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This paper investigates and considers factors which affect the success in solving a stand-alone geometrical problem by 182 students of the 7th and 8th grade of elementary school. The starting point for consideration is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura), utilising elementary- level geometry concepts. The task was presented as a textual problem, as a textual problem with an appropriate drawing, and as a task “wearing” a context. After data processing, the key factors affecting the process of problem solving were singled out: visualisation skills, detection and connection of concepts, symbolic notations, and problem-solving culture. The obtained results are the base of suggestions for changes in the geometry teaching – learning process.

The paper defines a special type of problem tasks and considers its didactic potential, as well as the success of students in solving the selected problem. The research instrument used is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura). The geometrical task is presented in three versions: as a verbal problem, as a verbal problem with a corresponding image and as a problem in context. The material analysed in the present paper was collected from 182 students in 7th and 8th grade of Croatian urban elementary schools. The didactic potential is considered from the aspect of use of mathematical concepts and connections. The success of students in problem-solving is considered from the aspect of implementation of the problem-solving process and producing correct answers, depending on the manner in which the tasks are set up. The results show that the stand-alone problem, as a special type of problem task, has considerable didactic potential. However, the students’ skills of discovering and connecting mathematical concepts and their properties are underdeveloped. In addition, the manner in which the tasks are set up considerably affects the process of solving the task and consequently the success of that process. Based on the results of the research, proposals are given for application of stand-alone problems in teaching mathematics.Key words: isolated problem; mathematical task; problem solving; problem evaluation. --- U radu se definira posebna vrsta problemskoga zadatka te se razmatra njegov didaktički potencijal kao i uspješnost učenika u rješavanju odabranoga problema. Instrument istraživanja je geometrijski zadatak s državne mature koji se postavlja u tri inačice: kao tekstualni problem, kao tekstualni problem uz odgovarajuću sliku te kao zadatak u kontekstu. U istraživanju je sudjelovalo 182 učenika 7. i 8. razreda hrvatskih gradskih osnovnih škola. Didaktički potencijal razmatra se s aspekta iskoristivosti matematičkih koncepata i veza, a uspješnost učenika u rješavanju problema razmatra se s aspekta provedbe procesa rješavanja i otkrivanja točnoga rješenja ovisno o načinu postavljanja zadatka. Rezultati pokazuju da promatrani problem kao posebna vrsta problemskoga zadatka ima veliki didaktički potencijal, ali da učenici imaju nedovoljno razvijene vještine otkrivanja i povezivanja matematičkih koncepata i njihovih svojstava. Osim toga, način postavljanja zadatka značajno utječe na proces rješavanja, a posljedično i na uspješnost određivanja rješenja. Na temelju rezultata daju se prijedlozi primjene opisane vrste problema u nastavi Matematike.Ključne riječi: izolirani problem; matematički zadatak; rješavanje problema; vrednovanje problema

U radu se obrađuje jedan stari, ali još uvijek aktualni, problem dijeljenja jednakostraničnog trokuta na najmanji broj dijelova tako da se preslagivanjem nastalih dijelova, bez preokretanja, može oblikovati kvadrat. Problem je poznat pod nazivom Haberdasherov problem, ali unatoč brojnim pokušajima, jedino do danas poznato valjano rješenje dao je engleski matematičar Dudeney, još 1903. godine. Uz opis konstrukcije i opravdanja valjanosti Dudeneyjeva rješenja, posebno se razmatra jedno rješenje koje pokazalo pogrešnim, a zapravo se radi samo o dobroj aproksimaciji kvadrata. Također, razmatra se klasa rješenja dijeljenja trokuta na dijelove koji se daju oblikovati u pravokutnik među kojima je i jedan kvadrat. Konačno, navode se i razlozi zašto su ovakvi i slični problemi korisni i prikladni za korištenje na svim razinama obrazovanja.

The paper defines a special type of problem tasks and considers its didactic potential, as well as the success of students in solving the selected problem. The research instrument used is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura). The geometrical task is presented in three versions: as a verbal problem, as a verbal problem with a corresponding image and as a problem in context. The material analysed in the present paper was collected from 182 students in 7th and 8th grade of Croatian urban elementary schools. The didactic potential is considered from the aspect of use of mathematical concepts and connections. The success of students in problem-solving is considered from the aspect of implementation of the problem-solving process and producing correct answers, depending on the manner in which the tasks are set up. The results show that the stand-alone problem, as a special type of problem task, has considerable didactic potential. However, the students’ skills of discovering and connecting mathematical concepts and their properties are underdeveloped. In addition, the manner in which the tasks are set up considerably affects the process of solving the task and consequently the success of that process. Based on the results of the research, proposals are given for application of stand-alone problems in teaching mathematics., U radu se definira posebna vrsta problemskoga zadatka te se razmatra njegov didaktički potencijal kao i uspješnost učenika u rješavanju odabranoga problema. Instrument istraživanja je geometrijski zadatak s državne mature koji se postavlja u tri inačice: kao tekstualni problem, kao tekstualni problem uz odgovarajuću sliku te kao zadatak u kontekstu. U istraživanju je sudjelovalo 182 učenika 7. i 8. razreda hrvatskih gradskih osnovnih škola. Didaktički potencijal razmatra se s aspekta iskoristivosti matematičkih koncepata i veza, a uspješnost učenika u rješavanju problema razmatra se s aspekta provedbe procesa rješavanja i otkrivanja točnoga rješenja ovisno o načinu postavljanja zadatka. Rezultati pokazuju da promatrani problem kao posebna vrsta problemskoga zadatka ima veliki didaktički potencijal, ali da učenici imaju nedovoljno razvijene vještine otkrivanja i povezivanja matematičkih koncepata i njihovih svojstava. Osim toga, način postavljanja zadatka značajno utječe na proces rješavanja, a posljedično i na uspješnost određivanja rješenja. Na temelju rezultata daju se prijedlozi primjene opisane vrste problema u nastavi Matematike.