Teaching Spatial Geometry in a Virtual WorldUsing Minecraft in Mathematics in Grade 56

Klaus-Tycho FoersterAalborg University

ktfoerstercsaaudk

AbstractmdashSpatial geometry is one of the fundamental mathe-matical building blocks of any engineering education However itis overshadowed by planar geometry in the curriculum betweenplayful early primary education and later analytical geometryleaving a multi-year gap where spatial geometry is absent atlarge Hence we investigate the usage of Minecraft as tool tohelp bridge said gap as the virtual worlds of Minecraft allowchildren to create three-dimensional objects in a constructive andalgorithmic way We study two learning scenarios in grade 56with 103 students reporting on our amp the childrensrsquo experiencesBased on our findings we believe Minecraft to be a valuablemathematical tool that can be easily used to augment the currentcurriculum

I INTRODUCTION AND MOTIVATION

Geometry is one of the cornerstones of mathematics andengineering education making it relevant for all stages ofteaching from kindergarten to graduate school In this contextspatial geometry is especially important for designing andunderstanding real-world phenomena and constructions Assuch it is included in most parts of the K-12 curriculum withspecial focus on early primary education and later analyticalgeometry In between it is however absent at large cf [1]Even worse spatial geometry has often been reduced tocalculations [2] mostly ignoring spatial concepts [3]

However students have more experiences with spatial thanplanar geometry after the first years of primary school aug-mented by their daily lives [4] Therefore a further explorationof spatial geometry should not be delayed until the end of thelower secondary [3] especially as the studentsrsquo susceptibilitygrows significantly around the age of 12 in particular forgirls [5]

Accordingly Maier [6] suggests that the concepts of spaceshould be developed by teaching spatial geometry As thecreation of real-world models can be time- and cost-intensiveLuig and Straesser [7] propose to use geometry software inthe classroom backed by Schumann [8] Especially for spatialgeometry the use of computers allows a transition to digitalPestalozzian ideals cf [9] giving more importance to theconcepts of solids [2] and allowing traditional learning goalsto be reached by new technologies [10]

There is no lack [11] of (interactive) 3D geometry soft-ware for use in schools eg Archimedes Geo3D [12] [13]GeoGebra [14] [15] or POV-Ray [16] [17]

The listed programs however suffer from the downside thatthey require the explicit use of a three-dimensional coordinate

Klaus-Tycho Foerster is supported by the Danish Villum Foundation

system making them harder to grasp than simple planargeometry According to Kadunz and Straesser the studentseven need to mentally construct a numerical image due to theuse of coordinates [1]

Fig 1 Spatial constructions in Archimedes Geo3D [12] require the explicituse of three-dimensional coordinates making it harder to intuitively use inthe context of elementary education

Elementary spatial geometry programs for children areuncommon cf [3] The only example of educational softwarethat we are aware of in this area is BAUWAS [18] [19] whereblocks can be placed in a 10times 10times 10 grid see Figure 2

Fig 2 Screenshot created with BAUWAS [18] Blocks can be createddeletedby simply clicking with the leftright mouse button

However we would like to have a program beyond the sizerestrictions of 10 blocks in every dimension allowing alsofor collaborative possibilities and an incentive for the studentsto use the program in their own free time [24] As thus inthis article we investigate the choice of the popular programMinecraft [20] to augment the the teaching of spatial geometryin grades 56 In particular we design and evaluate classroomstudies coming to the conclusion that Minecraft can be usedas a valuable tool to augment the concept of space

(a) Screenshot taken in Minecraft [20] (b) Uncanny valley effect from [21] (c) Screenshot in the CryENGINE 2 from [22]

Fig 3 The uncanny valley hypothesis first proposed by Mori cf [23] states that non-perfect depictions of humans are easily identified by an observercausing revulsion and feelings of discomfort On the other hand clearly artificial scenes such as in Figure 3a are more pleasant than the previously mentionedas there is no expectation of perfection Current video games need a high level of modification to bridge the uncanny valley as attempted in Figure 3c

A Overview of this Article

In the following Section II we start by covering (relatededucational work on) Minecraft and its deployment in theclassroom Then in Section III we discuss our goals andclassroom studies to augment the understanding of spatial ge-ometry in grades 56 followed by a qualitative and quantitativeassessment in Section IV also briefly covering a grade 12trial in Section V Lastly we conclude in Section VI with asummary and outlook on future work

II MINECRAFT

Minecraft is a first-person video game released in Novem-ber 2011 by the company Mojang with the main developersbeing Markus ldquoNotchrdquo Persson and Jen Bergensten [25] [26]In total more than 100 million copies of Minecraft were soldworldwide so far [27] with its immense popularity being thereason why Mojang was bought by Microsoft in late 2014 for25 billion dollars [28]

