Invited Paper - Virtual Environment: A Tool for Developing Students’ Abilities to Apply Mathematics to Real-life Problems

Abstract Virtual Environment: A Tool for Developing Students’ Abilities to Apply Mathematics to Real-life ProblemsIntroductionLong ago Freudenthal1 wrote “The huge majority of students are not able to apply theirmathematical classroom experiences, neither in the physics or chemistry school laboratory nor inthe most trivial situations of daily life” (p. 5). Freudenthal believed that mathematics wasneeded not by a few people, but by everybody and as such it should be taught to be useful foreverybody. He pointed out that the problem was not what kind of mathematics was taught, buthow it was taught. Even the fact that the teacher could apply mathematics himself did not implythat he knew how to use his knowledge in his teaching. It often happened that mathematiciansignored aspects of reality when doing mathematics.In recent research of 2007, Lesh & Zawojewski2 posed the same problem and asserted thatamong mathematics educators there was a common recognition, that a serious mismatch existedand was growing between the skills obtained at schools and the kind of understanding andabilities that were needed for success beyond school. Almost at the same time Ilyenkov3 wroteabout a current problem of ‘the practical application of knowledge to life’. Like Freudenthal in1968, Ilyenkov in 2009 saw the problem in how the subject was taught. He asserted that theattempts of some instructional theories to solve the problem by creating systems of rules of ‘howto apply knowledge to life’ impeded rather than helped. Moreover, Ilyenkov3 specified thesource of the problem, saying that ‘visual aid’ provided to students created only an illusion ofconcreteness of understanding because it was created independent of the activity of the student.That is, the decisive part of cognition - to go from the object to an abstract - remained outside ofstudent’s activity.The forty years old problem of teaching mathematics so that it would be connected with realityand useful for everybody appeared to be unresolved. “For all the talk about real-worldmathematics, it seems like we still don't get it”4.Many teachers and textbook writers have been working on the development of mathematicalschool tasks that resemble out-of-school situations. Palm and Burman5 reported that, in Finlandand Sweden, in many of the tasks encountered by students in school mathematics the situationdescribed in the task, was a situation from real-life. Such mathematical tasks, containingsituations from real-life, have been traditionally described by words and commonly referred tothe ‘word problems’. Word problems are firmly entrenched as a classroom tradition, particularlyin North American schools6, and yet, there has been long lasting debates about the reasons forthe lack of word problems’ effectiveness as a link between abstract mathematics and real-lifephenomena. Particularly, Gravemeijer7 noted that research on word problems has revealed thecomplex nature of the processes that lead to the lack of students’ activation of their real-worldknowledge. Palm8 stressed that in a large number of studies students did not pay much attentionto the realities of the situations described in the word problems. Gerofsky9, in turn, asserted thatword problems were unable to be faithful simulation of real-life tasks. She insightfully predictedthat there would appear new approaches based on new computer technologies.The contemporary computer technologies undoubtedly can provide much better than wordproblems simulations of real world situations in mathematical classrooms for connecting themathematical abstracts with out-of-school situations. The purpose of this study is to utilizecontemporary technologies, and particularly Virtual Environment (VE), for bringing the physicalworld into classrooms so that the students would be able to apply their mathematical knowledgeto the VE real-life problems and as such would develop their abilities to apply mathematics tobeyond schooling situations.Virtual Environment: A tool for simulation realityThe departure point in this study is the assumption that VE is a ‘reality’ for its users. In otherwords, VE is a contemporary technological tool which can ‘bring the reality’ into classroomsrepresenting the physical world situations with high degree of fidelity and immersion. Massara,Ancarani, Costabile, Moirano, & Ricotta10 claim that the immersion of the Second Life VEerases the difference between real and virtual worlds to the extent that, users’ psycho-physicalbehaviors in VR becomes consistent with real life. Meredith, Hussain, & Griffiths11 points outthat, investigators consider the Second Life VE as a synthetic world. Many “residents” of theSecond Life VE are escaping from their everyday real life into this synthetic world12 which inturn means that the VE synthetic world becomes a reality for VE users. The term ‘VirtualEnvironment’ is also known and widely used as ‘Virtual Reality’ (VR), which reflects its essenceof ‘reality’.Steuer13 asserts that “presence” and “telepresence” are fundamental for definition of VE. Heim14identified the following concepts of virtual environment: simulation, interaction, immersion,artificiality, telepresence, full-body immersion, and network communication. Among all theseconcepts the first three, simulation, interaction, immersion, are the most important for this studygoal of creating real-life in classrooms.Any VE is a simulation of the reality which in turn allows creating a variety of real-life situationsof