Mokytojų ugdymas 2011, 16(1) by Mokytojų ugdymas/Teacher Education

ŠIAULIŲ UNIVERSITETAS ŠIAULIAI UNIVERSITY

MOKYTOJŲ UGDYMAS TEACHER EDUCATION 2011

Nr. 16 (1)

Mokslo darbai

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Research Works

ISSN 1822-119X MOKYTOJŲ UGDYMAS

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TEACHER EDUCATION

Mokslo darbai

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Reasearch Works

Vyriausioji redaktorė

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Editor-in-Chief

Audronė Juodaitytė, prof. habil. dr. (socialiniai mokslai, edukologija), Šiaulių universitetas / Prof. Habil. Dr. (Social Sciences, Education Studies), Šiauliai University Mokslinė sekretorė

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Scientific Secretary

Ramutė Gaučaitė, doc. dr. (socialiniai mokslai, edukologija), Šiaulių universitetas / Assoc. Prof. Dr. (Social Sciences, Education Studies), Šiauliai University Redaktorių kolegija

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Editorial Board

Nada Babič, prof. habil. dr. (socialiniai mokslai, pedagogika, filosofija), Kroatijos Strosmajerio universitetas, Kroatija / Prof. Habil. Dr. (Social Sciences, Pedagogy, Philosophy), Strosmajer University, Croatia Marija Barkauskaitė, prof. habil. dr. (socialiniai mokslai, edukologija), Vilniaus pedagoginis universitetas, Lietuva / Prof. Habil. Dr. (Social Sciences, Education Studies), Vilnius Pedagogical University, Lithuania Inese Jurgena, prof. habil. dr. (socialiniai mokslai, edukologija), Rygos pedagogikos ir švietimo vadybos akademija, Latvija / Prof. Habil. Dr. (Social Sciences, Education Studies), Riga Teacher and Educational Managment Academy, Latvia Vytautas Gudonis, prof. habil. dr., Rusijos pedagoginių ir socialinių mokslų akademijos akademikas (socialiniai mokslai, psichologija), Šiaulių universitetas, Lietuva / Prof. Habil. Dr., Academician of Russian Academy of Pedagogical and Social Sciences (Social Sciences, Psychology), Šiauliai University, Lithuania Aušra Kazlauskienė, doc. dr. (socialiniai mokslai, edukologija), Šiaulių universitetas, Lietuva / Assoc. Prof. Dr. (Social Sciences, Education Studies), Šiauliai University, Lithuania Elena Jurašaitė-Harbison, dr. (socialiniai mokslai, edukologija), Hofstra universitetas, JAV / Dr. (Social Sciences, Education Studies), Hofstra University, USA Arija Karpova, prof. habil. dr. (socialiniai mokslai, socialinė ir asmenybės psichologija), Latvijos universitetas, Latvija / Prof.Habil. Dr. (Social Sciences, Social and Personality Psychology), University of Latvia, Latvia Glynn Kirkham, prof. habil. dr. (socialiniai mokslai, edukologija), Notingamo Trento universitetas, Didžioji Britanija / Prof. Habil. Dr. (Social Sciences, Education Studies), Nottingham Trent University, United Kindom Daina Liegeniece, prof. habil. dr. (socialiniai mokslai, edukologija), Liepojos universitetas, Latvia / Prof. Habil. Dr. (Social Sciences, Education Studies), Liepaja University, Latvia Gediminas Merkys, prof. habil. dr. (socialiniai mokslai, edukologija), Kauno technologijos universitetas, Lietuva / Prof. Habil. Dr. (Social Sciences, Education Studies), Kaunas University of Technology, Lithuania Jeffrey Mirel, prof. dr. (socialiniai mokslai, edukologija), Mičigano universitetas, JAV / Prof. Dr. (Social Sciences, Education Studies), The University of Michigan, USA Kęstutis Pukelis, prof. habil. dr. (socialiniai mokslai, edukologija), Vytauto Didžiojo universitetas, Lietuva / Prof. Habil. Dr. (Social Sciences, Education Studies), Vytautas Magnus University, Lithuania Alida Samuseviča, prof. dr. (socialiniai mokslai, edukologija), Liepojos universitetas, Latvia / Prof. Dr. (Social Sciences, Education Studies), Liepaja University, Latvia Ona Tijūnėlienė, prof. habil. dr. (socialiniai mokslai, edukologija), Klaipėdos universitetas, Lietuva / Prof. Habil. Dr. (Social Sciences, Education Studies), Klaipėda University, Lithuania Žurnalas leidžiamas nuo 2002 m. du kartus per metus birželio ir lapkričio mėn. Redakcijos adresas: Šiaulių universitetas Edukacinių tyrimų mokslinis centras P. Višinskio g. 25, LT-76351 Šiauliai, Lietuva Tel. (+ 370 41) 59 57 46 El. paštas mokytojuugdymas@ef.su.lt http://www.mokytojuugdymas.ef.su.lt Žurnalas referuojamas: tarptautinėje duomenų bazėje Index Copernicus (http://www.indexcopernicus.com) Education research complete (EBSCO) (http://www.ebscohost.com) © Šiaulių universitetas, 2011 ©  Edukacinių tyrimų mokslinis centras, 2011 ©  VšĮ Šiaulių universiteto leidykla, 2011

The journal is being published since 2002 twice per year (in June and November) Our address: Šiauliai University Scientific Centre of Educational Researches P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania Phone  (+370 41) 595746 E-mail: mokytojuugdymas@ef.su.lt http://www.mokytojuugdymas.ef.su.lt The journal is registered International Index Copernicus database http://www.indexcopernicus.com Education research complete (EBSCO) (http://www.ebscohost.com) © Šiauliai Univesity, 2011 © Scientific Centre of Educational Researches, 2011 © Publishing House of Šiauliai University, 2011

TURINYS

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CONTENTS

Pratarmė Foreword

Matematikos dalyko studijų kaita aukštojoje mokykloje: ugdomosios aplinkos ir studentų profesinis kompetentingumas The shift of learning mathematics in the higher education institution: educational enviroments and student’s professional competence Svetlana Asmuss, Natalja Budkina, Aleksandrs Šostaks

Naujos matematikos didaktikos prieigos aukštojoje mokykloje: būtinybės versus pakankamumo sąlyga new approaches of mathematics didactics in the higher education institution: the necessity condition versus the sufficiency condition Liudvikas Kaklauskas, Danutė Kaklauskienė

Virtualios mokymo(si) aplinkos įrankių panaudojimo matematikos mokymui analizė Analysis of Usage of Virtual Teaching/Learning Environment Tools for Teaching Mathematics Anda Zeidmane

Matematikos studijų svarba inžineriniame ugdyme: darbdavių vertinimas Importance of Math Studies in Engineering Education in Assessment of Employers matematikos mokymo mokykloje pedagoginės inovacijos: patirtys ir kaitos procesai pedagogical innovations of teaching mathematics at school: experiences and shift processes Kirsti Kislenko, Lea Lepmann

Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990–2010 m.) Changes in teachers’ approach, teaching MAthematics in estonian schools (1990-2010) Katrin Kokk Nacionalinio matematikos egzamino Estijoje lyginamoji analizė COMPARATIVE ANALYSIS OF THE NATIONAL EXAMINATION OF MATHEMATICS IN ESTONIA

Madis Lepik Įrodymais pagrįsto ugdymo taikymas mokant matematikos Estijos mokyklose THE ROLE OF PROOF IN ESTONIAN CURRICULA OF LOWER AND UPPER SECONDARY MATHEMATICS

besimokančiųjų skyrtybes įgalinančios matematikos didaktikos raiška MANIFESTATION OF MATHEMATICS DIDACTICS ENABLING LEARNERS’ DIFFERENCES

Viktorija Sičiūnienė, Janina Dargytė SILPNAI BESIMOKANČIŲJŲ BENDROJO LAVINIMO MOKYKLOS MOKINIŲ MATEMATINIŲ GEBĖJIMŲ DIAGNOSTIKA DIAGNOSTICS OF MATHEMATICS ABILITIES OF LOW ACHIEVERS IN A COMPREHENSIVE SCHOOL

4 Tatjana Bakanovienė 77 MATEMATIKAI GABIŲ VAIKŲ CHARAKTERINGŲ SAVYBIŲ RAIŠKA PAMOKOJE: PEDAGOGŲ PATIRTIES ANALIZĖ Manifestation of FEATURES CHARACTERISTICS TO CHILDREN GIFTED FOR MATHEMATICS: ANALYSIS OF TEACHERS’ EXPERIENCE AKTUALIJOS TOPICALITIES Sigita Turskienė

MATEMATIKOS MOKYMO IR MOKYTOJŲ RENGIMO IDĖJŲ SKLAIDA BEI PATIRTIS TARPTAUTINĖJE bALTIJOS ŠALIŲ KONFERENCIJOJE „mATEMATIKOS MOKYMAS: RETROSPEKTYVA IR PERSPEKTYVOS“ SPREAD AND EXPERIENCE OF MATHEMATICS TEACHING AND TEACHER TRAININS IDEAS IN THE INTERNATIONAL CONFERENCE OF BALTIC COUNTRIES “TEACHING MATHEMATICS: RETROSPECT AND PERSPECTIVES” Leidinio „Mokytojų ugdymas“ publikuojamų mokslo darbų reikalavimai Requirements for the Research Publications in the Issue “Teacher Education”

PRATARMĖ • FOREWORD

M�� okslo žurnalo „Mokytojų ugdymas“ 16-asis numeris skirtas matematikos didaktikos problemoms. Jame publikuojami straipsniai yra parengti mokslininkų iš trijų šalių: Estijos, Latvijos ir Lietuvos. Pagal tematiką straipsniai yra suskirstyti į tris skyrius: Matematikos dalyko studijų kaita aukštojoje mokykloje: ugdomosios aplinkos ir studentų profesinis kompetentingumas (I skyrius), Matematikos mokymo mokykloje pedagoginės inovacijos: patirtys ir kaitos procesai (II skyrius), Besimokančiųjų skirtybes įgalinančios matematikos didaktikos raiška (III skyrius). Pirmojo skyriaus Matematikos dalyko studijų kaita aukštojoje mokykloje: UGDOMOSIOS APLINKOS IR STUDENTŲ PROFESINIS KOMPETENTINGUMAS straipsnyje Naujos matematikos didaktikos prieigos aukštojoje mokykloje: būtinybės versus pakankamumo sąlyga autoriai (S. Asmuss, N. Budkina, A. Šostaks) analizuoja universiteto studentų patirtį sprendžiant matematikos dalyko studijų problemas. Tyrėjai nustatė, jog studentai dažnai nesuvokia būtinybės ir pakankamumo sąlygų prasmingumo ir tai pasireiškia sprendžiant matematikos uždavinius. Tyrėjai aprašo naują pedagoginio darbo patirtį, kai studentams sudaromos galimybės spręsti matematinio pobūdžio problemą, atkreipiant jų dėmesį į būtinybės ir pakankamumo sąlygų dermes. Kitame šio skyriaus straipsnyje Virtualios mokymo(si) aplinkos įrankių panaudojimo matematikos mokymui analizė (L. Kaklauskas, D. Kaklauskienė) pagrindžiama aukštojoje mokykloje e.studijų matematikos kurso virtualioji mokymosi aplinka Moodle. Nurodomi jos vertinimo įrankių kriterijai (16 standartinių ir 6 specializuoti). Autoriai teigia, jog būtent šie standartai ir specializuoti Moodle įrankiai geriausiai tinka matematikos mokymuisi virtualioje aplinkoje. Kitame šio skyriaus straipsnyje Matematikos studijų svarba inžineriniame ugdyme: darbdavių vertinimas A. Zeidmane analizuoja matematikos dalyko studijų pritaikomumo galimybes tokių inžinerinių gebėjimų ugdymuisi, kaip problemų formulavimo ir sprendimų įgūdžių gerinimo. Visa tai pagrindžiama empirinio tyrimo duomenimis, praturtinančiais autorės nurodyto matematikos studijų būdo reikšmingumą profesinio, inžinerinio studentų rengimo gerinimui aukštojoje mokykloje.

Issue 16 of the journal “Teacher Education” is dedicated to problems of didactics of mathematics. Articles published in this issue are prepared by scientists from three countries: Estonia, Latvia and Lithuania. According to the topics the articles are brought under three sections: The Shift of Learning Mathematics in the Higher Education Institution: Educational Environments and Students’ Professional Competence (Section 1), Pedagogical Innovations of Teaching Mathematics at School: Experiences and Shift Processes (Section 2), and Manifestation of Mathematics Didactics Enabling Learners’ Differences (Section 3). The authors of the article “New Approaches of Mathematics Didactics in the Higher Education Institution: the Necessity Condition versus the Sufficiency Condition” in Section 1 Educational Environments and Students’ Professional Competence (S.Asmuss, N.Budkina, A. Šostaks) analyse the experience of university students, solving problems while learning mathematics. The researchers identified that students often do not perceive the meaningfulness of necessity and sufficiency conditions and this manifests itself solving mathematical problems. The researchers describe new pedagogical work experience when students are given opportunities to solve a mathematical type problem, focusing their attention on coherence of necessity and sufficiency conditions. Another article in this section „Analysis of Usage of Virtual Teaching/Learning Environment Tools for Teaching Mathematics“ (L.Kaklauskas, D.Kaklauskienė) grounds virtual learning environment Moodle of e-learning the mathematics course in the higher education institution. Criteria of its assessment tools are indicated (16 standard and 6 specialised). The authors state that namely these standard and specialized Moodle tools suit learning mathematics in virtual environment best. Another article in this section “Importance of Math Studies in Engineering Education in Assessment of Employers” (A.Zeidmane) analyses applicability possibilities of learning mathematics for self-development of such engineering abilities as improvement of problem formulation and solution skills. All of it is grounded on the empirical survey and data, enriching the significance of the way of learning mathematics indicated by the author for improvement of professional and engineering training in the higher education institution.

Mokytojų ugdymas 2011

Antrajame skyriuje MATEMATIKOS MOKYMO MOKYKLOJE PEDAGOGINĖS INOVACIJOS: PATIRTYS IR KAITOS PROCESAI publikuojami trys straipsniai, kuriuose pateikiami mokyklinės matematikos didaktikai reikšmingi empirinių tyrimų rezultatai, bylojantys apie Estijos mokyklų matematikos mokymo ir nacionalinio matematikos egzamino problemas. Straipsnyje Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990 – 2010) autorės (K. Kislenko, L. Lepmann) remdamosi empiriniais duomenimis, surinktais per dešimt metų, teigia, kad Estijos mokyklose matematikos mokyme vyrauja du metodai: tradicinis ir netradicinis. Mokslininkės, norėdamos įrodyti, kuris metodas yra efektyvesnis, pasitelkia PISA 2009 m. rezultatus ir teigia, jog Estijos mokinių balai už tas matematikos užduotis, kurios reikalauja aukščiausio lygio mąstymo, pablogėjo. Todėl, jų nuomone, reikia taikyti netradicinius matematikos mokymo metodus. Straipsnyje Nacionalinio matematikos egzamino Estijoje lyginamoji analizė (K. Kokk) autorė atliko nacionalinę matematikos egzamino Estijoje organizavimo patirties penkerių metų analizę. Remdamasi konkrečiais rezultatais tyrėja suformulavo išvadas ir pateikė kai kurias kritines įžvalgas bei išryškino pasiekimus nacionalinio egzamino organizavimo srityje. Straipsnyje Įrodymais pagrįsto ugdymo taikymas mokant matematikos Estijos mokyklose (M. Lepik) autorė pademonstruoja, kaip Estijos vidurinių mokyklų programose, mokant matematikos dviem lygiais (žemesniu ir aukštesniu), taikomas įrodymais grįstas metodas. Ištirta mokinių įrodinėjimo gebėjimų sklaida. Straipsnio autorė teigia, jog šis metodas yra efektyvus, nes įrodinėjimo procesas pasidaro prasmingas ne tik mokytojui, bet ir pačiam besimokančiajam. Trečiajame skyriuje BESIMOKANČIŲJŲ SKIRTYBES ĮGALINANČIOS MATEMATIKOS DIDAKTIKOS RAIŠKA publikuojami du straipsniai. Straipsnyje Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika autorės (V. Sičiūnienė, J. Dargytė) analizuoja žemo matematinio lygio mokinių gebėjimų rezultatus, gautus pritaikius moderniosios testų teorijos dviejų parametrų modelį. Trejus metus jos tyrė nacionalinius mokinių pasiekimų duomenis ir atlikusios jų analizę teigia, jog ji yra vertinga, tobulinant matematikos dalyko programas, vadovėlius bei reflektuojant matematikos mokymo praktiką. Straipsnyje Matematikai gabių vaikų charakteringų savybių raiška pamokoje: pedagogų patirties analizė T. Bakanovienė apibūdina tas gabių

The second section Pedagogical Innovations of Teaching Mathematics at School: Experiences and Shift Processes contains three articles listing the results of empirical surveys that are significant for didactics of mathematics in schools. They point to problems of teaching mathematics and national mathematics examination in Estonian schools. Based on empirical data collected during ten years, the authors of the article “Changes in Teachers’ Approach, Teaching Mathematics in Estonian Schools (1990-2010)” (K.Kislenko, L.Lepmann) state that two methods are prevailing in teaching mathematics in Estonian schools: traditional and non-traditional. Seeking to prove which method is more efficient, the scientists employ PISA 2009 results and state that the scores of Estonian pupils for mathematical tasks that require the highest level of thinking have worsened. Therefore, in their opinion, non-traditional mathematics teaching methods have to be applied. The author of the article “Comparative Analysis of the National Examination of Mathematics in Estonia” (K.Kokk) carried out the analysis of experience of organising the national mathematics examination in Estonia during five years. Based on concrete results, the researcher formulated conclusions, provided certain critical insights and highlighted achievements in the area of organising the national examination. In the article “Application of Evidence-Based Education, Teaching Mathematics in Estonian Schools“ (M.Lepik) the author demonstrates how the evidence-based method is applied teaching mathematics at two levels (lower and upper) in Estonian secondary school curricula. The spread of pupils’ proving abilities has been researched. The author of the article enables to state that this method is efficient because the process of proving becomes meaningful not only for the teacher but also for the very learner. The third section Manifestation of Mathematics Didactics Enabling Learners’ Differences contains two articles. The authors of the article “Diagnostics of Mathematic Abilities of Low Achievers in a Comprehensive School“ (V.Sičiūnienė, J.Dargytė) analyse the results of the abilities of low achievers in mathematics, which were obtained having applied the two parameter model of test theory. During three years they were researching pupils’ national achievement data and upon completion of their analysis they state that that analysis is valuable for improvement of curricula, textbooks of mathematics and reflection on practice of teaching mathematics. T. Bakanovienė in her article “Manifestation of Features Characteristic to Children Gifted for Mathematics: Analysis of Teachers’ Experience” describes such

Pratarmė

vaikų charakterologines ypatybes, kurios reiškiasi matematikos pamokoje ir leidžia mokytojams identifikuoti matematikai gabių vaikų identiteto raišką ir, remiantis ja konstruoti bei vertinti pedagoginę patirtį. Žurnalo „Mokytojų ugdymas“ redaktorių kolegija tikisi, jog skaitytojus sudomins Estijos, Latvijos ir Lietuvos matematikos didaktikos srityje mokslininkų atlikti tyrimai, nes jie apima ne tik universitetų, bet ir bendrojo lavinimo mokyklų mokymo metodus, studentų bei moksleivių pasiekimus bei proteguoja šių šalių matematikos mokymo patirtį.

personality peculiarities of gifted children which manifest themselves in the mathematics lesson and enable teachers to identify manifestation of identity of children gifted for mathematics and, based on it, develop and evaluate pedagogical experience. The editorial board of the journal “Teacher Education” hopes that the readers will get interested in researches carried out by Estonian, Latvian and Lithuanian researchers in the field of didactics of mathematics because they encompass teaching methods that are employed not only in universities but also in comprehensive schools, students’ and pupils’ achievements in mathematics and favour the experience of teaching mathematics in these countries.

Vyriausioji redaktorė Prof. habil. dr. Audronė Juodaitytė

Editor-in-chief Prof. Habil. Dr. Audronė Juodaitytė

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 10–17 Teacher Education. 2011. No. 16 (1), 10–17

Svetlana Asmuss, Natalja Budkina, Aleksandrs Šostaks Latvijos universitetas, Latvija Rygos technikos universitetas, Latvija Latvijos universitetas, Latvija

• University of Latvia, Latvia

Riga Technical University, Latvia Latvijos universitetas, Latvia

Naujos matematikos didaktikos prieigos aukštojoje mokykloje: būtinoji versus pakankamoji sąlyga

new approaches of mathematics didactics in the higher education institution: the necessity condition versus the sufficiency condition

Anotacija Viena iš pagrindinių problemų, su kurioms susiduria studentai, pradedantys studijuoti universitete, yra ta, kad be didesnio informacijos, kurią reikia įsisavinti, kiekio, reikalaujama iš esmės tikslesnio samprotavimo, sprendžiant matematikos uždavinius. Konkrečiai studentas dažnai painioja būtinybės ir pakankamumo sąlygų sąvokų vartojimą. Šiose pastabose autoriai dalinasi savo patirtimi, bandant spręsti šią problemą, dėstant matematikos kursus universitete. Pagrindiniai žodžiai: matematinis teiginys, būtinoji sąlyga, pakankamoji sąlyga.

Abstract One of the main problems which encounter students starting studies at the university is in addition to larger volume of information to be digested is the request for a essentially more precise reasoning when dealing with mathematical statements. Specifically a student is often confused when using concepts of necessity and sufficiency conditions. In these notes the authors share their experiences when trying to solve this problem teaching mathematical courses at university. Key words: mathematical statement, necessary condition, sufficiency condition.

Introduction In the process of studying mathematical courses at the university students encounter various theorems, exercises and problems whose formulations contain different conditions. One of the principal tasks of a teacher, especially working with the first year students, is to train a student to distinguish clearly between • necessary conditions, • sufficient conditions, • criteria. Students have to get an accurate precise idea of the role of every condition in the statements of theorems and in the wordings of exercises and problems. When listening to a lecture in mathematics or reading mathematical textbooks a student constantly meets such expressions as • “A holds if and only if B is true”, • “if A is true then B holds”, • “A holds whenever B is true”, and other similar phrases. But do students always understand clearly the meaning of such expressions? Do they really feel the function and the role of such expressions in each mathematical statement they have

to deal with? Our experience of many years shows that unfortunately not. There are students, especially of the first year, who deliberately mix link constructions “if A is true then B holds” and “A holds whenever B is true” and others, thus obtaining false mathematical statements. Clearly this problem is directly related to the task of distinguishing between sufficient and necessary conditions in formulations of mathematical statements, and of understanding clearly the function of each one of necessary and sufficient conditions. Methodology The aim of this work is to discuss methodologically the role of necessary and sufficient conditions in mathematical reasoning in general and to illustrate their importance by examples. The methodology of research used in the article is based on the analysis of process of teaching mathematical courses. When carrying out this analysis the authors founded themselves on their experience in lecturing and training mathematical courses at the University of Latvia and at the Riga Technical University, the two largest universities in

Svetlana Asmuss, Natalja Budkina, Aleksandrs Šostaks Naujos matematikos didaktikos prieigos aukštojoje mokykloje: būtinybės sąlyga versus pakankamumo sąlyga

• it is raining if and only if there are clouds in the sky. Such and other appropriately formulated statements can be repeatedly used illustrating different formal mathematical statements in mathematical courses for the first year students. Criteria conditions Coming closer to the discussing of the essence of the problems outlined in the introduction we first linger on the so called tests, or criteria-type conditions. Actually a pupil encounters with statements of this type already on the level of a secondary, if not of a primary school. Unfortunately at that stage pupils usually do not realize such statements as criteria, but in the best case, as two separate statements. Criteria in mathematical courses at the secondary school To illustrate some cases when criteria conditions can be easily noticed in the standard programme of the secondary school we consider a couple of examples. 1. Divisibility conditions: • a natural number is divisible by 3 (resp. by 9) if and only if the sum of digits of this number is divisible by 3 (resp. by 9); • a natural number n is odd if and only if its square n 2 is odd. Usually pupils pay attention to the implication ⇐ in the first statement and the implication ⇒ in the second one. 2. Pythagorean theorem: a triangle is right if and only if the area of the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs. Usually pupils grasp only implication ⇒ in this statement or, in a better case, as two different theorems: the direct and the opposite one. 3. Right triangle with 30 0 angle: the hypotenuse of the right triangle is twice as long as the shorter leg if and only if the smallest angle of this triangle has 30 degrees. In this case the accent is usually stressed on the implication ⇐ . We feel it is a right moment to pay the reader’s attention also to the fact that the Pythagorean theorem has fundamental counterparts in the advanced courses of mathematics as well. In Algebra: • criterion of orthogonality of two vectors a and

Latvia. In order to solve the problem under discussion, the authors suggest the following methods: • visualization, the use of diagrams; • methods of mathematical logic, the use of quantifiers; • detailed analysis of a large number of specially selected examples. To illustrate these methods the authors consider many examples mainly taken from the university course of Mathematical Analysis. Besides, there are also some examples adopted from the mathematical courses at secondary school. Visualization of the problem In order to make students better feel the role of necessity and sufficient conditions in mathematical statements we would like to emphasize the role of their visualization. Here by visualization we note two different, but very useful in our opinion, methods which help students to approach the understanding this problem. The first one is to use diagrams and elementary logical symbols showing relations between statements. For example, illustrate • the statement “if A holds, then B is true” by the implication sign “ A ⇒ B ”; • the statement “A holds if and only if B holds” by the equivalence sign “ A ⇔ B ”; • the statement “A holds whenever B is valid” by the implication sign “ A ⇐ B ”; • the statement “if A is not valid then either B or C is true” by the diagram “ ¬A ⇒ ( B ∨ C ) ”; • the statement “if A and B hold then either C or D is true” by the diagram “ ( A & B ) ⇒ (C ∨ D) ”; etc. Such diagrams are an effective tool to help students feel the relations between different conditions which appear in mathematical statements and in their proofs. In our opinion the use of elementary logical symbols need not request from the students knowledge of the basics of the formal logic. By the second method we mean illustration of the role of different conditions by examples taken from everyday life. Although seemingly naïve, such visual examples promote a student to distinguish clearer between necessary, sufficient conditions and the criteria. Let a student see the difference between correct and wrong statements in the examples like: • if it is raining, there are clouds in the sky; • if there are clouds in the sky it is raining;

Mokytojų ugdymas 2011

b in an Euclidean space

a⊥b⇔ a

+ b

= a+b

In Functional Analysis: • the norm in a vector space X is defined by a scalar product if and only if the equality 2 2 2 2 a+b + a−b = 2 a +2 b holds for any vectors a, b ∈ X . Criteria conditions in Mathematical Analysis In the course of mathematical analysis many criteria are named after French mathematician Augusten Louis Cauchy (1789 – 1857). Among them the test for convergence of sequences the test for existence of a finite limit of a function, the tests for convergence of number and function series, the tests for uniform convergence of function sequences and function series, the test for convergence of improper integrals, etc., are repeatedly used.

Consider a non-negative non-increasing function f defined on the interval [1, ∞) . Then the series ∞

∑ f (n) converges if and only if the integral

n =1 ∞

∫ f ( x)dx 1

harmonic series

∑ an

k =n

∞

1 • harmonic series ∑ diverges since for n n =1

every N by choosing m=2n > n > N we have

∞

n =1

1 + 1 we have e

However the tendency to apply most tests only in one direction is dominating in the course of Mathematical Analysis, although the both directions could be useful. We illustrate this idea on the example of the integral test for convergence of number series:

∞

∫ f ( x)dx

converges ⇐

converges,

traditionally is used in this case. • On the other hand one can use the implication of the integral test in the opposite direction: ∞

∑

n n =1 2

converges since for every e > 0 by taking m > n>

∞

a n =1 n

∑

implication:

∑

∑ ak < e .

series

∫x

for a ≤ 1 is easily established. Thus the

harmonic

it is usually

whose convergence for a > 1 and divergence

This test traditionally is used in both directions: the direction “IF” and the direction “ONLY IF”. For example:

• generalized

a n =1 n

converges if and only if for every

1 1 1 1 1 + +L + > n⋅ = ; n n +1 2n 2n 2

∑

+∞

e > 0 there exists such a positive number N that if m > n > N then

∞

compared with the improper integral

∞

n =1

finite.

• To study the convergence of a generalized

Let us linger on the use of the Cauchy test for convergence of number series stating this as A series

converges ⇒

∞

∫ f ( x)dx 1

converges,to show the convergence of the improper integral on the right hand side the convergence of the number series on the left hand side can be easily proved by applying D’Alembert sufficiency condition. IF –THEN statements In mathematical courses one constantly encounters statements containing links of the type IF – THEN. The typical example of such statement is „if A is true then B holds” or in symbols

A⇒ B. Starting to discuss such statements we first of all emphasize that they can be viewed either as a

Svetlana Asmuss, Natalja Budkina, Aleksandrs Šostaks Naujos matematikos didaktikos prieigos aukštojoje mokykloje: būtinybės sąlyga versus pakankamumo sąlyga

sufficiency type condition or as a necessity type condition: • A is sufficient for B, • B is necessary for A. Note that when viewing a statement A ⇒ B as a necessity type condition, usually it is applied in the form of a contra positive implication:

¬B ⇒ ¬A .

Starting to discuss IF – THEN type statements we first accentuate on the necessity part. Necessity conditions As it was stated above the necessity type condition is usually formulated in the form of a contra positive statement ¬B ⇒ ¬A . Below we consider some concrete examples taken from the standard course of mathematical analysis to illustrate our idea. ∞

Convergence of series. If a number series ∑ a n n =1 converges, then lim a n = 0. n →∞

Usually this statement is used as the convergence

Uniform convergence of functional sequences and series. If a sequence of continuous functions converges uniformly on [a; b], then the limit function is also continuous. Again, applying this theorem in the way of a contra positive implication, in some cases we can conclude that a given sequence is not uniformly convergent:

f n ( x) = x n , x ∈ [0;1] ,  0, x ∈ [0; 1) the limit function is f ( x) =  , x =1  1,

• for the sequence

which is obviously discontinuous at the point

x = 1, therefore the sequence does not converge uniformly.

∞

If a functional series ∑ f n converges uniformly n =1

on a set E, then the functional sequence { f n } converges uniformly on E to 0. ∞

• Consider the functional series ∑ x n , n =1

x ∈ (0;1)

One can easily see that these series convergence

necessity condition, which is in the way of a contra

on the set (0,1). However, by taking x n =

positive implication lim a n : if lim a n does not exist

we get f n ( x n ) =

n →∞

or lim a n ≠ 0, then the series diverges. As a typical n →∞

application of this sign we recall the following two examples: • for

the

series

lim (−1) n

n →∞

diverges;

∞

∑ (−1)

n =1

n +1 n+3

limit

n +1 does not exist, so the series n+3

 2n 2 − 3   • for the series ∑  2  n =1 2n + 1  ∞

 2n − 3   lim  2 n →∞ 2 n + 1    2

the

1 − e 2

1 , and therefore can conclude 2

One more possibility to refute the uniform convergence of sequence is to use the theorem of a passage to a limit in integrals: if a sequence of continuous functions { f n } converges uniformly to f b

on [a; b] , then lim ∫ f n ( x ) dx = ∫ lim f n ( x ) dx .

the limit

n →∞

• Using this theorem we easily establish that the convergence of the sequence of functions f n ( x ) = nx (1 − x ) to the limit function 2

diverges.

that the series does not converge uniformly.

n →∞

≠ 0 , so the series

f ( x) = 0

is not uniform on the interval [0,1]. Indeed,

This necessary condition for convergence of number series can be used also as a sufficient condition for a convergence of sequences. For example: ∞ n! an and ∑ n n =1 n! n =1 n

∞

• the convergence of the series ∑

can be easily established by D’Alembert sign,

an n! and therefore lim =0 and lim n =0. n →∞ n! n →∞ n

taking into account that

∫ f ( x)dx = 2(n + 1) n

we have b

lim n →∞

∫ a

∫

f n ( x ) dx ≠ lim f n ( x ) dx = 0 a

n →∞

Sufficient conditions

Mokytojų ugdymas 2011

Starting to discuss sufficient conditions, that is conditions “if A holds then B holds” that are given as the implication A ⇒ B, we feel it important first to emphasize that the correct use of such type formulas and theorems is not possible without verification of all assumptions under which the results hold. Thus all conditions contained in the premise A should be verified. Our experience in teaching Mathematical Analysis and other mathematical courses shows that in order to teach students to verify sufficient conditions, a lecturer must: • formulate clearly all assumptions, under which the result is true; • emphasize the role of each one of the sufficient conditions; • consider examples of typical mistakes when the result is used incorrectly. Many students nowadays often concentrate on methods and techniques of calculations, manipulations with formulas and pay little attention to the concepts, conditions of the theorems, properties of the functions. As the result students often ignore precise reasoning and justification. It is very usual that a student does not bother himself to check whether all of the sufficient conditions are satisfied. Students often do not recognize the significance of everyone of these conditions or simply do not think about them at all. Therefore it is very important for teachers to formulate clearly all assumptions at implementation under which a result is correct. For a deeper understanding of the result it is helpful for students to have a list of certain typical examples, which satisfy all the assumptions except for one in the premise of the considered theorem without the conclusion being valid. Each example illustrates why the theorem will be invalid if the corresponding assumption is violated. In many cases a simple sketch of a graph is enough to create such an example. But there are some cases when examples illustrating the significance of sufficient conditions are not so elementary. Consider for example a theorem which should be well-known to every student in mathematics. Newton - Leibniz theorem (or formula): if a function f is integrable on a closed interval [a; b] and F is its antiderivative there then the Riemann integral of f can be evaluated by the formula .

∫ f ( x)dx = F (b) − F (a ) a

This simple example can illustrate how important it is to verify carefully all conditions appearing in the statement of the theorem. 3

Indeed, consider the integral

∫ x − 1 dx . „Applying” 0

Newton – Leibniz formula and taking function

1 one comes to x −1

ln x − 1 as the antiderivative of the wrong result: 3

∫ x − 1 dx = ln x − 1

3 0

= ln 2 − ln1 = ln 2

while actually the integral

∫ x − 1 dx

diverges.

As more complicated example we could consider the integral 1

d 

1

∫ dx  arctg x  dx

−1

By taking function arctg

1 as the antiderivative x

d  1  arctg  of one could conclude that dx  x 1

d §

1·

arctg ¸ dx . ³ dx ¨© x¹ 1

1· § ¨ arctg ¸ x¹ ©

1 1

S 2

However this is not correct: function arctg not the antiderivative of

1 is x

d  1  arctg  on the interval dx  x

[-1; 1] since the function arctg

1 is discontinuous x

at the point x = 0. Here is another example concerning Riemann integral. Consider two functions:

Svetlana Asmuss, Natalja Budkina, Aleksandrs Ĺ ostaks Naujos matematikos didaktikos prieigos aukĹĄtojoje mokykloje: bĹŤtinybÄ&#x2014;s sÄ&#x2026;lyga versus pakankamumo sÄ&#x2026;lyga

Then for this function

1 ďŁą ďŁ´ x 2 sin 2 , x â&#x2030; 0, F ( x) = ďŁ˛ x ďŁ´ďŁł 0, x = 0, 1 2 1 ďŁą ďŁ´ 2 x sin 2 â&#x2C6;&#x2019; cos 2 , x â&#x2030; 0, f ( x) = ďŁ˛ x x x ďŁ´ďŁł 0, x = 0.