Minecraft can be roughly summarized as a (collaborative)sandbox game where Lego-like blocks can be placed and re-moved Besides this building mode (coined ldquocreative modeldquo)there is also a ldquosurvival moderdquo available where the playersstart with an empty inventory and have to gathermine alldesired resources while the world is populated with friendlynon-player characters animals and hostile monsters

Even though Minecraft and its clones might like outdatedin terms of todayrsquos video game graphics its vintage lookdoes not hinder its popularity ndash it might even have helpedpopularize it cf Figure 3 Furthermore the procedural worldgeneration allows for all sorts of landscapes in a virtuallyunlimited environment as illustrated in Figure 4

Fig 4 Depiction of the maximum size of a Minecraft world assuming eachblock having a size of 1m3 from [29]

A Minecraft in Educational Use

The phenomenon Minecraft1 quickly found its way intoeducation and research as well Already shortly after itsrelease articles discussed the classroom use of Minecraftfor topics such as biology ecology physics chemistry andgeology and geography eg [30] In regards to programmingmodifications of Minecraft allow for interesting possibilitiesof visual coding with methods such as coding blocks [31] inCodeBlocks [32] or turtle coding [33] [34] in ComputerCraft-Edu [35] We refer to the survey of Nebel et al [25] for anencompassing literature review also covering further schoolsubjects

Specifically for mathematics Bos et al [26] let students(third grade) explore area and perimeter using Minecraftgiving also additional Minecraft mathematical activities forstudents in elementary education Eg reasoning with shapesgeometric measurements analyzing patterns or scaling Forthe age group 7-9 Al Washmi et al [36] also considermathematical activities in a modified version of Minecraftusing a pre-built world with tasks to perform by the studentsKoslashrhsen and Misfeldt [37] perform an ethnomathematicalstudy by monitoring seven 10-year olds in an afterschoolMinecraft program describing their findings in terms ofBishoprsquos categories [38] and elaborating on the collaborativeeffect of playing Related to spatial involvement Nguyen andRank [39] present a digital training regimen in Minecraftwhere they plan to evaluate mental rotation performance infuture work Lastly Floyd [40] reasons that ldquoMinecraft clearlypromotes the development of spatial abilitiesrdquo by comparingthe possible actions in Minecraft with related work in spatialreasoning skills development literature

B Deploying Minecraft for Classroom Studies

The full version of Minecraft is available for most recentversions of WindowsMacLinux and also (at a reduced price)for most gaming consoles and mobile operating systemsHowever the single client license cost of sim26 USD2 isprohibitive for most school budgets There is also an enhanced

1From now on we are going to use Minecraft as a synonym for Minecraft-like software similar to the use of Google as a verb ndash unless noted otherwise

2At httpsminecraftnetenstore price as of November 2016

(a) Screenshot taken in Minecraft [20] (b) Uncanny valley effect from [21] (c) Screenshot in the CryENGINE 2 from [22]

Fig 3 The uncanny valley hypothesis first proposed by Mori cf [23] states that non-perfect depictions of humans are easily identified by an observercausing revulsion and feelings of discomfort On the other hand clearly artificial scenes such as in Figure 3a are more pleasant than the previously mentionedas there is no expectation of perfection Current video games need a high level of modification to bridge the uncanny valley as attempted in Figure 3c

A Overview of this Article

In the following Section II we start by covering (relatededucational work on) Minecraft and its deployment in theclassroom Then in Section III we discuss our goals andclassroom studies to augment the understanding of spatial ge-ometry in grades 56 followed by a qualitative and quantitativeassessment in Section IV also briefly covering a grade 12trial in Section V Lastly we conclude in Section VI with asummary and outlook on future work

II MINECRAFT

Minecraft is a first-person video game released in Novem-ber 2011 by the company Mojang with the main developersbeing Markus ldquoNotchrdquo Persson and Jen Bergensten [25] [26]In total more than 100 million copies of Minecraft were soldworldwide so far [27] with its immense popularity being thereason why Mojang was bought by Microsoft in late 2014 for25 billion dollars [28]

Minecraft can be roughly summarized as a (collaborative)sandbox game where Lego-like blocks can be placed and re-moved Besides this building mode (coined ldquocreative modeldquo)there is also a ldquosurvival moderdquo available where the playersstart with an empty inventory and have to gathermine alldesired resources while the world is populated with friendlynon-player characters animals and hostile monsters

Even though Minecraft and its clones might like outdatedin terms of todayrsquos video game graphics its vintage lookdoes not hinder its popularity ndash it might even have helpedpopularize it cf Figure 3 Furthermore the procedural worldgeneration allows for all sorts of landscapes in a virtuallyunlimited environment as illustrated in Figure 4

Fig 4 Depiction of the maximum size of a Minecraft world assuming eachblock having a size of 1m3 from [29]