different scales within a restricted space (e.g., a room, a computer screen). The majority ofcurrent virtual environments are based on visual displays presented either on a computer screenor through special stereoscopic equipment giving the perception of 3-D depth. In this researchthe choice of a computer screen VE is based on the fact that among five senses constitutinghuman sensory perception (smell, sight, taste, touch, and hearing), the sight is the dominant one.It is believed that not less than 70 percent of all sensory information that is imported into humansbrains comes from what we see15,16. There are a variety of technologies, algorithms andcomputer languages for simulations of physical reality on computer screens. For example,Campbell17 mentioned 3D web-based graphic languages such as Virtual Reality ModelingLanguage (VRML), eXtensible 3D (X3D) graphics language; proprietary languages such asJava3D (a 3D extension of Java) and Shockwave 3D.Such advances as 3-D stereoscopic equipment and equipment providing aural/haptic informationshould result in stronger feeling of presence in the environment, which in turn is commonlydefined as subjective feeling of “being there” and mainly conceived as deriving from immersion,interaction, and social and narrative involvement with suitable technology18. Brooks19 definedVE from the viewpoint of VE experience, reflecting both immersion and action. That is, VEshouldn’t be defined only by a number of technical advances; it should be connected directlywith experiencing in it. Why presence in terms of immersion and action is so important forsimulating ‘reality’ in mathematical classrooms? Immersion and action/interaction can providestudents with the perception of the body action, which, in turn, creates an embodied sense of theaction including such important calculus concepts as motion and velocity. Liljedahl20 showedthat it is unfavorable task to achieve perception and reflection of "motion and change" usingtraditional static methods of teaching and learning. Particularly, he described the difficulty ofteaching motion in the absence of the embodiment of motion as well as the tension that wascreated between an embodied sense of motion and its static representations.The concept of presence in VE can provide the perception of and reflection on properties ofsimulated real objects. In different VEs the body immersion can be achieved by different means.Campbell17 noted, Indeed, in an important sense virtual environments in fact are quite real and embodied, but in different ways. First, those who are engaged in virtual worlds, such as SL for instance, are having experiences – real lived experiences. Secondly, they are embodied, albeit virtually, through their avatars. (p. 592).One of the effective ways of immersion achievement is a choice of either egocentric orallocentric view perspectives, depending on individual preferences. According to Berthoz21, thebrain uses two frames of reference for representing the position of objects: egocentric andallocentric. For example, the relationships between objects in a room for estimation thedistances and angles can be encoded either ‘egocentrically’, or ‘allocentrically’. In the first case,everything is related to yourself; a second way means of encoding spatial relationships betweenthe objects themselves or in relationship to a frame of reference external to your body (ibid).That is, the egocentric perspective gives a perception of ‘being’ within the VE and seeing objectsfrom the ‘first person’ view. The allocentric perspective is provided when an avatar is present inthe environment and a learner controls the avatar navigation.The VE concept of interaction allows students to manipulate and transform the simulatedobjects. The interactive function helps to overcome a major contradiction of traditional education- absence of correlation between schooled, passive knowledge and real object. According toIlyenkov3, one of the main reasons of absence of such correlation is that instead of a real-lifeobject students are given a ‘ready – made’ image of it as a substitute with no activity with theobject. As the result, the students encounter the object itself only outside of the school. Tall22wrote “It is possible to design enactive software to allow students to explore mathematical ideaswith the dual role of being both immediately appealing to students and also providingfoundational concepts on which the ideas can be built.” (p. 20). From the viewpoint of inactivity,VE is exactly the type of software Tall22 wrote about in 1991.Space is an additional background concept which is introduced in this study. Space connects VE,physical reality, and mathematics. It is a fundamental concept of geometry and calculus.Immersion, action, interaction – they all depend on our subjective perception of simulated space.Space and objects on computer screen of VEs have been modeled and programmed usingdifferent graphic languages. Campbell17 showed that the majority of contemporary computerlanguages are designed for programming 3D space and 3D objects in space. Actually, even oneof the earliest software package used for design VE objects, namely Computer-Aided Drafting(CAD) represented 3D objects in 3D Cartesian co-ordinate system. VEs with stereoscopicequipment, giving the perception of 3-D depth, provide better 3D perception.For simulating a real-life problem for this particular research we chose the Second Life VE.Many recent publications are devoted to Second Life VE, to its popularity andapplication10,11,12,17,23. Second Life is an accessible (http://www.secondlife.com) and easy usingVE. It has 3D computer graphics and high fidelity; it provides egocentric and allocentric viewperspectives, becoming a reality for its users.Realistic Mathematics EducationThe realistic essence of VE and the aim of this study of utilizing VE for ‘bringing reality intoclassroom’ for application corresponding mathematical knowledge, determined the studytheoretical framework. Realistic Mathematics Education (RME) is used as an appropriate theoryfor experimental design and data analysis. Particularly, the ways of students’ application ofmathematics to a simulated in the Second Life VE real-life optimal navigation problem areanalysed through the lenses of mathematizing and modelling.RME is a teaching and learning theory in mathematics education which is based onFreudenthal’s idea that mathematics must be connected to reality. The use of realistic contextsbecame one of the determining characteristics of this approach. The subject matter should beexperientially real to students24. In addition to being ‘experientially real’, the RME contextimplies that it also should be suitable for mathematizing, which in turn is a fundamentalcharacteristic of RME. Freudenthal1 wrote that, mathematics can best be learned bymathematizing. His primary focus was on mathematizing reality in the common sense meaningof the real world out there.Treffers25 formulated the idea of ‘progressive mathematizing’ as a sequence of two types ofmathematical activity – horizontal mathematizing and vertical mathematizing. Horizontal onedescribes transforming a problem field into a mathematical problem. Treffers25 suggests thathorizontal mathematizing is constituted by nonmathematical problem field or related to realworld situation. He affirmed that “in the horizontal component the way towards mathematics ispaved via model formation schematizing, symbolizing” (ibid, p. 247). Vertical mathematizing isgrounded on horizontal one and includes such activities as reasoning about abstracts, structures,generalization and formalizing within the mathematical system itself.This study also uses the RME modeling principle for data examining. The important feature ofRME models is that they should support progression in vertical mathematizing without blockingthe way back. Another requirement for models in RME is that they can be re-invented by thestudents on their own and should be easily adapted to new situations. According to Streefland(cited in Van den Heuvel-Panhuizen26), models can fulfill the bridging function between theinformal and the formal level: by shifting from a ‘model-of’ to a ‘model-for’. At first, the modelis a model-of a situation that is familiar to the students. By a process of generalizing andformalizing, the model eventually becomes an entity on its own. It becomes possible to use it asa model-for mathematical reasoning.Finally, the RME guided reinvention principle is adopted and used as a part of methodology.According to RME instructional design theory, the teacher provides guidance, playing a‘proactive role’ within the classroom setting. This study allows every student to decide whetherand to what extent s/he needs guidance which in turn is provided in the specially designedguiding-reflecting journal.Participants and MethodsTen students from Vancouver Templeton Secondary School ranging in age from 17 to 18 years,5 males and 5 females, participated in the study. They were at the end of AP calculus course andhad completed such topics as application and computation of derivatives. The students wereinformed about the goal of the research and that the experiments would be conducted in theschool’s Teacher’s room, outside of regular calculus class time and that each session would last60-90 minutes.The simulated in the Second Live VE setting includes a pond with shallow water, surrounded bybushes and trees (Figure 1). Figure 1. Simulated in the Second Life VE SettingThe environment is programmed so that walking/running speed on land is twice as fast aswalking/running speed in water. There are two green small round platforms: one platform islocated on land near the water’s edge, another is located in the water (Figure 1). The optimalnavigation VE Task is to find the path between the platforms which would minimize time oftravel. The setting is programmed to record time and distance traveled by land for each tripbetween the platforms. This information is indicated on white banners, one of which is shown in(Figure 1). The SL allows utilizing both egocentric and allocentric view perspectives.After each trip the student must transfer the data from the banners (time and distance traveled byland) into a specially designed guiding–reflecting journal, which is an integral methodologicalpart of the research design.The guiding–reflecting journal contains the instructions for VE activity followed by someinformation about water and land components of trips. The journal provides tables for 10forward and return trips, then the student has to record the best (minimal) time out of these tripswith corresponding value of distance traveled by land. Then the journal suggests the student totry to solve the problem mathematically with blank space for independent reasoning. At thispoint student chooses whether to develop his/her own model-of the situational problem or toaccept the model offered in the journal.The experimental design contains a few stages. The first stage is a so called ‘exploration trial’ asin (Mueller, Jackson, & Skelton27; Ross, Skelton, & Mueller28). This first stage is the students’free activities in the VE before they get