One can show that function f is the derivative of the function F at each point x, and so f has an antiderivative F. On the other hand since f is unbounded on the interval [-1; 1], so it is not integrable (in the sense of Riemann) on this interval. Many students prefer calculations to theoretical reasoning. Therefore we recommend to consider also the following instructive counterexamples concerning the use of Substitution Rule in

â&#x2C6;Ť dy â&#x2C6;Ť f ( x, y )dx â&#x2030; â&#x2C6;Ť dx â&#x2C6;Ť f ( x, y)dy although each integral is proper. Indeed, for 0 < y < 1 we have 1

â&#x2C6;Ť 0

f ( x, y ) dx = â&#x2C6;Ť y â&#x2C6;&#x2019;2 dx â&#x2C6;&#x2019; â&#x2C6;Ť x â&#x2C6;&#x2019;2 dx = 1 ,

and so

â&#x2C6;Ť dy â&#x2C6;Ť f ( x, y )dx = 1

On the other hand, for 0 < x < 1 we have

Integration.

â&#x2C6;Ť f ( x, y )dy = â&#x2C6;&#x2019; â&#x2C6;Ť x

â&#x20AC;˘ Assume that we have to calculate integral

â&#x2C6;Ťx

1 â&#x2C6;&#x2019; x 2 dx . The (often recommended in

and therefore

similar situations) substitution x = sin t leading

dy + â&#x2C6;Ť y â&#x2C6;&#x2019;2 dy = â&#x2C6;&#x2019;1 x

â&#x2C6;Ť dx â&#x2C6;Ť f ( x, y )dy = â&#x2C6;&#x2019;1 .

Easy counterexamples can be constructed to show the role of conditions in Cauchy Theorem. This theorem states that if f and g are continuous functions on a closed interval [a; b] , differentiable at least on (a; b) and besides g â&#x20AC;˛( x) â&#x2030; 0 for all x â&#x2C6;&#x2C6; (a; b) then there exists a point c â&#x2C6;&#x2C6; (a; b) such that

f â&#x20AC;˛(c) f (b) â&#x2C6;&#x2019; f ( a ) = â&#x20AC;˛ g (c ) g (b) â&#x2C6;&#x2019; g ( a )

at the first stage of the calculation to the 2

indefinite integral

â&#x2C6;Ť cos 3 td cos t is wrong in

this case because of the restriction sin t â&#x2030;¤ 1 . â&#x20AC;˘ To calculate

â&#x2C6;&#x2019;2

â&#x2C6;Ť R(sin x, cos x)dx , where

R(sin x, cos x) is a rational function of sin x and cos x , the substitution t = tg

is often 2 recommended. However it is not applicable 2p

in case of the integral

â&#x2C6;Ť 1 + 0, 5 cos x

. Indeed t â&#x2020;&#x2019; â&#x2C6;&#x2019;â&#x2C6;&#x17E;(t â&#x2020;&#x2019; +â&#x2C6;&#x17E;) when

x o S 0( x o S 0) The importance to take into account all conditions when calculating Double integrals can be illustrated by the following instructive example. Let

ďŁą y â&#x2C6;&#x2019;2 , 0 < x < y < 1, ďŁ´ â&#x2C6;&#x2019;2 f ( x, y ) = ďŁ˛ â&#x2C6;&#x2019; x , 0 < y < x < 1, ďŁ´0, for other points of square x â&#x2C6;&#x2C6; [0,1], y â&#x2C6;&#x2C6; [0,1] ďŁł

One can easily see that the conclusion of this theorem does not hold for example for functions 2 3 â&#x20AC;˘ f ( x) = x , g ( x) = x , x â&#x2C6;&#x2C6; [â&#x2C6;&#x2019;1;1] . 2 3 â&#x20AC;˘ f ( x) = x + x, g ( x) = x , x â&#x2C6;&#x2C6; [â&#x2C6;&#x2019;1;1] .

since the condition g â&#x20AC;˛( x) â&#x2030; 0 is not fulfilled for all x â&#x2C6;&#x2C6; (â&#x2C6;&#x2019;1,1) in this case. Many instructive examples can be constructed to illustrate the significance of every condition in the Lâ&#x20AC;&#x2122;Hopital Rule. Let

f and g be real-valued functions defined

and differentiable in a neighbourhood of a point a. Further, suppose that =0 and

lim f ( x) = lim g ( x)

xâ&#x2020;&#x2019;a

( f ' ) 2 ( x) + ( g ' ) 2 ( x) â&#x2030; 0 for all x â&#x2030; a

Mokytojų ugdymas 2011

from some neighbourhood of the point a. Then

lim

x →a

f ( x) f ' ( x) = lim provided that the latter limit g ( x ) x →a g ' ( x )

exists. When studying this topic student often use this rule mechanically and do not ponder whether this use is rightful. Quite often they do not verify the validity of all conditions under which the rule can be used.

1 x = lim x x sin 1 =0. lim x→0 sin x x→0 sin x x x 2 sin

Sufficient conditions need not be criteria Sometimes students wrongly use statements of the type „if A is true then B holds”or in symbols

Sometimes even the presence of the indeterminacy

0   is not verified. Therefore it is helpful to give 0

also such elementary examples as, say, the following:

e x + sin x − 1 . x→0 1 − cos x + sin( x − 1) lim

A⇒ B

as criteria. It is important to show that sufficient conditions need not be at the same time necessary, that they need not be criteria. We illustrate this by the following three examples. ∞

• Convergence of series. If a number series ∑ a n

By using formally the L’Hopital Rule we get:

converges then lim a n = 0. n →∞

e + sin x − 1 lim = x →0 1 − cos x + sin( x − 1)

lim

)′

+ sin x − 1

e x + cos x 2 = x→0 sin x + cos( x − 1) cos1 result

n+2

n =1

(n + 1) n

n+2

n →∞

incorrect

because

lim (1 − cos x + sin( x − 1) ) ≠ 0 and hence the use of

x →0

L’Hopital Rule is illegal. Note that the above limit can be easily calculated directly:

lim S n = +∞ . n →∞

• If the limit lim a n exists, then the sequence n →∞

{a n } is bounded.

e x + sin x − 1 lim = 0. x →0 1 − cos x + sin( x − 1)

The sequence {( −1) n }

It is worth mentioning also that non-existence

lim(−1) n does not exist.

of the limit

lim

x→a

f ' ( x) g ' ( x)

does not imply the non-

f ( x) existence of lim x →0 g ( x) .

n →∞

• Let f : (a; b) → R be a function and suppose

 x 2 sin 1 ′ 1 1 2 x sin − cos   x x x = lim lim  x →0

differentiable at x0 then f ′( x0 ) = 0 . The function f ( x) = x 3 has no local extremum at x0 =0 although the derivative f ′( x) = 3 x 2 is 0 at x= 0.

can easily see that the limit

(sin x )′

is bounded, but

that x0 ∈ (a; b) is a local extremum of f. If f is

1 x 2 sin x . One Indeed, consider the limit lim x→0 sin x

x →0

the limit

= 0 , but for k = 1, 2, K , n (n + 1) n 1 1 k+2 > ≥ , so ak = (k + 1) k k n n 1 k+2 Sn = ∑ ≥n = n and k =1 ( k + 1) k n lim

lim

This

∞

For the series ∑

(1 − cos x + sin( x − 1) )′

x →0

n =1

cos x

does not exist. But on the other hand

Conclusion Students starting studies at the university come across with a large number of theorems and tasks whose formulations include different type of conditions. Further, these conditions become apparent and show themselves in the proofs of theorems and in

Svetlana Asmuss, Natalja Budkina, Aleksandrs Šostaks Naujos matematikos didaktikos prieigos aukštojoje mokykloje: būtinybės sąlyga versus pakankamumo sąlyga

SVETLANA ASMUSS Matematikos mokslų daktarė, Latvijos universitetas, Fizikos ir matematikos fakultetatas. Moksliniai interesai: aproksimacijos teorija, splainų teorija.

PhD (Mathematics), University of Latvia, Faculty of Physics and Mathematics. Research interests: Approximation Theory, Spline Theory.

Address: Zellu Str. 8, Riga, LV-1002, Latvia E-mail: svetlana.asmuss@lu.lv

NATALJA BUDKINA the solutions of the tasks what requests the verification of different type of conditions. This situation is new (to compare with exposition of mathematical courses at the secondary school) for most of the students and

PhD (Mathematics), Riga Technical University Research interests: Approximation under inexact discrete information

Address: Kalku Str. 1, Riga, LV-1050, Latvia E-mail: natalja.budkina@rtu.lv

ALEKSANDARS ŠOSTAKS Habilituotas mokslų daktaras (matematika), Latvijos universitetas, Fizikos ir matematikos fakultetas. Moksliniai interesai: topologija ir geometrija, neapibrėžtos aibės ir neapibrėžtos struktūros.

Habil. Dr. (Mathematics), University of Latvia, Faculty of Physics and Mathematics. Research interests: Topology and Geometry, Fuzzy Sets and Fuzzy Structures.

Address: Zellu Str. 8, Riga, LV-1002, Latvia E-mail: aleksandrs.sostaks@lu.lv

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 18–32 Teacher Education. 2011. No. 16 (1), 18–32

Liudvikas Kaklauskas, Danutė Kaklauskienė Šiaulių universitetas • Šiauliai University

Virtualios mokymo(si) aplinkos įrankių panaudojimo matematikos mokymui analizė

Analysis of Usage of Virtual Teaching/Learning Environment Tools for Teaching Mathematics

Anotacija Analizei parinkta Moodle virtualioji mokymo(si) aplinka (VMA), turinti standartinius ir specializuotus įrankius. Pagal verbalinių dokumentų analizės ir ekspertinės apklausos rezultatus suformuoti septyni VMA įrankių vertinimo kriterijai. Nustatyta, kad matematikos mokymui aukštosios mokyklos e.studijų kurse geriausiai tinka naudoti 16 standartinių ir 6 specializuotus Moodle įrankius. Pagrindiniai žodžiai: e. studijos, virtuali mokymo(si) aplinka, Moodle, verbalinių dokumentų analizė, ekspertinė apklausa.

Abstract The analysis was carried out employing Moodle virtual teaching/learning environment, which has standard and specialized tools. Based on the results of the analysis of verbal documents and on expert interview results, seven assessment criteria of virtual teaching/learning environment tools were formed. It was identified that the most suitable tools for teaching mathematics in the e-studies course of higher education institutions are 16 standard and 6 specialized Moodle tools. Key words: e-learning, virtual teaching and learning environment, Moodle, analysis of verbal documents, expert survey.

Įvadas Taikant šiuolaikines informacines-kompiuterines technologijas mokymo procese galima sukurti mokymosi aplinką, ugdančią bendravimą, savarankiškumą, kūrybiškumą, kritinį mąstymą ir informacinę kultūrą. 1924 m. S. L. Pressey sukurtą mechaninę testavimo mašiną (Pressey, 1926) galima laikyti šiuolaikinių nuotolinių studijų užuomazga. Bihevioristinės psichologijos atstovas B. Skinner (Skinner, 1968) 1958 metais aprašė programuoto mokymo idėjas, kurias realizavo mechaninėje automatizuoto mokymo mašinoje. Toliau šias idėjas plėtojo Miller ir Rice, sukurdami sistemų organizavimo teoriją (Miller, Rice, 1967), Kaye ir Rumble (Kaye, Rumble, 1981) – distancinio mokymo sistemas. 1991 m. D. Keegan (Keegan, 1991) aprašė nuotolinio mokymosi schemą, kurią tobulino kiti mokslininkai. Lietuvoje nuotolinio mokymo organizavimo klausimus nagrinėja A. Targamadzė, E. Normantas, D. Rutkauskienė, A. Vidžiūnas (Targamadzė, Normantas, Rutkauskienė, Vidžiūnas, 1999), A. Volungevičienė (Volungevičienė, 2008) ir kiti mokslininkai. Nuo 2000 m. vis plačiau naudojama e. studijų są-

Introduction Application of modern information-computer technologies in the teaching process enables to create the learning environment that develops communication, independence, creativity, critical thinking and information culture. The mechanical testing machine created by S. L. Pressey in 1924 (Pressey 1926) can be treated as a rudiment of modern distance learning. In 1958, the representative of behaviouristic psychology B. Skinner (Skinner B. 1968) described the ideas of programmed teaching, which he later implemented in the mechanical automated teaching machine. These ideas were further developed by Miller and Rice, who created the systems theory of organizations (Miller, Rice 1967), and Kaye and Rumble (Kaye, Rumble 1981), who developed distance teaching systems. In 1991, D. Keegan (Keegan 1991) described the distance learning scheme, which was further improved by other scientists. In Lithuania issues of distance learning organization are analysed by A. Targamadzė, E. Normantas, D.Rutkauskienė, A. Vidžiūnas (Targamadzė, Normantas, Rutkauskienė, Vidžiūnas 1999), A. Volungevičienė (Volungevičienė 2008) and other scientists. Since 2000, the concept of e-learning is being

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

voka. E. studijas galima organizuoti įvairiai: naudojant spausdintas priemones, garso ir vaizdo įrašus, tinklalapius, kompiuterinius treniruoklius, vaizdo konferencijas, įvairialypės terpės aplinkas (Enciklopedinis kompiuterijos žodynas, 2011), mokymąsi susirašinėjant, virtualias mokymo aplinkas (toliau – VMA) (Enciklopedinis kompiuterijos žodynas, 2011) ir kitas analogiškas priemones. Remiantis D. Rutkauskienės (Rutkauskienė ir kt. 2003), A. Targamadzės (Targamandzė ir kt. 1999) ir kitų Lietuvos mokslininkų darbais galima įvardyti šias e. studijų patrauklumo priežastis: • mokytis gali bet kas, bet kada, bet kurioje vietoje, turinčioje interneto ryšį – namie, bibliotekoje, darbe ir pan.; • mokymas(is) pigesnis, nes nėra kelionės, auditorijos išlaikymo išlaidų; • pateikiama informacija yra nauja, nuosekli, nuolat atnaujinama ir tobulinama; • studijuojantieji gali iš karto pasiekti naujausią informaciją, kai tik jos prireikia; • mokymas yra lankstus, t. y. lengvai taikomas pagal asmens gebėjimus ir poreikius; • naudojamos bendravimo priemonės leidžia visiems diskutuoti vienu metu ir vienas kitam netrukdyti taip sudarant psichologinio komforto įspūdį; • sudėtingas mokymosi situacijas galima daug kartų kartoti; • vienodas turinys ir reikalavimai visiems studijuojantiesiems. Populiariausia priemonė, naudojama organizuojant e. studijas yra VMA. Standartinė VMA apima (žr. 1 pav.): priemones, skirtas mokomosios medžiagos perteikimui; įrankius interaktyviai (Enciklopedinis kompiuterijos žodynas, 2011) aplinkoje registruotų asmenų sąveikai; įrankius, užtikrinančius studijuojančiųjų bendravimą ir bendradarbiavimą; ataskaitas apie studijuojančiųjų pasiekimus ir aktyvumą aplinkoje; įrankius, leidžiančius administruoti vartotojus. Virtualias mokymo aplinkas galima suskirstyti į dvi grupes: mokymo(si) aplinka, realizuota kaip atviroji programinė įranga (Enciklopedinis kompiuterijos žodynas, 2011) ir mokymo(si) aplinka, realizuota kaip komercinė programinė įranga (Enciklopedinis kompiuterijos žodynas, 2011). Pasaulyje populiaresnės VMA realizuotos kaip atviroji programinė įranga, nes jas tobulinant ir plėtojant gali prisidėti kiekvienas. Tokias VMA galima lengvai pritaikyti savo reikmėms, papildyti trūkstamais moduliais ir pan. Komercinės VMA turi gerą aplinkų palaikymo sistemą, tačiau yra mažiau lanksčios. Kiekvienoje VMA yra interaktyvios sąveikos įrankiai, skirti studijuojančiųjų bendravimui ir bendradarbiavimui, tokie įrankiai vadinami studijų proceso aktyvinimo įrankiais.

increasingly used. E-learning can be organized in different ways: using printed aids, audio and video records, websites, computer- based teaching aids, video conferences, diverse media environments (Encyclopaedic Dictionary of Computer Terms 2011), learning by correspondence, virtual teaching environments (further referred to as VTE) (Encyclopaedic Dictionary of Computer Terms 2011) and other analogous aids. According to the works of D. Rutkauskienė (Rutkauskienė et al. 2003), A. Targamadzė (Targamandzė et al. 1999) and other Lithuanian scientists e-learning is attractive for the following reasons: • Learning is accessible to anyone, anytime and in any place with the Internet connection: at home, in the library, at work, etc.; • Learning and teaching is cheaper because there are no travel costs, lecture room upkeep costs; • Information provided is new, consistent, constantly updated, and improved; • Learners gave access to the newest information at once whenever they need it; • Teaching is flexible; i.e., easily adjusted to the person’s abilities and needs; • Employed means of communication enable everybody to discuss at the same time and not disturb each other, this way creating the impression of psychological comfort; • Difficult learning situations can be repeated many times; • The same content and requirements to all learners. The most popular means used organizing e-learning is virtual teaching environment (VTE). Standard VTE encompasses (see Fig. 1): means for conveying teaching materials; tools for interactive interaction (Encyclopaedic Dictionary of Computer Terms 2011) of persons registered in the environment; tools ensuring communication and cooperation between learners; reports on learners’ achievements and activeness in the environment; tools enabling to administer users. Virtual teaching environments can be distributed into two groups: teaching and learning environment implemented as open source software (Encyclopaedic Dictionary of Computer Terms 2011) and teaching and learning environment implemented as a commercial software (Encyclopaedic Dictionary of Computer Terms 2011). VTEs implemented as open source software are more popular in the world because everybody can contribute to their improvement and development. Such VTEs can be easily adjusted to one’s needs, supplemented with missing modules, etc. Commercial VTEs have a good system of maintaining environments but are less flexible. Every VTE contains interactive interaction tools for communication and cooperation between learners; such tools are called tools of activating the learning process.

Mokytojų ugdymas 2011

Teaching materials

VMA/ VTE Communication and cooperation

Reports

Interactive interaction

Administration

1 pav. VMA priemonės Fig. 1. Means of the VTE

Technologiniai e.studijų proceso organizavimo sprendimai VMA pateikia didelę įrankių, skirtų studijoms, įvairovę. Norint pasiekti kuo geresnių e. studijų rezultatų mokant matematikos aukštojoje mokykloje būtina taikyti naujas, šiuolaikiškas mokymo(si) metodikas, įvertinti siūlomų VMA įrankių tinkamumą dėstomajam dalykui. Jau 1992 m. studijų kokybės problematiką tyrinėjo P. Honey ir A. Mumford (Honey, Mumford, 1992). Jie bandė studentus skirstyti į grupes pagal jų pasirinktą studijavimo būdą ir atsižvelgiant į tai rengti studijų medžiagą. Adaptacijos metodų taikymo pedagoginius ypatumus bei problematiką, įvertinant studentų individualias savybes, analizavo S. Aboujaoude (Aboujaoude, 2011), A. Heraz ir C. Frasson (Heraz, Frasson, 2008), S. Chaffar, C. Frasson (Chaffar, Frasson, 2004). Šiuolaikiniam aukštosios mokyklos studentui svarbūs gebėjimai ir įgūdžiai, padedantys susiorientuoti ir išreikšti save kūrybinėje visuomenėje ir pasiekti geresnių rezultatų. Pasak S. Aboujaoude, mokymo ir bendravimo naujovės XXI a. teikia ne tik daug privalumų, tačiau kelia nemažai pavojų, todėl jas pasirenkant ir taikant būtina išsami analizė. Apžvelgus nūdienos mokslininkų darbus, vertinančius VMA įrankių naudojimą aukštosios mokyklos

Technological solutions of organizing the e-learning process in the VLE provide with a big diversity of tools for learning. To achieve better results of e-learning, teaching mathematics at the higher education institution, it is necessary to apply new modern teaching and learning methods, consider suitability of proposed VTE tools for the delivered subject. P. Honey and A. Mumford studied the problems of quality of studies already in 1992 (Honey, Mumford 1992). They tried to distribute students into groups according to their chosen way of studying and considering this, prepare course materials. Pedagogical peculiarities and problems of applying adjustment methods, considering students’ individual features, were analysed by S. Aboujaoude (Aboujaoude 2011), A. Heraz and C. Frasson (Heraz, Frasson 2008), S. Chaffar, C. Frasson (Chaffar, Frasson 2004). Modern student of the higher education institution finds it important to have abilities and skills, helping to orientate and express himself/herself in the creative society and achieve better results. According to S. Aboujaoude, teaching and communication novelties in the 21 century both provide many advantages and cause quite many dangers; therefore, it is necessary to carry out a comprehensive analysis, choosing and applying them. Having reviewed contemporary scientists’ works, evaluating usage of VTE tools for teaching

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

matematikos mokymui, nustatyta, kad tyrimuose dažniausiai apsiribojama bendraisiais įrankių panaudos e. studijoms tyrimais (Daugiamas, Taylor, 2003; Cole, Foster, 2007), analizuojamas jų panaudojimas kitų dalykų (ne matematikos) mokymui (Martin-Blas, Seranno-Fernandez 2008), aptariamas atskirų matematikos skyrių mokymas (Iglesias ir kt., 2008), nagrinėjamos pasirinktos Moodle įrankių grupės taikymas matematikos mokymui daugiau techniniu nei edukaciniu aspektu (Sangwin, 2005). Reikėtų pažymėti, kad standartinių ir specializuotų Moodle (Moodle 2011) aplinkos įrankių tinkamumo matematikos mokymui e. studijų kurse edukaciniai aspektai nėra analizuoti. Tikslas: įvertinti VMA Moodle įrankių tinkamumą matematikos mokymui aukštosios mokyklos e.studijų kurse.

mathematics at higher education institutions, it has been identified that most often researches are limited to general researches into usage of tools for e-learning (Daugiamas, Taylor 2003; Cole, Foster 2007), analysis of their usage for teaching other subjects (not mathematics) (Martin-Blas, Seranno-Fernandez 2008), analysis of teaching separate sections of mathematics (Iglesias et al. 2008), and analysis of application of the chosen Moodle tool group for teaching mathematics more in a technical than educational aspect (Sangwin 2005). It has to be noted that educational aspects of suitability of standard and specialized Moodle (Moodle 2011) environment tools for teaching mathematics in the e-course have not been analysed. The aim of the article: to evaluate suitability of VTE Moodle tools for teaching mathematics in the e-learning course at the higher educational institution.

VMA Moodle įrankių apžvalga Virtualių mokymo(si) aplinkų įvairovė išsamiai apžvelgta straipsnyje (Kaklauskas, Kaklauskienė, 2010). Lietuvoje dažniausiai naudojama VMA yra Moodle (Kaklauskas, Kaklauskienė, 2010). Moodle naudojama daugumoje šalies aukštųjų mokyklų bei sėkmingai diegiama ir į Lietuvos bendrojo lavinimo mokyklas. Aplinkoje realizuoti įrankiai visiškai tenkina e. studijų poreikius. Tai patvirtina ir kūrėjų tinklalapio duomenys (Moodle 2011). Remiantis informacija, pateikiama Moodle kūrėjų tinklalapyje, aplinka naudojama 209 pasaulio šalyse, registruoti ir patvirtinti 53592 tinklalapiai, kuriuose ji įdiegta ir naudojama mokymo(si) reikmėms, sukurta ir įregistruota 4636711 kursų su 44693924 vartotojų (Moodle 2011). Visavertė VMA turi apimti septynias standartizuotas priemones (Enciklopedinis kompiuterijos žodynas, 2011). Trumpai aptariamos priemonės, kurias siūlo Moodle VMA. • Bendravimo ir bendradarbiavimo priemonės. VMA Moodle tam skirti įrankiai: forumas, pokalbis, dialogas, pasirinkimas, kalendorius, apklausa (angl. Survey), socialinis forumas (angl. Wiki), žurnalas, internetinis dienoraštis (angl. Blog), žinutės, Youtube. • Besimokančiųjų ir mokytojo pristatymo sritys. VMA Moodle tam skirti įrankiai: kolegos – kurso dalyvių sąrašo peržiūra pagal jų paskutinio prisijungimo datą, prisijungę vartotojai – per paskutines 5 minutes prisijungusių vartotojų sąrašas. • Naudotojų registravimo priemonės. VMA Moodle galima rankiniu būdu įregistruoti vartotojus, tačiau yra galimybė įsiregistruoti į aplinką

Review of VTE Moodle Tools Diversity of virtual teaching and learning environments is comprehensively reviewed in the article (Kaklauskas, Kaklauskienė 2010). In Lithuania, the VTE that is most often used is Moodle (Kaklauskas, Kaklauskienė 2010). Moodle is used in the majority of Lithuanian higher education institutions and is successfully implemented in Lithuanian comprehensive schools. Tools implemented in the environment fully meet e-learning needs. This is also confirmed by the data of the designers’ website (Moodle 2011). Based on the information given on the Moodle designers’ website, the environment is being used in 209 countries of the world; there are 53592 registered and approved websites in which it is installed and used for teaching and learning needs, there are 4636711 designed and registered courses with 44693924 users (Moodle 2011). The full-fledged VTE has seven standardized means (Encyclopaedic Dictionary of Computer Terms 2011). We shall briefly discuss the means offered by the Moodle VTE. • Means of communication and cooperation. Tools in the VTE Moodle for this purpose: forum, conversation, dialogue, choice, calendar, survey, Wiki, journal, blog, messages, You Tube. • Areas of learners’ and the teacher’s presentation. VTE Moodle tools for this purpose: colleagues (review of the list of course participants by the last login date), online users (the list of online users who have logged in during the last 5 minutes). • Means for users’ registration. In VTE Moodle users can be registered manually but students themselves can also register in the environment. Manual registration can be centralized, applying

Mokytojų ugdymas 2011

ir patiems studentams. Rankinis registravimas gali būti centralizuotas, pritaikant bet kurį standartinį kompiuterių tinklo vartotojų indentifikavimo įrankį. Centralizuotas vartotojų registravimas pagerina vartotojų veiksmų kontrolę ir užtikrina didesnį jų saugumą VMA. Naudojant informacijos apie e. kurso vartotoją įrankį galima peržiūrėti kiekvieno kurso dalyvio paskelbtą asmeninę informaciją. Šis įrankis leidžia studijų proceso dalyviams tarpusavyje komunikuoti naudojant kitas komunikavimo priemones, pavyzdžiui, elektroninį paštą ar telefoną ir pan. Įsirašymo į kursą įrankis padeda apsaugoti autoriaus parengtą kursą nuo nepageidaujamų lankytojų. Apsaugai naudojamas įsirašymo raktas, kurį skelbia tik kursą dėstantis dėstytojas. • Ugdymo turinio tvarkymo priemonės. VMA Moodle ugdymo turiniui tvarkyti galima panaudoti tokius įrankius: kurso medžiaga arba resursas – tai įvairiausios mokymo medžiagos skelbimo galimybės, kai kūrėjas gali formuoti tinklalapį, skelbti bet kokio formato medžiagą skaitymui bei formuoti nuorodas į VMA bei interneto informacijos šaltinius; kurso peržiūra – visų ar dalies temų peržiūros įrankis kurso aplinkoje; žodynėlis – sąvokų, naudojamų kurse, sąrašas, kuris gali būti pildomas ir bendradarbiaujant su kurso dalyviais; duomenų bazė (angl. Database) – įrankis, naudojamas klasifikuotos medžiagos surinkimui ir jos vertinimui, kai į procesą įtraukiami ir kurso dalyviai; įvykiai – kurso svarbių įvykių skelbimas susiejant juos su kalendoriumi; paskutinės naujienos – kurso naujienų skelbimas. Reikėtų pažymėti, kad nuotoliniame kurse galima sėkmingai skelbti garso bei vaizdo medžiagą, keistis garso žinutėmis. • Užduočių rengimo ir apklausos organizavimo priemonės. VMA Moodle tam skirti šie įrankiai: kontrolinis – kurso dalyvių savikontrolės arba kontrolės priemonė, leidžianti automatiškai ar rankiniu būdu vertinti atsakymus pagal dėstytojo nurodytus kriterijus; užduotis – priemonė, naudojama tada, kai darbą atlikti virtualioje aplinkoje nėra galimybių, studijuojantysis jį atlieka namų kompiuteryje ar kitur ir prisega arba aprašo atlikto darbo rezultatus; virtualus seminaras (angl. Workshop) – tai Moodle seminaras, kai atsiskaitymo tvarką reglamentuoja dėstytojas, o rezultatus komentuoja ir vertina studentai bei dėstytojas; pamoka – amerikiečių psichologo B. F. Skinerio programuoto mokymo idėjas

any standard tool for identification of computer network users. Centralized registration of users improves users’ action control and ensures their higher safety in the VTE. Using the information tool about the e-course user, one can review every course participant’s announced personal information. This tool enables the participants of the study process to communicate with each other, using other communication means; e.g., e-mail or telephone, etc. The tool of logging in the course helps to protect the author’s prepared course from unwanted visitors. Protection is ensured by the login key which is announced by the teacher delivering the course. • Means for managing the content of education. To manage the content of VTE Moodle, the following tools can be used: course materials or resource (these are various possibilities of announcing teaching materials when the designer can form the website, announce materials in any format for reading and form references to the VTE and online information sources); course review (the tool for review of all or part of topics of the designed course in the course environment); glossary (the list of concepts used in the course, which can also be filled in cooperation with course participants); database (the tool used for collection and evaluation of classified materials, when course participants are also involved in the process); events (announcement of important course events, relating them to the calendar); the latest news (announcement of course news). It must be noted that in the distance course audio and video materials can be successfully announced, audio messages can be exchanged. • Means for task preparation and organization of tests. In VTE Moodle the following tools serve this purpose: the test (a tool for course participants’ self-control or control enabling to assess answers automatically or manually according to the criteria indicated by the teacher); the task (a tool, used when there are no possibilities to perform the work in the virtual environment, the learner carries it out on his home PC or elsewhere and attaches or describes the results of the carried out work); virtual workshop (a Moodle workshop when the order of accounting is regulated by the teacher whilst the results are commented and assessed by students and the teacher); the lesson (the tool implementing the American psychologist B. F. Skinner’s programmed teaching ideas); Hot Potatoes (a collection for preparation of crosswords and other interactive testing means)

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

realizuojantis įrankis; kryžiažodžių ir kitų interaktyvių testavimo priemonių rengimo rinkinys – Hot Potatoes (Hot Potatoes 2011). • Besimokančiųjų mokymosi ir pažangos stebėjimo priemonės. VMA Moodle skirti šie įrankiai: testo rezultatai – studijuojančiųjų apklausų peržiūros, eksporto į kitus formatus bei statistinių įverčių skaičiavimo įrankis; vartotojų aktyvumo analizės įrankis, leidžiantis įvertinti studentų aktyvumą e. kurse; virtuali studijuojančiųjų įvertinimų suvestinė – žurnalas. • Aplinkos sąsajos keitimo priemonės. Šios priemonės aplinkoje pasiekiamos tik administratoriui arba kūrėjui. Jų dėka galima keisti virtualaus kurso komponentų išdėstymą, keisti kurso dizaino šabloną, koreguoti bei papildyti aplinką savo logotipais bei kitomis kurso identifikavimo priemonėmis. Nuo 2011 m. platinama Moodle 2.0.1+ versija, kurioje realizuota daug naujų galimybių: • virtualių studijų centras – čia registruojami ir klasifikuojami kursai, kurie nebūtinai yra Moodle serveryje, o vartotojams, atsižvelgiant į jų teises, gali būti leidžiama skelbti, peržiūrėti, registruotis, persisiųsti bei atlikti kitus veiksmus su centre paskelbtais kursais (Moodle community hubs 2011); • VMA failų ir katalogų valdymas administratoriaus teisėmis (Moodle repositories 2011); • virtuali registruotų vartotojų duomenų saugykla – čia galima ne tik saugoti vartotojui reikalingą informaciją, bet ir ja dalytis su kitais kurso dalyviais (Moodle portfolio 2011); • kurso baigimo fiksavimas – čia, atsižvelgiant į kūrėjo nurodytus kriterijus, aprašoma e. kursų studijavimo tvarka. Pavyzdžiui, e.kursas A turi būti baigtas prieš pradedant studijuoti e. kursą B. (Moodle course completion 2011); • sąlyginis kurso priemonių naudojimas – čia, atsižvelgiant į studento pasiekimus, datą, aktyvumą ir kitus kriterijus, aprašomos įvairių kurso priemonių panaudojimo galimybės (Moodle conditional activities 2011); • studentų grupės registravimas į kursus – čia galima, atsižvelgiant į nurodytus kriterijus, automatiškai kurti studentų grupes ir jas registruoti į numatytus kursus (Moodle cohorts 2011); • interneto paslaugų integravimas į Moodle – tai priemonė, leidžianti kitas internetines sistemas įtraukti į Moodle aplinką (Moodle Web services 2011);

(Hot Potatoes 2011). • Means for observation of learners’ learning and progress. In VTE Moodle the following tools perform this function: test results (a tool for reviewing learners’ tests, exporting to other formats and computing statistical scores); the tool for analysing users’ activeness (it enables to assess students’ activeness in the e-learning course); the journal (a virtual summary of learners’ marks). • Means of changing the interface of the environment. Here tools are accessible in the environment only to the administrator or designer. They enable to make changes in the layout of the virtual course components, the course design template, correct and supplement the environment with one’s logotypes and other means of course identification. Since 2011, Moodle 2.0.1+ version has been disseminated, which has many new possibilities: • Virtual learning centre. Courses which are not necessarily in the Moodle server are registered and classified here, whilst users, considering their rights, can be allowed to announce, review, register, forward and carry out other actions with courses announced in the centre (Moodle community hubs 2011); • Management of VTE files and catalogues with administrator’s rights (Moodle repositories 2011); • Virtual storage of registered users’ data. Here information necessary for the user can be both stored and shared with other course participants (Moodle portfolio 2011); • Recording of the completion of the course. Here, considering the criteria indicated by the designer, the order of studying e-courses is described. For example, e-course A has to be completed before studying e-course B (Moodle course completion 2011); • Conditional usage of course aids. Here, considering the student’s achievements, date, activeness and other criteria, possibilities of using various course aids are described (Moodle conditional activities 2011); • Registration of the students’ group to courses. Here, considering the indicated criteria, students’ groups can be automatically created and registered to foreseen courses (Moodle cohorts 2011); • Integration of Internet services into Moodle. This is the means enabling to include other internet systems into the Moodle environment (Moodle Web services 2011);

Mokytojų ugdymas 2011

• komentarų palaikymas – šis įrankis leidžia komentuoti bet kurį Moodle puslapį (Moodle comments block 2011); • plagiato kontrolė – tai Moodle modulis su plagiato kontrolės Moodle VMA funkcija (Moodle plagiarism prevention 2011).

• Maintenance of comments. This tool enables to comment any Moodle page (Moodle comments block 2011); • Plagiarism control. This is a Moodle module with the Moodle VTE function for control of plagiarism (Moodle plagiarism prevention 2011).