A Minecraft in Educational Use

The phenomenon Minecraft1 quickly found its way intoeducation and research as well Already shortly after itsrelease articles discussed the classroom use of Minecraftfor topics such as biology ecology physics chemistry andgeology and geography eg [30] In regards to programmingmodifications of Minecraft allow for interesting possibilitiesof visual coding with methods such as coding blocks [31] inCodeBlocks [32] or turtle coding [33] [34] in ComputerCraft-Edu [35] We refer to the survey of Nebel et al [25] for anencompassing literature review also covering further schoolsubjects

Specifically for mathematics Bos et al [26] let students(third grade) explore area and perimeter using Minecraftgiving also additional Minecraft mathematical activities forstudents in elementary education Eg reasoning with shapesgeometric measurements analyzing patterns or scaling Forthe age group 7-9 Al Washmi et al [36] also considermathematical activities in a modified version of Minecraftusing a pre-built world with tasks to perform by the studentsKoslashrhsen and Misfeldt [37] perform an ethnomathematicalstudy by monitoring seven 10-year olds in an afterschoolMinecraft program describing their findings in terms ofBishoprsquos categories [38] and elaborating on the collaborativeeffect of playing Related to spatial involvement Nguyen andRank [39] present a digital training regimen in Minecraftwhere they plan to evaluate mental rotation performance infuture work Lastly Floyd [40] reasons that ldquoMinecraft clearlypromotes the development of spatial abilitiesrdquo by comparingthe possible actions in Minecraft with related work in spatialreasoning skills development literature

B Deploying Minecraft for Classroom Studies

The full version of Minecraft is available for most recentversions of WindowsMacLinux and also (at a reduced price)for most gaming consoles and mobile operating systemsHowever the single client license cost of sim26 USD2 isprohibitive for most school budgets There is also an enhanced

1From now on we are going to use Minecraft as a synonym for Minecraft-like software similar to the use of Google as a verb ndash unless noted otherwise

2At httpsminecraftnetenstore price as of November 2016

education version available with a more reasonable price of5 USD3 This version only runs on Windows 10 or macOSneither being available at the high school in Northern Germanywhere the teaching took place

As we only needed basic building functionalities and es-pecially wanted the students to also being able to continuetheir work at home we thus opted for free versions ofMinecraft In the first cross-sectional study we used MinecraftClassic [41] with a browser-based client As Minecraft Classicwas removed in Fall 2015 [41] we then used the free alter-native ClassiCube [42] combined with MCGalaxy [43] as aserver Both ClassiCube and MCGalaxy have low hardwarerequirements allowing them to run on dated school hardware(in our case from asymp 2010)

III SETTING OF THE CLASSROOM STUDIES

In this section we will describe our settingsdesign andgoals for our classroom studies extending our earlier pre-liminary technical report [44] We chose Minecraft for ourclassroom studies as we were looking a for software whichis 1) available in a free version 2) which students would usein their own time at home and 3) where the students cancollaborate4

We designed two classroom studies the first one focusingon the transfer between the plane amp space [45] and collabo-rative building [37] and the second one for implementing thesuggestion of scaling by Bos et al [26]

In total we evaluated our scenarios with 103 students 69for the first setting and 12 for the second setting also addinga control group of 22 students Roughly half of all studentswere girls The students were also provided with instructionsfor Minecraft see Figure 5

Fig 5 Instructions (in German) for the students on how to use MinecraftStudents in grade 56 barely made use of them after a short period while afew students in grade 12 still consulted them throughout the first hour

3At httpseducationminecraftnet price as of November 20164We refer to the work of Franke [24] who also elaborates on that the

final products of the studentsrsquo geometric work do not need to be graded ndash anapproach we will follow in this article

A Collaboration and Transfer from the Plane to Space

First in a cross-sectional study (69 students 1 class in grade56 each 1 group of students in grade 5 in extra-curricularactivities) we wanted to see if the students could transfer theirdrawn objects from the plane to the space of the virtual worldif they would have problems with spatial orientation in thefirst-person view of Minecraft if already an empty world isenough to spark creativity and if collaboration at class size isfeasible

The 4 lectures (2 times 2) were structured as follows After ashort introduction the students would try working in Minecrafton their own then work together with a partner on a projectdesigned together and lastly the whole class together wouldsketch a project for the remaining 2 lectures In these thestudents would all build on a project collaboratively Thestudents would also be given the option to play at home ina provided (yet empty) world In the following week thestudents would participate in a questionnaire

B Scaling of Constructions

Second in an experiment (12 students from grade 6 with acontrol group of 22 students in grade 6) we were interested inscaling constructions in Minecraft and its short-term impacton spatial skills

The 2 lectures (2times1) were structured as follows After a pre-questionnaire the students would sketch a desired constructionand then build it twice once at normal size and once atdoubled dimensions In the second lecture the scaling processwould be inverted scaling the dimensions down to half thesize concluded by a post-questionnaire The control groupwould be given one questionnaire to compare the two groups

IV EVALUATION OF THE CLASSROOM STUDIES

In this section we present our and the studentsrsquo experiencesduring the classroom studies We begin by describing thedetails of the first cross-sectional study in Section IV-Afollowed by the second experiment in Section IV-B

