<p>U&nbsp; doktorskoj&nbsp; disertaciji&nbsp; je&nbsp; prezentovano&nbsp; pedago&scaron;ko&nbsp; istraživanje&nbsp; koje&nbsp; se&nbsp; odnosi&nbsp; na&nbsp; teorijsko&nbsp; i eksperimentalno&nbsp; ispitivanje&nbsp; efekata&nbsp; primene&nbsp; metodskih&nbsp; pristupa&nbsp; zasnovanih&nbsp; na&nbsp; matematičkom modelovanju u obradi izvoda funkcije i njegove primene u visokom strukovnom obrazovanju. Na osnovu teorijskih principa na kojima je zasnovan proces matematičkog modelovanja, osmi&scaron;ljen je način za implementaciju modelovanja u nastavni proces i kreirani su modeli za realizaciju obrade sadržaja iz oblasti izvoda funkcije i njegove primene. Predložen je i originalni pristup matematičkom modelovanju koji se realizuje u računarskom okruženju i istaknute su sve prednosti novog pristupa<br />koje&nbsp; se&nbsp; tiču&nbsp; realizacije&nbsp; nastavnog&nbsp; procesa&nbsp; i&nbsp; rezultata&nbsp; učenja&nbsp; i&nbsp; poučavanja.&nbsp; Disertacija&nbsp; se&nbsp; bavi&nbsp; i savremenim trendovima u obrazovanju nastavnika i njegovom unapređenju. Poseban akcenat je stavljen&nbsp; na&nbsp; osmi&scaron;ljavanje&nbsp; elemenata&nbsp; obuke&nbsp; nastavnika&nbsp; za&nbsp; primenu&nbsp; matematičkog&nbsp; modelovanja&nbsp; u &scaron;kolskoj praksi.<br />Istraživanje o efektima primene matematičkog modelovanja je sprovedeno u dva ciklusa. Prvi ciklus istraživanja je realizovan u periodu od 2009-2014. godine sa četiri generacije studenata strukovnih studija. U eksperimentu je učestvovalo ukupno 555 studenata organizovanih u paralelne grupe. U prvom&nbsp; ciklusu&nbsp; istraživanja&nbsp; je&nbsp; praćen&nbsp; uticaj&nbsp; primene&nbsp; matematičkog&nbsp; modelovanja&nbsp; na&nbsp; postignuća studenata iz oblasti izvoda funkcije i njegove primene. Od instrumenata primenjenih za ispitivanje postignuća&nbsp; studenata&nbsp; su&nbsp; kori&scaron;ćeni&nbsp; testovi&nbsp; znanja&nbsp; (kolokvijum&nbsp; i&nbsp; ispit)&nbsp; i&nbsp; anketa&nbsp; koja&nbsp; je&nbsp; ispitivala stavove studenata o realizaciji nastave matematike i njenoj korisnosti.<br />Drugi&nbsp; ciklus&nbsp; istraživanja&nbsp; je&nbsp; sproveden&nbsp; &scaron;kolske&nbsp; 2016/2017.&nbsp; godine.&nbsp; Eksperiment&nbsp; je&nbsp; realizovan&nbsp; sa paralelnim grupama u kojima su učestvovala 204 studenta Visoke&nbsp; tehničke &scaron;kole&nbsp; strukovnih&nbsp; studija.U&nbsp; drugom&nbsp; ciklusu&nbsp; istraživanja&nbsp; su&nbsp; ispitani&nbsp; efekti&nbsp; primene&nbsp; novog&nbsp; pristupa&nbsp; matematičkom modelovanju&nbsp; u&nbsp; računarskom&nbsp; okruženju&nbsp; na&nbsp; znanja&nbsp; studenata&nbsp; iz&nbsp; oblasti&nbsp; izvoda&nbsp; funkcije&nbsp; i&nbsp; njegovih primena (eksperimentalna grupa) i dobijeni rezultati su&nbsp; upoređeni sa rezultatima koje su studenti ostvarili&nbsp; kada&nbsp; je&nbsp; nastava&nbsp; realizovana&nbsp; primenom&nbsp; tradicionalnog&nbsp; ciklusa&nbsp; modelovanja&nbsp; (kontrolna<br />grupa).Na&nbsp; osnovu&nbsp; rezultata&nbsp; oba&nbsp; pedago&scaron;ka&nbsp; istraživanja,&nbsp; utvrđeno&nbsp; je&nbsp; da&nbsp; realizacija&nbsp; nastave&nbsp; matematike primenom matematičkog modelovanja, kao i matematičkog modelovanja u računarskom okruženju, na obradu pojma izvoda funkcije i njegovih primena&nbsp; ima značajan uticaj na kvalitet znanja studenata i ostvarenost optimalnih rezultata u učenju, razumevanju nastavnih sadržaja i nji hovoj primeni na re&scaron;avanje problema iz ove oblasti.</p> / <p>In the doctoral dissertation, pedagogical research is presented, which refers to the theoretical and experimental examination of the effects of the application of methodological approaches based on mathematical modeling in the field of the&nbsp; derivative of the function and its application in college education.<br />Based on the theoretical principles on which the process of mathematical modeling is based, a way has been devised for the implementation of modeling in the teaching process and models have been created for the teaching process in the field of function &nbsp; derivative and its applications. An original approach to mathematical modeling that is realized in a computer environment is also proposed, and all the advantages of the new &nbsp; approach concerning the realization of the teaching process and the results of learning and teaching are highlighted. The dissertation also deals with modern trends in teacher education and its improvement. Special emphasis is placed on designing elements&nbsp; of&nbsp; teacher&nbsp; training&nbsp; for&nbsp; the&nbsp; application&nbsp; of&nbsp; mathematical&nbsp; modeling&nbsp; in&nbsp; school&nbsp; practice.