Razumevanje neskončnosti v osnovni šoli

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Language:	Slovenian
Year of publishing:	2013
Source:	Ljubljana
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[I. Gole]
UDC:	51:373.3(043.2)
COBISS:	9907273
Views:	835
Downloads:	197
Average score:	0 (0 votes)
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Secondary language:	English
Secondary title:	The understanding of infinity in primary school
Secondary abstract:	Infinity is a concept that occurs in a variety of discussions, not only in mathematical, but also in the philosophical, physical and religious ones. The concept is often referred to an infinite set with such a large number of items that that it is impossible to count them. A number of paradoxes occur in connection with infinity (Hotel Infinity, the Barber, Achilles and the tortoise, dichotomy, Galileo's paradox, etc), showing a contradiction, since it is difficult to understand something which opposes our intuition. This thesis deals with the understanding of infinity with primary school pupils. The pupils find infinity interesting, they think about it intuitively, linking it to various contexts, especially with everyday life. In mathematics they come across the concept of infinity in arithmetic (numbers, number of solutions to inequalities, etc.), geometry (line, plane, etc.), but their perception of this concept depends to a large extent on the type and content of tasks. The understanding of infinity cannot be explored in one, but in various contexts. Since the concept is so complex, the understanding is developed over time in students’ mental structures. Students' understanding and interpretation of the concept of infinity depends on the context in which the term is expressed (numeric, geometric), the type of an infinite set (infinitely many, infinitely close) and the way of presenting the problem. The empirical part presents the understanding of the concepts of infinity with primary school pupils. A test was designed which verified students' understanding of the concept of infinity in different contexts. The aim was to explore how the fifth and ninth graders understand the different types of infinity, the differences between them, and which type of tasks about infinity presents the most problems and which infinity the students used in solving various infinity problems. Another interest was to learn the differences in the understanding of infinity between the ninth graders, according to the homogeneous group (level) they attend in mathematics. The results showed that the ninth graders are more successful in solving the tasks than the fifth graders. The fifth graders, as well as the ninth graders were the most successful in the tasks that were connected to concrete examples from everyday life, while they had the most difficulties with the tasks that checked the application and understanding of the potential and actual infinity. It turns out that the understanding depends on the type of the task and the context in which it is introduced. Taking into account the results of research, knowledge of the different types of infinity and the stage of students’ development, some recommendations have been created for teachers to provide guidance for teaching selected topics on infinity in primary school.
Secondary keywords:	primary school;pupil;mathematics;osnovna šola;učenec;matematika;
File type:	application/pdf
Type (COBISS):	Master's thesis
Thesis comment:	Univ. v Ljubljani, Pedagoška fak.
Pages:	121 str.
Type (ePrints):	thesis
Title (ePrints):	The understanding of infinity in primary school
Keywords (ePrints):	neskončnost
Keywords (ePrints, secondary language):	infinity
Abstract (ePrints):	Neskončnost je pojem, ki se vedno znova pojavlja v najrazličnejših razpravah, ne le v matematičnih, temveč tudi v filozofskih, fizikalnih in verskih. Pojem pogosto označuje neštevno množico s tako velikim številom elementov, da jih ni mogoče prešteti. V zvezi z neskončnostjo se pojavljajo tudi številni paradoksi (Hotel neskončnost, Brivec, Ahil in želva, dihotomija, Galilejev paradoks idr.), ki kažejo na protislovje, saj je težko razumeti nekaj, kar nasprotuje naši intuiciji. V magistrskem delu obravnavamo razumevanje neskončnosti pri učencih v osnovni šoli. Učencem je neskončnost zanimiva, o njej razmišljajo intuitivno, povezujejo jo z različnimi konteksti, predvsem z vsakdanjim življenjem. Pri matematiki se srečajo s pojmom neskončno pri aritmetiki (števila, število rešitev pri neenačbah itd.), geometriji (premica, ravnina itn.), vendar pa je njihovo dojemanje tega pojma v veliki meri odvisno predvsem od vrste in vsebine nalog. Razumevanje neskončnosti ne moremo raziskovati samo v enem, temveč v različnih kontekstih. Ker je pojem zahteven, se razumevanje v miselnih strukturah učenca razvija