A First Classroom Cross-Sectional Study

Construction in Minecraft is performed from a first-personview ie from a higher level of abstraction as the (avatarrsquos)viewpoint is inside the to be performed task itself Notwith-standing we observed that all students except for one hadno problems orienting themselves in the virtual world ofMinecraft The students agreed with this observation cf Fig-ure 6d Similarly the students quickly mastered controllingthe avatar cf the caption of Figure 5 and were immideatelyable to construct their first ad-hoc projects alone

Afterwards the students sketched a small project in pairsand built them in Minecraft see the left four pictures inFigure 9 We observed that the students were able to transfertheir two-dimensional sketches to three-dimenstional buildingsquite well performing some minor modifications on the goThe sketches in grade 5 were still very planar (see Figures 7a7d) while the sketches in grade 6 were already more spatial(see Figures 7b 7e)

86

14

MinecraftPaper

(a) Would you rather draw on paperor build in Minecraft

95

5

MinecraftLego

(b) Would you rather build inMinecraft or Legobuilding blocks

51

49

YesNo

(c) Would you play Minecraft withyour parents

99No

(d) Did you have issues orientingyourself in the world of Minecraft

Fig 6 Evaluation after the cross-sectional study with n = 69 (rounded ) Regarding the question of playing with their own parents in Fig 6c some studentsanswering no noted that their parents would probably not see a relation between Minecraft and mathematics as something fun cannot be mathematics

(a) Sketch of a castle (grade 5) (b) Sketch of a cubic building (grade 6) (c) Outside of the collaboratively built cinema

(d) Construction of the castle above (e) Construction of the cubic building above (f) Inside the collaboratively built cinema

Fig 7 Examples of constructions from the cross-sectional study Especially the sketches in grade 6 were already more spatial

The results of the following questionnaire are shown inFigures 6 and 105 Most students responded that they preferredbuilding in Minecraft to alternatives such as drawing on paperor building with Legobuilding blocks see Figures 6a and 6b

For the conclusion of the first two lectures the studentsdiscussed what sort of construction they would like to collab-oratively build in the next two lectures and set up rules forcollaborative building (also for at home) The class in grade 5built a castle (cf Figure 8) the class in grade 6 built a parkand the extra-curricular group built a cinema see Figures 7fand 7c All groups took great care to collaboratively buildwithout upsetting others but their building plan was morespontaneously Students looked where they could help finish-ing building and then moved their avatar there to continueconstructing We suspect that for larger and more complexprojects we would need to guide the students beforehand tocreate a (modularized) building plan

While the constructions for the castle and the park focusedjust on the construction the extra-curricular group decidedto also host an event once their cinema was done Theyperformed a short play on the stage of the cinema (behindthe glass in Figure 7f) with some students while the otherstudents watched from their seats Especially the collaborativebuilding fascinated the students which also showed in their

5Eggeling [46] performed a study with six students using also the questionsin 6 amp 10 except for 6c She notes [46 p 63] that the studentsrsquo answersmostly correspond to ours drawn from our preliminary technical report [44]

interest to build in their free time from home (we did not givethe students any ldquohomeworkrdquo it was left to them as their owndecision) Roughly 80 of the students played on these worldsat least once from home more than half at least for 6 hoursin a two week timespan We initialized the small worlds tobe empty (flat) and noticed that interest in each world wentdown the ldquofullerrdquo the worlds became with interest rising againonce we proved new worlds We suspect that the fascinatingpart of Minecraft is not so much interacting with an alreadyexisting environment but rather the construction itself Oncethe construction is deemed finished it is not re-iterated orextended later but rather a new project is started

One further observation that surprised us is that we alsosaw no signs of vandalism in the freely accessible worldsand that we received no complaints from any student thattheir construction was destroyed We believe that this effectis rooted in the communal rule-generation eg in Figure 8and collaborative building in class We did not perform thesesteps in grade 12 observing some vandalism cf Section V

Fig 8 Notes (in German) from a student regarding collaborative play Firstshe noted that she would like to build a castle collaboratively but that thiscastle should belong to everyone Second she pointed out that everyone shouldrespect the constructions of others

(a) Collaborative building at home (grade 5)

(b) Collaborative building at home (grade 6)

Fig 9 We provided an empty world for the students to build at home foreach group in the cross-sectional study While the students in grade 5 mostlyconstructed (independent) buildings next to each other (Fig 9a) the world ingrade 6 was more organized (Fig 9b)

Similar to the transfer from paper to space in grades 5 and 6we observed that the building from home was more structuredin grade 6 leading to more organized virtual worlds see thetwo exemplary screenshots in Figure 9

Lastly according to the sudentsrsquo opinions they preferredcollaborative building where they could construct togetherwith the whole class cf Figure 10 We believe that this is oneof the great strengths and unique opportunities of Minecraftdeserving future investigations Minecraft allows for an easysetup of building together in large groups where all studentscan be immersed into the construction Without computer usesuch a task would be very demanding (money- and time-wise)and might not happen at all in everyday teaching As such webelieve Minecraft to be a useful additional tool for teachingspatial geometry in grades 5 and 6 which is also used bystudents in their free time