<br />The research on the effects of the application of mathematical modeling was conducted &nbsp; in two cycles. The first cycle of research was realized in the period from 2009-2014 with&nbsp; four generations of students. A total of 555 students organized in parallel groups participated in the experiment. In the&nbsp; first&nbsp; cycle&nbsp; of&nbsp; research,&nbsp; the&nbsp; influence&nbsp; of&nbsp; the&nbsp; application&nbsp; of&nbsp; mathematical&nbsp; modeling&nbsp; on&nbsp; the achievements of students in the field of function derivative and its applications was examined. Among the instruments used to examine students achievements, knowledge tests (colloquium and final&nbsp; exam)&nbsp; and&nbsp; a&nbsp; survey&nbsp; that&nbsp; examined&nbsp; students&#39;&nbsp; attitudes&nbsp; toward&nbsp; the&nbsp; teaching&nbsp; process&nbsp; and usefulness of mathematics were created and used.<br />The second cycle of research was conducted in the 2016/2017 school year. The experiment was realized&nbsp; with&nbsp; parallel&nbsp; groups&nbsp; in&nbsp; which&nbsp; participated&nbsp; 204&nbsp; students&nbsp; of&nbsp; the&nbsp; Technical&nbsp; College&nbsp; of Applied Sciences. In the second cycle of the research, the effects of applying a new approach to mathematical&nbsp; modeling&nbsp; in&nbsp; the&nbsp; computer&nbsp; environment&nbsp; on&nbsp; students&#39;&nbsp; knowledge&nbsp; of&nbsp; function derivative and its applications (experimental group) were examined and the obtained results were compared&nbsp; with&nbsp; the&nbsp; results&nbsp; achieved&nbsp; by&nbsp; students&nbsp; using&nbsp; traditional&nbsp; modeling&nbsp; cycle&nbsp; (the&nbsp; control group).<br />Based&nbsp; on&nbsp; the&nbsp; results&nbsp; obtained&nbsp; from&nbsp; both&nbsp; pedagogical&nbsp; researches,&nbsp; it&nbsp; was&nbsp; determined&nbsp; that&nbsp; the realization of teaching mathematics by applying mathematical modeling, as well as mathematical modeling&nbsp; in&nbsp; the&nbsp; computer&nbsp; environment,&nbsp; has&nbsp; a&nbsp; significant&nbsp; impact&nbsp; on&nbsp; the&nbsp; quality&nbsp; of&nbsp; students&#39; knowledge and the realization of optimal learning outcomes, and their application on solving the problems from this area.</p>

Philosophiae Doctor - PhD / This study attempts to understand how distance education practices influence the learning of calculus. Understanding student learning in a distance education environment is an important factor to consider in improving the learning experiences of those students who for one reason or the other opt not to study in conventional institutions of higher education. On one hand, understanding student learning may illuminate the influences that the learning environment has on student learning and on the other hand, it may inform on how learning experiences can be improved. The aim of this study is to acquire a deeper understanding of the diverse manner in which distance students learn calculus. Specific focus is also placed on how the distance education context of the Zimbabwe Open University (ZOU) influences student learning. The study describes a group of students' experiences of learning calculus in the ZOU distance education environment. The study also describes the students' learning styles and relates these to their mathematical understanding of certain calculus concepts. The specific content topics of "limit of function" and "derivative of function" are used to view achievement and performance, thereby indicating the distance students' mathematical understanding. The information processing learning theory is used as the theoretical framework for this study. The constructs of learning styles and mathematical understanding are used to illuminate the student's learning processes. The study used the Felder-Silverman learning styles model and Hiebert and Carpenter's notion of mathematical understanding to expound these constructs. The distance education environment of the B.Sc. Mathematics and Statistics (BSMS) programme at the ZOU provided the context of the study and an interpretive case study approach was adopted. A group of