MET:Learning Math in 3D simulations
Authored by David Wees for ETEC 510 65C (2010)
Areas of Math
What is powerful about using virtual realities in education is that "[t]here are also certain experiences and viewpoints which only virtual reality can provide" . Students can view things in the virtual world which are not possible for them to view in the real world. Given the high level of abstraction and difficulty present in mathematics, providing access to reduce the abstraction of these concepts. Even highly complex mathematical shapes can be represented.
For example, visualization of three dimensional shapes can be quite difficult for students as well. Although there are graphing packages which let students examine these figures but "[o]nly VR can let the student walk around on that math surface, climbing the peaks and valleys to see how the variables inter-relate" . As well, "...it possible to explore transformations [of the graphs] in a new way and get an intuitive sense for how a specific transformation works."
One area where mathematics can be found within a 3D simulator is the construction of the objects themselves in the environment. Students can construct buildings and other objects. The way students construct buildings helps them see the relationship between two dimensional shapes and three dimensional shapes. It is important to note that this type of spatial recognition is not taught in schools.
Bulmer (2002) says that "exploration of higher dimensional fractals requires a more sophisticated 3D user interface, more complex visualization techniques, and places higher demands on the capabilities of the graphics hardware." Virtual reality allows for this interface, and provides the additional benefit of allowing students to observe the fractals from any direction they wish. Some additional affordances need to exist in VR however so that students can zoom into the fractal and observe the self-repeating patterns at different scales. It is well known that most students have difficulty visualizing three dimensional objects drawn in two dimensional form, so one assumes it is equally difficulty to understand a three dimensional fractal. VR helps mitigate this problem.
It is also possible to represent chaotic attractors in VR, which "can still be conveyed with a significantly lower number of primitives and with careful design". Such a representation is a powerful tool for improving student understanding of a difficult topic. Unfortunately "[VR] is unlikely to be suitable for real time or animated fractal systems" because of limitations in computer processing time when rendering complex chaotic shapes in three dimensions.
Caprotti and Sepp (2007) suggest using Second Life, a popular online virtual reality to study economic transactions, geometric transformations, vector calculus, and mathematical scripting. As Caprotti (2007) says, "I am now devouring mathematical formulae as if they were chocolate cake, and for the first time I'm experiencing the joys of geometry." As he points out, interest in doing mathematics is renewed by the novelty of working in an electronic and interactive three dimensional space.
According to Deubel (2007), "[e]verything is created using prims, which are basic geometric solids that are put together and stretched to form a new object." Clearly the use of geometric solids and transformations of those solids into more complex shapes involves some visualization of advances mathematics, even if the VR viewer simplifies the process of doing those transformation, and adding the geometric shapes. This kind of visualization is normally extremely difficult for students.
Virtual worlds have the ability to foster the learning of both spatial visualization and spatial orientation; both are skills which are critical to understanding three dimensional geometry. Students can even transfer into geometric objects in some worlds, allowing them to virtually become a geometric solid.
Virtual reality systems can also allow for scripting or programming within the system  which is highly related to mathematics. As Feurzeig (1970) suggests that computer programming helps students development mathematical "literacy about the process of solving problems."
Virtual reality has been used to teach 9th grade mathematics Winn (1992) suggests that "A student's understanding of Algebra could be guided by the ways in which the rules of Algebra are programmed to act in the virtual world."
Obviously one of the greatest strengths of a virtual reality is that the students are not visiting the world alone. Instead, students can meet at a mathematical construct embedded in the virtual space to view and manipulate it in the context of a community of other learners. According to Dickey (2005), "[r]esearch in educational VR reveals that 3D interactive environments provide support for constructivist-based learning activities by allowing learners to interact directly with information from a first-person perspective." This social interaction within the virtual reality allows students to have authentic learning experiences "which closely mimic the real world."
Virtual worlds are definitely helpful for teaching math, provided the instructor has some understanding of how to guide the student through the virtual world.  In his paper on using Second Life in the classroom, Conklin suggests over 100 ways to use this new virtual world in our real world education. Of those ideas, many of them are mathematical in nature.This diversity of ideas for use is represented in the virtual realities themselves which allow for nearly infinite possibilities.