419

2057

alone

a few friends

a few classmates

whole class

Fig 10 Would you rather play Minecraft (with) Question posed to n = 69students from grade 56 after the cross-sectional study

B Second Classroom Experiment

In our smaller (12 students in grade 6) and shorter (2lectures) second classroom experiment we focused on the taskof scaling as an interpretation of multiplication in Minecraftcf Bos et al [26]

In the first lecture we started with a pre-questionnaireincluding one spatial geometry question cf Figure 11 Thenthe students first sketched a construction then built it atnormal size in 10 minutes and then tried to build it atdoubled scale in 20 minutes While the volume of a doubledconstruction (in all dimensions) is increased by 23 = 8 theouter hull block amount just increases by a factor of 22 = 4(ignoring constructions inside) As thus the students doublingthe constructions in all dimensions were not able to finish theirwork see eg the crane with missing features in Figure 12bor especially the house in Figure 12c Only one student wasable to completely finish his construction as he only scaledin two dimensions needing twice as many blocks in Figure12d This lead to discussions why scaling a construction by afactor of two is problematic in twice the construction time

One might think that reversing the process in half the timeshould be easy but there is the problem that Minecraft has anatomic block size Eg if a window has a size of one blockwhat size should it have when scaling the construction downThe students used two different approaches when scaling wasnot possible either ignore the feature such as in Figure 12g orscale the amount of the features as in Figure 12h Only fewstudents planned a construction that could be properly scaledas can be seen in Figure 12e

We concluded the two lectures with a post-questionnaire oftwo spatial geometry questions comparing the results with thesame questions asked in a control group of 22 students and theinitial pre-questionnaire As there was no delay in taking thequestions the effects have to be considered short-term effectsAs can be seen in Figure 12 both group behaved roughly thesame initially but the results of the students taking the twolectures improved strongly We would like to note that we didnot perform a classical mathematical lesson plan but just letthe students sketch and build in Minecraft as described

We believe that the propaedeutic algorithmic6 tasks per-formed in these two lectures resulted in the better results of thestudents We therefore think that the idea of scaling proposedfor Minecraft by Bos et al [26] is a promising conceptfor late elementary education warranting further exploratorystudies for its use regarding surface area and volume in themathematics curriculum

V FURTHER OBSERVATIONS IN GRADE 12

We also shortly introduced a class in grade 12 to Minecraftfor one hour and offered them as well the option to play athome on a world hosted by us We defer a detailed overviewof this study to a future article Notwithstanding there are twoobservations that we would like to mention

6We refer to the article [47] of Schmidt-Thieme for a general overview onthe importance of algorithms in (mathematical) education

(a) Never (4 CG5) (b) A little (1 CG5) (c) Few hrs (1 CG2) (d) Often (6 CG10)

Fig 11 Results in the second classroom experiment from the geometry questions in the questionnaire (prepost with 4 + 1 + 1 + 6 = 12 students) andthe control group (CG with 5 + 5 + 2 + 10 = 22 students) The captions denote how much the students played Minecraft before also showing how manystudents are in each group Responses in blue are correct incorrect ones are in orange The results before the lectures are roughly similar to the control group(except for question 1 in Figure 11a) but have improved to mostly correct results after the two lectures

(a) Sketch of the crane in Fig 12b (b) Construction of the crane fromFig 12a properly scaled

(c) Properly scaled house but toolarge to complete in time

(d) Properly scaled house in twodimensions but not in height

(e) Sketch of the construction inFig 12f each block outline shown (f) Properly scaled construction (g) Windows not properly scalable (h) Another take on scaling

Fig 12 Examples of constructions from the second classroom experiment The students all chose to build houses in the first lecture (top row) except for onestudent building a crane in Fig 12b In the second lecture all the students built constructions related to buildings ie merlons (battlements) in Fig 12f

1) Students in grade 12 that did not play Minecraft (orother first-person games) before had more problemsorienting themselves in the virtual world of Minecraftthan the corresponding group of students in grade 56They especially lacked the playful behavior of youngerchildren who accustomized themselves to the new en-vironment quickly Furthermore these students in grade12 also continued to consult the instructions in Figure 5from time to time unlike the students in grade 5 and 6

2) The students in grade 12 logged in at roughly similarlevels from home as the younger children but their col-laborative behavior was different Cooperative buildingwas limited most students built on their own Further-more some students actively damaged other buildingswhen nobody else was logged on ndash which did not happenin grade 56 Two female students noted that they weredisappointed by this behavior of their fellow classmateswhich they deemed inappropriate for their age

We did not study these occurrences in depth as we did notexpect them to happen beforehand However we believe thatfurther research in this area would definitely be interesting