Įrankių vertinimo kriterijai Taikant verbalinių dokumentų analizės metodą (angl. Content analysis), išanalizuoti darbai, aprašantys įrankių savybes, kurios yra tinkamiausios matematikos mokymui aukštosios mokyklos e.studijų kurse. Remiantis analizės rezultatais, atlikta septynių Šiaulių universiteto dėstytojų ekspertų ekspertinė apklausa. Ekspertinė apklausa vykdyta tiesioginio kontakto būdu. Parenkant apklausai ekspertus buvo atsižvelgta į du pagrindinius kriterijus: • dėsto universitete bet kurią matematinę discipliną ne mažiau nei trejus metus; • parengė (arba rengia) e. studijų kursą arba yra baigęs nuotolinių kursų rengimo mokymus ir turi jų baigimo pažymėjimą. 1986 m. D. Keegan (Keegan, 1991), remdamasis B. Holmberg (Holmberg, 1983) teorija, įvardijo šešis svarbiausius nuotolinio kurso medžiagos rengimo ypatumus. H. T. K. Yee, W. S. Luan, A. F. M. Ayub ir R. Mahmud, išanalizavę literatūrą, nurodė keturis veiksnius, lemiančius studentų e. mokymąsi: lytį, požiūrį į mokymąsi internete, naudojimo paprastumą ir naudingumo suvokimą (Yee ir kt., 2009). G. Smith ir D. Ferguson konstatavo, kad matematikos dėstytojai, rengdami medžiagą e.kursui, susiduria su matematinių išraiškų įterpimo problema, t. y. nepalaikoma matematinių išraiškų įterpimo kalba, todėl jas reikia įterpti kaip grafinius objektus (Smith, Ferguson, 2005). Pasak G. Albano, matematikos e. studijų modelis remiasi faktoriais, kurie priklauso trims skirtingiems lygmenims: kognityvinis lygmuo apima konkrečias sąvokas ir metodus; metakognityvinis lygmuo – mokymosi proceso savikontrolę; nekognityvinis lygmuo – įsitikinimus, emocijas, požiūrius ir visus emocinius aspektus, formuojančius besimokančiųjų kritinius sprendimus ir veiklas (Albano, 2006). V. James (2008) tyrimai parodė, kad verbalinė ir regimoji informacija, nepaisant jos perteikimo būdo, yra pagrindinis rodiklis, formuojantis studento žinias. Pasak P. Honey ir A. Mumford, studentai skirstomi į eksperimentatorius – linkusius iš karto pabandyti įgytas žinias praktikoje, registratorius – pirmiausia kaupiančius žinias ir tik po to jas apibendrinančius, pragmatikus – ieškančius naujų idėjų bei teoretikus – jiems aktualu patiems daryti apibendri-

Criteria for Assessment of Tools Applying the method of content analysis of verbal documents, the works describing the features of tools, which are most suitable for teaching mathematics in the e-learning course at the higher education institution, are analysed. Based on the results of the analysis, the expert survey of seven teachers-experts of iauliai University has been carried out. The expert survey was carried out in a direct contact way. Choosing experts for the survey, two main criteria have been considered: • the expert has not less than three year experience of teaching any mathematical subject at the university; • the expert has prepared or must be preparing an e-learning course or has completed training on distance course preparation and has the certificate certifying the completion. In 1986, based on B. Holmberg (Holmberg 1983) theory, D. Keegan (Keegan 1991) named six most important peculiarities of preparing distance course materials. Based on literature analysis, H. T. K. Yee, W. S. Luan, A. F. M. Ayub and R. Mahmud distinguished four factors influencing students’ e-learning: gender, attitude towards learning on the Internet, simplicity of usage and perception of usefulness (Yee et al. 2009). G. Smith and D. Ferguson stated that teachers of mathematics, preparing materials for the e-course, encounter the problem of inserting mathematical expressions; i.e., the language for inserting mathematical expressions is not maintained, that is why they have to be inserted as graphic objects (Smith, Ferguson 2005). According to G. Albano, the model of mathematics e-learning is based on factors which belong to three different levels: the cognitive level, which encompasses concrete concepts and methods; the meta-cognitive level, which encompasses self-control of the learning process; the non-cognitive level, which encompasses beliefs, emotions, attitudes and all emotional aspects, forming learners’ critical decisions and activities (Albano 2006). V. James’ (James 2008) researches demonstrated that verbal and visual information, independently of the way of its conveying, is the key indicator forming students’ knowledge. According to P. Honey and A. Mumford, students are distributed into

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

nimus, ieškoti medžiagos tarpusavio sąryšių (Honey, Mumford, 1992). E. studijos yra efektyvesnės, kai jose naudojamos: a) kognityvinės studentų savybės – samprata, kalba, mąstymas, dėmesys, atmintis, vaizduotė, sąvokų formavimas, panaudojant turimas žinias, intelektas; b) psichomotorinės žinių formavimo galimybės – garsas bei kitos tiesioginio poveikio priemonės; c) jausmai ir stereotipai (Solso, Maclin, Kimberly, 2007). Pasak amerikiečių mokslininko E. Tulvingo, žmogaus atmintis suskirstyta į: semantinę – čia saugoma verbalinė informacija, abstrakčios idėjos, tai kalbos suvokimo mechanizmas; epizodinę – saugančią informaciją apie konkrečius išgyvenimus, įvykius; procedūrinę (arba asociatyviąją) – saugančią informaciją apie įvykius, reakciją ir jų sąryšius (Tulving, 1972). E. studijų procese informacija dažniausiai įsimenama per semantinę atmintį. Reikėtų pažymėti, kad medžiaga įsimenama geriau, kai į informacijos įsiminimo procesą įtrauksime epizodinę atmintį, aktyviai veikiančią, kai taikomos interaktyvios aplikacijos nuotoliniame kurse. Tai gali būti judantys paveikslėliai, iškylantys langai, garsiniai signalai ir kiti netikėti ir malonūs įvykiai. Procedūrinė atmintis formuos sąryšius, sudarančius pateikiamos medžiagos ir tam tikrų įvykių studijuojant medžiagą asociacijas. Išnagrinėjus Lietuvoje spausdintus verbalinius e. studijų kursų rengimą apibūdinančius dokumentus, skirtus aukštųjų mokyklų dėstytojams, nurodyti šie sėkmingo mokymo(si) sąlygas užtikrinantys principai: vaizdumas, sąmoningumas ir aktyvumas, prieinamumas, sistemingumas ir nuoseklumas, žinių tvirtumas, teorijos ir praktikos ryšys (Rutkauskienė ir kt., 2003, Rutkauskienė ir kt., 2007). Verbalinių dokumentų analizės metodu nustatytos e. kurso bei įrankių savybės, turinčios įtakos matematikos mokymuisi aukštosios mokyklos e. studijų kurse, pateiktos atrinktiems ekspertams. Tiesioginio kontakto metu kiekvieno eksperto buvo paprašyta, remiantis savo asmenine patirtimi, suformuluoti ne mažiau nei penkis VMA įrankių vertinimo kriterijus. Apibendrinant apklausos rezultatus buvo taikomi du pagrindiniai kriterijai: a) įvardytą VMA įrankio vertinimo kriterijų galima priskirti bent vienam iš dokumentų analizės metodu nustatytų veiksnių; b) kriterijus atmetamas, jei jį nurodė tik vienas ekspertas. Įvertinus matematikos dėstytojų ekspertinės apklausos rezultatus buvo atrinkti šie VMA įrankių vertinimo kriterijai: • Matematinių išraiškų, parengtų panaudojant TEX (Tex users group 2011) dokumentų rengimo kalbą, palaikymas. TEX matematikų aka-

experimenters, who are inclined to try out the acquired knowledge in practice immediately, reflectors, who first accumulate knowledge and only then generalise it, pragmatists, who look for new ideas and theorists, who find it relevant to generalise themselves, look for interrelations of material (Honey, Mumford 1992). E-learning is more efficient when it employs: a) students’ cognitive features: conception, language, thinking, attention, memory, imagination, formation of concepts using existing knowledge, intellect; b) psycho-motoric possibilities of knowledge formation: sound and other means of direct impact; c) feelings and stereotypes (Solso, Maclin, Kimberly 2007). According to the American scientist-psychologist E. Tulving, human memory is subdivided into: semantic memory, storing verbal information, abstract ideas, which is the mechanism of language perception; episodic memory, storing information about concrete experiences, events; procedural or associative memory, storing information about events, reaction and links between them (Tulving 1972). Information in the e-learning process is most often memorised through semantic memory. It has to be noted that materials will be memorised better when we involve episodic memory into the process of memorising information; this memory actively operates when interactive applications are applied in the distance course. These can be moving pictures, emerging windows, audio signals and other unexpected and pleasant events. Procedural memory will form links, making up associations between the given material and certain events, studying the material. Having analysed printed verbal documents describing preparation of e-learning courses for teachers of higher education institutions, the following principles ensuring successful teaching and learning conditions are distinguished: picturesqueness, consciousness and activeness, accessibility, systematicity and consistency, firmness of knowledge, link between theory and practice (Rutkauskienė et al. 2003, Rutkauskienė et al. 2007). Employing the method of content analysis, the identified features of the e-course and tools influencing learning mathematics in the e-learning course at the higher education institution were submitted to chosen

experts. During the direct contact every expert was asked to form not less than five criteria for evaluating VTE tools, based on personal experience. Generalising the results of the survey, two main criteria were applied: a) named criterion for evaluating the VTE tool can be attributed at least to one of the factors, identified by the content analysis method; b) the criterion is rejected if it is

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•

• • • •

deminėje visuomenėje laikoma kaip geriausia priemonė, skirta sudėtingoms matematinėms išraiškoms pavaizduoti; Mokomosios medžiagos perteikimas diskrečiomis dalimis su kiekvienos dalies žinių įsisavinimo lygio įvertinimo galimybe, pagrįsta bihevioristinės psichologijos atstovo B. F. Skinerio programuoto mokymo idėjomis (Skinner 1965); Užduočių, atliktų naudojant matematines sistemas įdiegtas besimokančiųjų kompiuteriuose, pateikimo, įvertinimo ir komentavimo VMA galimybė; Interaktyvios sąveikos tarp studijuojančiųjų realizavimas; Interaktyvios grafinės informacijos perteikimas; Bendradarbiavimo bei žaidybinių elementų taikymas mokymo procese; Svarbiausių sąvokų ir apibrėžimų bei interaktyvių santraukų rengimas ir perteikimas.

Įrankių panaudojimo matematikos mokymui Moodle VMA tyrimas Siekiant atrinkti tinkamiausius įrankius aukštosios mokyklos matematikos mokymui e.studijų kurse buvo atlikta jų funkcinių galimybių lyginamoji analizė, taikant verbalinių dokumentų analizės metodą pagal septynis suformuotus vertinimo kriterijus. Generalinė aibė yra baigtinė, nes Moodle siūlo 40 standartinių ir 78 specializuotus įrankius, kurie yra diegiami kaip moduliai arba sistemos papildiniai (Moodle 2011). Įvertinti visų įrankių empiriniai aprašai, taikant verbalinių dokumentų analizės metodą, todėl tyrimo patikimumas 100 proc.. Tyrimas vykdytas dviem etapais. Pirmame etape pagal empirinius įrankių aprašus atmesti tie įrankiai, kurie nenaudotini matematikos mokymui aukštosios mokyklos e.kurse. Po pirmojo tyrimo etapo atrinkti 27 standartiniai VMA Moodle įrankiai ir 11 specializuotų. Antrame etape pagal suformuotus septynis VMA įrankių vertinimo kriterijus įvertinti atrinktų įrankių empiriniai aprašai ir savybės. Vertinant įrankių ir nurodytų kriterijų atitikimą buvo sudaryta atitikties (toliau – AT) procentinė skalė [0, 100]. Jei analizuojamas įrankis visiškai neatitinka kriterijaus, tai AT = 0 proc., jei pilnai atitinka – AT = 100 proc. Kaip tinkami matematikos mokymui aukštosios mo-

pointed out only by one expert. Having evaluated the results of mathematics teachers’ expert survey, the following evaluation criteria for VTE tools were selected:

• Maintenance of mathematical expressions, prepared employing the language for preparation of TEX (Tex users group 2011) documents. In the mathematicians’ academic community TEX is treated as the best means for representation of complicated mathematical expressions; • Conveyance of teaching materials by discrete parts with the possibility of evaluating the level of mastering knowledge of every part, grounded on programmed teaching ideas of the representative of behaviouristic psychology B. F. Skinner (Skinner 1965); • The possibility to submit, assess and comment tasks in the VTE, accomplished using mathematical systems, installed in learners’ computers; • Implementation of interactive interaction between learners; • Conveyance of interactive graphic information; • Application of cooperation and game elements in the teaching process; • Preparation and conveyance of the most important concepts, definitions and interactive summaries.

Research on Usage of Tools for Teaching Mathematics in the Moodle VTE Seeking to choose the most suitable tools for teaching mathematics in the e-learning course, comparative analysis of possibilities of their functions was carried out, applying the content analysis method according to seven formed evaluation criteria. The general set is finite because Moodle offers 40 standard tools and 78 specialized tools, which are installed as modules or supplements of the system (Moodle 2011). Empirical descriptions of all tools were considered, applying the content analysis method; therefore, reliability of the research is 100%. The research was carried out in two stages. In the first stage according to empirical descriptions of tools, such tools which should not be used for teaching mathematics in the e-learning course at the higher education institution were rejected. After the first stage of the research 27 standard VTE Moodle tools and 11 specialized tools were selected. In the second stage according to formed seven criteria for evaluation of VTE tools empirical descriptions and features of chosen tools were evaluated. Evaluating correspondence of tools to distinguished criteria, percentage scale of correspondence (further referred to

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

kyklos e.studijų kurse buvo atrinkti tie VMA Moodle įrankiai, kurių AT > 80proc.. Pirmą kriterijų tenkino šie standartiniai VMA įrankiai: resursas, žodynėlis (AT = 90 proc.), duomenų bazė (AT = 95 proc.), kontrolinis (AT = 97 proc.), virtualus seminaras (AT = 83 proc.), pamoka (AT = 91 proc.). Iš specializuotų įrankių atrinkti tik du: DragMath (AT = 100 proc.) (žr. 2 pav., a) – TEX intarpų rengimo įrankis, turintis grafinę vartotojo sąsąją; b) Formula (AT = 97 proc.) (žr. 2 pav., b) – interaktyvių klausimų rengimo priemonė su galimybe taikyti matematinės logikos taisykles vertinant studentų atsakymus.

as CO) was set [0, 100]. If the analysed tool completely does not correspond to the criterion, then CO=0%, if it completely corresponds, then CO=100%. Such VTE Moodle tools that have CO>80% were chosen as suitable for teaching mathematics in the e-learning course at the higher education institution. The following VTE standard tools met the first criterion: resource, glossary (CO=90%), database (CO=95%), test (CO=97%), virtual workshop (CO=83%), lesson (CO=91%). Only two specialized tools were selected: DragMath (CO=100%) (see Fig. 2, a) – the TEX tool for designing insertions, which has the user’s graphic link; Formula (AT=97%) (see Fig. 2, b) – the tool for preparing interactive questions with the possibility to apply logic rules, evaluating students’ answers.

a) DragMath

b) formula

2 pav. Specializuoti VMA Moodle įrankiai, naudojantys TEX Fig. 2. Specialized VTE Moodle Tools that Use TEX

Antrąjį kriterijų tenkino tik vienas standartinis VMA Moodle įrankis pamoka (AT = 100 proc.) (žr. 3 pav.), kurį naudojant galima parengti mokomąją medžiagą diskrečiomis dalimis su kiekvienos dalies žinių įvertinimo galimybe. Čia studentas gali pasirinkti individualų mokymosi tempą. Pagal nurodytus kriterijus sistema automatiškai parenka tolesnę mokymosi eigą: grąžina atgal, jei studento žinios nepakankamos; nukreipia į papildomą kursą, jei studentas turi nepakankamas bazines žinias, reikalingas šiai matematikos kurso daliai įsisavinti; pateikia kitą medžiagos dalį, jei studento įvertinimas atitiko numatytuosius kriterijus. Įrankis realizuoja bihevioristinės psichologijos teoretiko B. F. Skinerio programuoto mokymo idėjas, kurias vėliau plėtojo N. L. Gage ir D. C. Berliner (Gage, Berliner, 1994). Trečiąjį kriterijų tenkino tik vienas standartinis įrankis – užduotis (AT = 100 proc.). Naudojant šį įrankį studijuojantysis pateikia savo darbą, o dėstytojas jį vertina, gali komentuoti bei persiųsti studentui pataisyti surašęs savo pastabas.

The second criterion was met only by one standard VTE Moodle tool – the lesson (CO=100%) (see Fig. 3). Using it, one can prepare teaching materials in discrete parts with the possibility of assessing knowledge of every part. Here the student can choose individual pace of learning. According to set criteria, the system automatically chooses further process of learning: returns back if the student’s knowledge is insufficient; directs to the additional course if the student has insufficient basic knowledge necessary for mastering this part of the mathematics course; gives next part of the material if the student’s assessment corresponds to the foreseen criteria. The tool implements programmed teaching ideas of the theoretician of behaviouristic psychology B. F. Skinner, which were later developed by N. L. Gage and D. C. Berliner (Gage, Berliner 1994). The third criterion was met only by one standard tool – the task (CO=100%). Using this tool, the learner submits his/her work and the teacher assesses, can comment and forward it to the student to correct, having given his comments.

Mokytojų ugdymas 2011

3 pav. Pamokos (angl. Lesson) įrankio kriterijų aprašymo langas Fig. 3. The Window Describing Criteria of the Tool – the Lesson

Pagal ketvirtąjį kriterijų, atrinktos šios standartinės aktyvinimo priemonės: forumas (AT = 100 proc.) (žr. 4 pav.), pokalbis (AT = 95 proc.), virtualus seminaras (AT = 84 proc.), dialogas (AT = 81 proc.), socialinis forumas (AT = 89 proc.), žurnalas (AT = 81 proc.), žinutės (AT = 82 proc.). Reikėtų pažymėti, kad forume yra galimybė reitinguoti studijuojančiųjų pasisakymus. Tai suaktyvina jų procedūrinę atmintį, todėl geriau įsisavinama studijuojama medžiaga. Virtualių seminarų organizavimas sudaro galimybę studentams ne tik dalyvauti seminare, bet ir vertinti, analizuoti, komentuoti bei reitinguoti kitų kurso dalyvių pateiktą medžiagą. Iš specializuotų įrankių atrinkti du: NanoGong (AT = 98 proc.) (žr. 4 pav.) – įrankis, leidžiantis palikti garso žinutes kurse; Video (AT = 99 proc.) – įrankis, leidžiantis įkelti filmuotą medžiagą, tačiau jis turi ribotą interaktyvumą.

4 pav. NanoGong įrankis Fig. 4. The NanoGong Tool

According to the fourth criterion, the following standard activation means were selected: forum (CO=100%) (see Fig. 4), conversation (CO=95%), virtual workshop (CO=84%), dialogue (CO=81%), Wiki (CO=89%), journal (CO=81%), messages (CO=82%). It has to be noted that the forum provides with a possibility of rating learners’ comments; this activates learners’ procedural memory; therefore, studied materials are mastered better. Organisation of virtual workshops creates a possibility for students not only to take part in the workshop but also evaluate works of other course participants. Here everyone can review other course participants’ submitted materials, analyse them, comment and rank. Two specialized tools were selected: NanoGong (AT=98%) (see Fig. 4) – the tool enabling to leave audio messages in the course; Video (AT=99%) – the tool enabling to load filmed material but it has limited interactivity.

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

Penktąjį kriterijų visiškai tenkino tik viena specializuota programa GeoGebra (AT = 100 proc.) (žr. 5 pav.). Ji lengvai integruojama į VMA Moodle. Su šia programa galima parengti aukštosios mokyklos matematikos kurso teorinės medžiagos grafines interpretacijas, testų klausimus. Programa yra interaktyvi, todėl studentai gali modifikuoti dėstytojo pateiktą grafinę medžiagą pagal užduotyje nurodytus kriterijus, analizuoti pokyčius, vertinti savybes.

The fifth criterion was fully met by only one specialised software GeoGebra (CO=100%) (see Fig. 5). It is easily integrated into VTE Moodle. This software enables to prepare graphic interpretations, test questions of theoretical mathematics course of the higher education institution. The software is interactive; therefore, students can modify the teacher’s given graphic material according to the criteria indicated in the task, analyse changes, evaluate features.

5 pav. Užduoties rengimo langas, taikant GeoGebra programą Fig. 5. Window for Preparation of the Task, Applying the GeoGebra Software

Šeštąjį kriterijų tenkino šie standartiniai VMA Moodle įrankiai: duomenų bazė (AT = 86 proc.), žurnalas (AT = 82 proc.), socialinis forumas (angl. Wiki) (AT = 93 proc.), internetinis dienoraštis (angl. Blog) (AT = 90 proc.), pasirinkimas (AT = 84 proc.). Duomenų bazėje pirmiausia dėstytojas aprašo duomenų saugyklos struktūrą, po to renkama medžiaga, kurią gali komentuoti studentai. Žurnalas naudojamas tada, kai dėstytojas nori vykdyti individualią studentų apklausą. Šis įrankis individualizuoja mokymo procesą, padeda studijuojantiems išvengti kai kurių asmeninių psichologinių problemų, kai studentas nenori, kad bendrakursiai žinotų jo žinių ir gebėjimų lygį. Pasirinkimas – tai įrankis, leidžiantis organizuoti balsavimus dėl konsultacijos ar kitų dėstytojo planuojamų įvykių laiko ir pan. Pasirinkimo dėka studentas pasijunta kaip visavertis kurso dalyvis, kurio nuomonės visada yra atsiklausiama. Socialiniame forume kiekvienas gali išsakyti savo nuomonę, mintis bei rašyti komentarus. Internetinis dienoraštis – tai priemonė leidžianti studijuojančiajam susikurti e. kurse savo dienoraštį, susijusį su aukštosios mokyklos matematikos kurso tematika. Čia studentas gali „išsikrauti“, pasidalyti savo proble-

The following standard VLA Moodle tools met the sixth criterion: database (CO=86%), journal (CO=82%), Wiki (CO=93%), blog (CO=90%), choice (CO=84%). In the database first of all the teacher describes the structure of data storage, then material is collected, which can be commented by students. The journal is used when the teacher wants to test individual students. This tool individualizes the teaching process, helps learners to avoid certain personal psychological problems when the student does not want other students to know the level of his/ her knowledge and abilities. The choice is the tool enabling to organize voting regarding consultation or timing of other events planned by the teacher, etc. The choice enables the student to feel a full-fledged course participant, whose opinion is always considered. In Wiki everyone can impart his/her opinion, ideas and write comments. The blog is the means enabling the learner to create his/her diary in the e-course, related to the mathematics course topic of the higher education institution. Here the student can “unload”, share his/ her problems, discoveries and expectations with course participants and the teacher. From specialised

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momis, atradimais ir lūkesčiais su kurso dalyviais bei dėstytoju. Iš specializuotų įrankių geriausiai šį kriterijų atitiko Hot Potatoes (AT = 99 proc.) programa. Tai daugiafunkcinė priemonė, leidžianti VMA rengti interaktyvias užduotis, kuriose naudojami standartiniai apklausų komponentai ir žaidybiniai elementai: kryžiažodžiai (žr. 6 pav.), dėlionės. Naudojant Hot Potatoes parengtą aukštosios mokyklos matematikos e. kurso medžiagą per žaidimus aktyvinama studento asociatyvioji atmintis, taip gerindama studijuojamos medžiagos įsiminimą.

tools this criterion was best met by the Hot Potatoes (AT=99%) software. This is a multifunctional means enabling to prepare interactive tasks, which employ standard survey components and game elements, in the VLE: crosswords (see Fig. 6), puzzles. E-course mathematics materials of the higher education institution prepared with Hot Potatoes through games activate the student’s associative memory, this way

improving memorising of the studied material.

6 pav. VMA Moodle kryžiažodis, parengtas su HotPotatoes Fig. 6 VTE Moodle Crossword Prepared with Hot Potatoes

Septintąjį kriterijų tenkino tik vienas standartinis įrankis – žodynėlis (AT = 100 proc.). Jis gali būti naudojamas kurso sąvokų kartojimui. Žodynėlį gali pildyti tik dėstytojas arba dėstytojas ir studentai. Siekiant išvengti nesusipratimų, galima kontroliuoti studentų įvestų sąvokų teisingumą ir jas įtraukti į žodynėlį arba atmesti. Įrankių lyginamosios analizės rezultatai parodė, kad VMA Moodle turi pakankamai didelę standartinių įrankių aibę aukštosios mokyklos matematikos e. studijų kurso rengimui. Panaudojant specializuotus interaktyvios sąveikos įrankius aktyvuojama studijuojančiųjų semantinė, epizodinė bei procedūrinė atmintys, kurios padeda jam geriau įsisavinti pateikiamą matematikos kurso medžiagą. Reikėtų pažymėti, kad kai kurie Moodle VMA analizuoti įrankiai tenkino daugiau nei vieną analizės kriterijų.

Only one standard tool – the glossary – met the seventh criterion (AT=100%). It can be used for repeating the concepts of the course. The glossary can be filled in only by the teacher or by the teacher and students. In order to avoid misunderstandings, correctness of the concepts introduced by students can be monitored and they can be included in the glossary or rejected. The results of the comparative analysis of tools demonstrated that VTE Moodle has a sufficiently large set of standard tools for preparation of mathematics e-course of the higher education institution. Employing specialized tools of interactive interaction, semantic, episodic and procedural memories are activated, which help learners to master the given mathematics

Išvados • Remiantis verbalinių dokumentų analizės ir ekspertų ekspertinės apklausos rezultatais su-

Conclusions • Based on the results of content analysis and experts’ expert survey, seven assessment criteria

course materials better. It must be noted that certain tools analysed in Moodle VTE met more than one analysis criterion.

Liudvikas Kaklauskas, Danutė Kaklauskienė Virtualios mokymo (-si) aplinkos įrankių panaudojimo matematikos mokymui analizė

of VTE tools were formed. • Based on empirical descriptions of Moodle VTE tools, applying the content analysis method, 27 standard and 11 specialized tools have been selected from the finite set of tools, which are suitable for preparation of mathematics e-course of the higher education institution. Employing the method of comparative analysis, selected tools were evaluated according to seven assessment criteria. It was identified that 16 standard and 6 specialized Moodle VTE tools were most suitable for teaching mathematics e-course at the higher education institution.

formuoti septyni VMA įrankių vertinimo kriterijai. • Remiantis Moodle VMA įrankių empiriniais aprašais, taikant verbalinių dokumentų analizės metodą, iš baigtinės įrankių aibės atrinkti 27 standartiniai ir 11 specializuotų įrankių, tinkamų aukštosios mokyklos matematikos e. kurso rengimui. Naudojant lyginamosios analizės metodą, atrinkti įrankiai įvertinti pagal septynis vertinimo kriterijus. Nustatyta, kad matematikos mokymui aukštosios mokyklos e. studijų kurse geriausiai tinka naudoti 16 standartinių ir 6 specializuoti Moodle VMA įrankiai.

Literatūra

Aboujaoude, E. (2011). Virtually you. The dangerous powers of the e.personality. New York, W. W. Norton & Company, p. 350. Albano, G. (2006). A case study about mathematics and e.learning: first investigations. Proceedings CIEAEM 58 – SRNI, Czech Republic. Chaffar, S., Frasson, C. (2004). Inducing Optimal Emotional State for Learning in Intelligent Tutoring Systems. Lecture Notes in Computer Science, Vol. 3220, p. 7–21. Cole, J., Foster, H. (2007). Using moodle, 2nd edition. Sebastopol, O’Reilly, 2007, p. 282. Daugiamas, M., Taylor, P. (2003). Moodle: Using Learning Communities to Create an Open Source Course Management System. World Conference on Educational Multimedia, Hypermedia and Telecommunications (EDMEDIA), USA. Enciklopedinis kompiuterijos žodynas. 2011. [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: <http://www.likit.lt/ term/enciklo.html>. Gage, N. L., Berliner, D. C. (1994). Pedagoginė psichologija. Vilnius: Alma litera. Heraz, A., Frasson, C. (2008). Detecting the Affective Model of Interplay between Emotions and Learning by Measuring Learner’s Brainwaves. WECITS’2008. ITS2008. Montréal, Canada. Holmberg, B. (1983). Guided didactic conversation in distance education. In D. Stewart, D. Keegan, and B. Holmberg (eds.), Distance education: International perspectives (p. 114–122). London: Croom Helm. Honey, P., Mumford, A. (1992). The Manual of Learning Styles 3rd Ed. Maidenhead: Peter Honey. Hot potatoes. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://hotpot.uvic.ca/ >. Iglesias, C. E., Carbajo, A. G., Sastre Rosa, M. A. (2008). Interactive tools for Discrete Mathematics e-learning. Wseas Transactions on Advances in Engineering Education. Vol. 5, No 2, p. 97–103. Yee, H. T. K., Luan, W. S, Ayub, A. F. M ir Mahmud, R. A (2009). Review of the Literature: Determinants of Online

•

References

Learning Among Students. European Journal of Social Sciences, Vol.8, No 2. James, V. (2008), Memories and Studies. NuVision Publications, p. 156. Kaye, A., Rumble, G. (1981). Distance Teaching for Higher and Adult Education. London, Croom Helm. Kaklauskas, L., Kaklauskienė, D. (2010). E. studijų aktyvinimo priemonių įtaka studentų mokymosi rezultatams. Konferencijos medžiaga, Nr 1, p 61–67. Šiauliai: ŠLK. Keegan, D. (1991) Foundations of Distance Education. London and New York, Routledge. Martín-Blas, T., Serrano-Fernándeza, A. (2009). The role of new technologies in the learning process: Moodle as a teaching tool in Physics. Computers & Education Vol. 52, No 1, p. 35–44. Miller, E., Rice, A. (1967). Systems of Organization. The Control of Tasks and Sentient Boundaries. London, Tavistock. Moodle – A Free, Open Source Course Management System for Online Learning. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://moodle.org/stats/>. Moodle cohorts. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://docs.moodle.org/en/Cohorts >. Moodle comments block. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://docs.moodle.org/en/ Comments_block>. Moodle community hubs. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: <http://hub.moodle.org/>. Moodle conditional activities. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: <http://docs.moodle.org/ en/Conditional_activities >. Moodle Course completion.2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://docs.moodle.org/en/ Course_completion >. Moodle plagiarism prevention. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://docs.moodle. org/en/Plagiarism_Prevention >. Moodle portfolio. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga

Mokytojų ugdymas 2011

per internetą:<http://docs.moodle.org/en/Portfolios>. Moodle repositories. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: <http://docs.moodle.org/en/ Repositories>. Moodle Web Services. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą: < http://docs.moodle.org/en/Web_ Services >. Pressey, S. L. (1926). A simple apparatus which gives tests and scores – and teaches. School and Society. Vol. 23, p. 373–376. Rutkauskienė, D., Lenkevičius, A., Targamadzė, A., Volungevičienė, A., Pociūtė, E., Dėmenienė, A., Kelmienė , V. (2007). Nuotolinio mokymosi dėstytojo vadovas. Mokomoji knyga, p. 188. Kaunas: Technologija. Rutkauskienė, D., Targamadzė, A., Kovertaitė, V. R., Simonaitienė, B., Abarius, P., Mačiulis, M., Kulvietienė, R., Cibulskis, G., Kubiliūnas, R., Žvinienė, V. (2003). Nuotolinis mokymasis. Kaunas: Technologija. Rutkauskienė, D., Targamadzė, A., Kovertaitė, V. R., Simonaitienė, B., Abarius, P., Mačiulis, M., Kulvietienė, R., Cibulskis, G., Kubiliūnas, R., Žvinienė, V. (2003). Nuotolinis mokymasis, p. 256. Kaunas: Technologija. Sangwin, C. (2005). Serving mathematics in a distributed e.learning environment. MSOR Connections. Vol 5, No 2, p. 1–4.

Skiner, B. F. (1965). Review Lecture: The Technology of Teaching. Proceedings of the Royal Society of London. Series B, Biological Sciences Vol. 162, No. 989, p. 427443. Skinner, B.F. (1968). The technology of teaching. New York: Meredith. Smith, G., Ferguson, D. (2005). Student attrition in mathematics e-learning. Australasian Journal of Educational Technology, 21(3), p. 323–334. Solso, R., Maclin, O.H, Kimberly, M. (2007). Cognitive Psychology:Internationa Edition, New Jersey, Prentice Hall. Targamadzė, A., Normantas, E., Rutkauskienė, D., Vidžiūnas, A. (1999). Naujos distancinio švietimo galimybės. Kaunas: Technologija. Tex users group. 2011 [Žiūrėta 2011 m. gegužės 2 d.] Prieiga per internetą:< http://www.tug.org/>. Tulving, E. (1987). Episodic and semantic memory. New York, Organization of memory, Academic Press, p 381–403. Volungevičienė, A. (2008). Distance Learning and Teaching Design Model: Quality Assessment Factors and Dimensions. (Summary of Doctoral Dissertation, Vytauto Didžiojo universitetas, 2008).

Liudvikas Kaklauskas Vilniaus universiteto Matematikos ir informatikos instituto, informatikos krypties doktorantas, Šiaulių universiteto Informatikos katedros lektorius. Nuotolinių studijų centro techninio skyriaus vedėjas. Moksliniai interesai: fraktalinių procesų kompiuterių tinkluose stebėsenos ir valdymo metodų tyrimas, šiuolaikinių mokymo(si) metodų taikymas studijose.

Institute of Mathematics and Informatics of Vilnius University, doctoral student in informatics. Šiauliai University, Department of Informatics, lecturer, Distance Studies Centre, Head of the Technical Department. Research interests: research on monitoring and management methods of fractal processes in computer networks.

Address: Vasario 16 Str. 26, LT-76351 Šiauliai, Lithuania E-mail: liudas@fm.su.lt. Danutė Kaklauskienė Šiaulių universiteto Informacinių technologijų katedros lektorė. Moksliniai interesai: tekstų lingvistinė analizė, prognozavimo metodikų tyrimas, šiuolaikinių mokymo(si) metodų taikymas studijose.

Šiauliai University, Department of Information Technologies. Research interests: linguistic analysis of texts, research on forecasting methods, application of modern teaching and learning methods in the learning process.