students registered in a first year first semester calculus course were studied. Data were collected from students based in four ZOU regional centres; namely Harare, Mashonaland Central, Mashonaland West, and Masvingo. These regional centres were conveniently selected for the study on the basis of proximity and accessibility. A total sample of twenty six students was involved and data for the in-depth part of the study emanated from five students who were purposively selected to participate in interviews. The interviewees were selected on the basis of their performance in a written calculus test. Data for this study were collected through use of learning journals, learning styles preference questionnaires, calculus tests and interviews. The data on students' learning experiences were predominantly qualitative in nature though supported by some quantitative data. The data on learning styles and mathematical understanding were also qualitatively analysed and presented case by case for the five interviewees. The study established that in a distance education system, the type of learning environment has the potential to influence students' learning, both positively and negatively, of which the main contributing factor is the learning support system. The study found that the learning support system provided by the institution and distance educators can have an impact on student learning. With reference to the calculus course in the BSMS programme, the study identified specific aspects where the environment facilitated or deterred learning. The study also revealed that students have varied learning style preferences, and that the learning environment has the potential to impact on students' learning styles. Since learning styles occupy a central place when it comes to improving distance learning materials, the study further explored the relationship between the constructs of learning styles and mathematical understanding. The study revealed that students' learning styles can influence the students' mathematical understanding. Improving students' learning in a distance education environment rests mainly on improving the learning materials and the support systems. A carefully designed and well supported instructional distance learning package can facilitate learning. Implications of the findings point towards the improvement of the distance teaching processes through the improvement of learning materials and the learning support systems for the BSMS distance education programme.

<p>U doktorskoj disertaciji je prezentovano pedago&scaron;ko istraživanje koje se odnosi na teorijsko i eksperimentalno ispitivanje i proučavanje efekata primene kognitivno-vizuelnog pristupa zasnovanog na primeni grafičkih reprezentacija funkcija u obradi nastavnih sadržaja iz oblasti funkcija i njihovih primena. Na osnovu teorijskih principa kognitivno-vizuelnog pristupa, osmi&scaron;ljeni su i kreirani originalni didaktičko-metodički modeli za realizaciju sadržaja iz oblasti: matematičke analize (funkcije i izvod funkcije); problema sme&scaron;e (problemi me&scaron;anja rastvora); linearne funkcije i njene primene na re&scaron;avanje problema (problemi iz oblasti algebre, fizike i realnog okruženja). Ključni aspekti predloženog metodičkog pristupa su: vizuelizacija, vi&scaron;estruke reprezentacije, konceptualna znanja, postepeni prelazak ka vi&scaron;im nivoima apstrakcije u razvoju matematičkog m&scaron;ljenja, kao i planska i sistematska upotrebu računara sa odgovarajućom softverskom podr&scaron;kom. Okosnicu kognitivno-vizuelnog pristupa u obradi sadržaja matematičke analize i značajnu novinu u izučavanju izvoda funkcije predstavljaju zadaci sa grafičkim sadržajima i/ili zahtevima. Prikazane su metodičke mogućnosti ovih zadataka u kreiranju multireprezentativnog okruženja za učenje čija je polazna tačka formiranje i razvoj grafičkog razumevanja matematičkih koncepata. Takođe je izložena mogućnost klasifikacije zadataka sa grafičkim sadržajima i/ili zahtevima po različitim kriterijumima, na osnovu koje je data tipologija i razrađena metodika njihove primene u nastavnoj praksi.Sprovedeno je empirijsko istraživanje u okviru koga je u periodu od 2013. do 2015. godine realizovano pet pedago&scaron;kih eksperimenata sa paralelnim grupama u kojima su učestvovala 642 ispitanika (222 učenika četvrtog i 120 učenika prvog razreda gimnazije