VI SUMMARY AND OUTLOOK

We studied the usage of Minecraft as a tool to teach spatialgeometry in virtual worlds in grades 5 and 6 Specificallywe evaluated two classroom studies focusing on buildingpre-planned constructions cooperative building and scalingthree-dimensional constructions Based on our findings webelieve the appropriate classroom use of Minecraft to be anenriching experience regarding the concept of space at the endof primary education However we would like to emphasizethat we do not want to advocate for Minecraft replacing partsof the current education curricula Rather we see Minecraftas a useful tool to augment teaching in the curriculum ldquoTobe used when appropriate not to be applied when other toolsare more useful and not taught for its own sakerdquo [48]

The mentioned related work in Section II lists many morepossibilities for the further use of Minecraft We think thatespecially the usage of programmable agents in this context ispromising as it allows for visual debugging by observing theagentsrsquo actions in a virtual world ndash a topic we plan to explorefurther at undergraduate level [34]

Another future opportunity is the use of Virtual Reality (VR)for spatial geometry or concepts of space [49] [50] [51] withsome research ideas being already over 15 years old [52][53] Since recently Minecraft is also available in a VRversion [54] with Azmandian et al [55] already exploring thepossibilites of haptic targeting eg to build a castle [56] Oncethe VR hardware7 and appropriate software becomes widelyavailable to schools we anticipate exciting new directions foraugmented learning in the classroom

Lastly we would like to conclude with a statement of astudent in Figure 13 with which we wholeheartedly agree

Fig 13 ldquoThe whole game consists of mathrdquo ndash Response from a student ingrade 6 when asked about how Minecraft relates to mathematics

ACKNOWLEDGEMENTS

We would like to thank the anonymous reviewers of the8th IEEE Global Engineering Education Conference (IEEEEDUCON) for their helpful comments on our submission

REFERENCES

[1] G Kadunz and R Straesser Didaktik der Geometrie in der Sekun-darstufe I [Didactics of Geometry in Lower Secondary] HildesheimFranzbecker 2009

[2] M Ludwig and H-G Weigand Didaktik der Geometrie fuer die Sekun-darstufe I [Didactics of Geometry in Lower Secondary] SpektrumHeidelberg 2009 ch Konstruieren [Constructions] pp 55ndash80

[3] H-J Elschenbroich Mathematik-Didaktik [Didactics of Mathematics]Cornelsen Berlin 2003 ch Unterrichtsgestaltung mit Computerunter-stuetzung [Creating Teaching with Computer Support] pp 212ndash233

[4] M Ludwig ldquoVorlesungskapitel Propaedeutische Geometrie in Klasse5 und 6 [Lecture Chapter Propaedeutics of Geometry in Grades5 and 6]rdquo Available at httpwwwmathuni-frankfurtdesimludwigvorlesungenskriptendidgeoskript didgeohtml 2007

[5] B Andelfinger Geometrie [Geometry] Soest Landesinstitut fuerSchule und Weiterbildung 1988

[6] P H Maier Raeumliches Vorstellungsvermgen Ein theoretischer Abrissdes Phaenomens raeumliches Vorstellungsvermoegen [Spatial Imagina-tion A Theoretical Outline of the Phenomenon of Spatial Imagination]Auer Verlag 1999

[7] K Luig and R Straesser ldquoFoerderung ausgewaehlter Aspekte derRaumvorstellung mit dynamischer Geometriesoftware [Promotion ofSelected Aspects of Spatial Representation with Dynamic GeometrySoftware]rdquo in Beitraege zum Mathematikunterricht 2009

[8] H Schumann Raumgeometrie Unterricht mit Computerwerkzeu-gen[Space Geometry Teaching with Computer Tools] CornelsenVerlag 2001

[9] M Ludwig ldquoRaumgeometrie mit Kopf Herz Hand und Maus [SpaceGeometry with Head Heart Hand and Mouse]rdquo in Beitraege zumMathematikunterricht 2001

7Currently ranging from hundreds (Oculus Rift HTC Vive) to thousands(Microsoft HoloLens) of dollars for a single headset as of November 2016A less powerful alternative is to use modern smartphones as a display forvirtual reality eg Google CardboardDaydream or Samsung Gear VR

[10] H-G Weigand and T Weth Computer im Mathematikunterricht NeueWege zu alten Zielen [Computers in Mathematics Teaching New Waysto Old Goals] Spektrum Akademischer Verlag 2002

[11] (2016) List of interactive geometry software 3D Software [Online]Available httpsenwikipediaorgwikiList of interactive geometrysoftware3D Software

[12] A Goebel Archimedes Geo3D [Online] Available httpwwwraumgeometriededrupalen

[13] mdashmdash Experimentieren im Geometrieunterricht [Experimentation inGeometry Teaching] Franzbecker 2006 ch Dynamische Raumge-ometriesoftware - ein ideales Werkzeug zum Experimentieren in derGeometrie [Dynamic Spatial Geometry Software - an ideal Tool forExperimentation in Geometry] pp 27ndash36