Address: Vilniaus Str. 88, LT-76351 Šiauliai, Lithuania E-mail: d.kaklauskiene@gmail.com

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 33–39 Teacher Education. 2011. No. 16 (1), 33–39

Anda Zeidmane Latvijos Žemės Ūkio Universitetas, Jelgava, • Latvia University of Agriculture, Latvija Jelgava, Latvia

Matematikos studijų svarba inžineriniame ugdyme: darbdavių vertinimas

Importance of Math Studies in Engineering Education in Assessment of Employers

Anotacija Pastaruoju metu inžineriniame ugdyme vyksta dideli pokyčiai. Matematikos studijos ir tiesiogiai (matematika kaip priemonė spręsti ir skaičiuoti, atliekant įvairius uždavinius) ir netiesiogiai (matematika gerina inžinerinių problemų formulavimo ir sprendimo įgūdžius) gerina inžinieriams reikalingus rezultatus. Tyrimas buvo atliktas su 56 inžinieriais, analizuojant, kiek svarbi matematika jų dabartinime darbe. Šiame straipsnyje taip pat reziumuojamas mokymosi procesas ir rezultatai. Pagrindiniai žodžiai: matematikos studijos, inžinerinis ugdymas, darbdavių vertinimas

Abstract Nowadays engineering education undergo major changes. Mathematics studies have an impact on the development of the necessary outcomes for engineers’ both directly (Mathematics serves as a tool for solving and calculating various problems) and indirectly (Mathematics develop skills to formulate and solve engineering problems). The survey was carried out among 56 engineers about the importance of Mathematics in their present work and the learning process and results are summarized in this article. Key words: Mathematics studies, engineering education, employers’ assessment

Introduction Changes in engineering education are related to changes in manufacturing because technological processes, management systems, the nomenclature of goods, legal rules change. Nowadays engineering education undergo major changes that are mainly related to the changes in Engineering Practice [1]. One of the factors is increasing complexity, because engineering projects, products and systems are becoming more complex. The second factor is sustainability which becomes an important part of engineering practice because we cannot ignore the problem of climate change and global warming. Sustainability requires a critical consideration of the development of all engineering products. The third factor is working in cooperation. Examples of this include joint ventures, strategic alliances, and publicprivate partnerships. The objective is to gain improved outcomes by working together in collaboration rather than an adverse relationship. The fourth factor is risk management because engineering projects become more complex with more outcomes that are less predictable. This risk has dimensions of health and safety, commercial, technical, environmental, social

and political dimensions. The fifth factor is looking for opportunities to improve outcomes without increasing the level of resources used. There are continuous changes in the labor market. The system of higher education cannot provide an individual with the knowledge necessary during the whole professional life. On the other hand, after the completion of undergraduate studies, young specialists have to solve common problems: find a job, take the first steps in their carrier, adapt in a new environment, change work or even profession in case of a failure etc. It follows that the system of higher education has to be aware of the competences necessary for graduates that will make young specialists competitive in the labor market. Problem of Research The aims of the study are to identify the necessary outcomes of engineering education, explore the role of Mathematics in the engineering education, as well as to identify importance of Math studies in engineering education in assessment of employers. Materials Engineering education should focus not only

Mokytojų ugdymas 2011

on the development of professional competences – professional knowledge, professional skills and reflection but also on social competence: selfcompetence, co-operation and communication. Each engineering program has its own, slightly different standards, however, the basic engineering outcomes are common. For example, the outcomes required in civil engineering. The outcomes collectively prescribe the necessary depth and breadth of knowledge, skills, and attitudes required of an individual aspiring to enter the U.S. practice of civil engineering at the professional level (licensure) in the 21st century [2], [3]. As noted earlier, relative to today’s approach, tomorrow’s U.S. civil engineer will: • master more Mathematics, science, and engineering science fundamentals, • maintain technical breadth, • acquire broader exposure to humanities and social sciences, • gain additional professional practice breadth, and achieve greater technical depth, that is, specialization The recommended outcomes are eight technical outcomes and seven professional outcomes (see Figure 1).

this outcome. Engineering tools outcome means that specialists will be able to select and organize the relevant techniques, skills, and modern engineering tools to solve a complex problem. This includes the role and use of appropriate information technology, contemporary analysis and design methods, and applicable design codes and standards as practical problem-solving tools to complement knowledge of fundamental concepts. Engineering problem solving outcome means that specialists will be able to identify, formulate, and solve an engineering problem involving integration of general technical areas appropriate to civil engineering and evaluate the effectiveness of the solution. Assessing situations in order to identify engineering problems, formulate alternatives, and recommend feasible solutions is an important aspect of the professional responsibilities of a civil engineer. Design outcome means that specialists will be able to evaluate the design of a complex system, component, or process to ensure that it meets a client’s needs and accounts for all relevant constraints. Critical design methodology and process elements include problem definition, scope, analysis, risk assessment,

Outcomes for civil engineering

Technical Outcomes 1. Mathematics and science 2. Engineering tools 3. Engineering problem solving 4. Design 5. Experiments 6. Impact of engineering solutions 7. Project management, construction, and asset management 8. Specialization

Professional

Outcomes

9. Communication 10. Professional and ethical responsibility 11. Business, public policy, and public administration 12. Multidisciplinary 13. Leadership 14. Contemporary issues 15. Life-long learning

Fig. 1. Outcomes for entry into the practice of civil engineering at the professional level

Mathematics plays an important role in the development of technical outcomes, so they will be examined in more detail. Mathematics and science outcome means that specialists will be able to solve problems in Mathematics through differential equations, calculus-based physics, chemistry information technology, and one additional area of science. [4]. A technical core of knowledge and breadth of coverage in Mathematics and science is stressed in

environmental impact statements, creativity, iteration, regulations, codes, safety, security and constructability, sustainability and multiple objectives and various perspectives. Design experiences should be integrated throughout the professional component of the curriculum. Experiments outcome means that specialists will be able to evaluate the effectiveness of a designed experiment in meeting an ill-defined real world need. Impact of engineering solutions

Anda Zeidmane

Matematikos studijĹł svarba inĹžineriniame ugdyme: darbdaviĹł vertinimas

outcome means that specialists will be able to develop a complex engineering solution that appropriately accounts for the global, economic, environmental, and societal impacts of that solution. Project management, construction, and asset management outcome means that specialists will be able to develop management plans for a complex real-world engineering project. Specialization outcome means that specialists will be able to evaluate the design of a complex system or process or evaluate the validity of newly-created knowledge within a specialized area of civil engineering. Mathematics studies have an impact on the development of engineerâ&#x20AC;&#x2122;s necessary outcomes both directly and indirectly. First, Mathematics serves as a tool for solving and calculating various problems. This corresponds to the first technical competence shown in the Figure 1. But much greater is indirect impact of Mathematics which provides the other outcomes. Mathematics develop skills to formulate and solve engineering problems. Four parameters are necessary for developing ability to solve problems: 1) body of knowledge and three problem solving abilities: 2) the ability to formulate a problem in cognitive sphere; 3) the ability to construct the solution of a problem; 4) the ability to apply a constructive solution of a problem to the real situation. In Mathematics study process the skills are trained to directly apply the formal rules, which are sometimes quite abstract and complex, demanding to choose exactly that one which is necessary from the long list of known rules to complete the task, to build a sequence of applicable regulations. The latter means ability to design a solution plans which develop management skills. These skills are close to the practical arts and are obtained only by learning from good examples and solving many problems in the process. Formal mathematical formulation and modification of the identical do not make a Mathematical nature. However, not knowing the large amount of formula and without freedom of the identical transformation we cannot speak of the Mathematics learning, even more of the creativity in the field of Mathematics. One of the specific characters of the Mathematics is that Mathematics uses the language of symbols. Students need to combine a continuous unity of the verbal expression and the sub-language of special symbols for the solution of Mathematical problems (see Figure 2). This transformation also helps to move from qualitative to quantitative values â&#x20AC;&#x2039;â&#x20AC;&#x2039;and the calculation of problems. The second specific feature of Mathematics

is that Mathematics contains long chains of logic conclusions [5]. The problem in Mathematics study process is studentsâ&#x20AC;&#x2122; disability to make a long chain of logical conclusions in order to acquire large pieces of information. Dealing with reduced attrition (mental and physical) is one of the studentsâ&#x20AC;&#x2122; Mathematical ability components. Cognitive load may be reduced using a kind of â&#x20AC;&#x2DC;information quantaâ&#x20AC;&#x2DC;- it is a designation for a portion of information with a new symbol (see example in Figure 3.)

x 1 dx Âł x 1 2 x 1 (1 )

ÂŞ x 1 t o x 1 t2 Âş t 2 dt ... ÂŤ Âť 2Âł 3 t 1 ÂŤÂŹ x t 2 1 o dx 2tdt ÂťÂź (2)

Fig. 3. Using integral substitutions

[

]

x + 1 = t and the It is used for substitution integral (1) is transformed into an integral (2), which is much easier to integrate. Results Employersâ&#x20AC;&#x2122; point of view about the future engineers in each country could be different. The engineers from Latvia engaged in heat supply, heating, ventilation and air conditioning, gas supply, water supply and sewerage fields, i.e. practicing engineers and academic staff at the universities were questioned. The survey was carried out among 56 engineers - employers and employees about the importance of Mathematics in their present work and the learning process. Analyzing the results (see Figure 4), it is possible to conclude that real engineering problems are solved mostly applying integrals (36%), differential equations (24%), and derivative applications (23%). Engineers in manufacturing industries do not apply directly the skills of solving problems of higher Mathematics. Usually an engineering problem is defined, Mathematical equations are created, but IT software is used for calculations. Figure 5 shows the solution path of the engineering problems and the necessary knowledge of Mathematics in the problem solving phase. As employers, they recognize that they need new specialists with developed cognitive skills and with skills to learn new technologies fast. A normal situation in the study process would be for the university to teach young specialists cooperating with employers (see Figure 6.a). Employers often donâ&#x20AC;&#x2122;t cooperate with the university. They are ready to do universityâ&#x20AC;&#x2122;s functions and teach the young specialist themselves, because they are not

Mokytojų ugdymas 2011

Given a natural number. If the square of this number subtract the same number, the result is divisible by two without a remainder?

Problem definition in the English language The transition in the language of symbols

Indicate: n- natural number

Problem definition in the language of symbols

“ If (n2 – n) divide by 2 ?”

(n2 – n) = n· ( n - 1 )

Transformation in the language of symbols

„ If n is natural number then n and (n –1) are two following integers. Therefore, one of them is even, the other – odd. Even-numbered and odd multiplication is even. So it is divisible by 2 without remainder”

Judgements in the English language

Fig. 2. Example of Mathematics: problem solution using transition from English into language of symbols

35 30

25 20 15 10

5 0 Verctors

Matrix algebra

Derivative applications

Integrals

Fig. 3. Results of the question: ”Which knowledge of Mathematics is used in present work?”

Differential equations

Anda Zeidmane

Matematikos studijų svarba inžineriniame ugdyme: darbdavių vertinimas

Knowledge in engineering science

Define the problem (engineering)

Use language of symbols (engineering)

Indirect impact of Mathematics

Use mathematics regularities, to describe the interrelationships between values Skills to use IT programms

Direct impact of Mathematics

Knowledge in Math (what the programms done)

Use IT programs in Mathematics to perform Mathematical calculations

Knowledge in engineering science

Analysis of the results obtained

Fig. 4. Solution path of the engineering problems and mathematics impacts in this process

University

Young specialists

Employer

University

Young specialists

Employer

Fig. 5. Situation in the study process at universities (A – normal situation, B - the current situation)

Professor of the practical work

cooperation

Professor of MathCad Hand out homework

Students must solve on the paper, showing the process of solving step by step

Practical classes using MathCad programme check solutions themselves

Fig. 6. Integration of MathCad program in the Mathematics study process at LUA

Mokytojų ugdymas 2011

willing to share their newest technologies. At the same time, employers recognize that they had little idea of the usability of Mathematics during the study of Mathematics.

more Mathematic problems, students are interested in solving several variants. 2) Professor spend less time for checking homework and 3) the final tests show the improved scores. Relating to employers’ remark: ”Studying higher Mathematics, students get a small idea of its usability” the university has two major problems: • The number of ECTS has been reduced in Mathematics and there is no time to solve practical Mathematical tasks; • Mathematics teacher as nonprofessional should explain technical concepts which are taken out of context and therefore they are not well understandable for students. The solution will be to design the e-study materials which consist of the practical Mathematics tasks and insert links to relevant explanations of terms.

Discussion As regards labor market demand and IT increasingly rapid entry into our lives a compromise should be found in the study process of Mathematics between • the acquisition of the fundamental knowledge; • acquisition of know-how application of knowledge; • the use of IT software in the calculations. Many colleagues concentrate mainly on the education of “users”, because they think that the most important thing is to teach students to perform the necessary calculations. But then the question arises: “What is the main objective of the university?” Should universities teach students “how to do it?” or should they teach “why should it be done?” The Latvia University of Agriculture (LUA) has experience in the integration of IT in the study process of Mathematics. The acquisition of software MathCad is integrated in the study process accounting for 0.5 of contact lessons per week. Professor of the MathCad in cooperation with Professor of the practical work hand out homework to students. Students must solve on the paper, showing the process of solving step by step. One or two weeks later in the practical classes students check solutions themselves using MathCad programme. The final tests complete the study period of individual tasks and MathCad problems. The results are very good because: 1) students enjoy comparing the results of their individual tasks with the results obtained via MathCad and increasing the motivation to solve Literatūra Radcliffe D. Global Challenges facing Engineering Education: Opportunities for Innovation. Proceedings of the 35th International IGIP Symposium in cooperation with IEEE / ASEE / SEFI, 2006–09–18, Tallinn, Estonia P. 15-27 ASCE Levels of Achievement Subcommittee, Levels of Achievement Applicable to the Body of Knowledge Required for Entry Into the Practice of Civil Engineering at the Professional Level. Reston, VA: ASCE, 2005. ASCE Body of Knowledge Committee, Civil Engineering Body of Knowledge for the 21st Century: Preparing the

Conclusion • Regarding the employers’ specific needs, it is advised to review and refocus the Mathematics programs. • Young specialists should have fundamental basic knowledge and developed cognitive abilities. • To relate Mathematical studies to real life, it is advised to design the e-study materials which consist of the practical Mathematics tasks and insert links to relevant explanations of the terms. • Achievements of IT cannot be ignored - the skills of software application to calculate quickly and easily should be developed.

•

References Civil Engineering for the Future. Reston, VA: ASCE, 2004. Walesh S. G. Body of Knowledge for Civil Engineers:Essential for Success in the International Arena. Proceedings of the 35th International IGIP Symposium in cooperation with IEEE / ASEE / SEFI, 2006–09–18, Tallinn, Estonia P.163-178 Ястребов А.В. Дуалистические свойства математики и их отражение в процессе преподавания // Ярославский педагогический вестник. 2001. № 1. С. 48-53.

Anda Zeidmane

Matematikos studijų svarba inžineriniame ugdyme: darbdavių vertinimas

Anda Zeidmane Mokslo daktarė (Pedagogika), Latvijos žemės ūkio universitetas, Fizikos katedra.

PhD (Education) Latvia University of Agriculture, Depatment of Physics. Research interests:

Address: Liela Str. 2, LV-3001 Jelgava, Latvia E-mail: anda.zeidmane@llu.lv

matematikos mokymo mokykloje pedagoginÄ&#x2014;s inovacijos: patirtys ir kaitos procesai pedagogical innovations of teaching mathematics at school: experiences and shift processes

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 42–49 Teacher Education. 2011. No. 16 (1), 42–49

Kirsti KISLENKO, Lea LEPMANN Agder universitetas ir Tallinn universitetas, Estija • University of Agder & Tallinn University, Tartu universitetas, Estija Estonia University of Tartu, Estonia

Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990 – 2010 m.)

Changes in teachers’ approach, teaching MAthematics in estonian schools (1990 -2010)

Anotacija Šiame straipsnyje nagrinėjami Estijos matematikos mokytojų požiūriai, susiję su matematika ir matematikos mokymu ir mokymusi. Duomenys buvo surinkti, naudojant klausimyną su Likert skale, 1990aisiais ir 2010-aisiais. Analizėje lyginami šie du tyrimai. Pasirodo, kad mokytojai laikosi ir formalistinio, ir konstruktyvistinio mokymo požiūrių. Vėlesniame tyrime mokytojai mažiau akcentuoja uždavinių sprendimą ir labiau pratimus, kuriuos galima atlikti, naudojant konkrečius algoritmus. Atrodo, kad šis rezultatas susijęs su paskutinio PISA 2009 rezultatais, kuomet Estijos mokinių balai už matematikos užduotis, kurios reikalauja aukščiausio mąstymo lygio, pablogėjo. Pagrindiniai žodžiai: mokytojų požiūriai, lyginamasis tyrimas, tradicinis, formalistinis ir konstruktyvistinis požiūris.

Abstract In this article Estonian teachers’ beliefs about mathematics and mathematics teaching and learning are investigated. The data was collected through a Likert scale questionnaire in years 1990 and 2010, and the analysis concentrates on the comparison between these studies. It appears that amongst teachers both - formalistic and constructivist - views of teaching are supported. Teachers in the latest study put less emphasise on problem solving and more on exercises that can be solved using concrete algorithms. It seems that this result is connected to the last PISA2009 results where Estonia pupils’ scores of the mathematical tasks that demand the highest level of thinking had decreased. Key words: teachers’ beliefs, comparative study, traditional, formalist and constructivist view.

Introduction In the last decades the content and ways of teaching mathematics have changed. Based on the results of the survey called TALIS (Teaching and Learning International Survey) it is evident that teaching style has a big influence on pupils’ performance (OECD, 2009). Nevertheless, the researchers have not agreed upon the one and best way of teaching. The way of teaching depends on the cultural context as well as the traditions of the teachers, which are rather homogeneous inside of one country (Andrews & Hatch, 2000; Pehkonen & Lepmann, 1994). However, the studies carried out amongst future mathematics teachers have detected differences in teacher’ beliefs depending upon the school level they are going to teach. For example, Niss (2002) claims that Mathematics teachers in Denmark form a very

varied and heterogenous group. A further aspect of the cultural and institutional differences that exist in Danish mathematics education is that mathematics is perceived and treated so differently at the different levels that one can hardly speak of the same subject, even if it carries the same name throughout the system (Niss, 2002, p. 2). Teachers´ beliefs about the nature of mathematics do not only influence what is taught but also how it is taught (Andrews & Hatch, 2000). If one has an idea to change something in the teaching of mathematics, it is necessary to investigate what teachers understand by good teaching and what should be changed in it. The studies on teachers’ beliefs help us to identify the differences on their beliefs, and also the factors that can improve teaching and the development of a favourable learning environment for pupils. These studies aim

Kirsti KISLENKO, Lea LEPMANN Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990 – 2010 m.)

to help countries to review and develop policies in education to make the teaching more effective. Different profiles of teaching Teachers’ beliefs are clustered into belief systems, which are called their mathematical views or teaching profiles. In the following, some examples of different profiles are presented. Skemp (1978) proposed two quite different views that are readily transmitted to classroom practices: instrumental mathematics and relational mathematics. Instrumental mathematics is knowledge with a set of fixed plans and step-by-step procedures and can be compared to walking in a strange town knowing only a few useful routes. Relational knowledge of mathematics means that the learner possesses multiple relevant conceptual structures and therefore is able to construct several plans for performing a given task. It can be compared to constructing a cognitive map of the town in the walker’s mind. This view allowed a shift in focus from calculation techniques to a focus on mathematical thinking and understanding. Based on the interviews with teachers Törner (1997) describes the teachers using the following metaphors: • teacher as a stable boy – likes to feed pupils with interesting material that they can individually investigate; • teacher as a coach – how to use formulas are trained together with the pupils; • teacher as a waiter – serves nice examples of mathematical applications to prevent the difficulties of learning; • teacher as a kindergarten teacher – when holding pupils hands leads them through mathematics taking into account the aspect that the pupils need in their future lives; • teachers as a hunter - constantly ambushes for new ideas; • teachers as a conservative fighter – fights against the “modern” approaches that lead to loitering. Kuhn and Ball have identified at least four dominant and distinctive views of how mathematics should be taught (in Thompson, 1992): • Learner-focused: mathematics teaching that focuses on the learner’s personal construction of mathematical knowledge; • Content-focused mathematics with an emphasis on conceptual understanding: teaching that is driven by the content itself but which emphasises conceptual understanding; • Content-focused with emphasis on performance: teaching that emphasises student performance

and mastery of mathematical rules and procedures; • Classroom-focused: teaching based on knowledge about effective organised and structured teaching. Several research papers have reveal that in the case of mathematics teachers it is possible to differentiate four different views of mathematics and its structure: schematic oriented (“toolbox”), process oriented, system oriented and application oriented view (Grigutsch, Raatz & Törner 1997). Similar scales are used in COACTIV project in Germany, and these scales are mathematics as a system, mathematics as a process, mathematics as a toolbox, and Platonist view about mathematics (Bernack, Holzäpfel, Leuders & Renkl, 2010). In this paper the classification presented by Dionne (1984) is used, which introduces three perceptions of mathematics: • traditional view – mathematics is limited to calculations and following rules; • formalist view – stresses rigorous logic, proofs and exact use of language; • constructivist view – the pupil comes first and the emphasize is on pupil-centred learning methods and intuition. Beliefs and teaching practice Teachers´ belief research has tried to develop understandings about the relationship between teachers´ beliefs about mathematics teaching and learning and their classroom practices. The survey TALIS provides a comparative perspective on the conditions of teaching and learning. In years 2007-2008 the research was conducted in 24 OECD countries, and the participants were seventh to ninth grade teachers and the head of the schools. The aim of the study was to find out the factors that help to improve teaching and to endorse the development of a good learning environment (OECD, 2009). To clarify teachers beliefs about teaching and learning two standardized factor scales were constructed, which were the basis for the comparisons between the countries (Loogma, Ruus, Talts & Poom-Valickis, 2009): • traditional beliefs: the teaching is first and foremost the direct transmission of knowledge from the teacher to the pupil; • constructivist beliefs: the main emphasize is on the development of thinking and the understanding of the causal connections. Together with Island, Austria, Australia, Denmark and Belgium Estonia was the strongest supporting

Mokytojų ugdymas 2011

country in constructivist beliefs (Loogma et al., 2009). Although the constructivist views were the most supported in most countries, there could be found differences in the structures of beliefs. The correlation analysis showed that in some countries (such as Australia, Austria, Island) the teachers tended to “choose sides”, meaning that they significantly more chose one view over of the other. In some countries, these two types of views occurred together as there was a strong positive correlation between these two perspectives (especially in Asia and Central and South America). The strong positive correlation was also detected in the former Soviet Union countries, except Estonia. In Estonia the correlation coefficient was positive but statistically non-significant. Which means that even though Estonian teachers believed more in a constructivist way of teaching they did not directly contrast this view to the direct transformation of knowledge, and could therefore believe in the combination of these two views. The statements about the teaching practice in the TALIS-study were summed to three standardized factors, which were: • structured teaching practices (the teacher explicitly announces the learning goals to the pupils; the teacher checks pupils’ homework, and working books; the teacher asks questions to the pupils for controlling if they have understood the subject); • the practices oriented to the pupils (the pupils do problem-solving in small groups; exercises that are solved in the lessons are differentiated based on pupils’ abilities; the pupils themselves model the activities and themes in the lesson); • the practices supporting innovation and creative activities (the pupils carry out longer projects that can be accomplished not faster than a week, write essays, debate). Generally, in all TALIS countries there could be found a relationship between teachers’ beliefs and their classroom practices. In particular, teachers, who employ pupil-oriented practices, were more likely to be those who believe in a constructivist view of teaching. Therefore, teachers who believed that pupils should be active participants in the learning process tended to follow this in his/her practice. On the other hand, there could not be found a consistent pattern of the association between teachers’ beliefs and more structured lessons and teaching (OECD, 2009). Estonian teachers’ pedagogical views in TALIS-

study were moderately forward-looking and modern. The teachers believed in constructivist way of teaching but at the same time use traditional teaching practices i.e. highly structured ordinary lesson in their classrooms. Little emphasis was put on pupils’ wishes and individuality, such as learning pace or difficulties, and there were few opportunities to the pupils for personal initiative, investigative activities, creative works, project works, debates, etc. The assumption that traditional view of teaching (transformation of knowledge) is only related to the traditional teaching practice (structured teaching practices) did not find a proof in Estonian case. Compared to the others the teachers of mathematics and foreign languages use the most the structured teaching practices, and the least the practices supporting innovation and creative activities (Loogma et al., 2009). It has also been pointed out that mathematics teachers’ beliefs are related to their pupils abilities, as for example, the teachers who teach pupils with higher abilities are more in a favour of constructivist view than the teachers who teaches pupils with lower mathematical abilities (Jukk & Lepmann, 2007). Research design In 1990 the study on Estonian teachers beliefs was carried out for comparing Estonian and Finnish teachers of mathematics (Pehkonen & Lepmann, 1994). In 1997 the new curriculum was implemented in Estonia that permitted more freedom to the teacher to choose the content and pose goals for the teaching. Therefore, new data collection was conducted in 1997 for detecting the changes in teachers’ beliefs, and for clarifying how well teacher beliefs match with “reform” ideas of mathematics teaching (Lepmann, 1998). In both studies the strong support to constructivist way of teaching was concluded. It was also noted that teachers’ beliefs had stayed rather stable. Therefore, the change in the curriculum had not significantly influenced teachers’ beliefs. Also, the mathematical rigor and the emphasis on pupils’ high achievement in mathematics were important for the teachers. In 2009 a new cross-cultural survey of mathematics teachers beliefs was initiated. The objectives of this study were: • to construct an instrument that can, in crossculturally valid ways, measure teachers’ beliefs concerning mathematics and mathematics teaching and learning; • to use the instrument for an explorative study of mathematics teacher beliefs structures in

Kirsti KISLENKO, Lea LEPMANN Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990 – 2010 m.)

Baltic and Scandinavian countries and compare possible cross-cultural differences. In this article a study that is a part of this crosscultural survey of mathematics teachers are presented from the perspective of the comparison. The study focuses on mathematics teachers working in lower secondary school (grades seven to nine). Data collection took place in winter 2010 and 241 teachers responded to a Likert scale questionnaire (1 = “fully agree”, …, 5 = “fully disagree”). This analysis concentrates on the comparison between the studies carried out in two different years: 1990 and 2010 (the results from year 1997 are presented for illustration) that all use similar statements in their questionnaires. The grouping of the statements is done taking Dionne’s (1984) work as a basis (who distinguished three different perceptions of mathematics: traditional, formalist and constructivist) as mentioned earlier. The definition of belief is adapted from the work of Erkki Pehkonen (2003) who considers belief as individual subjective knowledge. A short description of the participants is presented

Research results The statistically significant differences in 0,05 level between teachers’ beliefs in 1990 and 2010 appeared in 58% of statements (14 statements out of 24). The differences were determined using 2-tailed independent t-test. As expected, constructivist way of teaching is now supported most strongly ( x 1,99 ), as well as formalistic ( x 2,19 ). Traditional way of teaching is supported the least ( x 2,72 ). The directions in the changes are negative in traditional and formalistic view; the changes in constructivist view go to both directions. In the following tables, the mean results of every statement in every category are presented, and the statistically significant (0,05 level) differences are marked with “*”. It is obvious that in all three surveys the teachers emphasize the importance of logical reasoning, exact use of language and systematizing. The support to practice the use of symbols and carry out proofs has reduced.

in Table 1.

Table 1. The overall description of the participants Year The number of participants Average age Average number of years of teaching Qualification as a mathematics teacher

1990 106 43 17 74%

1997 118 44 18 64%

2010 241 47 22 84%

Table 2. The formalist view Item number Item wording 20 Mathematics teaching should emphasize logical reasoning 1 One has to pay attention to the exact use of language (e.g. one should distinguish between an angle and the magnitude of an angle, between a decimal number and a decimal notation) 12*

In teaching, one should proceed systematically above all

18*

Abstraction practice should be stressed in mathematics

7* 11*

The proof of the Pythagorean theorem has to be worked in a mathematics lesson In particular, the use of mathematical symbols should be practiced

Working with exact proof forms is an essential objective of mathematics teaching

The irrationality of the number √2 has to be proved

year 1990 1,42

year 1997 1,47

year 2010 1,50

1,48

1,67

1,57

1,3

1,39

1,61

2,4

2,48

2,14

1,81

1,87

2,16

2,08 2,61

1,86 2,78

2,49 2,92

3,42

3,13

3,14

Mokytoj킬 ugdymas 2011

1990

1997

2010

4 3,5 3 2,5 2 1,5 1 0,5 0 20

12*

18*

11*

Fig. 1. The formalist view

Table 3. The traditional view Item number 13* 6* 19* 2* 17* 14 16*

Item wording The learning of central computing techniques (e.g. applying formulas) must be stressed

year year year 1990 1997 2010 1,39 1,52 1,76

In mathematics teaching, one has to practice much above all

1,6

1,88

Above all mathematical knowledge, such as facts and results, should be taught

1,88 1,98 2,84

In a mathematics lesson, there should be more emphasis on the practicing phase than on the 2,29 2,39 2,97 introductory and explanatory phase As often as possible such routine tasks should be solved where the use of the known 3,59 3,22 2,99 procedure will surely lead to the result Pupils should above all get the right answer when solving tasks 2,89 2,88 3,04 A pupil need not necessarily understand each reasoning and procedure

3,15

2,8

3,43

In this view statistically significant differences were found in almost every statement. The support to teach facts

1990

1997

2010

4 3,5 3 2,5 2 1,5 1 0,5 0 13*

Fig. 1. The Traditional view

19*

17*

16*

Kirsti KISLENKO, Lea LEPMANN Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990 – 2010 m.)

and practice a lot of exercises have decreased, and the support to understanding increased. Nevertheless, teachers put more emphasise on solving simple routine tasks (statement 17), and fewer attention to the exercise

that demand independent thinking (statement 24, see the following table). Item number

Item wording

Table 4. The constructivist view 21 22 24* 23 15* 10* 5* 9 3

Pupils should develop as many different ways as possible of finding solutions, and in teaching they should be discussed Pupils should formulate tasks and questions themselves, and then work on them

1,58

1,59

1,64

1,79

1,8

1,72

As often as possible, the teacher should deal with tasks in which pupils have to think first and for which it is not enough to merely use calculation procedures In assessment (classroom performance), above all the presented solutions of the tasks should be taken into account Above all the teacher should try to get pupils involved in intensive discussions

1,44

1,47

1,78

1,76

1,61

1,89

2,42

2,23

1,9

As often as possible, pupils should work using concrete materials (e.g. cardboard 1,32 models) Sometimes teaching should be realized as project-oriented (beyond subject limits), and 2,79 prerequisites for it should be created. (An example of the project: to buy and maintain an aquarium.) In mathematics teaching, learning games should be used 2,27

1,36

1,95

3,1

2,21

2,19

2,27

Mathematics has to be taught as an open system that will develop via hypotheses and cul-de-sacs

2,47

2,58

1990

1997

2,69

2010

3,5 3 2,5 2 1,5 1 0,5 0 21

Fig. 1. The constructivist view

24*

15*

10*

Mokytojų ugdymas 2011

The support on using social forms of learning, such as project work and class discussion, has increased. At the same time, the importance to use concrete materials has decreased. Moreover, the development of pupils’ independent thinking via more complicated problem solving have lost some of its popularity amongst teachers.

asked in a part a questionnaire that did not belong in this analysis). Thus, the actual picture in the classroom might not directly refer to teacher’ espoused believes (which was also the case in TALIS-study). Compared to the previous studies teachers put less emphasise on solving more complicated problems that demand independent thinking, and higher importance is given to exercises that can be solved using concrete algorithms. This result was also visible in Estonians pupils’ mathematical performance in PISA 2006 and 2009 tests where the scores of the tasks that demanded the highest level of thinking decreased (Puksand, Lepmann & Henno, 2010). One of the explanations for this found tendency could be the fact that the number of mathematics lessons per week has reduced over the years. Therefore, guiding teachers to use problem solving using less time is a problem that needs to be addressed. Because Wong and colleagues (2009) point out that the effectiveness of teaching via problem solving has positive impact on pupils learning (Wong, Lee, Kaur, Foong & Ng, 2009). The analyses of all three years shows that the differences between teachers’ beliefs in years 1990 and 1997 are significantly lower than the differences between years 1990 and 2010. Therefore, one could claim that teachers’ beliefs are fairly stable and the change in them takes time.

Summary and discussion When taking ten most agreed upon statements as a basis then in year 2010 Estonian teachers of mathematics can be described as follows. In mathematics teaching the most important is to carry out logical reasoning using correct mathematics language and systematic progressing. For exercises different solving procedures should be found, and the pupils should be leaded to construct exercises by themselves. Mastering the computing techniques in every exercise is emphasized. Nevertheless, figuring out problems related with everyday life and problem solving are also considered important. Above all the teachers should try to get pupils involved in intensive discussions, and when assessing pupils work the solutions of the tasks and not the answers should be taken into account. In all three studies the ideas of the formalistic teaching are valued. The rigour of mathematics as well as pupils high achieving in the subject are very important to the teachers. However, one can see an increase in valuing pupil-centred teaching and social forms of learning. Nevertheless, when asking teachers how often does he/she usually asks pupils to work together in small groups more than 8 teachers out from 10 answered either never or in some lessons (this was Literatūra Andrews, P., & Hatch, G. (2000). A comparison of Hungarian and English teachers´ conceptions of mathematics and its teaching. Educational Studies in Mathematics, 43, 31– 64. Bernack, K., Holzäpfel, L., Leuders, T., & Renkl, A. (2010). Initiating change on pre-service teachers´ beliefs in a reflexive problem solving course. In K. Kislenko (Ed.), Current state of research on mathematical beliefs XVI, (pp. 27–42). Tallinn, Estonia: Institute of Mathematics and Natural Sciences, Tallinn University. Dionne, J. (1984). The perception of mathematics among elementary school teachers. In J. Moser (Ed.), Proceedings of the 6th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp.223–228). Madison (WI): University of Wisconsin. Grigutsch, S., Raatz, U., & Törner, G. (1998). Einstellungen gegenüber Mathematik bei Mathematiklehrern. Journal für

Acknowledgements The study was supported by the European Social Fund Programme EDUKO (grant 1.2.0302.09-0004 and 30.2-4/549).

•

References Mathematikdidaktik, 19, 3– 45. Jukk, H., Lepmann, L., & Lepmann, T. (2007). Teachers´ beliefs about the cognitive and application-oriented competencies in school mathematics. Teaching mathematics: retrospective and perspectives. Proceedings of VIII international conference (pp. 103 – 108). Riga, Latvia: Macibu gramata. Lepmann, L. (1998). Changes in teachers´ mathematical conceptions in 1990 – 1997. In T. Breitreig & G. Brekke (Eds.), Theory into practice in Mathematics Education. Proceedings of NORMA 98 (pp. 179 – 185). Kristiansand: Agder College Research Series No 13. Loogma, K., Ruus, V.-R., Talts, L., & Poom-Valickis, K. (2009). Õpetaja professionaalsus ning tõhusama õpetamisja õppimiskeskkonna loomine. OECD rahvusvahelise õpetamise ja õppimise uuringu TALIS tulemused. Tallinna Ülikooli Haridusuuringute Keskus. Available online at:

Kirsti KISLENKO, Lea LEPMANN Mokytojų požiūrio kaita mokant matematikos Estijos mokyklose (1990 – 2010 m.)

http://ester.nlib.ee/record=b2507871*est. Niss, M. (2002). Mathematical competencies and the learning of mathematics: the Danish KOM project. Available online at: http://www7.nationalacademies.org/ mseb/mathematical_competencies_and_the_learning_of_ mathematics.pdf. OECD (2009). Creating Effective Teaching and Learning Environments. First Results from TALIS - Teaching and Learning International Survey. Available online at: http:// www.oecd.org/edu/talis/firstresults. Pehkonen, E. (2003). Læreres og elevers oppfatninger som en skjult faktor i matematikkundervisningen. (Teachers’ and pupils’ conceptions as a hidden factor in mathematics teaching). In B. Grevholm (Ed.), Matematikk for skolen (pp. 154-181). Bergen, Norway: Fagbokforlaget. Pehkonen, E., & Lepmann, L. (1994). Teachers’ conceptions about mathematics teaching in comparison (Estonia Finland). In M. Ahtee, & E. Pehkonen (Eds.), Constructivist

viewpoints for school teaching and learning in mathematics and science (pp. 105-110). Helsinki: University of Helsinki, Department of Teacher Education. Research Report 131. Puksand, H., Lepmann, & T. Henno, I. (2010). PISA 2009 – Eesti tulemused. Tallinn: Haridus- ja Teadusministeerium. Thompson, A.G. (1992) Teachers ´ beliefs and conceptions: A Synthesis of the Research. In: D. Grows (Ed), Handbook of Research on Mathematics Teaching and Learning (pp. 127-146). New York: Maxmillian. Törner, G. (1997). Methodische Überlegungen zur BeliefsForschung und einige inhaltliche Beobachtungen. Beiträge zum Mathematikunterricht 1997. (pp. 494–497). Hildesheim: Franzbecker. Wong, K. Y., Lee, P.Y., Kaur, B., Foong, P.Y., & Ng, S. F. (Eds) (2009). Mathematics education. The Singapore journey. Series on mathematics education (2). Singapore: World Scientific Publishing Co. Pte. Ltd.

KIRSTI KISLENKO

Doktorantė (Matematikos pedagogika) ir projekto “Mokytojų profesinis tobulėjimas ir palaikymas” mokslinė bendradarbė. Moksliniai interesai: emocinė sritys, mokinių ir mokytojų požiūriai į matematiką, vidurinis ir aukštesnysis vidurinis ugdymas.

PhD student in Mathematics Education and researcher in the project “Teachers’ professional development and its supporting” Institute of Mathematics and Natural Sciences, Tallinn University. Research interests: affective domain, pupils’ and teachers’ beliefs about mathematics, secondary and upper secondary education

Address: Narva road 25, 10120 Tallinn, Estonia E-mail: kirstik@tlu.ee

LEA LEPMANN

Mokslų daktaras (Matematikos pedagogika), Tartu universiteto matematikos instituto docentas. Moksliniai interesai: mokytojų požiūriai, matematikos mokytojų ugdymas, mokyklinės matematikos mokymo programa.

PhD in Mathematics Education, Associate Professor, Institute of Mathematics, University of Tartu. Research interests: teachers´ beliefs, mathematics teacher education, school mathematics curriculum.