prirodno-matematičkog smera, 180 studenata osnovnih studija hemije i 120 studenata prve godine osnovnih studija fizike). Za potrebe istraživanja osmi&scaron;ljeni su i izrađeni nastavni materijali koji su omogućili učenicima/studentima eksperimentalnih grupa da izučavaju funkcije i njihove primene u multireprezentativnom i vizuelno-dinamičkom računarskom okruženju. Instrumenti primenjeni u istraživanjima bili su inicijalni i finalni testovi znanja. Evaluacija ostvarenih efekata primene kognitivno-vizuelnog pristupa, izvr&scaron;ena je poređenjem postignuća učenika/studenata eksperimentalnih (kognitivno-vizuelnih) i kontrolnih (tradicionalnih) grupa na testovima znanja. Na osnovu rezultata sprovedenih pedago&scaron;kih eksperimenata, utvrđeno je da primena kognitivno- -vizuelnog pristupa zasnovanog na grafičkom i dinamičkom prikazu funkcije u prezentovanju matematičkih sadržaja i re&scaron;avanju problema, u računarskom okruženju, ima značajan uticaj na kvalitet znanja učenika/studenata i ostvarenost optimalnih rezultata u učenju i razumevanju nastavnih sadržaja iz oblasti funkcija i njihovih primena na re&scaron;avanje problema.</p> / <p>In the doctoral dissertation a pedagogical research related to theoretical and experimental study of effects of applying a cognitive-visual approach based on the use of graphic representations of functions in interpretation of teaching contents from the field of functions and their applications is presented. On the basis of theoretical principles of cognitive-visual approach, original didacticmethodical models for the realization of contents were designed and created in the field: of mathematical analysis (functions and function derivative); problem of mixture (problems of mixing solutions); linear functions and its applications to solve problems (problems in the field of algebra, physics and the real environment). Key aspects of the proposed methodical approach are: visualization, multiple representations, conceptual knowledge, gradual transition to higher levels of abstraction in the development of mathematical thinking, as well as the planned and systematic use of computers with appropriate software support. The background of cognitive-visual approach in interpretation of mathematical analysis content and a significant novelty in the study of function derivative represent the problems with graphical contents and/or requirements. The methodological possibilities of these problems in creation of a multi-representative learning environment whose starting point is the formation and development of a graphical understanding of mathematical concepts are presented. The possibility of problems classification with graphical contents and/or requests according to different criteria is presented, on the basis of which, the typology is proposed and methodology for their application in teaching practice is elaborated. During the period 2013-2015 an empirical research was&nbsp; conducted within which five pedagogical experiments were realized with parallel groups involving a total of 642 respondents (222 students of the fourth and 120 first-year high school students major in science, 180 students of bachelor chemistry studies and 120 students of the&nbsp; first year of bachelor physics studies). For the purpose of the research, teaching materials that enabled students of the experimental groups to study the functions and&nbsp; their applications in a multi-representative and visual-dynamic computing environment were created. The instruments used in the research were initial and final tests of knowledge. The realized effects of the application of cognitive-visual approach was evaluated by comparing the achievements of high school students/ students of experimental (cognitive-visual) and control (traditional) groups on knowledge tests. Based on the results of the conducted pedagogical experiments, it has been established that the application of the cognitive-visual approach based on the graphical and dynamic presentation of the function in teaching mathematical contents and in problem solving in a computer environment has a significant impact on the quality of high school students&rsquo;/students&rsquo; knowledge and on the achievement of optimal learning outcomes and understanding of teaching contents from the field of functions and their application to solving problems.</p>