[14] M Hohenwarter et al GeoGebra [Online] Available httpswwwgeogebraorg

[15] M Hohenwarter and J Preiner ldquoDynamic mathematics with geogebrardquoAMC vol 10 p 12 2007

[16] POV-Ray [Online] Available httpwwwpovrayorg[17] POV Team and others ldquoPersistency of vision ray tracer (pov-ray)rdquo

Version 10 Tech Rep 1991[18] BAUWAS [Online] Available httpwwwbicsbeschuledeson

machmitswbauwasindexhtm[19] H Meschenmoser Computergestuetzte Konstruktion und Visualisierung

das Konstruktionsprogramm BAUWAS [Computer-Assisted Constructionand Visualization the Construction Program BAUWAS] BerlinMachmit-Verlag 1999

[20] M Persson and J Bergensten ldquoMinecraftrdquo Stockholm Sweden MojangAB 2011 [Online] Available httpminecraftnet

[21] Smurrayinchester (2007) Mori Uncanny Valley CC BY-SA 30 httpscreativecommonsorglicensesby-sa30legalcode [Online] AvailablehttpscommonswikimediaorgwikiFileMori Uncanny Valleysvg

[22] inCrysis (2009) Guide to the game Crysis - Screenshots and ToD art[Online] Available httpsakamsincrysis

[23] M Mori K F MacDorman and N Kageki ldquoThe uncannyvalley [from the field]rdquo IEEE Robotics Automation Magazinevol 19 no 2 pp 98ndash100 June 2012 [Online] Availablehttpdoiacmorg1101109MRA20122192811

[24] M Franke Didaktik der Geometrie In der Grundschule [Didactics ofGeometry in Primary School] Spektrum Akademischer Verlag 2007

[25] G D R Steve Nebel Sascha Schneider ldquoMining learning and craftingscientific experiments A literature review on the use of minecraft ineducation and researchrdquo Journal of Educational Technology amp Societyvol 19 no 2 pp 355ndash366 2016

[26] B Bos L Wilder M Cook and R OrsquoDonnell ldquoLearningmathematics through Minecraftrdquo Teaching Children Mathematicsvol 21 no 1 pp 56ndash59 2014 [Online] Available httpdxdoiorg105951teacchilmath2110056

[27] O Hill (2016 June) Wersquove sold Minecraft many many timesLook Mojang [Online] Available httpsmojangcom201606weve-sold-minecraft-many-many-times-look

[28] J Pagliery (2014 September) Microsoft buys Minecraft for $25billion CNN [Online] Available httpmoneycnncom20140915technologyminecraft-microsoft

[29] C Huang and M Huang (2012) The Scale of the Universe 2 [Online]Available httphtwinsnetscale2

[30] D Short ldquoTeaching scientific concepts using a virtual world -minecraftrdquo Teaching Science-the Journal of the Australian ScienceTeachers Association vol 58 no 3 p 55 2012

[31] C Zorn C A Wingrave E Charbonneau and J J L Jr ldquoExploringminecraft as a conduit for increasing interest in programmingrdquo inProceedings of the 8th International Conference on the Foundationsof Digital Games FDG 2013 Chania Crete Greece May 14-17 2013 G N Yannakakis E Aarseth K Joslashrgensen andJ C Lester Eds Society for the Advancement of the Scienceof Digital Games 2013 pp 352ndash359 [Online] Available httpwwwfdg2013orgprogrampaperspaper46 zorn etalpdf

[32] CodeBlocks (2012) [Online] Available httpsdevbukkitorgbukkit-pluginscodeblocks

[33] D Saito H Washizaki and Y Fukazawa ldquoInfluence of the programmingenvironment on programming educationrdquo in Proceedings of the 2016ACM Conference on Innovation and Technology in Computer ScienceEducation ser ITiCSE rsquo16 New York NY USA ACM 2016 pp 354ndash354 [Online] Available httpdoiacmorg10114528994152925477

(a) Collaborative building at home (grade 5)

(b) Collaborative building at home (grade 6)

Fig 9 We provided an empty world for the students to build at home foreach group in the cross-sectional study While the students in grade 5 mostlyconstructed (independent) buildings next to each other (Fig 9a) the world ingrade 6 was more organized (Fig 9b)

Similar to the transfer from paper to space in grades 5 and 6we observed that the building from home was more structuredin grade 6 leading to more organized virtual worlds see thetwo exemplary screenshots in Figure 9

Lastly according to the sudentsrsquo opinions they preferredcollaborative building where they could construct togetherwith the whole class cf Figure 10 We believe that this is oneof the great strengths and unique opportunities of Minecraftdeserving future investigations Minecraft allows for an easysetup of building together in large groups where all studentscan be immersed into the construction Without computer usesuch a task would be very demanding (money- and time-wise)and might not happen at all in everyday teaching As such webelieve Minecraft to be a useful additional tool for teachingspatial geometry in grades 5 and 6 which is also used bystudents in their free time

