Giftedness is a multi-dimensional phenomenon that results from the interaction of three groups of human characteristics: above-average abilities, personality traits that contribute to the development of giftedness and a high degree of creativity. The results of the three characteristics in an individual are above average results in one or more areas. Giftedness needs to be identificial on time and the school as a place of education, should take care of gifted students in a way that they are adequately taught, encouraged and monitored. The main focus of my graduate thesis is mathematical giftedness. The theoretical part deals with characteristics of mathematically gifted children, the role of teachers in developing of mathematical giftedness and adequate method of teaching - guided by learning mathematics through exploration. Mathematically gifted children and children of average intellectual abilities participated in a qualitative research. This particular example illustrates which abilities and qualities are shown during math problem solving. By giving the same test to children of average intellectual abilities and the gifted ones the difference in approach, abilities and characteristics between them can clearly be seen.