Address: J. Liivi 2, 50409 Tartu, Estonia E-mail: Lea.Lepmann@ut.ee

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 50–55 Teacher Education. 2011. No. 16 (1), 50–55

Katrin KOKK Tartu universitetas a • University of Tartu

Nacionalinio matematikos egzamino Estijoje lyginamoji analizė

COMPARATIVE ANALYSIS OF THE NATIONAL EXAMINATION OF MATHEMATICS IN ESTONIA

Anotacija Straipsnyje pateikiama istorinė Estijos nacionalinio matematikos egzamino organizavimo apžvalga nuo 1997-ųjų iki 2011-ųjų. Nagrinėjami egzamino organizavimo, dalyvavimo, rezultatų ir turinio klausimai. Pagrindiniai žodžiai: išorinė kontrolė, nacionalinis matematikos egzaminas, matematinis ugdymas, istorija.

Abstract In the article has been given a historical overview of the organisation of the National Examination of Mathematics in Estonia in the years 1997–2011. The problems of the organisation, participation, results and the examination content are dealt with. Keywords: external control, the National Examination of Mathematics, mathematics education, history.

Introduction The quality evaluation of general education in Estonia (the so-called external control) is the task of the National Examination and Qualification Centre (http:// www.ekk.edu.ee/). The external evaluation is carried out based on the achievement papers for the 3rd and 6th classes, the administration of the secondary school final examinations and the national examinations of upper secondary school as well as the coordination of international educational studies (PISA, TIMSS).

syllabus and allow feedback to the learner at the level of in-depth treatment of the wide course syllabus. At the National Examination basic knowledge and skills are checked. At the examination two parallel versions are used, the examination takes place in two parts with the duration of 4–4.5 hours. Between the parts of the examination there is a 45-minute intervall. The student can get 50 points as a maximum for each part of the examination. With the tasks of the firts part of the National Examination of Mathematics the presence of the basic knowledge and skills in each subject course of the upper secondary school and the application of the knowledge and skills in real life situations are assessed. With the tasks of the second part of the examination it is assessed how structured and organised the examinee’s knowledge is, how well he/she can apply the obtained knowledge in the case of non-routine tasks, what the examinee’s preparation is like for continuing the studies on the following level. To a certain extent, the year 2002 in the history of the national examinations became a breakthrough. By that time the dissatisfaction of society with the level of general education in mathematics had reached its peak because • the number of compulsory classes of mathematics had decreased year after year in secondary and upper secondary school; • the programmes were overloaded and favoured

The organisation of the National Examination of Mathematics The national examinations in mathematics for upper secondary schools have been organised since the school year 1997–1998 when a new curriculum was introduced in Estonian schools. This was preceded by an experimental national examination carried out on two different levels (branches of humanities and realities) in the spring of 1997. However, institutions of higher education did not accept this examination as an entrance examination. In 1998, as a decision of educational policy, the national school-leaving examinations at upper secondary schools were unified with the entrance examinations to higher education institutions. This decision led to the creation of onelevel national examinations that are based on the subject content of the compulsory narrow course

Katrin KOKK Nacionalinio matematikos egzamino Estijoje lyginamoji analizė

non-profound studying; • in the society as a whole the negative attitude to studying sciences was prevailing (the cult of the so-called soft subjects was characteristic in studies); • the established requirement for passing the national examination – one point – did not motivate a big number of students to study well (the examination could be passed „without any efforts“); • the university departments had big problems with admitting capable students to science specialities; • in the new curriculum, to be launched in the futuure, only the level of narrow mathematics was fixed. To alleviate the situation, according to the new curriculum adopted in the year 2002 the new pass rate for the National Examination of Mathematics was established with 20 points. The participation in the National Examination Further we shall observe how during the last fifteen years the number of students, who passed the National Examination of Mathematics, has changed, i.e. we shall see how popular the National Examination of Mathematics is among school-leavers (see: Table 1). We can see in the table that the number of schoolleavers who chose the National Examination of Mathematics has constantly decreased year after year. In the first year when the examination was organized

the 33% participation was understandable to a certain extent (this national examination was not recognized by higher education institutions as an entrance examination and it was easier to pass it as a school examination). It is worrying that today we have fallen almost to the same level (38–39%). There are several reasons for it. • The students (having chosen a very narrow course) consider the examination in mathematics hard. After the examinations which were considered especially hard (2006–2007) we see the immediate decline in the number of participants by 5–6 percentage points. • The establishment of the 20-point pass rate (2002) has repelled the students whose aim was only to participate in the examination. • In the conditions of the decrease of the overall number of students the higher institutions agree to admit these students who instead of the mathematics examination have passed some other examination in realities or natural sciences. • The specialities connected with engineering and sciences are not popular. The results of the National Examination Observing the mean points obtained at the examination in different years, we can see that in the results there are two different levels – the reselts before and after establishing the 20-point passing rate (see: Table 1).

Table 1. The rates of participation of students who chose the National Examination of Mathematics (NEM) and the students who failed and the mean points of the examinees

Year

The school-leavers who chose NEM (%)

Arithmetic means of results

Students getting less than 20 points (%)

Experimental 1997 1998 1999 2000

33 59 65 58

51,1 32,1 43,2 39,2

3,0 38,8 21,2 27,8

2001

39,2

2002 Experimental 2003 2004 2005 2006 2007 2008 2009 2010

54 56 55 52 49 43 38 38 39

53,4 53,0 (17,9) 56,2 52,0 50,7 49,3 58,2 52,2 58,4

28,0 5,6 9,4 5,2 7,9 7,3 5,7 5,3 6,8 7,7

Mokytojų ugdymas 2010

In 2002 the mean result of the students who passed the examination rose by 14.2 percentage points and the number of students who failed with less than 20 points fell from 28% to 5.6%. The examination tasks were not assessed as easier in that year but preparation for the examination was more serious and a certain part of students did not choose the examination (the number of participants decreased by 4 percentage points). As assessed by the teachers, the examinations were more difficult in 1998, especially in 2006 and 2007. It is also shown by the respective means (32.1, 50.7 and 49.3). The teachers’ dissatisfaction with the great number of examination tasks and the complicated character of some tasks culminated in 2007 when the teachers made a public address to the Minister of Education. The table vividly demonstrates the results – after the year with bad examination results the mean result has become better step by step (1999 – 43.2 and 2008 – 58.2 ) which speaks of easier examination tasks in these years. It is somewhat suprising that in comparing the results of the students from the schools with the Estonian language as a language of instruction and the schools with Russian, the earlier better results of Russian schools (in 1997 higher by 12 percentage ponts) has step by step been replaced by better results of Estonian schools (in 2010 higher by 6 percentage points). Also, in the TIMSS and the PISA research surveys the results of Russian students have been worse although the same textbooks and curricula are used in both schools. When looking at the students’ beliefs, it has been revealed that Russian students devote significantly less attention to the tasks of practical content than Estonian students (Lepmann, 2000, p 35). In the recent years, in the tasks of the national examination much importance has been given to the skill of applying knowledge and skill of solving real life tasks. Evidently it is one of the reasons for the difference in results. About the content of the National Examination of Mathematics When analyzing the content of examinations, we shall, first of all, observe the distribution of compulsory and elective tasks in different years. In Table 2 obligatory tasks are marked in bold, a thicker vertical line separates different parts of the examination. We can see that in the first years students had more freedom for the choice of tasks, since 2002 only the last task offers choice in which one of the options of choice is a task in stereometry. Consequently, earlier it was possible to avoid some themes in the curriculum by choice. For example, in 2001 it was possible to choose between

six sets of different tasks and in one set it was possible to eliminate even three themes. The recent years are characterized by the fact that the examination checks all the themes of the curriculum. In comparison with the initial years the examination has become step by step richer in content. In the years 2003–2006 a big number of tasks in different themes has increased the amount of knowledge checked at the examination. In the last years we can see the decrease in the number of tasks but the so-called joint tasks containing the material of several themes have been added. This increases the complicated character of tasks and a bigger working load appears because many students have difficulties in connecting different themes. The examinations of 2005–2007 were extremely labour-intensive where primarily the best students had little time for solving tasks. Under the pressure of the teachers’ opinion in 2008 half an hour of working time was added to the examination but it did not decrease the working load. Consequently, in the recent years, both the working load and the time limit of the examination have increased. When comparing the examination points given to the task of a certain theme, for the same knowledge in the earlier work the students received more points. For example, the assessment of the tasks of the theory of probability. When in the year 2001 the application of the Bernoulli formula was given 8 points, in the year 2004 only 2 points. To earn 15 points in 1997, it was necessary to find the probability of one event, in the year 1998 of two events and in the year 1999 the probability of already three events. Table 3 also vividly demonstrates the growth of the working load where the frequency of occurrence of equations in algebra and systems of equations is described. In different years we can see the increase of the number of equations and the growth of complexity while algebra’s proportion as a whole has not significantly increased at the examination. Summing up the above said, we must admit that the students successfully passing the present national examination must be better than earlier students. The need for standardizing the results of the national examinations has been discussed for years. Since 2007 procentile ranks have been used giving additional information to examination points (http://www.ekk. edu.ee/kiirelt-leitav/statistika/protsentiilastakutevastavustabelid-riigieksamite-tulemustele). The procentile rank shows from how many percent of results the given result is stroger or equal in the

Katrin KOKK Nacionalinio matematikos egzamino Estijoje lyginamoji analizė

Table 2. The maximal points of the tasks in the National Examination of Mathematics, the division of compulsory and elective tasks

Year

1997 H/R

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

10 obligatory tasks 10 elective tasks (obligation to choose 2 tasks out of 3) 10 elective tasks (obligation to choose 1 task out of 2) 10 elective tasks (obligation to choose 4 tasks out of 6)

given subject in the given study year. Until today in admitting students to higher education institutions, only the points obtained at a particular examination are taken into consideration. The national examination providing guidelines for carrying out the process of studies Beside the role of the assessor of the quality of studies, the national examination shapes and directs the process of studies. The examination of mathematics very strongly fulfils this role. To quarantee further success for the students and admittance to higher education schools, the teachers in their teaching are very strongly oriented to the solution of the tasks of the national examination. That is why the proving side of mathematics suffers. Since 2002 the weight of tasks checking creativity has increased at the national examination. Also, at the national examination it is considered important that the student should know how

to apply his/her knowledge in solving real life tasks. Since 2005 more than 70% of the tasks in the second part of the examination check the subject knowledge at the highest, i.e. application skills level. Also, the national examination is a good means for introducing new trends. With the help of the examination teaching the theory of probability at schools at the end of the 1990s was strengthened, the students’ knowledge in trigonometry was improved and more attention was paid to the development of functional reading skills. Summary In the Republic of Estonia the system of organising national examinations has been functioning already for 14 years, being universal and with reasonable organisational costs. On the other hand, we have several problems. • The level of narrow course subject content is not sufficient for successful passing the present

Mokytojų ugdymas 2010

Table 3. The frequency of occurrence of different types of equations in algebra and the of equations at the national examinations

Year

Linear equation

1+3

Quadratic equation

2+1 4+3 1+2

2+2 2+4

3+2

3+1

6 1

Cubic equation

1+1

Fractional equation Equation with a parameter

2+1 1+1 3+1 1

1 1

1+3

2 1 2

1 1

System of equation 1 1

• linear • containing a quadratic equation • containing a fractional equation • containing a cubic equation • more complicated Sum total

2 1

1 1 4

national examination. It forces teachers to intensify the instruction within a narrow course and increases the pupils’ study load. • Due to the significance of National Examination results for the admission to a higher education establishment, school evaluation and school ranking, teachers tend to focus primarily on preparing for the National Examination of Mathematics. • The National Examination of Mathematics is considered difficult (the average result of the examination stands at the bottom of the subject list) and this scares students to choose the mathematics examination. • Teachers have criticized the National Examination of Mathematics saying that it is excessively labor-intensive (too little time for the students to solve the problems) and in some years the learning material that is not compulsory for everybody has been included in the problems. No deep analysis has been conducted on how voluminous and complex the national examination papers are. • The competition between higher education establishments to attract students makes it necessary to change the admission terms (replacement of the compulsory National Examination in Mathematics with another

2 1 1 10

subject) or organise their own entrance tests that in turn, has influence on the selection of national examinations. • The students who have not passed the mathematics examination often lack systematic and orderly knowledge in mathematics that creates difficulties in their higher education school studies. • In higher education schools the students drop out mostly in the fields related to information technology (IT), engineering and precise sciences. The reason for this is considered to be insufficient knowledge of mathematics. The new curriculum that will be introduced in the school year 2010–2011 will give more importance to mathematical education. According to this curriculum, a student must pass the compulsory National Examination of Mathematics that can be performed on two different levels. The administration of compulsory national examination to be introduced in 2014 is still under way, but according to the Minister of Education, the primary goal for national examinations must be qualifying for school graduation.

Katrin KOKK Nacionalinio matematikos egzamino Estijoje lyginamoji analizė

Literatūra Lepmann, L. (2000). Eesti ja vene õpilaste arusaamad matemaatikaõpetusest. Koolimatemaatika XXVII. Tartu: Tartu Ülikooli kirjastus, 33–37. Matemaatika katseline riigieksam. (1996). Uudelepp, H. (koost). Haridusministeeriumi Teataja 1996. Tallinn: Nivano. Matemaatika: katseline riigieksam. (1997). Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsioonikeskus. Tallinn. Matemaatika riigieksam 1998. (1998). Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsioonikeskus. Tallinn. Matemaatika riigieksam 1999. (1999). Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsioonikeskus. Tallinn. Matemaatika riigieksam 2000. Riigieksam 2001. (2000).

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References Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsiooni keskus. Tallinn. Matemaatika riigieksam 2001. (2001). Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsioonikeskus. Tallinn: REKK trükikoda. Matemaatika riigieksam 2002. (2002). Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsioonikeskus. Tallinn: REKK trükikoda. Matemaatika riigieksam 2003. Riigieksam 2004. (2004). Uudelepp, H. (koost). Riiklik Eksami- ja Kvalifikatsiooni keskus. Tallinn: REKK trükikoda. Statistika (s.a.). Last used 1.05.2011, address http://www. ekk.edu.ee/kiirelt-leitav/statistika.

KATRIN KOKK

Gamtos mokslų magistrė (Matematika), Tartu universiteto Matematikos instituto lektorė. Moksliniai interesai: nacionalinis matematikos egzaminas, lygių tyrimai, mokymo programa.

Master of Science (Mathematics), MSc Lecturer at the University of Tartu, Insitute of Mathematics; Research Interests: the national examination of mathematics, level researches, curricula.

Address: PJ. Liivi 2 – 223, Tartu 50409, Estonia E-mail: kkokk@ut.ee

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 56–61 Teacher Education. 2011. No. 16 (1), 56–61

Madis LEPIK Talino universitetas, Estija • Tallinn University, Estonia

Įrodymais pagrįsto ugdymo taikymas mokant matematikos Estijos mokyklose

THE ROLE OF PROOF IN ESTONIAN CURRICULA OF LOWER AND UPPER SECONDARY MATHEMATICS

Anotacija Šiame pranešime analizuojamas įrodymas ir įrodinėjimas Estijos vidurinės mokyklos mokymo programoje, mokant matematikos žemesniu ir aukštesniu lygiu. Buvo stengiamasi aptarti ne tik paruoštus įrodymus bet ir procesus, gerinančius mokinių įrodinėjimo gebėjimus ir veiklas, kurių tikslas yra prasmingas mokymasis įrodinėti. Analizė parodė, kad Estijos mokymo programoje įrodymas ir įrodinėjimas aptariami išsamiai ir tiksliai, į programą įtraukta daug įrodymų ir išvedimų. Tuo pačiu metu nėra akcentuojamas geras supratimas, kas yra įrodymas, ir įrodinėjimo įprasminimas. Pagrindiniai žodžiai: matematinis įrodymas, matematinis pagrindimas, mokymo programa, privalomoji mokykla, vidurinė mokykla.

Abstract This paper reports an analysis how proof and proving are dealt with in the Estonian lower and upper secondary mathematics curricula. It was attempted to consider not just only ready made proofs but also processes developing pupils’ proving abilities and activities aiming at meaningful learning about proving. The analysis shows that Estonian curricula discuss proof and proving explicitly and rigorously, numerous proofs and derivations are included. At the same time developing good understanding of proof and making proving meaningful is not stressed. Keywords: mathematical proof, mathematical reasoning, curriculum, cumpulsory school, secondary school.

Introduction Changing role of proof in mathematics education Proof and proving are central to doing mathematics. However, it is less clear what role proof should play and what status it should be afforded in school mathematics curriculum. The role of proof in school mathematics has varied during the last four decades in many countries. While in the nineteen seventies there was a worldwide Bourbaki inspired attempt to align mathematics teaching in the school with a formal conception of proof, later on there has been a significant reorientation towards communication and understanding in classroom practice (Mariotti, 2006). After a period with less focus on proof and proving, many countries are now revising their curriculum and give a more prominent place for these topics again (e.g. ICMI 19, 2009). For example, in the United States, proof has recently got a central position in Principles and Standards (NCTM, 2000, p 342):

“Reasoning and Proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied.” In spite of clear internationally followed trends there are obviously cultural differences with respect of how proof and proving are dealt with in school mathematics in various countries. Proof in Estonian school mathematics In Estonia during Soviet period mathematics and sciences were considered to be as priority subjects. It was reflected in the number of lessons devoted to these subjects and in high demands in terms of the content. So, mathematical rigor, exact use of language, deductive approaches and mathematical reasoning were strongly stressed in Estonian secondary mathematics education until 1990s. Also working with exact proof forms was

Madis LEPIK Įrodymais pagrįsto ugdymo taikymas mokant matematikos Estijos mokyklose

an essential objective of mathematics teaching. In 1991, Estonia regained independence, and started also democratization in education. The changes in mathematics syllabi were inspired by “mathematics for all” ideas. The number of lessons was remarkably decreased and the content was cut, wider and narrower course of school mathematics were introduced. After around two decades with more students in upper secondary level and less emphasis on mathematics and natural sciences, recent reform efforts have elevated the role of these subjects again. Also proof is expected to play a more prominent role at the upper-secondary mathematics again. These ideas are reflected in the new national curriculum, accepted in 2010 and will be put into action during the next couple of years (Gümnaasium, 2010; Põhikooli, 2010). Purpose of the study In this paper we present an analysis of how proof, proving and some proof-related items are stated in Estonian lower and upper secondary school mathematics curriculum. The main analytic task in this paper is to describe the types of statements related to proof and proving in the curricular documents and analyze how explicitly these aspects are articulated in curricula. Hence, we attempted to identify the statements that somehow aim to enhance students’ knowledge about proof and proving, not simply the statements that explicitly deal with these concepts. The meaning of proof in the teaching/learning of mathematics A lot of research articles have been published in the field of mathematics education during the last three decades concerning students’ difficulties with proof and proving, new approaches to proof, and the importance of proof and reasoning for learning mathematics (for reviews, see ICMI, 2009; Mariotti, 2006; Stylianou et al, 2009). Many researchers have discussed the functions of proof in mathematics and the relevance of these functions to the teaching of mathematics (e.g. Bell, 1976; de Villiers, 1990; Hanna, 2000; Hemmi, 2006). Proof has several functions that are intertwined in various ways to each others. Verification refers to the validation of the truth of the statement according to the rules of reasoning accepted by the mathematics community, whereas conviction is the personal experience concerning the truth of a statement. Explanation provides insights in different manners

into why something is true. Understanding is the personal experience of this. Communication refers to the critical debate that proofs and proving enhances among mathematicians when they communicate their results to each other. Systematization refers to the organization of various results into a deductive system of axioms, major concepts and theorems. It helps for example to find circular arguments and other shortcomings in mathematical reasoning. The functions of aesthetic and intellectual challenge refer to personal experiences when working with proving and proofs. With the function of discovery de Villiers (1990) refers to the way in which mathematicians, when proving statements, deductively discover for example that they can prove something more general than the original statement. The teachers state that proof can exercise logical reasoning skills, the learning of routines useful when structuring the solutions of problems and in acquiring the mathematical language, i.e. abilities needed in problem solving. Mathematics educators sometimes claim that students do not see any meaning in proving statements in mathematics. Hanna and de Villiers (2008, p.330) point out the importance of developmental proof that has three major features: • Proof and proving in school curricula have the potential to provide a long-term link with the discipline of proof shared by mathematicians. • Proof and proving can provide a way of thinking that deepens mathematical understanding and the broader nature of human reasoning. • Proof and proving are at once foundational and complex, and should be gradually developed starting in the early grades. According to the theory, possibility to various kinds of participation enhances pupils’ experience of meaning. Participatory approach means that proofs are not only considered as ready-made products, but the construction process of them is taken into account. It also means that students have opportunities to train actively and to take part in activities needed in proving process. In this study we attempt to analyze what possibilities the studied curricula offer to teachers in organizing various kinds of activities that allow different kinds of pupils’ participation in processes involving proof. Methodology of the study

Mokytojų ugdymas 2011

As a tool for explorative content analysis of curriculum the set of proof- related categories developed in our previous study (Hemmi et al, in press) was used. The following categories were used: • Proofs: Theorems and results that are expected to be proved or derived. This category includes elements where proof or deductive reasoning is explicitly mentioned as well as specific theorems, rules, formulas and results which are, according to the curriculum, expected to be proved or derived. • Knowledge about proof and proving: This category consists of elements referring to metalevel features of mathematical knowledge. All aspects concerning the nature and elements of proof and proving mentioned are collected into this category (for example the meaning of proof and proving in mathematics, the role and features of definitions, axioms and proofs, the logical structure of some proof, proof techniques, etc.). • Proof- related issues: Taking into consideration the complexity of the idea of proving it was decided to consider also curricular statements implicitly dealing with the processes needed for proving (adapted the broader view on developing proving abilities) and included also the following types of curricular statements. • Exercising mathematical thinking and logical reasoning: All curricular statements referring to the development of mathematical or logical thinking skills in a general level are collected here. • Other types of justification or evaluation: Elements referring to types of justification or argumentation other than a proof, like justification based on visual or concrete models. Elements which refer to evaluation of either students’ own reasoning or evaluation of ready-made reasoning are also included here. • Students’ investigations: Elements referring to investigative activities, like finding similarities, regularities, dependencies or causalities, classifying and making generalizations (some kind of reference to a justification of the investigations is still required). So, while analyzing the curricular texts we were searching for proof-related statements in it and classified them into the categories above. In the

analysis, the attention was also paid to how clearly and explicitly topics of proof and proving are stated in the curricula. Results The analysis presented here is based on the new national curricula, accepted in 2010, which will be put into action during the next couple of years (Põhikooli riiklik õppekava, 2010; Gümnaasiumi riiklik õppekava, 2010). The curriculum for compulsory school mathematics is 12 pages and the curriculum for upper secondary mathematics is 27 pages long. Both consist of general part and subject syllabi. The general part includes the description of the subject and the list of general competences to be attained during the studies. Subject syllabi presents the list of content and related learning outcomes. In the general part of the curriculum mathematical competence as one obligatory learning outcome is defined, and a list of intended learning outcomes which form mathematical competence is presented. According to the description given in the text, mathematical competence involves, among others things, the ability to reason logically, justify, and prove results. The new national curriculum for upper secondary school mathematics prescribes two alternatives for mathematical studies: long and short mathematics. These options differ remarkably in amount, content and also in their intended manner of presentation. Short mathematics, consisting of 8 courses (35 hours each), should be descriptive, and based on inductive reasoning. Long mathematics, consisting of 14 courses, is meant for those who plan to continue their studies in areas where mathematics has an important role (technology, natural science, etc.). In the description of the short mathematics stream, proving is not mentioned directly, but the curriculum requires that presentations should be descriptive and based on inductive reasoning and that justifications of presented results should be based on intuition and analogy. In the description of the long mathematics stream, validating the statements by deductive reasoning and proving is stated as an important goal. There are also relatively high demands in terms of mathematical rigor. Long mathematics deals with the terms and methods necessary to understand the nature of mathematics as a science. In the following the analysis of both curricula is presented on the bases of categories described above. In case of upper secondary curriculum we limit only with long mathematics stream. Proof and proving

Madis LEPIK Įrodymais pagrįsto ugdymo taikymas mokant matematikos Estijos mokyklose

Firstly proof and proving are explicitly mentioned in the description of the content for grades 7..9. Special topic on the meaning of mathematical proving is prescribed. The notion of proof is intended to be introduced in the context of geometry; triangle congruent properties, the proofs of Thales and Pythagorean theorems are included into the list of content to be covered. It is stated that by the end of the lower secondary school the pupils should be able to prove some theorems and have developed primary proving skills. On the upper secondary level students are expected to learn numerous proofs and derivations and to reproduce them. A large number of specific proofs and derivations of formulas are explicitly stated in the text of the syllabi: relations between sine, cosine and tangent, reduction formulas, trigonometric formulas like sin(α+β), sin2α etc., sine and cosine rules, equations of a line, plane, hyperbola, parabola, and circle, formulas for the general terms and sums of arithmetical and geometrical progressions, derivatives of functions, derivative formulas for sums, differences, products and quotients of functions, elements of a triangle and their properties, congruent and similar triangles, properties of similar triangles and polygons, parallelograms and their properties, midlines of trapezoids, circles, angles at their circumference and Thales’ theorem. In general, most of the formulas, properties and rules are intended to be introduced by deriving or proving. Students should also be able to solve proving tasks themselves. Knowledge about proof and proving As one of the general aims of learning mathematics the ability to discover regularities and develop generalizations is stated. As the learning outcomes for grades 4-6 it is stated that pupils should learn to analyse and describe mathematical objects and to classify them on the basis of several characteristics; to use different ways of presenting mathematical information and swich from one to another. In connection to upper secondary mathematics it is stated that pupils should be able to see relations between different mathematical concepts and create a system of concepts. Knowledge about proof and proving is presented already in lower secondary curriculum. Special topic on the meaning of mathematical proving is prescribed, where the nature of mathematical proving should be presented and pupils’ proving skills developed (grades 7-9). It is stated that by the end of grade 9 the pupils

should be able to describe the meanings of the notions like theorem, assumption, assertion and proof, and have developed primary proving skills. Upper secondary curriculum doesn’t include issues about proof and proving, proof techniques or specific logical aspects are not mentioned either. It seems that one expects that the students already know what a proof; an assumption, an assertion and so on are, as these notions have been studied in lower secondary school. Now, they are not focused on. It is assumed that pupils will learn proving by reproducing proofs exposed to them. Exercising mathematical thinking and logical reasoning This category is clearly represented in both analysed curricula. Among general goals it is stated that studies in mathematics should pay special attention to the development of logical and creative thinking of pupils. The keywords like explaining, argumentating, analyzing, classifying and logical reasoning are included into the description of mathematics content for all school levels. The curriculum states that secondary graduates should be able to discuss in a logical manner and formalize their mathematical trains of thought. The required environment for this purpose is created by dealing with formal mathematical terms, symbols, properties and relations. Every secondary school graduate should be able to reason logically and creatively, be able to pose mathematical hypotheses, and justify them. Other types of justification and evaluation Pupils’ ability to justify their reasoning, evaluate and validate the results is stressed as an important learning outcome on both school levels. One can found statements like: Pupils are intended to learn to justify their reasoning and validate the results (grades 4-6), formulate hypotheses and validate them (grades 7-9), skills to justify and validate the results are important outcomes of studies (grades 10-12). Meaningful learning, pupils ability to argumentate and explain is stressed on the general level. At the same time it is not stated how these goals should be attained, so it leaves to the teachers to decide the required level of rigor and methods of justification. Students’ investigations

Mokytojų ugdymas 2011

This category is represented weakly in Estonian curriculum. Only couple of general statements like pupils investigate and model mathematical processes; develop generalizations and discuss them in a logical manner (grades 6-9) or pupils build up mathematical hypotheses, verify and prove them (grades 10-12) could be identified in the content description. At the same time the development of pupils’ logical and creative thinking is declared to be as one important goal of mathematical education. Discussion and conclusions Estonian mathematics education has longstanding traditions of including proof in the curriculum. It was supported also by analysis carried out above. Both analysed in this paper curricula with slightly different wording, state as a general goal that students should practice logical reasoning. Validating statements by deductive reasoning and proving is pointed out as an important goal in it. Creative and logical thinking and also students’ primary ability to prove mathematical statements are seen as important outcomes of studies. One can conclude that proof, proving and derivation are explicitly presented in the curricular statements and lots of theorems to be proved and formulas to be derived are directly listed in the curriculum. At the same time earlier research shows that many aspects of proof remain invisible for students and even at the university level, many students struggle with elementary questions concerning the logical structure of proof, proof techniques and knowledge about the meaning of proof in mathematics (Hemmi, 2008; Schoenfeld, 1991). In Estonian lower secondary curriculum special topic on the meaning of proving is prescribed and development of primary proving skills is prescribed. On the contrary in upper-secondary curriculum development of knowledge about proof and proving is not prescribed. It is assumed that pupils will learn proving by reproducing ready-made proofs exposed to them. Obviously, the necessary condition for good quality of teaching of proof would be a good understanding of proof and proving techniques. One may conclude that it is very difficult to develop a good understanding of the idea of proof via reproducing of ready- made proofs. Although problem solving and discovery learning approaches have been recommended

by researchers for so long, at least in respect of proof and proving analyzed curricula continue the tradition of teacher centered strategies. For the pupils proof is above all a problem of meaning and educators have to device teaching context which make proof meaningful to them. The problem is that the pupils don’t necessarily encounter the metalevel reasoning behind proof and the nature of logical deduction. So, the role of instruction should be to make pupils familiar with the models of mathematical reasoning and the structures of mathematics. Teaching should be organised in a way that it inspires pupils themselves to ask questions, make assumptions, draw conclusions and also to justify and verify their results. Investigations and the need to validate the results are the most natural way to recognize the need for mathematical proving (Schalkwijk et al, 2001). Unfortunately, as was shown by the analysis above, the related category was clearly underrepresented in Estonian curricula. To sum up one could conclude, that from all functions of proof only two are clearly stressed in the curricula. For curriculum authors the main function of proving seems to be the verification of the given conjunctures. Proofs should be used to convince the pupils in the truth of the presented mathematical statements. Another strongly represented function is communicative function of proof. Proof seems to be recognized as most natural way to communicate mathematical results to the pupils. Other functions of proof (explanatory, discovery etc.) are not stressed. Both analyzed curricula are relatively brief and leave much space for interpretations. In this situation the textbooks prove to be the key components that actually determine the implementation of the curriculum, the level and depth of the mathematical content to be taught. So, to understand how the curricular ideas are implemented into school practice, it is important to study textbooks and also classroom practices. That is why the next step of our proof project will include textbook analyzes. Acknowledgements

The study was partly supported by the European Social

Fund Programme Eduko (grant no. 1.2.0302.090004). Bell, A. W. (1976). A study of pupils’ proof – explanations

Madis LEPIK Įrodymais pagrįsto ugdymo taikymas mokant matematikos Estijos mokyklose

Literatūra in mathematical situations. Educational Studies in Mathematics, 7, 23–40. Gümnaasiumi riiklik õppekava. [Estonian national curriculum for upper-secondary schools] Retrived from https://www.riigiteataja.ee/akt/114012011002 De Villers, M. (1990). The role and functions of proof in mathematics. Pythagoras, South Africa, 23,17-24. Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies of Mathematics, 44, 5-23. Hanna, G., & de Villiers, M. (2008). ICMI Study 19: Proof and proving in mathematics education. ZDM – The International Journal on Mathematics Education, 40(2), 329-336. Hemmi, K. (2006). Approaching proof in a community of mathematical practice. Doctoral thesis. Stockholm University, Department of Mathematics. Hemmi, K. (2008). Students’ encounter with proof – the condition of transparency. ZDM – The International Journal on Mathematics Education. The Special Issue on Proof, 40, 413-426. Hemmi, K., Lepik, M., Viholainen, A. (in press). Proof and reasoning in Estonian, Finnish and Swedish mathematics curricula. In: Annual Symposium of Finnish Mathematics and Science Education Research Association. University of Tampere. ICMI 19 (2009). ICMI Study 19: Proof and proving in

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References mathematics education. Discussion document. Mariotti, A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173-204). Rotterdam/Taipei: Sense Publishers. NCTM (2000). National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston. Põhikooli riiklik õppekava. [Estonian national curriculum for comprehensiv school] Retrived from https://www. riigiteataja.ee/akt/13273133 Schalkwijk, L. V., Bergen, T. & Rooij, A. V. (2001). Learning to prove by investigations. Educational Studies in Mathematics, 43, 293–311. Schoenfeld, A. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In D. N. Perkins, J. Segal, & J. Voss (Eds.), Informal reasoning and education (pp. 311343). Hillsdale, NJ: Erlbaum. Stylianou, D., Blanton, M., & Knuth, E. (2009). Teaching and Learning Proof Across the Grades. A K-16 Perspective. New York, NY: Routledge.

Madis Lepik Matematikos mokslų daktaras (Matematikos pedagogika), Talino universiteto Matematikos katedros docentas. Moksliniai interesai: mokytojų profesinis tobulėjimas, matematikos mokytojų rengimas, įrodymas matematikos pedagogikoje, vadovėlių pedagogika.

PhD in Mathematics Education Associate Professor of Department of Mathematics, Tallinn University. Research interests: teacher professional development, mathematics teacher education, proof in mathematics education, textbook pedagogy.

