**Abstract**

In this work we introduce a learning system based on Virtual Reality and Augmented Reality for studying analytical-geometric structures that are part of the curriculum in mathematics and physics in high school classes. We believe that an immersive study environment has several advantages with respect to traditional twodimensional environments (such as a book or the simple screen of a PC or even a tablet), such as the spatial understanding of the concepts exposed, more peripheral awareness and moreover an evident decreasing in the information dispersion phenomenon. This does not mean that our teaching proposal has to substitute the traditional approaches, but it can be seen as a robust tool to support learning. In the first phase of our research we have sought to understand which mathematical objects and which tools could have been used to enhance the teaching of mathematics, in order to demonstrate that the use of Virtual Reality and Augmented Reality techniques significantly improves the level of understanding of the mathematical subject being studied by the students. The system which provides for the integration of two machine levels, hardware and software, was subsequently tested by a representative sample of students who then provided feedback through a questionnaire.

**Keywords:** immersive learning, virtual reality, augmented reality, Unity3D, blender, usability, accessibility

#### **1. Introduction**

Virtual Reality (VR) and Augmented Reality (AR) are technologies that since their inception have sometimes suffered fluctuating fates, sometimes due to the lack of suitable low-cost hardware, sometimes due to the inherent complexity of the technologies adopted. With the advent of mobile technologies, their fates have radically changed and today we have both low-cost hardware and software approaches that make them widely usable in many areas of modern life. Nowadays, we find VR and AR everywhere, in manifold applications: from entertainment [1, 2], teaching [3, 4], tourism [5, 6], manufacturing [7, 8], networking and communications [9, 10], microelectronic and high performances hardware industries [11, 12], e-commerce [13], medicine [14–19].

In this work we are going to focus on the adoption of VR and AR technologies teaching mathematics, analyzing the feasibility of the process and analyzing the usability of the implemented software platform.

During our work, we were guided by a specific goal: the possibility of giving a real and visual form to the abstract objects of mathematics. This represents a further development in the visual representation of mathematical concepts, which in the course of history has been evolving from the primitive use representations of simple counting objects, such as the tally sticks, through the elegant structures of the symbolic algebra of the seventeenth century, to the imposing constructions of mathematical analysis and modern geometry, to get to the current and amazing views of numerical analysis through computer graphics. In this context, we have set out to investigate the possibility of enhancing students' understanding of the concept of link between an algebraic-set structure and its geometric representation on an orthogonal Cartesian space (concept best known with the name of function).

In the first phase of the work, our attention has been directed to a certain number of functions, that are used in senior high-school classes, such as the representation of trajectories in the Cartesian plane and simply surfaces in the three-dimensional space.

The very first proposed functions are as follows:

*y* ¼ cos *x* The **cosine function** is well known to students who currently use it to solve trigonometric, analytical and geometric problems. Moreover, it is also present in all fields of physics in order to model the behavior of several phenomena in mechanics, thermodynamics, optics and electrodynamics, just to limit our analysis to some main cases [20]. In **Figure 1** is shown the VR representation of such function.

*<sup>z</sup>* <sup>¼</sup> *<sup>e</sup>*�*<sup>x</sup>* The **exponential function** is one of the most important functions in mathematical analysis, whose applications span to different fields of knowledge and is well studied in high school. Exponential functions can be used in many contexts, such as the compound interest in finance, to study the evolution of the population growth and the radioactive decay [21]. In **Figure 2** is shown the VR representation of the named function.

*<sup>z</sup>* <sup>¼</sup> ln *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> ð Þ This function is an example of **bi-dimensional logarithm**, useful to describe astrophysical objects. In **Figure 3** is shown the VR representation of the named function.

*<sup>z</sup>* <sup>¼</sup> sin *<sup>x</sup>*2þ*y*<sup>2</sup> ð Þ *<sup>x</sup>*2þ*y*<sup>2</sup> This function is an example of **bi-dimensional dumped sine**, useful to describe objects in fluid dynamics, electronics and telecommunications. In **Figure 4** is shown the VR representation of the function.

**Figure 1.** *Cosine wave plot.*

*Learning Mathematics in an Immersive Way DOI: http://dx.doi.org/10.5772/intechopen.96533*

**Figure 2.** *Exponential function plot.*

**Figure 3.** *Bi-dimensional natural logarithm plot.*

**Figure 4.** *Dumped sine plot.*
