## 3 Materials and Methods
### Conducted Experiments and Research Group  
The presented method enables the development of a set of exercises for the students. The experiments encompassed a set of 60 exercises covering the following topics:  
* Geometry (flat geometric figures and their properties). * Elements of mathematical analysis (including properties of various types of functions). * Various types of charts (including bar charts and pie charts). The prepared set of 60 exercises from the above-mentioned areas (20 exercises from each area) was prepared according to the following assumptions:  
* Each of the exercises contained from a few to a dozen graphic elements;
* The size of the image elements was similar, e.g., from 2 to 8 cm;
* In the case of mathematical analysis exercises, the content concerned linear, quadratic, and trigonometric functions;
* The number of bars in the bar charts is between 10 and 25. The above assumptions result from the limitations of tactile image exploration with the use of hands and fingers by blind students, which was the subject of our previous research [41]. The research group consisted of 46 participants with blindness, who could solve math exercises at a high-school level. The group consisted of 19 women and 27 men, aged 15-19, with a median age of 17 years and a standard deviation of 1.21. The following inclusion criteria were adopted in the studies:  
* The degree of vision loss was significant, with more than 90% of vision loss;
* The student is in secondary school and completed a mathematics course at a primary school level;
* There were no mental disabilities;
* The ability to use the tablet computer, and the audio-tactile interface and interpret the tactile image, was assessed by a teacher having several years of experience in teaching the blind. The authors confirm that all methods were carried out in accordance with the relevant guidelines and regulations. Research carried out is not a medical experiment. All data are fully anonymized and conform to the ethical principles. In the context of the sense of touch, we analyzed the ratio of the number of taps that were successful to the total number attempted, considering all the stages of the exercise and whether the object is included in the completeness list. Initially, following previous research [41], the user begins with the default activation area size. If the number of successful taps is below the set threshold, the activation area increases to properly recognize the gesture for a given graphic primitive in a tactile image. The activation area serves as a global parameter (applied to all primitives) for a given user. Moreover, the mechanism also allows for adjustment of the time intervals between gestures. This modification is made during the initial configuration, in which the user learns how to perform exemplary gestures. The following gestures are verified: touch, 1-tap, 2-taps, 3-taps, and long press. In the context of the sense of hearing, the focus was on the verification of the proposed mechanisms of presentation. First, the mechanisms of audio presentation of overlapping objects were evaluated, including:* The depth map creation mechanism,
* The prioritization of objects in the image,
* The detectability by the long-hold gesture. Another tested mechanism included navigation modes. The students solved the exercise in three presented navigation modes: without navigation, with the activation of only selected elements, and with navigation in a specific direction. The average time for reaching a given object, defined in the list of objects necessary to solve the current stage of the exercise, was determined. The test procedure involved dividing the developed group of exercises into two parts, to enable an assessment of the impact of the proposed feedback mechanisms in the context of touch and hearing. Both parts contained exercises from all the prepared types. In the first part, no feedback mechanisms were applied in either sense. Furthermore, in the audio reproduction tests, no mechanisms were used for detecting objects lying underneath.