Address: Narva Road 25 10120 Tallinn Estonia E-mail: mlepik@tlu.ee

besimokančiųjų skyrtybes įgalinančios matematikos didaktikos raiška

MANIFESTATION OF MATHEMATICS DIDACTICS ENABLING LEARNERS’ DIFFERENCES

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 64–76 Teacher Education. 2011. No. 16 (1), 64–76

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Vilniaus pedagoginis universitetas • Vilnius Pedagogical University Nacionalinis egzaminų centras National Examination Centre

SILPNAI BESIMOKANČIŲJŲ BENDROJO LAVINIMO MOKYKLOS MOKINIŲ MATEMATINIŲ GEBĖJIMŲ DIAGNOSTIKA Anotacija Straipsnyje pateikiami žemų matematinių gebėjimų mokinių analizės rezultatai, gauti pritaikius moderniosios testų teorijos dviejų parametrų modelį 2005–2008 m. Lietuvos nacionalinių mokinių pasiekimų tyrimų duomenims ir atlikus gautų duomenų kiekybinę ir kokybinę analizę. Rezultatai vertingi planuojant tolesnius mokinių pasiekimų tyrimus, tobulinant programas ir vadovėlius bei apmąstant ugdymo praktiką. Pagrindiniai žodžiai: žemų matematinių gebėjimų mokiniai, matematinė kompetencija, Bendroji matematikos programa, nacionaliniai mokinių pasiekimų tyrimai. Įvadas Naujose pradinės ir pagrindinės mokyklos bendrosiose programose (2008) daug kalbama apie būtinybę diferencijuoti ir individualizuoti ugdymo turinį (plačiąja prasme), kad įvairių polinkių, poreikių ir gebėjimų mokiniai norėtų ir pajėgtų išsiugdyti įvairias kompetencijas (Pradinio ir pagrindinio ugdymo bendrosios programos, 2008). Bendroji matematikos programa yra sudėtinė Bendrųjų programų dalis. Joje, kaip ir visų kitų dalykų programose, pirmą kartą šalies istorijoje pateikti mokinių pasiekimų aprašai trimis lygiais – patenkinamo, pagrindinio ir aukštesniojo. Manoma, kad patenkinamą mokinių pasiekimų lygį turėtų pasiekti dauguma mokinių, kurių matematiniai gebėjimai yra žemi. Rengdamos šiuos mokinių pasiekimų aprašus dalykų programų darbo grupės studijavo kitų šalių programas, taip pat atsižvelgė į savo šalies mokytojų ir kitų švietimo srities specialistų išsakytą nuomonę (Apie atnaujintų Pradinio ir pagrindinio ugdymo bendrųjų programų įgyvendinimą, 2008). Kalbant apie Bendrąją programą (matematikos dalyko), daugiausia neaiškumų ir nesutarimų iškilo dėl paten-

DIAGNOSTICS OF MATHEMATICS ABILITIES OF LOW ACHIEVERS IN A COMPREHENSIVE SCHOOL

Abstract

The article presents the results of the analysis of pupils with low abilities in mathematics, which were obtained having applied a two-parameter model of the modern test theory for the data of surveys on Lithuanian national pupils’ attainments, dated 2005–2008, and having carried out qualitative and quantitative analysis of the obtained data. The results are valuable, planning further surveys on pupils’ attainments, improving curricula and textbooks and considering educational practice. Key words: pupils with low abilities in mathematics, mathematics competency, general mathematics curriculum, national surveys on pupils’ attainments. Introduction New General Curricula for primary and basic schools (2008) elaborate on the necessity to differentiate and individualize the content of education (in a broad sense) so that pupils of various inclinations, needs and abilities want and are able to self-develop various competencies [2]. The General Curriculum of Mathematics is a constituent of General Curricula. Like in the curricula of all other subjects, for the first time in the country’s history it contains descriptions of pupils’ achievements at three levels: satisfactory, basic and advanced. It is considered that the satisfactory level of pupils’ achievements should be reached by the majority of pupils whose mathematical abilities are low. Preparing these descriptions of pupils’ achievements, working groups of subject curricula were studying other country’s curricula and also considered the opinion expressed by teachers and specialists in other fields of education of Lithuania[1]. As to the General Curriculum of Mathematics, most of obscurities and disagreements were caused by the requirements for the satisfactory level of pupils’ achievements [2, p. 818–

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika

kinamo mokinių pasiekimų lygio reikalavimų [2, p. 818–820, 834–836, 855–858]. Šios diskusijos vyksta iki šiol: vieniems patenkinamo mokinių pasiekimų lygio reikalavimai atrodo per aukšti, kitiems – per žemi, tačiau tyrimais pagrįstų argumentų diskusijose stokojama. Tyrimo objektas – mokinių, kurių matematiniai gebėjimai yra žemi, pasiekimai. Tyrimo tikslas – pasitelkus 2005–2008 m. Lietuvos nacionalinių mokinių pasiekimų tyrimų duomenis, ištirti ir aprašyti IV, VI, VIII ir X klasių mokinių, kurių matematiniai gebėjimai yra žemi, matematinę kompetenciją ir pateikti rekomendacijas, kaip galima būtų panaudoti atlikto tyrimo rezultatus. Tyrimo uždaviniai: • Atrinkti uždavinius iš 2005–2008 m. nacionalinių tyrimų testų, kurie identifikuotų dviejų mokinių grupių – 15-os ir 25-ių proc. silpniausiai besimokančių – matematinius gebėjimus IV, VI, VIII ir X klasėse. • Analizuojant atrinktus uždavinius ir juos grupuojant, sukurti nurodytų mokinių grupių gebėjimų aprašus ir juos palyginus aprašyti silpnai matematiką besimokančių mokinių gebėjimus pagal matematinės veiklos sritis ir pagal kognityvines gebėjimų grupes. • Įvertinti silpnai matematiką besimokančių mokinių nuostatas dėl matematikos dalyko, lyginant jas su kitų mokinių, dalyvavusių 2005–2008 m. nacionaliniuose tyrimuose, nuostatomis. • Pateikti išvadas ir rekomendacijas. Tyrimo eiga ir rezultatai Uždavinių atranka. Tyrimui atlikti buvo panaudotos šalies nacionalinių mokinių pasiekimų tyrimų, kurie buvo vykdomi Švietimo ir mokslo ministerijos užsakymu 2005–2008 m., duomenų bazės (Nacionaliniai moksleivių pasiekimų tyrimai). Pirmoje lentelėje pateikta informacija apie nacionaliniuose mokinių pasiekimų tyrimuose matematikos testus ir anketas pildžiusių mokinių skaičius.

820, 834–836, 855–858]. These discussions continue to take place till now: some find the requirements for the satisfactory level of pupils’ achievements too high, others, too low, but the discussions lack research based arguments. Thus, we have decided to elaborate on this problem. Research subject: achievements of pupils with low abilities in mathematics. Research aim: based on data of research on Lithuanian national pupils’ achievements, dated 2005– 2008, to research and describe mathematic competency of pupils of the 4, 6, 8 and 10 forms with low abilities in mathematics and provide recommendations how the results of the carried out research can be used. Research objectives: • To select problems from national research tests, dated 2005 and 2008, which would identify the abilities of two groups of pupils: the group of 15 per cent of the weakest in mathematics and the group of 25 per cent of the weakest in mathematics in the 4, 6, 8 and 10 forms. • Analysing selected problems and grouping them, to create descriptions of abilities of our distinguished groups of pupils and, having compared them, to describe observed abilities of pupils who are weak in mathematics by mathematic activity areas and by cognitive ability groups. • To evaluate attitudes of pupils who are weak in mathematics towards mathematics, comparing them with attitudes of other pupils who took part in national researches in 2005 - 2008. • To provide conclusions and recommendations. Research Process and Results Selection of problems. The research employed the data bases of researches on Lithuanian national pupils’ achievements, which were carried out under the order of the Ministry of Education and Science in 2005 - 2008 [4]. Table 1 shows information about mathematics tests in national researches on pupils’ achievements and numbers of pupils who filled in the questionnaires.

1 lentelė. Matematikos testus ir anketas pildžiusių mokinių skaičius Table 1. Numbers of Pupils who Solved Mathematics Tests and Filled in Questionnaires Matematikos testus sprendusių mokinių skaičius Matematikos anketas pildžiusių Number of pupils who solved mathematics tests mokinių skaičius Metai Klasė Number of pupils who filled in Year Form 1 testas 2 testas 3 testas questionnaires Test 1 Test 2 Test 3 2005 IV ir VIII 908 ir 713 928 ir 740 866 ir 693 2702 ir 2146 2006 VI ir X 687 ir 639 671 ir 635 674 ir 649 2032 ir 1923 2007 IV ir VIII 1304 ir 717 1298 ir 705 1292 ir 677 2548 ir 705 2008 VI ir X 1065 ir 648 1101 ir 626 1044 ir 651 1101 ir 626

Mokytojų ugdymas 2011

Norimoms mokinių grupėms – 15-os proc. ir 25ių proc. silpniausiai matematiką besimokančių mokinių – identifikuoti ir šioms grupėms uždavinius atrenkant buvo taikytas moderniosios testų teorijos (IRT) dviejų parametrų modelis (Sičiūnienė, Dargytė, 2010). Tuo tikslu buvo apskaičiuotos gebėjimų skalės (~ N (0, 1)) reikšmės, atitinkančios 15 ir 25 procentilį. Šios reikšmės, identifikuojančios žemų gebėjimų mokinius, atitinkamai buvo -1,0365 ir -0,6745. Antrasis IRT analizės parametras – skiriamoji geba – buvo laikomas geru, jeigu viršijo 0,5. Antroje lentelėje pateikta, kiek buvo atrinkta uždavinių, tenkinančių abu šiuos kriterijus, taip pat nurodyta, kiek papildomai uždavinių gebėjo atlikti 25-ių proc. grupės mokiniai, lyginant su 15-os proc. grupės mokiniais. .

In order to identify the desirable pupils’ groups – the group of 15 per cent of the weakest pupils in mathematics (the 15 per cent group) and the group of 25 per cent of the weakest pupils in mathematics (the 25 per cent group) – and to select problems solved by these groups, the two parameter model of the Modern Test Theory (MTT) was applied [3], [5]. To achieve this, values of the ability scale were calculated (~ N(0 , 1)), corresponding to the 15 and 25 percentile. These values, identifying pupils with low abilities correspondingly were -1,0365 and -0,6745. The second parameter of the MTT analysis – resolution – was considered good if it exceeded 0,5. The second table presents the number of selected problems that met both of these criteria and also distinguishes the number of additional problems that pupils of the 25 per cent group were able to solve, compared with the pupils of the 15 per cent group.

2 lentelė. Visų spręstų uždavinių bei atrinktų uždavinių skaičius Table 2. The Number of All Solved Problems and of Selected Problems Klasė Form

Visų uždavinių skaičius Number of all problems 15 proc. silpniausiai besimokančių mokinių grupę identifikuojančių uždavinių skaičius Number of problems identifying 15 per cent of the weakest pupils 25 proc. silpniausiai besimokančių mokinių grupę identifikuojančių uždavinių skaičius Number of problems identifying 25 per cent of the weakest pupils

Mokinių gebėjimų aprašai. Ieškant sąlyčio su Bendrosiomis matematikos programomis (2008), kiekvieną amžiaus grupę apibūdinantys uždaviniai buvo grupuojami pagal programose aprašytas veiklos sritis. Priskiriant atrinktus uždavinius vienai ar kitai veiklos sričiai, buvo remiamasi 3-ioje lentelėje pateiktais aprašais.

VIII

181

173

189

164

49 (27 proc.)

29 (17 proc.)

21 (11 proc.)

33 (20 proc.)

49 + 15 (35 proc.)

29 + 20 (28 proc.)

21 + 20 (22 proc.)

33 + 15 (29 proc.)

Descriptions of pupils’ abilities. Looking for the link with General Curricula of Mathematics (2008), problems describing every age group were grouped according to activity areas described in curricula. Attribution of selected problems to one or another activity area was based on descriptions given in Table 3.

3 lentelė. Atrinktų uždavinių grupavimas pagal veiklos sritis Table 3.Grouping of Selected Problems by Activity Areas Veiklos sritys pagal Bendrąją programą Activity areas according to the General Curriculum 1. Skaičiai ir skaičiavimai Numbers and calculus 2. Reiškiniai, lygtys, nelygybės, sistemos Expressions, equations, inequalities, systems

Trumpas aprašas Short description Skaičiaus sandara, skaitymas, rašymas, pavaizdavimas. Skaičių palyginimas, apvalinimas, tarpusavio ryšiai. Veiksmai su skaičiais. Composition of the number, reading, writing, representation. Comparison of numbers, rounding, correlations. Operations with numbers. Reiškinių pertvarkymas, lygčių, nelygybių, sistemų sprendimas. Situacijų modeliavimas reiškiniais, lygtimis, sistemomis. Rearrangement of expressions, solution of equations, inequalities and systems. Modelling of situations by expressions, equations, systems.

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika

3 lentelės tęsinys Table 3 (continued) Veiklos sritys pagal Bendrąją programą Activity areas according to the General Curriculum 3. Funkcijos ir sąryšiai Functions and their links 4. Geometrija Geometry

5. Matai ir matavimai Measures and measuring 6. Statistika, tikimybės Statistics, probabilities

Trumpas aprašas Short description Naudojimasis lentelėmis, grafikais, formulėmis. Dėsningumų paieška. Funkcijų modelių ir savybių taikymas. Using tables, graphs, formula. Search for regularities. Application of models and features of functions. Plokštumos figūrų pažinimas ir jų savybių taikymas. Erdvės figūrų pažinimas ir jų elementų apskaičiavimas. Lygumo, panašumo, simetrijos ir trigonometrinių sąryšių taikymas. Cognition of figures of the plane and application of their features. Cognition of space figures and computation of their elements. Application of equality, similarity, symmetry and trigonometric links. Mato jausmas (objektų parametrų nustatymas matuojant ir ,,iš akies“). Perimetro, ploto, tūrio, kampų sumos formulių taikymas. The feeling of the measure (identification of parameters of objects, measuring “by eye”). Application of formulas for the perimeter, area, volume, the sum of angles Dažnių lentelių, diagramų skaitymas. Duomenų interpretavimas, vertinimas ir išvadų darymas. Klasikinio tikimybės apibrėžimo taikymas. Reading frequency tables and diagrams. Data interpretation, evaluation and making conclusions. Application of the classical definition of probability.

Kuriant mokinių gebėjimų aprašus buvo vadovautasi tokiomis nuostatomis: • bet kurį gebėjimą apibūdinanti aprėptis turi būti pagrįsta ne mažiau kaip dviem pagal kriterijus atrinktais uždaviniais; • IV, VI, VIII ir X klasių 15-os proc. mokinių grupės ir 25-ių proc. grupės gebėjimus apibūdinančios aprėptys atitinkamuose aprašuose modeliuojamos taip, kad galima būtų įžvelgti mokinių gebėjimų pokyčius. Kadangi straipsnio apimtis neleidžia pateikti išsamių abiejų grupių mokinių gebėjimų aprašų, todėl pateikiami tik jų fragmentai (žr. 4 ir 5 lent.). Nagrinėjant šiuos fragmentus, pažymėtina, kad visi žemesnėse klasėse apibūdinti gebėjimai buvo nurodyti ir aukštesnėse klasėse, t. y. kiekvienoje aukštesnėje klasėje yra pateikti tik naujai toje amžiaus grupėje papildomai susiformavę mokinių gebėjimai. Pavyzdžiui, paaiškėjus, kad ir IV klasės, ir visų aukštesnių klasių mokiniai geba palyginti natūraliuosius skaičius, šis gebėjimas buvo aprašytas tik IV klasėje ir nebekartojamas aukštesnių klasių aprašuose. Interpretuojant gautus duomenis pažymėtina, kad nacionaliniuose mokinių pasiekimų tyrimuose buvo kelti įvairūs tikslai, neskiriant atskiro dėmesio vienai ar kitai mokinių grupei, dėl ko testuose panaudoti uždaviniai neapėmė matematikos programos nei pa-

Descriptions of pupils’ abilities are based on the following approaches: • Coverage describing any ability has to be based on not less than two problems selected according to our criteria; • Coverages describing abilities of 15 and 25 per cent pupils’ groups of the 4, 6, 8 and 10 forms are modelled in corresponding descriptions in such a way that it can be possible to envisage changes in pupils’ abilities. Because the volume of the article is too small to submit complete descriptions of pupils’ abilities of both groups, only their fragments are given (see Tables 4 and 5). Analysing these fragments, it is necessary to bear in mind that all abilities described in lower forms were observed in higher forms too; i.e., in every higher form only the abilities that were newly formed in that age group are given. For example, having found out that pupils of both the 4 and all higher forms were able to compare natural numbers, this ability was described only in the fourth form and was not repeated in the descriptions of higher forms. Interpreting obtained data, it has to be born in mind that in national researches on pupils’ achievements various aims were raised, not focusing on one or another pupils’ group separately; for this reason problems used in the tests have not covered the mathematics curriculum either by activity areas of mathematics or

Mokytojų ugdymas 2011

gal matematikos veiklos sritis, nei pagal kognityvinių gebėjimų grupes, nei pagal numatomus mokinių pasiekimų lygius. Tai leido tik aprašyti stebimus silpnų mokinių gebėjimus šiame etape nekeliant sau tikslo jų palyginti su Bendrojoje programoje pateiktais patenkinamo mokinių pasiekimų lygio reikalavimais ar palyginti juos su matematiką geriau mokančių mokinių gebėjimais. Išanalizavus sudarytus mokinių gebėjimų aprašus, paaiškėjo, kad ir 15-os proc. grupės, ir 25-ių proc. grupės mokiniai turi nemažai vienodai išlavintų gebėjimų, tačiau kiekvienoje amžiaus grupėje yra ir tokių gebėjimų, kuriais nepasižymi 15-os proc. grupės mokiniai. Labai svarbu pabrėžti, kad 15-os proc. grupės mokiniai šiuos trūkstamus gebėjimus vis dėl to įgyja po 1–2 metų, t. y. kai pereina į kitą amžiaus grupę. Ši tendencija matoma visose klasėse. Pavyzdžiui, 5-oje lentelėje 25-ių prog. grupės gebėjimų aprašo fragmente kursyvu išskirti gebėjimai, būdingi tik 25-ių proc. grupės mokiniams, o 4-oje lentelėje matoma, kad šiuos gebėjimus 15-os proc. grupės mokiniai įgyja būdami kitoje amžiaus grupėje.

by groups of cognitive abilities or by foreseen levels of pupils’ achievements. This enabled us only to describe observed abilities of weak pupils in this stage, not aiming to compare them with the requirements for the satisfactory level of pupils’ achievements given in the General Curriculum of Mathematics or to compare them with the abilities of pupils who know mathematics better. Having analysed the drawn up descriptions of pupils’ abilities, it was found that pupils of the 15 and 25 per cent group had quite many abilities that were equally developed but every age group also had such abilities which were not characteristic to the pupils of the 15 per cent group. It is very important to emphasize that pupils of the 15 per cent group anyway acquire these abilities after 1 or 2 years; i.e. when they pass to another age group. This tendency is observed in all forms. For example, in Table 5 in the fragment of the description of abilities of the 25 per cent group the text in italics shows abilities characteristic only to the 25 per cent pupils’ group, whilst in Table 4 we see that pupils of the 15 per cent group acquire these abilities when they are in another age group.

4 lentelė. 15-os proc. mokinių grupės gebėjimų aprašo fragmentas Table 4. Fragment of the Description of Abilities of the 15 Per Cent Pupils’ Group IV klasė 4 form

Natūraliuosius skaičius (iki 10 000) užrašo žodžiais ir skaitmenimis. Palygina natūraliuosius skaičius. Put down natural numbers (up to 10 000) in words and digits. Compare natural numbers.

VI klasė 6 form

Natūraliuosius skaičius pažymi skaičių tiesėje. Palygina dešimtainius skaičius, kurie turi vienodą skaitmenų skaičių po kablelio. Note natural numbers in the drawn number line. Compare decimal numbers that have the same number of digits after the comma.

Žodžiais, schemomis pateiktas situacijas aprašo paprastosiomis trupmenomis ir atvirkščiai. Describe situations given in words, schemes in common fractions and vice versa.

VIII klasė 8 form

Dešimtainius skaičius pažymi skaičių tiesėje. Palygina dešimtainius skaičius ir suapvalina juos nurodytu tikslumu. Atpažįsta sveikuosius skaičius, pažymi juos skaičių tiesėje. Note decimal numbers in the drawn number line. Compare decimal numbers and round them according to indicated accuracy. Recognise whole numbers, note them in the number line.

Įvairias situacijas aprašo paprastosiomis trupmenomis. Skaičių tiesėje pažymi trupmeninius skaičius. Describe various situations in common fractions. Note fractional numbers in the number line.

X klasė 10 form

Palygina paprastąsias trupmenas, kurių vardikliai vienodi. Note decimal numbers in the drawn number line. Compare decimal numbers and round them according to indicated accuracy. Recognise whole numbers, note them in the number line.

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika

5 lentelė. 25-ių proc. mokinių grupės gebėjimų aprašo fragmentas Table 5. Fragment of the Description of Abilities of the 25 Per Cent Pupils’ Group IV klasė 4 form Natūraliuosius skaičius (iki 10 000) užrašo žodžiais ir skaitmenimis. Palygina natūraliuosius skaičius. Put down natural numbers (up to 10 000) in words and digits. Compare natural numbers.

Žodžiais, schemomis pateiktas situacijas aprašo paprastosiomis trupmenomis ir atvirkščiai. Describe situations given in words, schemes in common fractions and vice versa.

VI klasė 6 form Natūraliuosius skaičius pažymi skaičių tiesėje. Palygina dešimtainius skaičius, kurie turi vienodą skaitmenų skaičių po kablelio. Atpažįsta sveikuosius skaičius, pažymi juos skaičių tiesėje. Note natural numbers in the drawn number line. Compare decimal numbers that have the same number of digits after the comma. Recognise whole numbers, note them in the number line. Įvairias situacijas aprašo paprastosiomis trupmenomis. Skaičių tiesėje pažymi trupmeninius skaičius. Describe various situations in common fractions. Note fractional numbers in the number line.

Rezultatų palyginimas pagal matematinės veiklos sritis. Turimi duomenys leidžia teigti, kad 25-ių proc. grupės IV klasės mokiniai gana gerai įsisavinę natūraliuosius skaičius: geba juos užrašyti žodžiais ir skaitmenimis, palyginti, sudėti ir atimti, padauginti ir padalyti iš vienaženklio skaičiaus. Jie prisilaiko veiksmų atlikimo tvarkos, sprendžia elementarius praktinio turinio uždavinius su mažais natūraliaisiais ar matiniais dešimtainiais skaičiais. Šios grupės mokiniai geba nustatyti dėsningumą ir juo pasinaudoti ieškodami trūkstamo sekos elemento, atsirinkti tiesioginę informaciją iš grafiko, kalendoriaus, diagramos, skaityti nesudėtingas skales. Šie mokiniai taip pat geba nesudėtungas situacijas aprašyti paprastosiomis trupmenomis ir atvirkščiai. Panašius gebėjimus demonstruoja ir 15-os proc. grupės ketvirtokai, tik, skirtingai nuo savo bendraamžių, jie nesupranta ,,paprastosios trupmenos“ sąvokos. Ko gero, ši aplinkybė stabdo ir tolesnį šio gebėjimo vystymąsi, nes 15-os proc. grupės mokinių gebėjimai operuoti paprastosiomis trupmenomis nuolat atsilieka nuo bendraamžių ir net X klasėje yra labai menki (žr. 4 lent.). Nustatyta, kad 25-ių proc. grupės VI klasės mokiniai išmoksta: skaičių tiesėje pažymėti dešimtainius ir sveikuosius skaičius, sudėti ir atimti dešimtainius

VIII klasė 8 form Dešimtainius skaičius pažymi skaičių tiesėje. Palygina dešimtainius skaičius ir suapvalina juos nurodytu tikslumu. Note decimal numbers in the drawn number line. Compare decimal numbers and round them according to indicated accuracy.

Palygina paprastąsias trupmenas, kurių vardikliai vienodi. Compare common fractions with the same denominators.

X klasė 10 form

Suvokia ir taiko skirtingais būdais užrašytų skaičių ryšius. Perceive and apply links between numbers written down in different ways.

Comparison of results by mathematic activity areas. Our possessed data enable to state that the 4 form pupils of the 25 per cent group have mastered natural numbers quite well: they are able to write them down in words and in digits, compare, add, subtract, multiply and divide by one-digit numbers. They follow the order of doing operations, solve the simplest problems of practical content with small natural or concrete decimal numbers. Pupils of this group are able to identify the simplest regularity and use it looking for the missing element in the sequence and to select direct information from the graph, calendar, diagram, read simple scales. These pupils are also able to describe the simplest situations in common fractions and vice versa. Similar abilities are also demonstrated by the fourth form pupils of the 15 per cent group but differently from their peers they do not understand the concept of the “common fraction”. Most probably, this circumstance stops further development of this ability because the abilities of the 15 per cent pupils’ group to operate common fractions always lag behind their peers’ abilities and even when pupils reach the 10 form they are very poor (see Table 4). It has been identified that the 6 form pupils of the 25 per cent group learn the following: can note decimal and whole numbers in the drawn number line,

Mokytojų ugdymas 2011

skaičius, padauginti ir padalyti dešimtainį skaičių iš vienaženklio skaičiaus, palyginti paprastąsias trupmenas su vienodais vardikliais bei spręsti nesudėtingas lygtis. Jie sprendžia uždavinius, kurių sąlygose yra ir netiesiogiai veiksmą nusakančių žodžių ar tenka atlikti iki trijų sprendimo žingsnių. 15-os proc. grupės mokiniai iš bendraamžių išsiskiria tuo, kad neįstengia išspręsti net elementariausių uždavinių, kurių sąlygose vartojami matematiniai terminai: ,,dalmuo“, ,,sandauga“, ,,sveikasis skaičius“. Jie nesprendžia lygčių, nesupranta, kas yra reiškinys ir kaip yra apskaičiuojama jo reikšmė, negeba palyginti trupmenų, nors jau išmoksta paprastosiomis trupmenomis aprašyti situacijas. Aštuntos klasės 25-ių proc. grupės mokiniai jau geba: suapvalinti skaičius nurodytu tikslumu, sudėti ir atimti trupmenas su vienodais vardikliais, padauginti ir padalyti trupmenas, susieti paprastųjų trupmenų dešimtainę ir trupmeninę formas, skaičių pakelti natūraliuoju laipsniu ir ištraukti kvadratinę šaknį, taikyti laipsnio su natūraliuoju rodikliu savybes. Jie sprendžia procentų, dalies ir visumos radimo uždavinius, apskaičiuoja raidinių reiškinių skaitines reikšmes, sprendžia paprastas lygtis. 15-os proc. grupės mokiniai, skirtingai nei jų bendraamžiai, negeba pritaikyti laipsnio ir kvadratinės šaknies apibrėžimų. Dešimtos klasės 25-ių proc. grupės mokiniai atpažįsta, užrašo, įvairiais būdais pavaizduoja nedidelius racionaliuosius skaičius, suvokia skirtingais būdais užrašytų skaičių ryšius, apskaičiuoja skaitinių reiškinių su laipsniais ir šaknimis reikšmes, nesudėtingais atvejais taiko laipsnių ir šaknų savybes, atpažįsta parabolę, taiko paprastos kvadratinės lygties sprendinių radimo formulę, patikrina, ar skaičių pora yra tiesinių lygčių sistemos sprendinys. Dešimtoje klasėje ryškesnių skirtumų tarp tirtų grupių gebėjimų nepavyko nustatyti, nes atrinkta mažai uždavinių. Geometrijos ir matavimų srityje nė vienoje klasėje nepavyko gauti išsamesnio gebėjimų aprašo: net 25-ių proc. grupės mokiniai iš jiems pasiūlytų uždavinių pajėgė atlikti tik pačius paprasčiausius, t. y. tokius, kuriuose buvo prašoma atpažinti plokštumos ar erdvės figūrą, įvardyti aplinkos daiktų formą, taikyti gretimų ilgio matavimo vienetų ryšius, skaityti paprastas skales. Kuriant abiejų mokinių grupių aprašus, kitiems teiginiams apie šių grupių mokinių gebėjimus patvirtinti ar paneigti tinkamų uždavinių pritrūko. Rezultatų palyginimas pagal kognityvinių gebėjimų grupes. Bendrojoje programoje mokinių gebėjimų reikalavimai aprašomi ne tik pagal veiklos sritis,

add and subtract decimal numbers, multiply and divide a decimal number by a one-digit number, compare common fractions with the same denominators and solve the simplest equations. They solve problems that have words that indirectly direct towards the operation or pupils have to make up to three steps to solve the problem. Pupils of the 15 per cent group single out from their peers by the inability to solve even the simplest problems that contain mathematical terms: “quotient”, “product”, “whole number”. They do not solve equations, do not understand what the expression is and how its value is computed, and are not able to compare fractions, though they already master to describe the simplest situations by common fractions. The eighth form pupils of the 25 per cent group are already able to: round numbers according to the indicated accuracy, add and subtract fractions with the same denominators, multiply and divide fractions, relate decimal and fractional forms of the simplest fractions, raise the number to a natural power and extract a square root of the number, apply features of the power with the natural index. They solve simple problems of finding percentages, proportions and unit, compute numerical values of simple literal expressions, solve simple equations. Pupils of the 15 per cent group differently from their peers are not able to apply the definitions of power and square root. The tenth form pupils of the 25 per cent group recognise, put down and represent in various ways small rational numbers, perceive links between numbers written in different ways, compute values of numerical expressions with powers and roots, in the simplest cases apply features of powers and roots, recognise parabola, apply the formula for finding answers of the simple quadratic equation, check whether the pair of numbers is the solution to the system of linear equations. We failed to identify more distinct differences in the tenth form between the abilities of our surveyed groups due to small number of selected problems. In the area of geometry and measuring we could not obtain a more exhaustive description of abilities in any of the forms: here even the pupils of the 25 per cent group were able to solve just the simplest problems; i.e., such problems where it was requested to recognise the figure of the plane or space or name the shape of surrounding objects, apply links between adjacent length measurement units, read simple scales. Creating descriptions for both distinguished pupils’ groups, we were short of problems suitable to confirm or deny other statements about abilities of pupils of these groups. Comparison of results by cognitive ability groups.

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika

bet ir pagal bendruosius dalyko gebėjimus: žinias ir jų supratimą, matematinį komunikavimą, matematinį mąstymą ir problemų sprendimą. Mokinių gebėjimai tirti pagal tris kognityvinių gebėjimų grupes, kurios iš esmės atspindi visus bendruosius dalyko gebėjimus, aprašytus Bendrojoje matematikos programoje [2, p. 792–794]. Kas buvo turima omeny, priskiriant atrinktus uždavinius vienai ar kitai kognityvinių gebėjimų grupei, pateikiama 6 lentelėje. Išanalizavus ir palyginus abiejų grupių mokinių gebėjimus, paaiškėjo, kad 15-os proc. grupės mokinių žinios akivaizdžiai silpnesnės nei 25-ių proc. grupės (pagal kognityvinių gebėjimų grupę ,,Žinios ir supratimas“). Gebėjimas taikyti turimas žinias, turint omenyje tas, kuriomis disponuoja atitinkamo amžiaus mokiniai, abiejose nagrinėtose grupėse panašūs. Šis pastebėjimas tinka visų amžiaus tarpsnių mokiniams. Tai rodo, kad žinių įgijimas ir jų taikymas – abipusis ir nedalomas procesas.

Requirements for pupils’ abilities in the General Curriculum of Mathematics are described not only by activity areas but also by general abilities of the subject: knowledge and its understanding, mathematical communication, mathematical thinking and solution of problems. In our work we investigated pupils’ abilities according to three groups of cognitive abilities which basically reflect all general abilities of the subject that are described in the General Curriculum of Mathematics [2, p. 792–794]. Table 6 presents what was born in mind, attributing selected problems to one or another group of cognitive abilities. Having analysed and compared pupils’ abilities of both our distinguished groups in this respect, it was found that knowledge of the 15 per cent pupils’ group was visibly weaker than of the 25 per cent pupils’ group (according to the group of cognitive abilities “Knowledge and understanding”). The abilities to apply possessed knowledge, bearing in mind knowledge that is possessed by pupils of the corresponding age, are similar in both analysed groups. This notice suits to pupils of all age groups. This demonstrates that acquisition and application of knowledge is a two-way and undivided process.

6 lentelė. Atrinktų uždavinių grupavimas pagal kognityvinius gebėjimus Table 6. Grouping of Selected Problems by Cognitive Abilities Gebėjimai pagal Bendrąją programą Abilities according to the General Curriculum

Trumpas aprašas Short Description

1. Žinios ir supratimas Knowledge and understanding

Pagrindinių matematinių sąvokų ir žymenų supratimas / vartojimas. Matematinių objektų atpažinimas. Standartinių procedūrų atlikimas ir standartinis jų derinimas. Informacijos iš diagramų, lentelių, grafikų atrinkimas. Paprastų skalių skaitymas. Objekto priskyrimas tam tikrai kategorijai pagal nurodytą požymį. Matavimo vienetų taikymas. Atsakymas į tiesioginius klausimus. Understanding/usage of main mathematical concepts and notation. Recognition of mathematical objects. Carrying out standard procedures and their standard adjustment. Selection of information from diagrams, tables, graphs. Reading of simple scales. Attribution of the object to a certain category according to the indicated feature. Application of measurement units. The answer to direct questions.

2. Taikymas Application

Įprastų uždavinių sprendimas (panašių į tuos, su kuriais mokiniai, manoma, buvo susidūrę mokykloje). Uždaviniui spręsti tinkamos strategijos iš žinomų sprendimo strategijų ir būdų pasirinkimas. Matematinės informacijos ir duomenų pateikimas diagramomis, lentelėmis, schemomis ir graﬁkais. Objektų tarpusavio ryšių suvokimas ir taikymas. Tinkamo modelio uždaviniui spręsti sukūrimas. Uždavinio sprendimo pagrindimas ir užrašymas. Naudojimasis duomenimis iš schemų, lentelių, graﬁkų. Solution of usual problems (similar to those that are thought to have been encountered by pupils at school). Choice of a suitable strategy for solving the problem out of the known strategies and ways. Presentation of mathematical information and data in diagrams, tables, schemes and graphs. Perception and application of links between objects. Creation of a suitable model for solving the problem. Grounding and writing down the solution of the problem. Usage of data from schemes, tables and graphs.

Mokytojų ugdymas 2011

6 lentelės tęsinys Table 6 (continued) Gebėjimai pagal Bendrąją programą

Abilities according to the General Curriculum 3. Problemų sprendimas Solution of problems

Trumpas aprašas Short Description

Objektų ryšių matematinėse situacijose nustatymas, skirtingai pateiktų duomenų suderinimas. Proporcinio mąstymo taikymas. Objektų, situacijos išskaidymas į dalis. Pagrįstų išvadų darymas. Įvairių matematinių procedūrų ir idėjų taikymas, derinimas nepažįstamame arba sudėtingame kontekste. Apibendrinimų formulavimas, dėsningumų nustatymas. Objekto priskyrimas tam tikrai grupei. Problemų sprendimo strategijų taikymas nestandartinėse situacijose. Identification of links between objects in mathematical situations, adjustment of differently given data. Application of proportional thinking. Division of objects and the situation into parts. Making grounded conclusions. Application of various mathematical procedures and ideas and adjustment in an unfamiliar or complicated context. Formulation of generalisations, identification of regularities. Attribution of the object to a certain group. Application of strategies for solving problems in non-standard situations.

