC GEOMETRY and vec-tor calculus in various geometric tasks and to propose their possible solutions. The aim was realized in three parts: theoretical, experimental and applications. Following the con-clusions of experimental part we proposed new strategies in ANALYTIC GEOMETRY and VECTOR algebra that can help to eliminate the students' problems in solving mathematical prob-lems not only from ANALYTIC GEOMETRY and VECTOR algebra .

2 R sum . Le but de notre travail tait d crire la situation de l enseignement de la g om trie analytique et du calcul vectoriel aux coles secondaires en Slovaquie, sp cifier des pro-bl mes possibles que cause l application des m thodes de la g om trie analytique et du calcul vectoriel dans diff rents exercices g om triques et proposer leur possibles solu-tions. Les objectifs du travail on a r alis dans trois phases: th orique, exp rimentale et applique. En raison de nos r sultats implicites de la phase exp rimentale on a propos dans la g om trie analytique et dans le calcul vectoriel des strat gies nouvelles qui peu-vent liminer des probl mes des tudiants r soudre des exercices diff rents math matiques, non seulement de la g om trie analytique et du calcul vectoriel.

3 Zusammenfassung. Wir haben uns als Ziel unserer Arbeit den Zustands des Unterrichtens der analytischen Geometrie und der Vektorrechnung an Mittelschulen in der Slowakei und die Spezifikation m glicher Probleme der Sch ler mit der Applikation der Methoden der analytischen Geometrie und der Vektorrechnung in verschiedenen geometrischen Aufgaben zu beschreiben und die Aufl sung der Probleme vorzuschlagen gesetzt. Die Ziele der Arbeit wurden in drei Phasen realisiert: in der theoretischen, der experimentellen und der Applika-tionsphase. Anhand der Resultate der experimentellen Phase wurden neue Strategien in der analytischen Geometrie und der Vektoralgebra vorgeschlagen. Diese k nnen der Abschaffung der 1 This paper was created also with support of project Socrates Comenius n.

4 106663 CP 1 2002 1 IT COMENIUS C21 M. REGECOV 24 Probleme der Sch ler beim L sen verschiedener mathematischer Aufgaben, nicht nur aus der analytischen Geometrie und der Vektorrechnung, behilflich seien. Riassunto. Obiettivo del nostro lavoro stato quello di descrivere la situazione dell insegna-mento della geometria analitica e del calcolo vettoriale nelle scuole secondarie della Repubblica Slovacca, per individuare e precisare quei possibili problemi dovuti all applicazione dei metodi della geometria analitica e del calcolo vettoriale nei vari scopi che pu avere la geometria, quindi di proporre le possibili soluzioni. Il tutto stato realizzato in tre parti: teoretica, speri-mentale e applicativa. Seguendo le conclusioni della parte speri-mentale abbiamo proposto nuove strategie per la geometria analitica e per l algebra vettoriale che pu aiutare ad eliminare le difficolt degli studenti nel risolvere problemi matematici non soltanto di geometria analitica ed algebra vettoriale.

5 Abstrakt. Cie om na ej pr ce bolo pop sa stav vyu ovania analytickej geometrie a vektorov ho po tu na strednej kole v Slovenskej republike, pecifikova mo n prob-l my iakov s aplik ciou met d analytickej geometrie a vektorov ho po tu v r znych geometrick ch loh ch a navrhn ich mo n rie enie. Ciele pr ce sme realizovali v troch f zach: teoretickej, experiment lnej a aplika nej. Na z klade v sledkov vypl vaj cich z experiment lnej asti pr ce sme navrhli nov strat gie v analytickej geometrii a vekto-rovej algebre, ktor m u napom c odstr neniu probl mov iakov v rie en r znych ma-tematick ch loh, nielen loh z analytickej geometrie a vektorovej algebry. Key words: ANALYTIC GEOMETRY , VECTOR calculus , theory of didactic situations, actual situation of TEACHING of GEOMETRY in Slovak schools, problems in TEACHING of ANALYTIC GEOMETRY and VECTOR calculus , solving of those problems, statistical software 1 INTRODUCTION Two millennium ago GEOMETRY was a perfect and unique mathematical disci-pline which was founded on the axiomatic base.