v daljšem obdobju. Učenčevo razumevanje in razlaga pojma neskončnosti je odvisno od konteksta, v katerem je pojem izražen (številski, geometrijski), od vrste neskončne množice (neskončno veliko, neskončno mnogo, neskončno blizu) in glede na način predstavitve problema. Teoretični del se tako na podlagi raziskav različnih avtorjev osredotoča na zgodovino razvoja pojma neskončnost, vrste neskončnosti (števna, geometrijska, potencialna in aktualna), neskončnost v osnovni šoli pri pouku matematike in na raziskovanje učenčevih idej o neskončnosti. V empiričnem delu je predstavljena raziskava o razumevanju pojmov neskončnosti pri učencih v osnovni šoli. Oblikovali smo preizkus znanja, s katerim smo preverjali učenčevo razumevanje pojma neskončnost v različnih kontekstih. Zanimalo nas je, kako petošolci in devetošolci razumejo različne vrste neskončnosti, kakšne so razlike med njimi, pri kateri vrsti nalog iz neskončnosti imajo največ težav ter katero neskončnost so uporabili učenci pri reševanju različnih problemov iz neskončnosti. Prav tako nas je zanimalo, kakšne so razlike v razumevanju neskončnosti med devetošolci glede na homogeno skupino (nivo), ki jo obiskujejo pri matematiki. Rezultati so pokazali, da so devetošolci pri reševanju nalog uspešnejši od petošolcev. Tako petošolci, kot tudi devetošolci, so bili najuspešnejši pri nalogah, ki so se navezovale na konkretne primere iz vsakdanjega življenja, medtem ko so imeli največ težav pri nalogah, ki so preverjale uporabo in razumevanje potencialne ter aktualne neskončnosti. Izkazalo se je, da je razumevanje odvisno od vrste naloge in od konteksta, v katerega je vpeljana naloga. Ob upoštevanju rezultatov raziskave, poznavanju različnih vrst neskončnosti in razvojni stopnji učencev smo oblikovali priporočila za učitelje, ki predstavljajo smernice za poučevanje izbranih tem o neskončnosti v osnovni šoli.
Abstract (ePrints, secondary language):	Infinity is a concept that occurs in a variety of discussions, not only in mathematical, but also in the philosophical, physical and religious ones. The concept is often referred to an infinite set with such a large number of items that that it is impossible to count them. A number of paradoxes occur in connection with infinity (Hotel Infinity, the Barber, Achilles and the tortoise, dichotomy, Galileo's paradox, etc), showing a contradiction, since it is difficult to understand something which opposes our intuition. This thesis deals with the understanding of infinity with primary school pupils. The pupils find infinity interesting, they think about it intuitively, linking it to various contexts, especially with everyday life. In mathematics they come across the concept of infinity in arithmetic (numbers, number of solutions to inequalities, etc.), geometry (line, plane, etc.), but their perception of this concept depends to a large extent on the type and content of tasks. The understanding of infinity cannot be explored in one, but in various contexts. Since the concept is so complex, the understanding is developed over time in students’ mental structures. Students' understanding and interpretation of the concept of infinity depends on the context in which the term is expressed (numeric, geometric), the type of an infinite set (infinitely many, infinitely close) and the way of presenting the problem. The empirical part presents the understanding of the concepts of infinity with primary school pupils. A test was designed which verified students' understanding of the concept of infinity in different contexts. The aim was to explore how the fifth and ninth graders understand the different types of infinity, the differences between them, and which type of tasks about infinity presents the most problems and which infinity the students used in solving various infinity problems. Another interest was to learn the differences in the understanding of infinity between the ninth graders, according to the homogeneous group (level) they attend in mathematics. The results showed that the ninth graders are more successful in solving the tasks than the fifth graders. The fifth graders, as well as the ninth graders were the most successful in the tasks that were connected to concrete examples from everyday life, while they had the most difficulties with the tasks that checked the application and understanding of the potential and actual infinity. It turns out that the understanding depends on the type of the task and the context in which it is introduced. Taking into account the results of research, knowledge of the different types of infinity and the stage of students’ development, some recommendations have been created for teachers to provide guidance for teaching selected topics on infinity in primary school.
Keywords (ePrints, secondary language):	infinity
ID:	8311910