the journal with instructions for the VE Task. Theexploration trial is with unlimited duration until the student feels comfortable in the environmentand announces that s/he is ready to start the next stage. This first stage allows students toexplore the pond with its shallow water and to feel the speed difference on land and in water.The students have an opportunity to try egocentric and allocentric view perspectives and tochoose the preferable one. Altogether, the goal of exploration trial is to let students get feeling of‘being’ in the environment before starting the next, second, stage of the designed study which isthe optimal navigation VE Task. At the beginning of the second stage students receive theguiding–reflecting journals with instructions. Their optimal navigation in VE is accompanied byworking with the journal. The third stage of the designed study is mathematizing the VE activitywhich implies the journal work only. The final, forth, stage of the experimental design refers tocompleting the journal’s questionnaire.The data is drawn from 3 sources: video recording of students’ mathematizing in the guiding-reflecting journals; screen capturing of their VE activities; guiding-reflecting journals.Results and discussionsThe activities of five students, named Kenneth, Jason, Nick, Kate and Ann, were analysed. Thesefive students were chosen out of ten participants because they performed five different ways ofmathematizing which in turn allowed exploring the differences.Table 1 bellow demonstrates data which were integrated so that to consider connections betweenthe duration of exploration trial, computer game experience, and the first trip strategy. The lastcolumn of the table shows the students’ graphical models-of the situational problem. Thiscolumn allows seeing whether the best time trip strategy impacted the construction of themodels-of the situational problem.Analysis of the two first columns shows that the longer computer game experience, the less timestudents required for exploration trial. Particularly, Jason and Kate had the longest computergame experiences and spent the shortest time for the exploration trial. Their first trip strategy ofincreasing land distance was determined by the information that speed on land is faster thanspeed in water which they received from the journal. Kenneth, Nick, and Ann spent longer timefor exploration trial, and navigated in both media; they also knew about speed difference fromthe journal but all of them chose the shortest distance between the platforms strategy. Table 1. Integrated Data of Five Participants’ Activities.Comparison of the two last columns shows that the empirical best time trip strategies determinedthe graphical models-of the situational problem. Particularly, Kenneth’s and Nick’s best timetrips were the trips of balanced land and water distances (with water distances between theirminimal and maximal values) which, in turn allowed them to create models-of their empiricalactivity transferrable to model-for further mathematical development. Kenneth then appliedcalculus as the mathematical development of his model-of the situational problem. Nick usedgeometrical approach and was very persistent in developing his own model; at some point hepreferred to accept and develop the journal model. All the other participants’ best time trips wereeither maximum or minimum water distances which determined their right rectangle graphicalmodel’s-of their empirical activity and were not subjects to mathematical development for thereason of their stability and absence of variables to explore. They all accepted the journal model.Ann was the only student who was not able to follow the journal guidance. She neededadditional detailed explanation of how to apply calculus to her VE activity. After suchexplanation she exclaimed, “Now it all makes sense!” meaning the material she was taught in hercalculus course. This is an important point. This means that if students in the classroom areprovided the opportunity to apply mathematics to their own VE activity, the learning materialmakes more sense to them.The last two columns of Table 1 demonstrate the robustness of empirical knowledge obtainedfrom activity in VE: in all five cases best time trips determined horizontal mathematizing ofconstruction graphical modes-of the situational problem.A background assumption which was made at the beginning of the research was that VEtechnology, and particularly the Second Life VE provides simulation on the computer screensclose to the reality; and the real-life problems simulated in VE can be considered as the problemsof real physical world. The fact that all five students, Kenneth, Jason, Nick, Kate, and Ann,developed their models-of the particular situational problem on the basis of their empiricalactivity in VE suggests that the Second Life VE indeed provides simulation close to reality. Assuch it can be used for developing students’ abilities to apply mathematics to real-life problems.Bibliography1. Freudenthal, H. (1968). Why to teach mathematics so as to be useful? Educational Studies in Mathematics, 1, 3–8.2. Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. K. Lester (Ed.). Second Handbook of research on mathematics teaching and learning, (pp. 763-805). Greenwich, CT: Information Age Publishing.3. Ilyenkov, E.V. (2009). The Ideal in Human Activity. New York: Lawrence & Wishart.4. Stocker, D. (2006). Re-thinking real-world mathematics. For the Learning of Mathematics, 26(2), 29-30.5. Palm, T., & Burman, L. (2004). 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