# Mixed, Augmented and Virtual, Reality Applied to the Teaching of Mathematics for Architects
## 1 Mixed Reality Applied to Training  
Amongst the so-called emerging technologies that have recently been coming closer to the educational field, with true potential for training, \"Augmented Reality\" (AR) and \"Virtual Reality\" (VR) [1, 2] stand out. They offer different possibilities for their implementation in training: fostering mobile, ubiquitous learning; removing unnecessary information that may prevent the observation of the significant one; enriching the information about reality to make its understanding easier; favoring the perception of the most relevant elements, offering faithfulness in the representation of objects; creating multimedia training scenarios, establishing active and interactive environments for learning; increasing students' motivation; looking at objects from various points of view; adding value to the information supplied to students; and allowing the latter to become immersed in a totally controlled special training context. Despite this long list of possibilities associated with AR and VR, no unanimous consensus exists on their definition, which has to do with the rapid developments that have taken place within the context of these technologies. Different paths can be followed to distinguish AR from VR; one of them possibly being to place both of them along the reality-virtuality continuum, where AR would stand closer to the real context, while VR would be at one of the ends, with \"Augmented Virtual Reality\" or \"Mixed Reality\"--which brings together elements from AR and VR--positioned halfway between them. It is likewise possible to contrast AR and VR, and therefore we can say that, in AR, reality combines with informative elements located in technological devices so that a new reality can be attained, whereas VR places the subject within an immersive technological context that detaches the former from reality [3]. What AR pursues is to mix reality with the virtual sphere, or expressed differently, it enables the user to interact with the physical and digital worlds;instead, VR paves the way for users to access an alternative immersion world simulated by the computer in which different types of sensorial experiences take place and where subjects have the chance to interact with the environment as if they were inside it. In turn, the _Tecnologico de Monterrey_(2017) recently performed a study that included a comparison between Augmented Reality (AR) (addition of virtual information to physical elements), Virtual Reality (VR) (immersion in a digital world), Mixed Reality (MR) (combination of AR and VR) and the Internet of Things (IoT) (objects with sensors or chips that interact in the real world) (Table 1). What we are trying to highlight is that AR and VR propose different ways to interact with reality; whereas VR takes the subject to an unreal world and has as its aim for the subject to become immersed in it, AR adds information to the real world. When we talk about MR, both interaction options are found inside an object, which is the case at hand. Although educational experiences have been developed with AR to a greater extent than VR, amongst other reasons because the former does not require special devices to interact with objects, a diversity of experiences are actually beginning to appear, which combine them both so that they can be applied to different scientific fields: engineering and architecture (2017); language learning (2017); education sciences (2017); chemistry (2017); natural sciences (2017); biology (2017); geography (2017); medicine (2017; 2017); emergency protocol learning (2017); or physical activities (2017). Despite the scarcity of research studies devoted to the implementation of AR and VR in mathematics and in mathematical thinking, those carried out have turned out to be useful in a variety of ways: to prove their effectiveness for competence acquisition, as well as to improve performance and the attitudes towards mathematics learning (2017) and to favor the acquisition of mathematical knowledge for students with limitations in that area (2017).