6 It seems nowadays that among all of classical parts of mathematics only GEOMETRY isn t axiomatic . Even though this statement of English mathematician Zeemann is dated in 1970, its actuality and analogy with recent TEACHING of mathematics at secondary school is amazing. As since it elapsed more than three decades, the new millennium did not bring the remarkable changes in TEACHING of mathematics. Henceforth, GEOMETRY belongs to the most problematic parts of secondary school mathematics, students are not able to experiment and to utilize their knowledge from the different parts of mathemat-ics in solving the geometrical problems, they prefer arithmetic and algebraic ap-proach (methods of calculus in GEOMETRY ) and methods of construction in compari-son with methods of ANALYTIC GEOMETRY and VECTOR calculus , their space imagination is not sufficiently developed.

7 These facts lead us to focus our attention on the teach-ing of GEOMETRY , especially on ANALYTIC GEOMETRY and VECTOR calculus at secondary schools. TEACHING OF ANALYTIC GEOMETRY AND VECTOR calculus 25We chose the area of ANALYTIC GEOMETRY and VECTOR calculus also by the rea-son of understatement the methods that this area proposes for solving the geo-metrical tasks from secondary school mathematics. From our point of view the TEACHING of ANALYTIC GEOMETRY and VECTOR calculus at secondary schools is prac-ticed deficiently, isolated from the other parts of mathematics even thought ana-lytic GEOMETRY is one of mathematical discipline that supports the formation of student s thinking. In the same manner is also underestimated, in our opinion, the importance of VECTOR calculus that was based on ideas from mechanics, optics and geography and from the other different areas.

8 Even thought the VECTOR calculus at secondary schools is only auxiliary tool for TEACHING the ANALYTIC GEOMETRY , the teachers are oblivious to possibilities of its separated application in solving some geometrical tasks. Mentioned facts as well as our positive experience with TEACHING of VECTOR calculus and ANALYTIC GEOMETRY by methods different from classical methods of Slovak educational system (methods from French secondary school TEACHING ) formed our ideas to improve the TEACHING of ANALYTIC GEOMETRY and VECTOR calcu -lus at secondary schools in the Slovak Republic. The aim of our work was to describe situation in TEACHING of ANALYTIC geome-try and VECTOR calculus at secondary schools in the Slovak Republic, to specify possible problems those are due to the application of methods of ANALYTIC geome-try and VECTOR calculus in various geometric tasks and to propose their possible solution.

9 This aim we realized in following phases: 1. Historical and epistemological development of ANALYTIC GEOMETRY and vec-tor calculus . 2. Survey of actual situation and historical background of TEACHING of ANALYTIC GEOMETRY and VECTOR calculus considering textbooks used in Slovakia. 3. Analysis of actual mathematics French textbooks of secondary schools and their comparison with Slovak textbooks. 4. Actual situation and specification of problems in the TEACHING of ANALYTIC GEOMETRY and VECTOR calculus at secondary schools. 5. Proposals of possible problems solutions and compilation of tasks collec-tion those are leading to enhancement of TEACHING of ANALYTIC GEOMETRY and VECTOR calculus at Slovak secondary schools with specific recommended strategies of their solution.

10 6. Verification of the selected tasks efficiency from our tasks collection by comparing the abilities and capacities of students that took part in experi-mental TEACHING (with using of our propositions and strategies) and students that participated in the TEACHING with using of methods and processes, which are typical for Slovak secondary schools. M. REGECOV 26 2 THEORETICAL FRAMEWORK Analyse of particular didactical problem in educational process from differ-ent aspects is a research objective for many didactic schools in the world. One of them is the French didactic school represented by Guy Brousseau, Yves Chevallard and Anna Sierpinska. Research base of the theory of didactic situa-tions issued from this didactic school is analysis of problem in particular levels of didactic situations (Brousseau, 1998; Chevallard, 1992; Sierpinska, 2001).


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