Apibendrinus tirtų mokinių grupių Taikymo gebėjimus, galima daryti išvadą, kad IV–VI klasių mokiniai geba išspręsti iki 2 žingsnių, o VIII–X klasių – iki 4 žingsnių sudėtingumo uždavinius. Nustatyti tokie IV–X klasių mokinių, kurių matematiniai gebėjimai yra menki, sprendžiamų uždavinių požymiai: • uždavinio pateikimo forma yra mokiniui įprasta (tokia kaip daugumos uždavinių vadovėliuose); • duomenys – nedideli, ,,patogūs“ natūralieji skaičiai; • informacija, reikalinga uždaviniui išspręsti, tiesiogiai pateikta uždavinio sąlygoje; • uždavinio kontekstas arba praktinis ir artimas mokinių patirčiai, arba įprastas matematinis (dažnai ugdymo praktikoje vartojamas), bet pats uždavinys yra labai paprastas (vieno žingsnio); • klausimai tiesioginiai, reikalaujantys padaryti logines išvadas. Silpnų mokinių Problemų sprendimo gebėjimų nepavyko aprašyti, nes nepakako jiems apibūdinti tinkamų uždavinių. Kartu svarbu pažymėti, kad abiejų grupių mokiniai neįstengė įveikti net ir paprastų uždavinių, kuriuos sprendžiant reikėjo turimas žinias pritaikyti mokiniui neįprastame kontekste, todėl galima teigti, kad Problemų sprendimo gebėjimais, aprašytais pagal pateiktus požymius (žr. 6 lent.), silpnai matematiką išmanantys mokiniai nepasižymi. Mokinių nuostatos. Bendrojoje programoje aprašyti ne tik mokinių žinių ir gebėjimų, bet ir nuostatų reikalavimai, o ši trijų dedamųjų visuma įvardijama kaip mokinių matematinė kompetencija (Sičiūnienė, Dargytė, 2010, p. 792). Nuostatos dėl matematikos Programoje aprašytos 4.6.5. pastraipoje Teigiama, kad jos yra pozityvios, jeigu mokinys domisi matematika,

Having generalised “Application” abilities of investigated pupils’ groups, it can be concluded that pupils of the 4 - 6 forms are able to solve problems requiring up to two steps, whilst of the 8 - 10 forms, up to four steps. The following features of problems solved by pupils of the 4 - 10 forms with low abilities in mathematics were identified: • the form of presenting the problem to the pupil is usual (as in the majority of textbooks of problems); • data are not large and natural numbers are “convenient”; • information necessary for solution of the problem is directly given in the condition of the problem; • the context of the problem is either practical and close to the pupils’ experience or usual mathematical (often used in the educational practice) but the very problem is very simple (one-step) ; • questions are direct, requiring to make simplest logical conclusions. We failed to describe the abilities of “Solution of problems” of weak pupils due to insufficient number of problems suitable for them. It is also important to note that pupils of our both distinguished groups did not succeed to cope with even simple problems, the solution of which required application of possessed knowledge in the unusual context; therefore, it can be stated that pupils weak in mathematics cannot be characterised by the abilities of “Solution of problems”, described according to the features given by us (see Table 6). Pupils’ attitudes. The General Curriculum of Mathematics describes requirements for both pupils’ knowledge, abilities and their attitudes, and this whole of three constituents is called pupils’ mathematical

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika

62,0

X klasơ / form

20,9 53,0

VIII klasơ / form

32,8 67,6

VI klasơ / form

56,3 86,9

IV klasơ / form

74,4

20%

40%

60%

80%

100%

Kiti mokiniai / Other pupils Žemǐ matematiniǐ gebơjimǐ mokiniai (25 proc.) / Pupils with low abilities in mathematics (25 per cent)

Pav. 1. Mokinių, pritariančių teiginiams, dalis, proc. Fig. 1. Share of Pupils Approving the Statements (per cent)

Tu esi gabus matematikai You Are Gifted in Mathematics X klasơ / form VIII klasơ / form VI klasơ / form

Matematikos mokytis Ƴdomu It Is Interesting to Learn Mathematics

54,4

14,4

X klasơ / form

41,5

9,3 27,4

IV klasơ / form

VIII klasơ / form

47,5

72,8

Tu mơgsti matematikos pamokas You Like Mathematics Lessons X klasơ / form VIII klasơ / form VI klasơ / form IV klasơ / form

54,8

21,1 33,9

39,5

Tau patinka sprĊsti matematikos uždavinius You Enjoy Solving Mathematics Problems

VIII klasơ / form

60,2 53,2

86,2 78,5

IV klasơ / form

X klasơ / form

50,2

20,8 31,3

83,7

20% 40% 60% 80% 100%

- kiti mokiniai / other pupils - žemǐ matematiniǐ gebơjimǐ mokiniai (25 proc.) / pupils with low abilities in mathematics(25 per cent) Pav. 2. Mokinių, pritariančių teiginiams, dalis, proc. Fig. 2. Share of Pupils Approving the Statements (per cent)

48,5

82,2 72,1

IV klasơ / form 0%

56,2

58,9 55,3

VI klasơ / form 72,1

54,6 64,7 59,0

VI klasơ / form

52,4

59,4

30,5

Mokytojų ugdymas 2011

aktyviai dalyvauja mokymosi procese, pasitiki savo jėgomis, siūlo originalias idėjas ir jų įgyvendinimo būdus, jaučia atsakomybę už savo ir kitų daromą pažangą, noriai padeda kitiems mokytis bei vertina įgyjamas matematikos žinias ir gebėjimus [2, p. 794]. Siekiant įvertinti silpnų gebėjimų mokinių nuostatas, buvo išnagrinėta jų atsakymų į anketos klausimus statistika. Paaiškėjo, kad 15-os proc. grupės mokinių nuostatos, kaip ir buvo tikėtasi, yra menkesnės negu 25-ių proc. grupės mokinių, tačiau šie skirtumai nėra dideli. Norint įvertinti prastų matematinių gebėjimų mokinių nuostatas, 25-ių proc. grupės mokinių nuostatos buvo palygintos su kitų tyrime dalyvavusių mokinių nuostatomis. Paaiškėjo, kad visose amžiaus grupėse silpnų matematinių gebėjimų ir kitų mokinių nuostatų skirtumai ženklūs ir statistikai reikšmingi (žr. 1 pav. ir 2 pav.). Taip pat svarbu pažymėti, kad menkų matematinių gebėjimų ir kitų mokinių nuostatų skirtumai su mokinių amžiumi tik didėja: IV klasėje matematika patiko 74 proc. silpnų gebėjimų mokinių (87 proc. likusių mokinių), o X klasėje matematika patiko tik 21 proc. silpnų gebėjimų mokinių (62 proc. likusių mokinių). Daugiau nei pusei prastų matematinių gebėjimų mokinių, besimokančių pagrindinėje mokykloje, matematika nebepatinka. Tai rodo, kad silpnų matematinių gebėjimų mokinių ugdymo praktikoje esama labai rimtų spragų ir problemų. Mokinių atsakymai ir į kitus su mokinių nuostatomis susijusius klausimus šią išvadą patvirtina (žr. 2 pav.).

Išvados • Pritaikius IRT modelį iš 2005–2008 m. nacionalinių tyrimų užduočių banko pavyko atrinkti 132 užduotis, identifikuojančias 15-os proc. grupės mokinių silpniausius matematinius gebėjimus, ir 202 užduotis, identifikuojančias 25-ių proc. grupės mokinių silpniausius matematinius gebėjimus IV, VI, VIII ir X klasėse. • Remiantis atrinktomis užduotimis, sukurtos aprėptys įgalino apibūdinti tirtų mokinių kognityvinius gebėjimus pagal Žinias ir supratimą ir Taikymus, o Problemų sprendimo gebėjimams apibūdinti tinkamų užduočių nepakako. • Nustatyta, jog 15-os proc. grupės mokiniai kur kas prasčiau nei 25-ių proc. grupės mokiniai įsisavinę įvairius matematikos terminus ir žymenis, tačiau ,,Taikymo“ grupės uždavinius abiejų

competency [3, p. 792]. Attitudes towards mathematics in the curriculum are described in paragraph No. 4.6.5. It is stated that they are positive if the pupil is interested in mathematics, takes active part in the learning process, is self-reliant in mathematics, proposes original. ideas and ways of their implementation, feels responsibility for his/her and other pupils’ progress, willingly helps others to learn and appreciates acquired knowledge and abilities of mathematics [2, p. 794]. In order to evaluate weak pupils’ attitudes, statistics of their answers to questionnaire questions was analysed. It was found that attitudes of the 15 per cent pupils’ group as expected were lower than of the pupils of the 25 per cent group but these differences are not big. In order to evaluate attitudes of pupils with low abilities in mathematics, the attitudes of the 25 per cent group pupils were compared with the attitudes of other pupils who took part in the research. It was found that in all age groups differences in attitudes of pupils with low abilities in mathematics and other pupils were considerable and statistically significant (see Fig. 1 and Fig. 2). It is also important to note that differences in attitudes of pupils with low abilities in mathematics and other pupils increase with age: if in the 4 form 74 per cent of weak pupils liked mathematics (87 per cent of the remaining pupils), in the 10 form only 21 per cent of weak pupils liked mathematics (62 per cent of the remaining pupils). More than half of pupils with low abilities in mathematics learning in the basic school stop liking mathematics. This demonstrates that the practice of educating pupils with low abilities in mathematics has many serious gaps and problems. Pupils’ answers to other questions related to pupils’ attitudes only confirm this conclusion (see Fig. 2).

Conclusions

• Having applied the MTT model, from the task bank of national researches 2005-2008 we succeeded to select 132 tasks identifying the abilities of 15 per cent of the weakest pupils in mathematics and 202 tasks identifying the abilities of 25 per cent of the weakest pupils in mathematics in the 4, 6, 8 and 10 forms. • Based on the selected tasks, formed coverages enabled to describe cognitive abilities of investigated pupils according to “Knowledge and understanding” and “Applications”, whilst there was an insufficient number of tasks suitable for describing abilities of “Solution of problems”. • It was identified that pupils of the 15 per cent group considerably more poorly than pupils of the 25 per cent group have mastered

Viktorija SIČIŪNIENĖ, Janina DARGYTĖ Silpnai besimokančiųjų bendrojo lavinimo mokyklos mokinių matematinių gebėjimų diagnostika

mokinių grupių mokiniai sprendžia panašiai. • Nustatyti tokie žemų matematinių gebėjimų mokinių sprendžiamų uždavinių požymiai: duomenys – maži, ,,patogūs“ skaičiai, reikalinga uždaviniui išspręsti informacija tiesiogiai pateikta uždavinio sąlygoje, kontekstas įprastas, artimas mokinių patirčiai, klausimai tiesioginiai, reikalaujantys padaryti logines išvadas. Taip pat nustatyta, kad aukštesniųjų klasių mokiniai geba spręsti sudėtingesnius, bet visais minėtais požymiais pasižyminčius uždavinius. • Didžiausius 15-os ir 25-ių proc. grupių gebėjimų skirtumus pavyko nustatyti skaičiavimo ir algebros srityse: 15-os proc. grupės mokiniai ypač sunkiai įsisavina paprastąsias trupmenas, laipsnius ir lygčių sprendimą. Geometrijos ir matavimo srities gebėjimų išsamiau nepavyko aprašyti, nes nepakako jiems apibūdinti tinkamų užduočių. • Ir 15-os proc. grupės, ir 25-ių proc. grupės mokiniai turi nemažai vienodai išlavintų gebėjimų, tačiau kiekvienoje amžiaus grupėje yra ir tokių gebėjimų, kuriais nepasižymi 15-os proc. grupės mokiniai. Pastarieji trūkstamus gebėjimus vis dėl to įgyja po 1–2 metų, t. y. kai pereina į kitą amžiaus grupę. • Nustatyta, kad kuo silpnesni mokinių matematiniai gebėjimai, tuo jų nuostatos matematikos atžvilgiu žemesnės. • Menkų matematinių gebėjimų mokinių nuostatos dėl matematikos dalyko, jiems besimokant pagrindinėje mokykloje, pozityvumo sparčiai mažėja. • Apibendrinus visus gautus rezultatus galima teigti, kad silpnų matematinių gebėjimų mokinių ugdymo praktikoje esama išties rimtų spragų ir problemų. Atlikto tyrimo rezultatai gali būti panaudoti ne tik planuojant tolesnius mokinių pasiekimų tyrimus, tobulinant esamas programas ir vadovėlius, bet ir apmąstant, kaip galima būtų patobulinti mokinių matematinį ugdymą visais lygmenimis, turint omeny didį tikslą – padėti mokiniui ne tik įgyti matematikos žinių ir gebėjimų, bet formuoti į mokymosi sėkmę ir matematikos mokymosi prasmingumą orientuotas nuostatas ir bendruosius ugdymo tikslus atitinkančią vertybių sistemą [2, p. 791].

various mathematical terms and notation, but the problems of the “Application” group are similarly solved by pupils of both groups. • The following features of problems solved by pupils with low abilities in mathematics have been identified: small, “convenient” numbers, information necessary for solution of the problem is directly given in the condition of the problem, the context is usual and close to pupils’ experience, questions are direct, requiring to make the simplest logical conclusions. It has also been identified that pupils of higher forms are able to solve problems that are more complex but have all said features. • The largest differences of abilities in the 15 and 25 per cent group were identified in the areas of computation and algebra: pupils of the 15 per cent group particularly hardly master common fractions, powers and solution of equations. We have not succeeded to describe the abilities in the areas of geometry and measuring more comprehensively because there were not enough problems that were suitable for their description. • Both the pupils of the 15 per cent group and of the 25 per cent group have quite many equally developed abilities but in every age group such abilities were distinguished which are not characteristic to the pupils of the 15 per cent group. The latter acquire the missing abilities after 1 - 2 years; i.e., when they pass to another age group. • It has been identified that the lower the pupils’ mathematic abilities the lower their attitudes towards mathematics. • Positiveness of attitudes of pupils with low mathematic abilities towards mathematics while they are learning at the basic school is quickly decreasing. • Having generalised all obtained results, it can be stated that there are really serious gaps and problems in the practice of educating pupils who are weak in mathematics. The results of the research can be used not only planning further researches on pupils’ achievements, improving existing curricula and textbooks but also considering how pupils’ mathematic education could be improved at various levels, bearing in mind the big aim that we have set for ourselves: to help the pupil both to acquire knowledge and abilities of mathematics and form attitudes oriented to success of learning and meaningfulness of learning mathematics as well as the system of values corresponding to general aims of education [2, p. 791].

Mokytojų ugdymas 2011

Literatūra

Apie atnaujintų Pradinio ir pagrindinio ugdymo bendrųjų programų įgyvendinimą. (2008). Vilnius: Švietimo aprūpinimo centras. Pradinio ir pagrindinio ugdymo bendrosios programos. (2008). Patvirtinta Lietuvos Respublikos švietimo ir mokslo ministro 2008 m. rugpjūčio 26 d. įsakymu Nr. ISAK-2433. Sičiūnienė, V., Dargytė, J. (2010). Teaching mathematics to gifted learners in Lithuanian basic school: the analysis of the

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References

situation and prospects. Teaching mathematics: retrospective and perspectives: 11th international conference Proceedings (p. 83–91). Latvia: Daugavpils. Nacionaliniai mokinių pasiekimų tyrimai. www.pedagogika. lt/index.php?628923849 (.) http://www.r-project.org/ (Statistinės programos R internetinis puslapis.)

Viktorija SIČIŪNIENĖ Socialinių mokslų (edukologijos) daktarė, Vilniaus pedagoginio universiteto Matematikos didaktikos katedros docentė. Moksliniai interesai: matematikos didaktika, mokinių vertinimo technologijos, švietimo stebėsena.

Doctor of Social Sciences (Education Studies). Associate Professor of Department of Mathematics Didactics of Vilnius Pedagogical University. Research interests: didactics of mathematics, pupils’ assessment technologies, monitoring of education.

Address: Studentų Str. 39, LT-08106 Vilnius, Lithuania E-mail: viktorija.siciuniene@vpu.lt

Janina DARGYTĖ Nacionalinio egzaminų centro mokinių pasiekimų tyrimų ir analizės skyriaus metodininkė. Moksliniai interesai: kiekybiniai tyrimai ir jų taikymas.

Teacher methodologist of the Department of Student Achievement REsearch and Analysis of the National Examination Centre. Research interests: Quantitative researches and their application.

Address: M. Katkaus Str. 44, LT-09217 Vilnius, Lithuania E-mail: janina.dargyte@nec.lt

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 77–86 Teacher Education. 2011. No. 16 (1), 77–86

Tatjana Bakanovienė Šiaulių universitetas • Šiauliai University

MATEMATIKAI GABIŲ VAIKŲ CHARAKTERINGŲ SAVYBIŲ RAIŠKA PAMOKOJE: PEDAGOGŲ PATIRTIES ANALIZĖ

Anotacija Straipsnyje pristatomi kokybinio tyrimo, kurio metu buvo siekiama analizuojant pedagogų matematikai gabių vaikų ugdymo patirtį, atskleisti šiems vaikams charakteringų savybių raišką pamokoje, rezultatai. Per tyrimą pedagogai nurodė pagrindinę matematikai gabių vaikų atpažinimo procedūrą – stebėjimą, kurio metu stebima charakteringų savybių raiška. Informantai tyrimo metu teigė, kad matematikai gabiems vaikams būdingos įvairios savybės, susietos tiek su matematinės veiklos ypatumais, tiek su asmens savybėmis. Pagrindiniai žodžiai: matematikai gabūs vaikai, gabumai, atpažinimas. Įvadas Gabių vaikų ugdymas iki šiol yra aktualus mokslinių diskusijų objektas. Tiek Lietuvoje, tiek kitose užsienio šalyse šis klausimas nagrinėjamas įvairiais aspektais: socialiniu – pripažįstant, kad tai yra kiekvienos visuomenės turtas (Šimelionienė, �� 2008�� ), reikšmingas kiekvienai valstybei; pedagoginiu – pripažįstant gabių vaikų išskirtinius edukacinius poreikius, kurių tenkinimo bendrojo lavinimo mokyklose sistema iki šiol nepakankamai išnagrinėta; psichologiniu – pripažįstant, kad gabūs vaikai išsiskiria savo psichologiniais procesais (Лейтес, 1996) ir tai sąlygoja jų veiklos ypatumus. Šiame kontekste išryškėja ir pedagogų pasirengimo ugdyti gabius vaikus problema. N. Leites (Лейтес, 1996) teigimu, JAV apie 30 proc. iš mokyklų pašalintų už nepažangumą ar mokyklos nelankančių vaikų sudaro gabūs ir itin gabūs vaikai, kurie negavo atitinkamo ugdymo. Pasak E. Ščeblanovos (Щебланова, 2003), panašus tyrimo re-

Manifestation of Features Characteristic to Mathematically Gifted Children in the Lesson: Analysis of Teachers’ Experience Abstract

The article presents the results of the qualitative research. Analysing teachers’ experience of educating children gifted for mathematics, the research aimed to disclose the manifestation of these children’s characteristic features in the lesson. During the research teachers indicated the main procedure of recognising mathematically gifted children: observation, during which the manifestation of characteristic features is observed. During the research informants pointed out that mathematically gifted children are characterised by various features related both to peculiarities of mathematic activities and personal qualities. Key words: mathematically gifted children, talents, recognition. Introduction Up till now the problem of educating gifted children has been a relevant subject of scientific discussions. Both in Lithuania and other foreign countries the later problem is being analysed in various dimensions: social, acknowledging that this is the asset of every society (Šimelionienė, 2008), the potential of which is significant to every state; pedagogical, acknowledging such children’s exceptional educational needs, meeting of which in the system of comprehensive education has not been sufficiently analysed; and psychological, acknowledging that gifted children distinguish themselves by their psychological processes (Лейтес Н., 1996) and this determines the peculiarities of their activities. The problem of teachers’ preparation to educate gifted children actualises in this context as well. According to N. Leites (Лейтес Н., 1996), about 30 per cent of children suspended from school for underachievement or non-attendance in the USA are

Mokytojų ugdymas 2011

zultatai paskelbti ir Kanadoje, Vokietijoje, Vengrijoje ir kitose šalyse. Taip pat B. Graffam (2006), išanalizavęs Nacionalinės gabių vaikų asociacijos pateikiamas publikacijas, rado tik vieną šaltinį, kuriame analizuojamos gabaus vaiko mokytojo asmenybės savybės bei profesinės kompetencijos. Todėl būtini tyrimai, analizuojantys mokytojų pasirengimą gabių vaikų ugdymui. Vienas svarbiausių tyrimų klausimų, ar pedagogams yra būtinas specialus pasirengimas ugdyti tokius vaikus. Pedagogo pasirengimą kaip reikšmingą ugdant gabius vaikus nurodo ir S. Y. Lee ir kt. (2004), D. B. McCoach, D. Sietle (2007), kurie teigia, kad mokytojas, turintis specialųjį pasirengimą dirbti su gabiais vaikais arba darbo patirties su jais, turi palankesnį požiūrį į ugdymą, siekia jį tinkamai organizuoti ir labiau patenkinti mokinių edukacinius poreikius. Būtent pedagogo pasirengimą identifikuojant bei ugdant gabius vaikus kaip reikšmingą priežastį, dėl kurios pedagogai atsisako juos atpažinti ir ugdyti, nurodo ir S. Ojanen, J. Freeman (1994) (cit. Freeman, 2001). Pasak N. Leites (Лейтес, 1996), neturintis tinkamo pasirengimo pedagogas susiduria su tokiomis problemomis: • negeba atpažinti gabius vaikus, nežino jų ypatumų, savybių; • nepasirengę dirbti su gabiais vaikais mokytojai yra abejingi gabių vaikų problemoms (kartais jų tiesiog nesupranta); • kartais nepasirengę dirbti mokytojai yra priešiškai nusiteikę gabių vaikų atžvilgiu: jie gali kitų mokinių akivaizdoje pakenkti mokytojo autoritetui; • dažnai tokie mokytojai taiko kiekybinius gabaus vaiko darbo organizavimo metodus, mažiau dėmesio skirdami ugdymo proceso kokybei (t. y. gabaus vaiko ugdymą organizuoja didindami užduočių kiekį, bet neatsižvelgia į jų kokybę). Kitas reikšmingas pedagogų pasirengimo ugdyti gabius vaikus komponentas – dalykinis jų pasirengimas. A. Rudienės (2004), D. Genienės ir kt. (Генене, 2005) atlikti tyrimai rodo, kad dabartinių pradinių klasių matematinis raštingumas yra nepakankamas, todėl ir kyla klausimas, ar jie bus pajėgūs patenkinti gabių vaikų edukacinius poreikius. Daugelis tyrėjų (Narkevičienė, 2007; Van Tassel-Baska, 2007; Milgram, 1991 ir kt.) kaip reikšmingą efektyvaus gabaus vaiko savybę nurodo intelektualumą. Reikšminga ir tai, kad akcentuojami ir mokytojų profesinio pasirengimo reikalavimai, kurie nusako, kad mokytojas turi išmanyti gabių vaikų ypatu-

gifted or particularly gifted children who have not received corresponding education. According to E. Scheblanova (Щебланова, 2003), similar survey results were also announced in Canada, Germany, Hungary and other countries. Having analysed publications of the National Association for Gifted Children, B. Graffam (2006) also found only one source, which analyses personal qualities and professional competences of the gifted child’s teacher. Therefore, there is a need of researches analysing teachers’ preparation to educate gifted children. Analysing teachers’ preparation to educate gifted children, the following key question is given: Is special preparation necessary for teachers to educate such children? S. Y. Lee et al. (2004), D. B. McCoach, D. Sietle (2007) also indicate the teacher’s preparation as significant for organising efficient education of gifted children; they state that the teacher who has special preparation to work with gifted children or work experience with them has a more favourable approach towards their education, seeks to organise it appropriately and meet their educational needs better. S. Ojanen, J. Freeman (1994) (qtd. in Freeman, 2001) also state that namely the teacher’s preparation to identify and educate gifted children is a significant reason determining teachers’ refusal to recognise and educate them. According to N. Leites N. (Лейтес Н., 1996), the teacher who has no appropriate preparation encounters the following problems: • cannot recognise gifted children, does not know their peculiarities, features; • teachers who are not prepared to work with gifted children are indifferent to problems of gifted children (sometimes they simply do not understand them); • sometimes teachers who are not prepared to work with gifted children are hostile to gifted children: they can harm the teacher’s authority in other pupils’ presence; • often such teachers apply quantitative methods of organising the gifted child’s work, paying less attention to the quality of the educational process (i.e., they organise the gifted child’s education increasing the volume of tasks but disregarding their quality). Another significant component of teachers’ preparation to educate gifted children is their subject based preparation. Researches carried out by A. Rudienė (2004), D. Genienė et al. (Генене, 2005) indicate that mathematical literacy in today’s primary classes is insufficient; therefore, there arises a question if teachers will be capable to meet gifted children’s educational needs. Many researchers (Narkevičienė,

Tatjana Bakanovienė Matematikai gabių vaikų charakteringų savybių raiška pamokoje: pedagogų patirties analizė

mus ir mokėti juos atpažinti, organizuoti gabių vaikų ugdymo(-si) veiklą. Abi šios veiklos (ir atpažinimas, ir ugdymo(-si) organizavimas) nėra labai lengvos, dažnai mokytojai daro klaidų. Mokslininkų teigimu, išryškėja teorinių žinių ir atitinkamo praktinio patyrimo būtinybė (Narkevičienė, 2007 ). J. Van Tassel-Baska ir kt. (2007) nagrinėdami efektyvių gabių vaikų ugdymo programų pricipus nurodo, kad modelio efektyvumui, t. y. efektyviam įgyvendinimui įtakos turi mokytojo imlumas, jo gebėjimas įgyti žinių, būtinų organizuoti gabaus vaiko ugdymą. Teorinių žinių reikšmingumas vėl suponuoja mintį dėl profesinio pasirengimo dirbti su gabiais vaikais. Todėl žinios apie gabumus ir gabius vaikus turėtų būti svarbus pedagogų rengimo sistemos komponentas. Remiantis mokslinių šaltinių analize, pažymėtina, kad reikšminga sėkmingo gabių vaikų ugdymo sąlyga yra laikoma pastarųjų atpažinimas. Gabių vaikų atpažinimas, vadinamoji identifikacija, iki šiol yra aktualus mokslinių tyrimų objektas. Šio proceso problemiškumas pripažįstamas įvairiose srityse. Mokslininkai siekia išspręsti gabumų identifikavimui taikomų diagnostinių procedūrų, jos tikslų problemas. Pedagogams praktikams aktualu išspręsti gabių vaikų identifikavimo problemas, kuriant praktiškai realizuojamus modelius. Tačiau reikėtų pažymėti, kad dažnai mokslininkų identifikavimo problemos nagrinėjimas, neatitinka praktinių pedagogų poreikių. Kaip vieną pagrindinių gabių vaikų atpažinimo instrumentų pedagoginėje veikloje galima laikyti siūlomus gabiems vaikams būdingų savybių aprašus (Белошистая ir kt., 2005). Mokslinėje literatūroje gerai žinomas B. Clarko (1992) ir M. Seagoe (1972) gabiam vaikui būdingų bruožų (savybių) aprašas, kuriame nurodomos tokios savybės, kaip greitas informacijos išmokimas ir įsiminimas, smalsumas, intelektinių žinių troškimas, vidinė motyvacija ir kt. Taip pat mokslinėje literatūroje aptinkami ir savybių aprašai, kurie sudaryti atsižvelgiant į gabumų sritis. A. Matiuškin (Матюшкин, 2006) sudarydamas rekomendacijas tėvams bei mokytojams dėl gabių vaikų atpažinimo pateikia gabiam vaikui būdingų savybių sąrašą, suskirsto jas pagal gabumų sritis: kūrybinis potencialas, intelektinis potencialas, menininko, mokslininko, aktoriaus, techniko, sportininko, lyderio talentai. Remiantis mokslininko rekomendacijomis intelektiniam potencialui budingos tokios savybės: • Vaikas greitai išmoksta naują medžiagą. Greitai įsimena išgirstą ar perskaitytą informaciją. • Geba rasti skirtumus ir panašumus, nustato ryšius tarp priežasties ir pasekmės; geba suprasti

2007, Van Tassel-Baska, 2007; Milgram, 1991 et al.) indicate intellectuality as a significant feature of a gifted child. It is also significant that requirements for the teacher’s professional preparation are distinguished as well; they outline that the teacher has to know gifted children’s peculiarities and be able to recognise them, organize gifted children’s (self-)educational activities. Both of these activities (recognition and organisation of (self-)education) are not very easy, teachers often make mistakes. According to the researchers, the necessity of theoretical knowledge and corresponding practical experience shows up (Narkevičienė, 2007 ). Analysing the principles of gifted children’s curricula, J. Van Tassel-Baska et al. (2007) indicate that the efficiency of the model (i.e., effective implementation) is influenced by the teacher’s receptivity, his/her ability to acquire knowledge, necessary for organising the gifted child’s education. Significance of theoretical knowledge again presupposes the idea about professional preparation to work with gifted children. Therefore, knowledge about talents and gifted children should be an important component of the teacher training system. Based on the analysis of the above-mentioned scientific sources, it can be noticed that it is maintained that a significant condition of gifted children’s successful education is their recognition. Recognition of gifted children, the so-called identification, up till now remains a relevant subject of scientific researches. Problematicity of this process is acknowledged at various levels and areas. Researchers seek to solve problems of diagnostic procedures applied for identification of talents and their aims. Teachers practicians find it relevant to solve gifted children’s identification problems, creating practically implemented models. However, it is often the case that often identification problems analysed by researchers do not correspond to teachers’ practical needs to solve this problem. One of the key instruments for recognising gifted children, which is considered significant in teaching activities, is proposed descriptions of gifted children’s features (Белошистая et al., 2005). The description of gifted children’s characteristic features by B. Clark (1992) and M. Seagoe (1972) is well-known in scientific literature. It indicates such features as fast acquisition and memorising of information, inquisitiveness, pursuit for intellectual knowledge, internal motivation, etc. In scientific literature you can also find descriptions of features that are compiled considering the areas of talents. Compiling recommendations for parents and teachers on recognition of gifted children, Matiushkin (Матюшкин, 2006) provides a list of features characteristic to a gifted child and groups them by areas: creative potential, intellectual potential, the artist’s

Mokytojų ugdymas 2011

esmę, potekstę (tai, apie ką nesakoma, bet turima omenyje). • Lengvai išmoksta ir mėgsta skaityti. Turi turtingą žodyną ir lengvai vartoja pagal paskirtį naujus žodžius. • Daug išmano apie problemas pagal gabumų sitys bei įvykius, apie kurios jų bendraamžiai nežino arba nesupranta. • Aiškiai mąsto, samprotauja. Geba pagrįsti savo samprotavimus arba iliustruoti juos faktais ir pavyzdžiais. Matematikai gabiems vaikams būdingas savybes, kaip svarbius matematinių gabumų struktūros komponentus savo darbuose pateikia ir plačiai analizuoja V. Kruteckij (Крутецкий, 1968). Mokslininko nurodytos matematikai gabiems vaikams būdingos savybės pateikiamos 1 lentelėje.

talent, musical talent, scientific talent, literary talent, actor’s talent, technical talent, sport talent, leader’s talent. Based on the scientist’s recommendations, the following features are characteristic to intellectual potential: • The child learns new materials quickly. He/she quickly memorises heard or read information. • Is able to find differences and similarities, identifies links between the reason and the consequence; is able to understand the essence, implied sense (what is not said but is born in mind). • Easily learns and likes to read. Has rich vocabulary and finds it easy to use new words to the purpose. • Knows a lot about problems and events that are unknown or not understood by his/her peers. • Thinks and reasons clearly. Is able to ground his/her considerations or illustrate them by facts and examples. Features characteristic to mathematically gifted children, presenting them as components characteristic to the structure of mathematic talents, are presented and thoroughly analysed by V. Kruteckij (Крутецкий, 1968). The scientist’s indicated features characteristic to mathematically gifted children are given in Table 1.

1 lentelė. Matematinių gabumų struktūra Table 1. Structure of Mathematical Talents Uždavinio sprendimo etapas Stage of problem solving Matematinės informacijos gavimas Processing mathematical information

Komponento Nr. No. of component

Gebėjimai Abilities Gebėjimas formalizuotai suvokti matematinę medžiagą, formalios uždavinio struktūros suvokimas The ability to perceive mathematical materials in a formalized way; perception of the formal structure of the problem

Matematinės informacijos apdorojimas Processing mathematical information

Komponento tipas Type of component

Privalomi komponentai Obligatory components

Gebėjimas logiškai mąstyti. Gebėjimas mąstyti matematiniais simboliais The ability to think logically. The ability to think in mathematical symbols Gebėjimas greitai ir plačiai apibendrinti matematinius objektus, santykius ir veiksmus The ability to generalise mathematical subjects, relations and actions quickly and widely Gebėjimas „sutraukti“ matematinio samprotavimo procesą į atitinkamų veiksmų sistemą. Gebėjimas mąstyti „sutrauktomis“ struktūromis The ability to “contract” the process of mathematical reasoning into the system of corresponding operations. The ability to think in “contracted” structures. Mąstymo proceso ir matematinės veiklos lankstumas Flexibility of thinking process and of mathematical activity

Tatjana Bakanovienė Matematikai gabių vaikų charakteringų savybių raiška pamokoje: pedagogų patirties analizė

1 lentelės tęsinys Table 1 (Continued)

Matematinės informacijos apdorojimas Processing mathematical information

Komponento Nr. No. of component

Bendras sintetinis komponentas General synthetic component

10 11 12 13 14

Gebėjimai Abilities Uždavinio sprendimo racionalumo, aiškumo, paprastumo ir ekonomiškumo siekimas Pursuit of rationality, clearness, simplicity and economy of problem solving Gebėjimas greitai ir laisvai pakreipti mąstymo procesą, pereiti nuo tiesioginio prie atvirkštinio veiksmo ir atvirkščiai The ability to direct thinking process quickly and easily, pass from direct to reverse operation and vice versa Matematinė atmintis (apibendrinta atmintis matematiniams reiškiniams, tipinėms charakteristikoms, mąstymo schemoms, uždavinių sprendimo algoritmams) Mathematical memory (generalised memory for mathematical expressions, typical characteristics, thinking schemes, algorithms of problems solving) Matematinis mąstymo būdas Mathematical way of thinking

Matematinės informacijos saugojimas Storing mathematical information

Neutralūs komponentai Neutral components

Komponento tipas Type of component

Neprivalomi komponentai Non-obligatory components

Uždavinio sprendimo etapas Stage of problem solving

Mąstymo greitumas Quickness of thinking Skaičiavimo įgūdžiai (gebėjimas greitai ir tiksliai apskaičiuoti) Computation skills (the ability to calculate quickly and accurately) Gebėjimas atsiminti skaičius, formules The ability to remember numbers, formulas Gebėjimas konstruoti erdvinius vaizdinius The ability to construct spatial images Gebėjimas įsivaizduoti vaizdžiai abstrakčius matematinius santykius The ability to imagine abstract mathematical relations

V. Kruteckio (Крутецкий, 1968) teigimu, matematiniams gabumams būdingi privalomi ir neprivalomi komponentai, kurie nurodomi atsižvelgiant į uždavinių sprendimo etapus ir gali būti pastebimi pedagogams jų praktinėje veikloje. Tikslas – analizuojant pedagogų matematikai gabių vaikų ugdymo patirtį atskleisti tokių vaikų charakteringų savybių raišką pamokoje. Tyrimo organizavimas ir imtis Tyrimas buvo atliekamas taikant pusiau struktūruotą interviu su giluminio interviu elementais, t. y. siekiant atskleisti tyrimo dalyvio asmeninę poziciją, nuostatas, požiūrį (Bitinas, 2006). Per interviu pedagogams pateiktus klausimus galima suskirstyti į tokius diagnostinius konstruktus: informaciją apie profesinę veiklą, patirtį, išsilavinimą; matematikai gabių vaikų atpažinimo (identifikacijos) patirtį ir sistemą; ugdymo organizavimo patirtį (ugdymo technologijų pasirinkimas, ugdymo proceso valdymo ypatumai); mokytojų pasirengimą, jų didaktinius poreikius ugdant matematikai gabius vaikus;

According to V. Kruteckij (Крутецкий, 1968), mathematical talents are characterised by obligatory and non-obligatory components, which are distinguished depending on stages of problem solving and can be noticed by teachers in their practical activities. Aim – analysing teachers’ experience of educating mathematically gifted children, to disclose manifestation of features characteristic to mathematically gifted children in the lesson. Organisation of the Research and Sample The research was carried out applying a semistructured interview with elements of the in-depth interview; i.e., seeking to disclose personal standpoint, approaches, attitude of the research participant (Bitinas, 2006). Questions given to the teachers during the interview can be grouped into the following diagnostic constructs: information about professional activities, experience, education; experience and system of recognising (identifying) mathematically gifted children; experience of organising education of MGC (choice of education technologies, peculiarities of managing

Mokytojų ugdymas 2011

rekomendacijas ir patarimus įstaigoms, rengiančioms mokytojus arba teikiančioms kvalifikacijos kėlimo paslaugas. Šiame straipsnyje yra aptariamas tik matematikai gabių vaikų atpažinimo, atskleidžiant charakteringų savybių raišką, aspektas. Interviu dalyvių imtis sudaryta remiantis tyrimo tikslu ir dalyviams numatytais kriterijais (Rupšienė, 2007; Bitinas ir kt., 2008), t. y. taikant tikslinę kriterijumi grindžiamą atranką. Sudarant tyrimo imtį dalyviams buvo iškelti kriterijai: turi būti pradinės ar pagrindinės mokyklos mokytojas, turintis darbo su matematikai gabiais vaikais patirties (darbo patirtis su MGV buvo vertinima pagal tai, kiek kartų mokytojo ugdytiniai, dalyvavę 4–5 klasių šalies matematikos olimpiadose, tapdavo nugalėtojais. Siekiant išvengti atsitiktinio patekimo į imtį buvo remtasi kriterijumi, kad pedagogo ugdytiniai tapdavo nugalėtojais daugiau nei 5 kartus). Tyrimo duomenų analizė vykdoma remiantis turinio analizės (content) metodu.

the educational process); teachers’ preparation for education of MGC; teachers’ didactic needs educating MGC; recommendations and advice for institutions training teachers or providing in-service training services. This article deals only with the aspect of organising education for MGC. The sample of interview participants was drawn up based on the research aim and criteria established for participants (Rupšienė, 2007; Bitinas et al., 2008); i.e., applying targeted criteria-based selection. Drawing up the research sample, the participants had to meet the following criteria: they had to be teachers of a primary or a basic school with experience of teaching MGC. Work experience with MGC was evaluated according to the number of times when the teacher’s learners who took part in national mathematics olympiads of the 4-5 forms became winners. Seeking to avoid accidental getting into the sample, the selection was based on the criterion that the teacher’s learners used to become winners more than 5 times. The analysis of research data is carried out based on the method of content analysis.

