## 5 Discussion and Conclusions  
As the literature shows, a significant difficulty in learning mathematics for blind individuals is low self-esteem, motivation, poor learning progress, and difficulties in consolidating acquired skills. The results of the current study indicate an improvement in learning outcomes and, at the same time, an increased number of interactions between students. The students also supported their peers by explaining the material that they had already mastered. Thus, we can assume that the increased number of interactions allows students to be more actively involved (overcoming low motivation), overcome the barrier of poor results, and better consolidate the material. Additionally, we noticed that an additional advantage of the developed method was supporting students' self-study. In the traditional teaching method using Braille textbooks, teachers have to work individually with each student, engaging in dialog and encouraging interaction. Additionally, they need to verify the student's answers written in Braille, which requires the teacher's knowledge of Braille notation and can be pretty laborious. In comparison, the developed method also requires the teacher to spend time preparing the course, but this effort is a one-time occurrence, and the materials can be reused multiple times. The teacher also gains the ability to assess the student's learning progress with detailed identification of challenging issues for individual students. Furthermore, by utilizing feedback mechanisms such as the knowledge vector and the error vector, each student has a personalized automatic selection of subsequent exercises. It relieves the teacher of the need to individually assess the knowledge of each student, allowing them to focus on the most challenging aspects of the material. Referring to the limitations of the proposed method, the evaluation included a limited number of students divided into two groups. We decided to limit the scope of the educational material to a selected course in mathematical analysis at a high school level. The evaluation of the developed method focused on alternative descriptions in the Polish language. It would be worthwhile to compare students' results in classes using different languages. To sum up, the most valuable features of the developed method are the algorithmic selection of material (represented in the form of a directional graph of concepts), the preparation process (exercise decomposition into stages), and enhancement in the learning process (feedback elements: knowledge vector and error vector used during graph navigation, reflecting the learning process). Our approach, including the above stages, provides new opportunities, such as the following:  
* Standardization of the educational material and the phases of its acquisition;
* The possibility of a quantitative assessment of the student's knowledge;
* The possibility of a detailed assessment of challenging elements;
* The ability to easily compare the knowledge of students;
* The ability to choose lesson materials corresponding to the level of knowledge of students in the class;
* The possibility of cooperation between students via individual consultations;
* The development of various ways of navigation and presentation of the exercise content, as well as the formula itself, in visual forms and alternative audio presentations, depending on the stage of the exercise and its context;
* Semi-automatic generation of alternative presentations of mathematical formulas, including additional structural information based on the TEX notation, which is a standard used for writing mathematical expressions; the choice of widely used TEXnotation serves as a bridge between the commonly used presentation and the proposed alternative presentation. The developed method of alternative presentation can be used in different languages. It requires adapting the alternative descriptions to the rules of a specific language, while the TEX syntax modifications remain unchanged.