Integrali funkcija jedne varijable u srednjoškolskoj nastavi matematike

master's thesis

• Author: Lucija Brekalo
• Mentor: Željka Milin Šipuš (mentor)
• Committee member: Željka Milin Šipuš (predsjednik povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-12, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu prezentirali smo glavne konceptualne okosnice jednog od najvažnijih pojmova u matematici, pojma integrala funkcija jedne varijable. Matematičku pozadinu integralnog računa funkcija jedne varijable, pojmove neodređenog i određenog integrala, sagledali smo iz različitih perspektiva na različitim razinama obrazovanja. Koncentrirali smo se na jedan od krucijalnih teorema infinitezimalnog računa, Osnovni (fundamentalni) teorem infinitezimalnog računa, čiji su integrali funkcija jedne varijable jedan dio. U sklopu Osnovnog teorema naglasili smo i pojmovne veze između brzine promjene i ukupne količine promjene, tj. da je određeni integral brzine promjene funkcije (ovisnosti dviju veličina) na nekom intervalu jednak ukupnoj promjeni vrijednosti te funkcije na tom promatranom intervalu. Ta veza omogućuje razvoj konceptualnog razumijevanja elementarnih pojmova integralnog računa i njegove primjene u različitim disciplinama koje koriste matematiku (npr. fizika, ekonomija), kao i u situacijama u svijetu oko nas. Time se implicira da je integralni račun itekako značajan dio srednjoškolske nastave. Osim toga, cilj ovog diplomskog rada je prikazati različite metodičke realizacije koncepata integralnog računa u udžbeničkoj sveučilišnoj i srednjoškolskoj literaturi u Hrvatskoj, kao i pojedine primjere metodičkih aktivnosti za učenike i studente iz europske i svjetske literature. Time naglašavamo raznolike i široke mogućnosti razvoja konceptualnog razumijevanja integralnog računa na svim razinama obrazovanja, a osobito u srednjoškolskoj nastavi matematike.

Abstract (english)

In this graduate thesis we have presented the main conceptual frameworks for one of the most important mathematical concept, integral of functions of a single variable. The mathematical background of (definite and indefinite) integral of a function of a single variable is overviewed from different perspectives at different levels of education. We have concentrated on one of the crucial theorems of the calculus within which integrals of functions of a single variable are a part, The fundamental theorem of calculus (FTC) and the underlying concepts. Together with the aforementioned ideas as our main goal, we have tried to emphasize the conceptual links of the rate of change and the accumulated change of the quantity, that is that a certain integral rate of change of a function (dependence of two quantities) is equal to the total accumulated change in function value on an observed interval. Such resonance allows us to develop conceptual understanding of the underlying concepts of the integral calculus and its application in various disciplines that use mathematics (e.g. physics, economy) and in situations in world around us, implying that the concept of integral is a significant part of the high school education. Besides, the goal of this thesis was to present different didactical realizations of the concept of integral calculus from university and high school textbooks in Croatia, and examples of didactical activities for pupils and students from European and world literature. In this way we highlight the diverse and broad possibilities of development of conceptual understanding of the calculus at all levels of education, especially in high school mathematics education.

• Keywords: nastava matematike; srednja škola; integral funkcija jedne varijable; infinitezimalni račun; integralni račun
• Keywords (english): teaching mathematics; high school; middle school; integral of functions of a single variable; fundamental theorem of calculus; integral calculus
• Language: croatian
• URN:NBN: urn:nbn:hr:217:744321
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-13 10:11:08