Tyrimo rezultatai Per tyrimą buvo siekiama analizuojant pedagogų ugdymo patirtį atskleisti matematikai gabių vaikų charakteringų savybių raišką, kurių pagrindu vyksta jų atpažinimas. Informantai prieš pradėdami nusakyti matematikai gabiems vaikams būdingas savybes gana vieningai (net be prašymo) nurodė pagrindinį pastarųjų atpažinimo būdą – jų veiklos pamokoje stebėjimą („Man svarbiausia jo darbo per matematikos pamoką stebėjimas“ [Simona], ,,<...> man labai svarbus rodiklis yra vaiko darbas per pamoką“ [Daiva], ,,<...> pateikiu vaikui nestandartinį uždavinį ir žiūriu, kaip jis sprendžia“ [Vida]). Iš dalyvių teiginių matyti, kad gabių vaikų savybėms atpažinti būtinos tam tikros diagnostinės procedūros. Pedagogų išvardytas matematikai gabiems vaikams būdingas savybes galima suskirstyti į skirtingas grupes pagal jų prasmes: vienos jų nusako vaiko asmens savybes, kitos atspindi jų veiklos pamokoje ypatumus. Tačiau atsižvelgiant į matematinės veiklos specifiką nurodytos savybės išryškino dvi pagrindines kategorijas.

Research Results Analysing teachers’ educational experience, the research aimed to disclose manifestation of features characteristic to mathematically gifted children in the lesson, which is the basis of their recognition. Before starting to outline features characteristic to mathematically gifted children, the informants quite unanimously (even without a request) indicated the main way of recognising them: observation of their activity in the lesson (“As to me, it is most important for me to observe his work during lessons” [Simona], ,”<...> a very important indicator for me is the child’s work during the lesson” [Daiva], “,<...> I give a non-standard problem to the child and watch how he solves it” [Vida]). These statements imparted by the participants enable to notice that certain diagnostic procedures are necessary in order to recognise features of gifted children: they enable to highlight the latter. Teachers’ listed features characteristic of MGC can be grouped to different groups by their meanings: some of them outline the child’s personal features, others reflect peculiarities of their activities in the lesson. However, considering the specificity of mathematical activity, indicated features highlighted two main categories.

Tatjana Bakanovienė Matematikai gabių vaikų charakteringų savybių raiška pamokoje: pedagogų patirties analizė

2 lentelė. Kategorijos Matematikai gabaus vaiko savybės loginė schema Table 2. Logical Scheme of the Category Features of the Mathematically Gifted Child Subkategorijų pavadinimai Names of categories

Savybės, priskiriamos matematiniams gabumams Features that are attributed to mathematical talents

Savybės, priskiriamos bendriesiems gabumams Features that are attributed to general talents

Teiginių pavyzdžiai Examples of statements Greitai suvokia sąlygas, ypač tekstinių uždavinių. Geba greitai rasti sprendimus. Jie gali atlikti ne vieną veiksmą mintinai Gali tekste rasti loginius elementus Quickly perceive conditions, particularly of textual problems. Are able to find solutions quickly. They can perform several operations calculating mentally. Can find logical elements in the text. Domisi ir matematika, ir kitais panašiais dalykais. Pasižymi darbštumu. Kruopščiai atlieka užduotis. Are interested in mathematics and other similar things. Distinguish themselves as diligent children. They perform tasks accurately.

Subkategorija Savybės, priskiriamos matematiniams gabumams. Informantai apibūdindami matematikai gabiems vaikams būdingas savybes nurodė tokias, kaip greitumas, loginis mąstymas, netradiciniai uždavinių sprendimo metodai bei informacijos apdorojimo ypatumai. Reikėtų pažymėti, kad visi dalyviai nurodo vieną bendrą matematikai gabiam vaikui charakteringą bruožą – greitumą matematinėje veikloje. Vieni jų nusako, kad tokie vaikai „greitai suvokia sąlygas, ypač tekstinių uždavinių“ [Jurgita], ,,<...> greitai suvokia dėstomąją medžiagą“ [Simona], ,,<...> jie labai greitai pagauna idėją“ [Laura], ,,<...> aiškinant naują temą jie labai greitai perpranta tam tikras taisykles, algoritmus“ [Laura]. Kiti šią savybę sieja su vaikų praktiniais gebėjimais spręsti matematinius uždavinius: „Geba greitai rasti sprendimus“ [Jurgita], ,,<...> programinius uždavinius turėtų spręsti greičiau nei visa klasė“ [Janina], ,,<...> pasižymi greita orientacija bet kokioje situacijoje“, [Milda], ,,<…> ir šiaip dažniausiai greitai sprendžia uždavinius” [Laura], ,,<…> manau, labai svarbus spartumas, kaip greitai atlieka užduotis“ [Simona]. Dar viena daugumos dalyvių nurodytų savybių – loginis mąstymas ieškant uždavinių sprendimų arba juos pagrindžiant: „Geba tekste rasti loginius elementus“ [Jurgita], ,,<…> dar, manau, reikia pažymėti loginio mąstymo svarbą tokiam vaikui [Janina], ,,<...> tokie vaikai labai mėgsta loginius uždavinius ir teisingai juos išsprendžia“ [Janina], ,,<…> vaikui ypatingas loginis mąstymas kaip pagrindas” [Milda]. Taip pat, kaip nurodo D. Kiseliova, A. Kiseliovas (2004), matematikai gabūs vaikai dažnai pasirenka ne-

“Features attributed to mathematical talents”. Describing features characteristic to mathematically gifted children, informants indicated such features as quickness, logical thinking, non-traditional problem solving methods and information processing peculiarities. It should be noted that all participants indicate one common feature characteristic the mathematically gifted child: quickness in the mathematical activity. Some of them say that such children “perceive conditions quickly, particularly of textual problems” [Jurgita], ,”<...> quickly perceive delivered materials” [Simona], “<...> they grasp the idea very quickly” [Laura], ,”<...> explaining a new topic, they grasp certain rules, algorithms very quickly”[Laura]. Others relate this feature to children’s practical abilities to solve mathematical problems: “Are able to find solutions quickly”[Jurgita], ,”<...> should solve curricula problems quicker than the rest of the class” [Janina], “,<...> distinguish themselves by quick orientation in any situation”, [Milda], “<…> and otherwise most often solve problems quickly” [Laura], ,”<…> I think quickness is very important, how quickly they perform tasks” [Simona]. One more feature indicated by the majority of participants is logical thinking, looking for problem solutions and grounding them: “Are able to find logical elements in the text” [Jurgita], “,<…> again I think the importance of logical thinking for such child has to be noted” [Janina], “,<...> such children like logical problems very much and solve them correctly”[Janina], “,<…> the child has specific logical thinking as a basis”[Milda]. Also, as indicated by D. Kiseliova, A. Kiseliovas (2004), mathematically gifted children often choose

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tradicinius uždavinių sprendimų algoritmus. Dėl savo kognytivinės raidos ypatumų jie gali demonstruoti ypatingus sprendimo algoritmus. Pasak autoriaus, šabloniškų, tipinių uždavinių šie vaikai nemėgsta. Informantai taip pat nurodė šias savybes: „Pats mėgina kurti uždavinių sprendimo algoritmus“[Jurgita], ,,<...> tekstinius, probleminius ar loginius uždavinius sprendžia įdomiai” [Laura], ,,<…> dažnai netgi pasirenka labai netradicinius, bet teisingus uždavinių sprendimo metodus“ [Laura], ,,<...> tokie vaikai labai mėgsta ne programinius uždavinius, o tuos, kur reikia daugiau mąstyti [Daiva], ,,<...> „kartais net jų idėjos būna netradiciškos“ [Daiva]. V. Kruteckis (Крутецкий, 1968) apibūdindamas matematinių gabumų struktūrą nurodo tam tikrus matematikai gabių vaikų matematinės informacijos apdorojimo ypatumus. Tokius bruožus pažymėjo tyrime dalyvavę informantai, tačiau tai jie apibūdina skirtingais teiginiais. Vieni nusako operacijų ypatumus, susietus su gebėjimu operuoti matematine simbolika ir reiškiniais mintyse: „Jie labai daug ką gali atlikti mintinai ir dažnai tai būna ne vienas veiksmas“ [Daiva]. Kiti akcentuoja mąstymo išskirtinumą: ,,<…> mokėjimas nuosekliai mąstyti“ [Simona], ,,<...> pirmiausia atkreipiu dėmesį į vaiko mąstymą, kaip jis samprotauja, kaip dėlioja savo mintis“ [Vida], ,,<...> šiaip galima sakyti, kad jų darbas skiriasi nuo kitų vaikų, dažnai jie galvoja visai kitaip nei kiti vaikai“, „kitas vaikas niekada, atrodo, nesugalvotų taip išspręsti, o jie išsprendžia ir dar moka paaiškinti, kodėl taip padarė“[Daiva]. Subkategorija Savybės, priskirtos bendriesiems gabumams. Apibendrinant gabių vaikų išskirtinius bruožus buvo pastebėta, kad nors matematiniai gabumai ir priskiriami prie specifinių gabumų, tačiau matematikai gabūs vaikai gali pasižymėti ir bendriesiems gabumams būdingomis savybėmis (Clarko, 1992 ir Seagoe, 1972, cit. Webb, 2000; Wetty cit. Jakavičius, 1996, �� Матюшкин, 2006�� ir kt.�� ). Tyrime dalyvavę mokytojai laikosi panašaus požiūrio ir kaip būdingas savybes pripažino domėjimąsi matematika, darbštumą, kruopštumą. Šias savybes pedagogai apibūdina tokiais teiginiais: „Jie labai demonstruoja savo susižavėjimą matematiniais uždaviniais ir jiems kuo sunkesni uždaviniai, tuo įdomiau“ [Simona], ,,<...> domėjimasis ir matematika, ir kitais panašiais dalykais“ [Milda], ,,<...> ir dar jie turi būti suinteresuoti matematika“ [Laura], ,,<...> jeigu tik turi norą, jeigu jiems tai patinka, tai ir dirba“ [Laura]. Kalbant apie domėjimąsi dalyku, vertėtų pažymėti, kad dalyviai kaip savybę, būdingą gabiam vaikui, priskiria neįprastą darbštumą, kruopštumą matematinėje veikloje. Informantai tai

non-traditional algorithms for problem solving. Due to peculiarities of their cognitive activities they can demonstrate particular solution algorithms. According to the author, these children do not like stereotyped typical problems. Informants also indicated the following features: “he tries to create algorithms of problem solution himself” [Jurgita], “,<...> solves textual, problematic or logical problems in an interesting way”[Laura], ,”<…> often even chooses very non-traditional but correct methods of problem solving” [Laura], ,”<...> such children love not curriculum problems but the ones where they have to think more” [Daiva], “,<...> sometimes even their ideas are non-traditional” [Daiva]. Describing the structure of mathematical abilities, V. Kruteckij (Крутецкий, 1968) indicates certain peculiarities of processing mathematical information of mathematically gifted children. Such features were noted by informants who attended the research but they describe this using different statements. Some of them outline peculiarities of operations, related to the ability to operate mathematic symbols and expressions mentally: “they can do many things mentally and often more than one operation” [Daiva]. Others emphasize exceptionality of thinking: “,<…> the ability to think consistently” [Simona], ,”<...> first of all I pay attention to the child’s thinking, how he reasons and sets his thoughts” [Vida], “,<...> otherwise we can say that their work differs from other children’s; they often think completely differently than other children”, “it seems that another child would never think of solving this way and they solve and even are able to explain why they did so” [Daiva]. Subcategory “Features attributed to general talents”. Su sum up exceptional features of gifted children, it was noticed that although mathematical talents are attributed to specific talents, mathematically gifted children can distinguish themselves by features that are characteristic to general talents (B. Clark, 1992 and M. Seagoe, 1972, qtd. in Webb, 2000; P. Wetty qtd. in Jakavičius 1996, Матюшкин, 2006 et al.). Teachers who took part in the research follow a similar approach and acknowledged interest in mathematics, diligence and accuracy as characteristic features. Teachers describe these features by the following statements: “They demonstrate their fancy for mathematical problems a lot and the more difficult problems are, the more interesting it is for them”[Simona], ,”<...> interest in both mathematics and other similar subjects” [Milda], “,<...> and also they have to be interested in mathematics” [Laura], “,<...> if they only have a wish, if the enjoy it, they work” [Laura]. Speaking about interest in the subject, it is worth noting that the participants attribute unusual diligence, accuracy in

Tatjana Bakanovienė Matematikai gabių vaikų charakteringų savybių raiška pamokoje: pedagogų patirties analizė

apibūdina tokiais teiginiais: „Gabiam vaikui dar yra būtinas darbštumas, man atrodo, kad kokie 50 proc. sėkmės būtent tai ir nulemia“, „ir labai svarbus toks darbštumas, kuris leidžia kruopščiai dirbti ir pritaikyti tai, kas jau yra žinoma [Vida], ,,<...> pasižymi jie ir darbštumu...“ [Laura], ,,<...> na ir dar čia labai svarbus vaiko darbštumas, kruopštumas“ [Janina], ,,<...> darbštumas yra labai svarbus“ [Milda].

mathematical activities as a feature characteristic to a gifted child. Informants describe this by the following statements: “Diligence is also necessary for the gifted child; I think namely this determines some 50 per cent of success”, “and such diligence is very important, which enables to work accurately and apply what is already known” [Vida], ,”<...> they also distinguish themselves by diligence…”[Laura], ,”<...> well and also here the child’s diligence, accuracy are very important” [Janina], ,”<...> diligence is very important” [Milda].

Išvados • Dažniausiai pedagogai atpažindami matematikai gabių vaikų charakteringas savybes naudojasi praktinės veiklos pamokoje stebėjimu kaip jų identifikavimo procedūra. • Pedagogai matematikai gabiems vaikams priskiria įvairias savybes, kurias galima suskirstyti į dvi grupes: 1) priklausančias matematinių gabumų struktūrai ir 2) priklausančias bendrųjų gabumų struktūrai. Kaip reikšmingos laikomos savybės, atspindinčios matematinės veiklos specifiškumą – tai greitumas matematinėje veikloje, o iš bendrųjų savybių paminėtinas darbštumas. • Šio tyrimo rezultatai leidžia teigti, kad dauguma pedagogų matematinius gabumus laiko specifiniais gabumais, tačiau jiems būdingi bendrųjų gabumų bruožai.

Conclusions • Teachers most often use observation of practical activities in the lesson as an identification procedure for recognising features characteristic to mathematically gifted children. • Teachers attribute various features to mathematically gifted children, which can be grouped in two groups: 1) belonging to the structure of mathematic abilities and 2) belonging to the structure of general abilities. It is maintained that significant features are the ones reflecting the specificity of mathematical activities – quickness in mathematical activities – and diligence, distinguished from general features. • The results of this research enable to state that the majority of teachers consider mathematical talents as specific talents but mathematically gifted children are also characterised by features of general talents.

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Century. Gifted Child Quarterly , 51, 182–200 [Žiūrėta 2008 m. liepos 29 d.] Prieiga per internetą: <http://gcq. sagepub.com/cgi/content/refs/51/2/182>. Webb, J. T. (2000). Misdiagmosis and dual diagnosis of the gifted children. Americal Psychological Association Annual Convention. Washington [Žiūrėta 2008 m. rugsėjo 20 d.] Prieiga per internetą: <www.gifted.org/articles_conseling/ Webb_MisdiagnosisAndDualDiagnosisofGiftedChildren. pdf>. Генене, Д., Киселева, Д., Киселев, A. (2005). Математические достижения будущих учителей начальной школы: tarptautinės mokslinės konferencijos medžiaga (76–84). Liepāja : Liepājas Pedagogijas akadēmija.

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TATJANA bAKANOVIENĖ Socialinių mokslų (edukologijos) daktarė, Šiaulių universiteto Ugdymo sistemų katedros lektorė. Moksliniai interesai: gabių vaikų ugdymas, informacinės komunikacinės technologijos ugdymo procese.

Doctor of Social Sciences (Education Studies), Lector of the Department of Education Systems of Šiauliai University. Research interests: education of gifted children, information comunication technologies in education.

Address: P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania E-mail: atani78@yahoo.com

AKTUALIJOS TOPICALITIES

Mokytojų ugdymas 2011

ISNN 1822−119X

Mokytojų ugdymas. 2011. Nr. 16 (1), 88–92 Teacher Education. 2011. Nr. 16 (1), 88–92

Sigita TURSKIENĖ Šiaulių universitetas • Šiauliai University

MATEMATIKOS MOKYMO IR MOKYTOJŲ RENGIMO IDĖJŲ SKLAIDA BEI PATIRTIS TARPTAUTINĖJE BALTIJOS ŠALIŲ KONFERENCIJOJE „MATEMATIKOS MOKYMAS: RETROSPEKTYVA IR PERSPEKTYVOS“

Matematikos mokymo ir matematikos mokytojų rengimas, jų tobulinimo galimybės bei kaita kasmet analizuojami Baltijos šalių tarptautinėje mokslinėje konferencijoje. Su praėjusių konferencijų tikslais, programomis, nuotraukomis galima susipažinti internete adresu: http://math.distance.su.lt/. 2011 m. gegužės 5–6 d. Šiaulių universitete buvo organizuota konferencija Matematikos mokymas: retrospektyva ir perspektyvos (Teaching mathematics: Retrospective and Perspectives). Konferenciją organizavo Šiaulių universiteto Nuotolinių studijų centras, Edukologijos fakultetas bei Matematikos ir informatikos fakultetas. Šios konferencijos sudėtinės dalys buvo šios mokytojų ir mokinių metodinės-praktinės konferencijos: • Respublikinė mokslinė, metodinė-praktinė matematikos mokytojų konferencija Matematinės edukacijos realijos ir perspektyvos nūdienos mokykloje, organizuota Šiaulių Romuvos gimnazijoje. • Kasmetinė respublikinė 11-oji moksleivių matematikos, informacinių technologijų ir ekonomikos darbai bei projektai moksleivių konferencija, organizuota Šiaulių m. S. Daukanto gimnazijoje. Konferencijoje pranešimus skaitė dėstytojai, mokslininkai, tyrėjai, matematikos mokytojai iš Lietuvos, Latvijos, Estijos, Suomijos.

Spread and Experience of Teaching Mathematics and Teacher Training Ideas at the International Conference of Baltic Countries “Teaching Mathematics: Retrospect and Perspectives” Teaching mathematics and training mathematics teachers, improvement possibilities and changes are annually discussed at the international scientific conference of Baltic countries. Aims, programmes and photos of previous conferences are available at the address: http://math.distance.su.lt/. On 5-6 May 2011 the conference “Teaching Mathematics: Retrospect and Perspectives” was organised at Šiauliai university. The conference was organised by the Distance Studies Centre, the Faculty of Education Studies and the Faculty of Mathematics and Informatics of Šiauliai University. This conference consisted of the following teachers’ and pupils’ methodological-practical conferences: • National scientific, methodological-practical mathematics teachers’ conference “Realia of Teaching Mathematics and Perspectives in the Contemporary School” was organised in Šiauliai Romuva gymnasium. • Annual national pupils’ conference “11 Pupils’ Works and Projects in Mathematics, Information Technologies and Economics” was organised in Šiauliai S. Daukantas’ gymnasium. Papers at the conference were read by university teachers, scientists, researchers, mathematics teachers from Lithuania, Latvia, Estonia and Finland. Scientific discussions took place in the plenary meeting, teachers’ conference and 6 sections according to different problem areas: information

Sigita TURSKIENĖ Matematikos mokymo ir mokytojų rengimo idėjų sklaida bei patirtis tarptautinėje baltijos šalių konferencijoje „matematikos mokymas: retrospektyva ir perspektyvos“

Mokslinės diskusijos vyko plenariniame posėdyje, mokytojų konferencijoje ir 6 sekcijose pagal skirtingas problemines sritis: informacinės ir komunikacinės technologijos mokant matematikos, matematikos mokymas aukštojoje mokykloje, matematikos mokytojų rengimas, matematikos mokytojų kompetencija, baigiamieji matematikos egzaminai ir matematikos olimpiados, matematikos mokymas mokykloje. Plenariniame konferencijos posėdyje pagrindinis dėmesys buvo skiriamas Lietuvos, Latvijos ir Estijos pagrindiniams pasiekimams ir problemoms matematikos mokytojų rengime aptarti. Pranešime Šiaulių universitetas ir mokyklinės matematikos literatūra doc. A. Bakštys bei doc. R. Macaitienė apibendrino ir susistemino Šiaulių universiteto mokslininkų indėlį į mokyklinės matematikos metodinės literatūros rengimą Lietuvoje. Šiaulių universitete matematikos metodikos pradininkais buvo doc. Juozas Revuckas (1923–1991) ir prof. Bronius Balčytis, doc. doc. Vaclovas Viruišis. Aktyvūs šioje srityje ir doc. A. Bakštys, prof. A. Kiseliovas. Prof. A. Ažubalis plačiai pristatė prof. B. Balčyčio mokslinius pasiekimus matematikos mokymo srityje. Prof .B. Balčytis 1965 m. parengė pirmąjį vadovėlį. Tai buvo originalus lietuviškas vadovėlis, ko neturėjo kitos Tarybų Sąjungos respublikos. 1971 ir 1972 m. pasirodė antros ir trečios klasių vadovėliai. Ketvirtos klasės vadovėlio pirmoji laida išėjo 1989 m. Vėliau buvo spausdinami papildyti ir patobulinti prof. B. Balčyčio vadovėliai, kurie išversti į rusų ir lenkų kalbas. Drauge su vadovėliais buvo leidžiami pratybų sąsiuviniai ir mokytojo knygos skirtingoms klasėms. 1995–2001 m. buvo išleistos I–IV mokiniams papildomos sunkesnės užduotys. M. Lepik ir A. Pipere nagrinėjo Baltijos ir Skandinavijos šalių žvalgomąjį tyrimą dėl matematikos mokytojų nuostatų dėstant dalyką. Jie Pateikė kelis preliminarius rezultatus dėl Estijos ir Latvijos matematikos mokytojų nuostatų palyginimo. G. Lāce analizavo pradinių klasių mokytojo matematikos mokymo kompetenciją, remdamasi Latvijos pradinės mokyklos mokytojų matematikos mokymo kompetencijos kokybiniu įvertinimu. Buvo pateiktos tobulintinos kompetencijų galimybės. Taikant šiuolaikines informacines ir komunikacines technologijas (IKT) mokymo procese sukuriama mokymosi aplinka, ugdanti bendravimą, savarankiškumą, kritinį mąstymą, kūrybiškumą, o kartu ir informacinę kultūrą. L. Kaklauskas analizavo virtualios mokymosi

and communication technologies for teaching mathematics, teaching mathematics in the higher education institution, training mathematics teachers, mathematics teachers’ competency, final mathematics examinations and mathematics olympiads, teaching mathematics at school. The plenary meeting of the conference focused on discussions about main achievements and problems, training mathematics teachers in Lithuania, Latvia and Estonia. Assoc. Prof. A.Bakštys and Assoc. Prof. R.Macaitienė in their paper Šiauliai University and Literature on School Mathematics generalised and systematized the contribution of the scientists of Šiauliai University to writing school mathematics books in Lithuania. The originators of methodology of mathematics at Šiauliai university were Assoc. Prof. Juozas Revuckas (1923 - 1991) and Prof. Bronius Balčytis. Active workers in this area were Assoc. Prof. V. Viruišis, Assoc. Prof. A.Bakštys and Prof. A.Kiseliovas. Prof. A.Ažubalis very widely introduced Prof. B.Balčytis’ scientific achievements in teaching mathematics. In 1965, Prof. B. Balčytis wrote the first textbook. This was an original Lithuanian textbook, not found in other republics of the former Soviet Union. In 1971 and 1972, textbooks for the second and the third forms were published. The first issue of the textbook for the fourth form was published in 1989. Prof. B. Balčytis continued to publish supplemented and improved textbooks. They were translated to the Russian and Polish languages. Alongside with textbooks, practice notebooks and teacher’s books for separate forms were published. Between 1995 and 2001, additional more difficult tasks were issued for the first-fourth form pupils. M.Lepik and A.Pipere analysed the exploratory survey of Baltic and Scandinavian countries on approaches of mathematics teachers towards teaching mathematics and their behaviour in the class. Several preliminary results of comparing Estonian and Latvian mathematics teachers’ approaches were given. Based on qualitative assessment, G. Lāce analysed teaching competency of primary class mathematics teachers. Possibilities for competency improvement were given. Application of modern information and communication technologies (ICT) in the teaching process creates a learning environment that develops communication, independence, critical thinking, creativity and at the same time informational culture.

Mokytojų ugdymas 2011

aplinkos (VMA) Moodle įrankių tinkamumą matematikos mokymui aukštosios mokyklos e. studijose. Taikant verbalinių dokumentų analizės metodą iš Moodle VMA baigtinės įrankių aibės buvo atrinkti 27 standartiniai ir 11 specializuotų įrankių, tinkančių aukštosios mokyklos matematikos e. studijų kursui rengti. Atlikus atrinktų standartinių ir specializuotų VMA įrankių tyrimą pagal septynis vertinimo kriterijus, nustatyta, kad matematikos mokymui aukštosios mokyklos e. studijų kurse geriausiai tinka naudoti 20 standartinių ir 6 specializuoti VMA Moodle įrankiai. Mokslinę diskusiją pratęsė prof. A. Baškienė pranešimu apie VMA Moodle ir vaizdo konferencijų įrangos panaudojimo patirtį aukštosios mokyklos geometrijos kurse. Analizuojant IKT taikymą matematikos mokyme buvo aktualizuojamos mokymosi kompiuterių tinkle (prof. J. Lipeikienė) galimybės. Doc. S. Turskienė ir K. Kiriliauskaitė analizavo diferencialinių lygčių ir jų sistemų sprendimo kompiuterine matematikos sistema (KMS) MAPLE galimybes, atskleidžiant pagrindinių komandų sintaksę, paskirtį, panašumus bei skirtumus. Tai palengvina KMS pasirinkimą praktiniams uždaviniams spręsti ir diferencialinių lygčių sprendimo teorijai mokyti(is). Konferencijoje nemažas dėmesys buvo skirtas matematikos mokymo aukštojoje mokykloje problemoms analizuoti (O. Panova, R. Kudžma, E. Kirjackas, J. Buls, A. Vintere ir kt.). Prof. E. Stankus siūlė tikimybių teorijos sąvokas aiškinti ekonominio pobūdžio uždaviniais ir pavyzdžiais, nes dėstymo patirtis rodo, kad tikimybių teorijos sąvokos yra sunkiai įsisavinamos tiek moksleivių, tiek studentų. Literatūros šaltinių analizę apie skirtuminių lygčių panaudojimą bei sprendimą pateikė I. Bula. Nemaža dalis konferencijos pranešimų buvo skirta matematikos mokymo kompetencijai tirti ir ugdyti (A. Šuste, H. Lapina, I. Kaldo, E. Gingulis, A. Kiseliovas ir D. Kiseliova ir kt.). Pastaraisiais metais ypatingas dėmesys visų Baltijos šalių švietimo sistemoje skiriamas gabiems vaikams ugdyti. I.Donielienė ir P.Grebeničenkaitė atliko Lietuvos respublikinių 6–8 klasių moksleivių matematikos olimpiadų, kasmet organizuojamų Šiaulių universitete, užduočių turinio kokybinę analizę ir pristatė įdomesnius uždavinių sprendimo atvejus. Analizę grindė užduočių kokybės statistiniais parametrais. K. Kokk pristatė Estijos nacionalinio matematikos egzamino raidą ir ypatumus. A. Cibulis nagrinėjo matematikos

L.Kaklauskas analysed suitability of the virtual learning environment (VLE) Moodle tools for teaching mathematics in the e-learning course of the higher education institution. Applying the method of analysis of verbal documents, 27 standard and 11 specialized tools suitable for preparing mathematics e-learning course in the higher education institution were chosen from the finite set of Moddle VLE tools. Having carried out a survey on standard and specialized tools of VLE according to seven assessment criteria, it was identified that 20 standard and 6 specialized VLE Moodle tools suite best for teaching mathematics in the e-learning course of the higher education institution. The scientific discussion was continued by Prof. A.Baškienė’s paper on the experience of using VLE Moodle and video conference facilities in the course of geometry of the higher education institution. Analysing application of the VLE in teaching mathematics, the possibilities of learning in the computer network were actualised (Prof. J.Lipeikienė). Assoc. Prof. S.Turskienė and K.Kiriliauskaitė analysed possibilities for solving differential equations and their systems by means of MAPLE computerassisted mathematical system (CMS), disclosing syntax, purpose, similarities and differences of main commands. This facilitates the choice of CMS for solving problems and teaching/learning theory of solving differential equations. Considerable attention was paid for analysing teaching mathematics in the higher education institution (O.Panova, R.Kudžma, E.Kirjackas, J.Buls, A.Vintere et al.). Prof. E.Stankus proposed to explain the concepts of the probability theory, employing economic type tasks and examples because teaching experience shows that the concepts of the probability theory are difficult to master both for pupils and students. The analysis of literature sources on usage of differential equations and solution was given by I. Bula. Quite a significant part of conference papers was dedicated to surveying and development of the competency of teaching mathematics (A.Љuste, H.Lapina, I.Kaldo, E.Gingulis, A.Kiseliovas and D.Kiseliova et al). Recently particular attention has been paid to development of gifted children in the systems of education of all Baltic states. I.Donielienė and P.Grebeničenkaitė carried out qualitative analysis of the content of tasks of Lithuanian Mathematics Olympiads for the 6-8 forms, which are annually organised at Šiauliai University, and presented cases of more

Sigita TURSKIENĖ Matematikos mokymo ir mokytojų rengimo idėjų sklaida bei patirtis tarptautinėje baltijos šalių konferencijoje „matematikos mokymas: retrospektyva ir perspektyvos“

olimpiados uždavinių formulavimo problemą. Daug dėmesio sulaukė A. Bukio pranešimas apie kasmetinę nacionalinę moksleivių konferenciją vykstančią Šiaulių miesto S. Daukanto gimnazijoje. Pranešėjas apžvelgė konferencijų raidą, dalyvių statistiką, pranešimų įvairovę. Konferencijos dalyviai lankėsi mokinių konferencijoje. Matematikos mokytojų rengimo problemas nagrinėjo A. Shapkova, K. Kislenko, T. Bakanovienė, L. Kvedere, D. Bonka ir kt. Matematikos mokymo mokykloje sekcijoje buvo aptarti matematikos mokymo metodai, teoremų įrodymų vaidmuo, miniprojektų organizavimo principai, matematinės indukcijos metodo grafinės interpretacijos galimybės ir kt. L. Tomėnienė nagrinėjo specialiųjų ugdymo poreikių vaikų matematikos mokymo organizavimą. Konferencijos dalyvių pranešimų tezės išspausdintos leidinyje (Th 12TH international conference „Teaching matehematics:retrospective and perspectives“, 2011). Apibendrinimas Konferencijoje vykusiose mokslinėse-metodinėse diskusijose pasidalyta gerąja patirtimi su kitų Baltijos šalių mokslininkais, mokytojais ekspertais, atsakyta į pagrindinius šiuo metu keliamus klausimus: kokio matematikos mokytojo šiandien reikia žinių visuomenėje? Kokias kompetencijas turi turėti matematikos mokytojas? Kokias IKT diegti matematikos mokymo procese? Kaip ugdyti gabius matematikai vaikus? Pagrindiniuose pranešimuose buvo diskutuojama apie šiuolaikines matematikos mokytojų rengimo tendencijas ir problemas Baltijos šalyse, siūlomi būdai tobulinti mokytojų rengimą Lietuvoje ir kitose šalyse. Nemažai dėmesio buvo skiriama IKT panaudojimo galimybėms matematikos mokyme. Apibendrinant konferencijoje skaitytų pranešimų tematiką, galima pažymėti konferencijos dalyvių naują mąstymą, aukštą probleminį pranešimų lygį, konstatuoti mokslinių idėjų naujumą, brandumą bei naudingumą jomis keičiantis tiek Lietuvos, tiek ir kitų Baltijos šalių mokslininkams.

interesting solutions of problems. The analysis was grounded on statistical parameters of quality of tasks. K.Kokk presented the development and peculiarities of Estonian national mathematics examination. A.Cibulis analysed the problem of formulating tasks for Mathematics Olympiads. A.Bukys’ paper about the annual national pupils’ conference, which takes place in Љiauliai S.Daukantas’ gymnasium, received much attention. He presenter reviewed the development of conferences, statistics of participants and diversity of papers. Conference participants visited pupils’ conference. Problems of training mathematics teachers were analysed by A.Shapkova, K.Kislenko, T. Bakanovienė, L. Kvedere, D.Bonka, etc. Methods of teaching mathematics, the role of proving theorems, principles of organising miniprojects, graphic interpretation possibilities of the mathematic induction method, etc were discussed in the section Teaching Mathematics at School. L. Tomėnienė analysed organisation of teaching mathematics to special needs children. The theses of conference participants are printed in the publication of the 12 International Conference “Teaching Mathematics: Retrospect and Perspectives”, 2011. Generalisation Scientific-methodological conference discussions enabled to share good experience with the researchers of other Baltic states, teachers experts, answer relevant questions like: What kind of mathematics teacher is needed in the knowledge society today? What competences must the mathematics teacher have? What ICT are to be implemented in the process of teaching mathematics? How to develop children gifted for mathematics? Main papers discussed modern tendencies and problems of training mathematics teachers in the Baltic countries, offered ways for improving teacher training in Lithuania and other countries. Considerable attention was paid to usage of ICT in teaching mathematics. To sum up the topics of the papers read at the conference, one can note new thinking of conference participants, high problematicity of papers and state the novelty, maturity and usefulness of scientific ideas in exchanges between scientists of Lithuania and other Baltic countries.

Mokytojų ugdymas 2011

Literatūra

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References

Th 12TH international conference „Teaching matehematics:retrospective and perspectives“ Konferencijos tezės, sud. Gruslytė M., 2011. Šiauliai: Šiaulių universiteto leidykla. /ISBN 978-609-430-070-7/.

SIGITA TURSKIENĖ Technologijos mokslų daktarė, Šiaulių universiteto informatikos katedros docentė. Moksliniai interesai: informacinių ir komunikacinių technologijų taikymas matematikos mokyme, e. studijų sistemos ir jų modeliai.

Doctor of Technology Science, Associate Professor of Department of Informatics of Šiauliai University. Research interests: application of information and communication technologies in teaching mathematics, e-learning systems and their models.

Address: Vasario 16 Str. 26 Str. 25, LT-76351 Šiauliai, Lithuania E-mail: sigita@fm.su.lt

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Research Works

Vyriausioji redaktorė / Editor-in-chief Audronė Juodaitytė Redaktorė / Edited by Birutė Kucinienė (lietuvių k. / Lithuanian), Asta Čiudarienė (anglų k. / English) Maketavo / Mocked-up by Tatjana Bakanovienė

SL 843. 2011-XX-XX. Leidyb. apsk. l. 11,9. Tiražas 100. Užsakymas 118. Išleido VšĮ Šiaulių universiteto leidykla, Vilniaus g. 88, LT-76285 Šiauliai. El. p. leidykla@cr.su.lt, tel. (8 ~ 41) 59 57 90, faks. (8 ~ 41) 52 09 80. Interneto svetainė http://leidykla.su.lt Spausdino UAB „Šiaulių knygrišykla“, P. Lukšio g. 9 A, LT-76207 Šiauliai. SL 843. XX-XX-2011. Publishing in Quires 11,9. Edition 100. Order 118.