The computer, if used more effectively, could bring advances that would improve mathematical education dramatically, not least with its ability to calculate quickly and display moving graphics. There is a gap between research results of the enthusiastic innovators in the field of information technology and the current weak integration of the use of computers into mathematics teaching. This paper examines what exactly the real potentials of using some mathematics computer software are to support mathematics teaching and learning in graph-oriented problems, more specifically we try to estimate the value added impact of computer use in the mathematics learning process. While electronic computation has been used by mathematicians for five decades, it has been in the hands of teachers and learners for at most three decades but the real breakthrough of decentralised and personalised micro-computer-based computing has been widely available for less than two decades. And it is the latter facility that has brought the greatest promise for computers in mathematics education. That computational aids overall do a better job of holding students' mathematical interest and challenging them to use their intellectual power to mathematical achievement than do traditional static media is unquestionable. The real question needing investigation concerns the circumstances where each is appropriate. A case study enabled a specification of advantages and obstacles of using computers in graph-oriented questions. Individual students' interviews revealed two less able students' reactions, difficulties and misinterpretations while using computers in mathematics learning. Among research outcomes is that the mathematical achievement of the two students observed improved and this makes teaching with computers an overriding priority for each defined teaching method. This paper may not have been realised without the valuable help of the Hungarian Eötvös State Grant.