- Bell, J. & Fogler, H. (1995). The investigation and application of virtual reality as an educational tool. In Proceedings of the American Society for Engineering Education. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.111.8371&rep=rep1&type=pdf.
- Sourin, A., Sourina, O., Wei, L. & Gagnon, P. (2009). Visual immersive haptic mathematics in shared virtual spaces. Transactions on Computational Science III, LNCS5300, Springer 5300, 2009. Retrieved from http://www.springerlink.com/index/5q16n147wk06822w.pdf. Cite error: Invalid
<ref>tag; name "sourin" defined multiple times with different content Cite error: Invalid
<ref>tag; name "sourin" defined multiple times with different content
- Taxen, G. & Naeve, A. (2001). CyberMath: A shared virtual environment for mathematics exploration. Center for User Oriented IT Design, Royal Institute of Technology, Technical Report CID-129, Stockholm, Sweden. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.13.1715&rep=rep1&type=pdf.
- Kwon, O., Kim, S. & Kim, Y. (2002). Enhancing spatial visualization through Virtual Reality (VR) on the web: Software design and impact analysis. Journal of Computers in Mathematics and Science Teaching 21 (1), 17--32. Retrieved from http://www.editlib.org/index.cfm/files/paper_10769.pdf?fuseaction=Reader.DownloadFullText&paper_id=10769.
- Bulmer, M. (2002). Teaching And Learning Mathematics With Virtual Worlds. In Proceedings of the 2nd International Conference on the Teaching of Mathematics (at the undergraduate level). Wiley. Retrieved from http://www.math.uoc.gr/~ictm2/Proceedings/pap42.pdf.
- Caprotti, O. & Sepp, M. (2007). Mathematics Education in Second Life. In Sixth Open Classroom Conference on real learning in virtual worlds. Retrieved from http://jem-thematic.net/files_private/EDEN_CaprottiSeppala.pdf.
- Deubel, P. (2007). Virtual worlds: A next generation for instruction delivery. Journal of Instruction Delivery Systems 21 (2). Retrieved from http://www.ct4me.net/Deubel_JIDS%20Volume%2021_2007Publication%20.pdf.
- Yeh, A. & Nason, R. (2004). Vrmath: A 3d microworld for learning 3d geometry. In World Conference on Educational Multimedia, Hypermedia & Telecommunications, Lugano, Switzerland. . Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.87.6815&rep=rep1&type=pdf.
- Bricken, M. (1991). Virtual reality learning environments: potentials and challenges. ACM SIGGRAPH Computer Graphics 25 (3), 178--184. Retrieved from http://portal.acm.org/ft_gateway.cfm?id=126657&type=pdf&coll=GUIDE&dl=GUIDE,ACM&CFID=79073725&CFTOKEN=37631068.
- Feurzeig, W., Papert, S., Bloom, M., Grant, R. & Solomon, C. (1970). Programming-languages as a conceptual framework for teaching mathematics. ACM SIGCUE Outlook 4 (2), 13--17. Retrieved from http://beyondbitsandatoms.stanford.edu/readings/class3/Feurzeig-1969-Programming%20languages%20as%20a%20conceptual.pdf.
- Winn, W. & Bricken, W. (1992). Designing Virtual Worlds for Use in Mathematics Education. . Retrieved from http://www.wbricken.com/pdfs/03words/03education/03iconic-math/07worlds-for-math.pdf.
- Dickey, M. (2005). Three-dimensional virtual worlds and distance learning: two case studies of Active Worlds as a medium for distance education. British Journal of Educational Technology 36 (3), 439--451. Retrieved from http://mchel.com/Papers/BJET_36_3_2005.pdf.
- Stone, S. (2009). Instructors' Perceptions of Three-Dimensional (3D) Virtual Worlds: Instructional Use, Implementation, and Benefits for Adult Learners. Unpublished doctoral disseration. Retrieved from http://www.lib.ncsu.edu/theses/available/etd-08152009-114119/unrestricted/etd.pdf.
- Conklin, M. (2005). 101 uses for Second Life in the college classroom. Games, Learning and Society 1. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.133.9588&rep=rep1&type=pdf.