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1 Introduction

The emergent technology of Augmented Reality (AR) brings the opportunity to transform the way we interact with Mathematics. Dealing with mathematical knowledge involves dealing with symbolic representations necessarily. Numerical, algebraic and graphical are the standard representations, and their exchange is an important issue for the learning of Mathematics.

We consider the graphical representation as the one that requires more cognitive effort in order to understand and explain a behavior. We believe that a well-conceived graphical representation can give students’ mind a stable support in order to deal with other mathematical representations. With that in mind, our proposal is to elaborate visual approaches for the students in order to foster their interaction with mathematical knowledge.

We believe that problems related to the learning of Math are associated with how we conceive this science and how it is reflected in the teaching process. Aiming to change the perception of students about mathematical knowledge, we propose working on a new pedagogy to approach students with this revised knowledge [1, 2, 3].

We share the point of view about digital technology offering the opportunity to create new forms of symbol-focused experience. Particularly, the idea of co-action guides our design process in the AR application. We strongly look for an understanding about the way digital technology can be effectively integrated into mainstream education and become an active participant of the cognitive process [4, 5].

We work at Monterrey Institute of Technology and Higher Education, a private educational institution in Mexico. Our educational model cares for the integration of technology in the learning process. Our particular concern is Mathematics Education, and as part of an innovation program, we have been combining the use of emerging digital technologies and educational methods in order to nourish a positive attitude from students for the learning of Mathematics.

Today we identify ourselves as TEAM, Tecnología Educativa para el Aprendizaje de las Matemáticas. We took on the challenge of creating a multidiscipline team that could lead an innovation process for the learning of Calculus in our Campus.

Technology changes the way we learn, and students in this millennium are already prepared to live learning experiences through visual and gestural interaction. As TEAM we are confident of the advantages on this new constantly-developing paradigm. We seek to transform the learning of Mathematics through an enjoyable interaction with technology. Augmented Reality technology has been the first choice to begin our research about its advantages.

2 The AR Application: Towards a Visual and Tangible Mathematics

Thinking in traditional content of the three Mathematics courses for Engineering, we identified spatial visualization as the mathematical skill that deserves to be explored by AR.

The application considers three levels, named: From 2D to 3D, Solids of Revolution and 3D Surfaces, which can be associated with Mathematics I, II, and III in College. Below we describe each level and its features.

2.1 First Level: From 2D to 3D

The first part of the application studies the transition from a 2D curve to a 3D surface; through ‘accumulation in time-space’ of different curves. It starts with a known curve-form where different graphical effects will take place as shown in Table 1.

Table 1. Different graphical simulations for level 1
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The curves will be shaped by the graphical effect that corresponds to the presence of the parameter k in the algebraic expression.

The curve images are originally in 2D, but an animation occurs with the graphical effect simulating a 3D surface. Successive curves are placed in parallel planes situated so near in such a way that the surface begins to take its own shape. With the effect of the parameter (k) and simultaneously the “motion of time” through the successive copies in parallel planes, the 3D visualization takes place. Figure 1 illustrates this process.

Fig. 1.
figure 1

Different frames of the simulation in level 1

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2.2 Second Level: Solids of Revolution

The second level of the application considers four curves behavior, as shown in Table 2. UI contains buttons leading to a total of 24 simulations.

Table 2. Different graphical simulations for level 2
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Each curve behavior includes both the case with rotation of the curve performed around the x-axis and also around the y-axis. In each case, the visualization of the solid and the method for volume calculation acknowledged as ‘disk’ and ‘shell’, are simulated with AR technology.

Taking advantage of the AR possibilities, we included the visualization of the solid of revolution before the introduction to the methods for calculating its volume. We decided to produce a simulation in order to mentally construct the solid, performing a natural cognitive process to visualize it. Figure 2 illustrates part of the visualization simulation.

Fig. 2.
figure 2

Different frames of the simulation in level 2

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2.3 Level 3: 3D Surfaces

The third and last level of the AR app consists of 3D surfaces, seen as the graph of a two-variable function. A variety of functions with different features is presented in order to illustrate the kind of behavior that is possible in a 3D space. Table 3 shows the functions considered, as well as intersections with the surface produced by planes which are parallel to the XY, XZ, YZ planes.

Table 3. Different graphical simulations for level 3
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The innovation we introduced in this level is the simulation of a process now performed from 3D to 2D. It consists of a return to the 2nd dimension when the surface in 3D is affected by the intervention of parallel planes to the coordinate planes XY, YZ and XZ. This visual process of cutting and rebuilding the surface promotes a cognitive evocation of a new way to visualize the surface. Figure 3 shows part of the simulation process.

Fig. 3.
figure 3

Different frames of the simulation in level 3

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3 Concluding Remarks

As TEAM we care for the actual changes that education is experiencing. It is not difficult to recognize that the university model of education, remaining for hundreds of years, nowadays is being challenged. Top Educational Institutions are looking for innovation. Reinvention about the usefulness of time and space for the teaching and learning event makes us aware of the great potential of emergent technologies.

We take part of this reinvention for the impact in the learning of Mathematics. Here we presented the possibility to offer students a visual and tangible Math. They can interact with it inside and outside the classroom. But mostly it has been our main concern to conceive an AR App that performs a simulation to be adopted cognitively. Spatial visualization is not a mathematical topic considered in the syllabus, and we are not stating it should be. Instead, we are identifying mathematical skills that make a great difference when learning Mathematics, and try to integrate digital technology in order to develop those skills. This way, mathematical knowledge should be better understood and because of this, a better attitude for its learning should eventually appear among the students.