Neformalna vs. formalna matematika

master's thesis

• Author: Dominik Čulo
• Mentor: Franka Miriam Bruckler (mentor)
• Committee member: Franka Miriam Bruckler (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U prvom poglavlju ovoga rada pokazali smo da prirodni i matematički jezik različito interpretiraju istinu, logičke veznike, definicije, pa čak i čitave rečenice (usporedite implikature u govoru i implikacije u matematici). Time smo pokazali da površno razumijevanje određenih matematičkih pojmova može dovesti do oslanjanja na njihovo značenje u svakodnevnom govoru, a time i na razvoj miskoncepcija (npr.pojam približavanja u objašnjenju limesa). U drugom poglavlju smo se usredotočili na matematičko razmišljanje i dokaze te smo dali nekoliko primjera različitih vrsta dokaza. Iako su primjeri jednostavni, jasno ilustriraju rigoroznost matematičkog dokaza u usporedbi s naivnom argumentacijom. U trećem poglavlju smo pobliže promotrili specifične koncepte koji su poznati i nematematičarima (broj, beskonačnost, pravac i vjerojatnost), te načine na koje je matematika generalizirala i formalizirala intuitivne koncepte. Taj odmak od intuicije ka apstraktnom možda otežava razumijevanje tih koncepata u kontekstu matematike. Naposljetku, u četvrtom smo poglavlju analizirali nekoliko čestih miskoncepcija -- iluziju linearnosti, probleme koji proizlaze iz pogrešnog shvaćanja limesa i vjerojatnosti, te varljivo korištenje statistike. Zajedničko svima jest da proizlaze iz oslanjanja na intuiciju ili na neprimjereno generaliziranje već naučene matematike.

Abstract (english)

In the first chapter of this thesis we have shown that the natural and mathematical language interpret truth, logical connectors, definitions, and even whole sentences differently (compare implicatures in speech with implications in mathematics). With this, we have shown that superficial understanding of certain mathematical terms may lead to reliance on their meanings in everyday speech, and with this, to the development of misconceptions (e.g. the phrase ”getting closer” in the explanation of limits). In the second chapter, we focused on mathematical thinking and proofs, and we gave a few examples of different types of proofs. Even though the examples are simple, they clearly illustrate the rigor of mathematical proof in comparison to naive argumentation. In the third chapter we took a closer look at specific concepts that are familiar to nonmathematicians (number, infinity, line, and probability), and ways in which mathematics generalised and formalised intuitive concepts. This departure from intuition towards the abstract may make understanding those concepts in the context of mathematics more difficult. Finally, in the fourth chapter we analysed several common misconceptions – the illusion of linearity, problems that arise from misunderstanding limits and probability, and misuse of statistics. Common to all of them is that they arise from relying on intuition or improper generalisation of previously learned mathematics.

• Keywords: prirodni jezik; matematički jezik; broj; beskonačnost; pravac; vjerojatnost; iluziju linearnosti; limesi; statistika
• Keywords (english): natural language; mathematical language; number; infinity; line; probability; illusion of linearity; limits; statistics
• Language: croatian
• URN:NBN: urn:nbn:hr:217:875274
• 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: 2019-03-28 11:03:51