419

2057

alone

a few friends

a few classmates

whole class

Fig 10 Would you rather play Minecraft (with) Question posed to n = 69students from grade 56 after the cross-sectional study

B Second Classroom Experiment

In our smaller (12 students in grade 6) and shorter (2lectures) second classroom experiment we focused on the taskof scaling as an interpretation of multiplication in Minecraftcf Bos et al [26]

In the first lecture we started with a pre-questionnaireincluding one spatial geometry question cf Figure 11 Thenthe students first sketched a construction then built it atnormal size in 10 minutes and then tried to build it atdoubled scale in 20 minutes While the volume of a doubledconstruction (in all dimensions) is increased by 23 = 8 theouter hull block amount just increases by a factor of 22 = 4(ignoring constructions inside) As thus the students doublingthe constructions in all dimensions were not able to finish theirwork see eg the crane with missing features in Figure 12bor especially the house in Figure 12c Only one student wasable to completely finish his construction as he only scaledin two dimensions needing twice as many blocks in Figure12d This lead to discussions why scaling a construction by afactor of two is problematic in twice the construction time

One might think that reversing the process in half the timeshould be easy but there is the problem that Minecraft has anatomic block size Eg if a window has a size of one blockwhat size should it have when scaling the construction downThe students used two different approaches when scaling wasnot possible either ignore the feature such as in Figure 12g orscale the amount of the features as in Figure 12h Only fewstudents planned a construction that could be properly scaledas can be seen in Figure 12e

We concluded the two lectures with a post-questionnaire oftwo spatial geometry questions comparing the results with thesame questions asked in a control group of 22 students and theinitial pre-questionnaire As there was no delay in taking thequestions the effects have to be considered short-term effectsAs can be seen in Figure 12 both group behaved roughly thesame initially but the results of the students taking the twolectures improved strongly We would like to note that we didnot perform a classical mathematical lesson plan but just letthe students sketch and build in Minecraft as described

We believe that the propaedeutic algorithmic6 tasks per-formed in these two lectures resulted in the better results of thestudents We therefore think that the idea of scaling proposedfor Minecraft by Bos et al [26] is a promising conceptfor late elementary education warranting further exploratorystudies for its use regarding surface area and volume in themathematics curriculum

V FURTHER OBSERVATIONS IN GRADE 12

We also shortly introduced a class in grade 12 to Minecraftfor one hour and offered them as well the option to play athome on a world hosted by us We defer a detailed overviewof this study to a future article Notwithstanding there are twoobservations that we would like to mention

6We refer to the article [47] of Schmidt-Thieme for a general overview onthe importance of algorithms in (mathematical) education

(a) Never (4 CG5) (b) A little (1 CG5) (c) Few hrs (1 CG2) (d) Often (6 CG10)

Fig 11 Results in the second classroom experiment from the geometry questions in the questionnaire (prepost with 4 + 1 + 1 + 6 = 12 students) andthe control group (CG with 5 + 5 + 2 + 10 = 22 students) The captions denote how much the students played Minecraft before also showing how manystudents are in each group Responses in blue are correct incorrect ones are in orange The results before the lectures are roughly similar to the control group(except for question 1 in Figure 11a) but have improved to mostly correct results after the two lectures

(a) Sketch of the crane in Fig 12b (b) Construction of the crane fromFig 12a properly scaled

(c) Properly scaled house but toolarge to complete in time

(d) Properly scaled house in twodimensions but not in height

(e) Sketch of the construction inFig 12f each block outline shown (f) Properly scaled construction (g) Windows not properly scalable (h) Another take on scaling

Fig 12 Examples of constructions from the second classroom experiment The students all chose to build houses in the first lecture (top row) except for onestudent building a crane in Fig 12b In the second lecture all the students built constructions related to buildings ie merlons (battlements) in Fig 12f

1) Students in grade 12 that did not play Minecraft (orother first-person games) before had more problemsorienting themselves in the virtual world of Minecraftthan the corresponding group of students in grade 56They especially lacked the playful behavior of youngerchildren who accustomized themselves to the new en-vironment quickly Furthermore these students in grade12 also continued to consult the instructions in Figure 5from time to time unlike the students in grade 5 and 6

2) The students in grade 12 logged in at roughly similarlevels from home as the younger children but their col-laborative behavior was different Cooperative buildingwas limited most students built on their own Further-more some students actively damaged other buildingswhen nobody else was logged on ndash which did not happenin grade 56 Two female students noted that they weredisappointed by this behavior of their fellow classmateswhich they deemed inappropriate for their age

We did not study these occurrences in depth as we did notexpect them to happen beforehand However we believe thatfurther research in this area would definitely be interesting

VI SUMMARY AND OUTLOOK

We studied the usage of Minecraft as a tool to teach spatialgeometry in virtual worlds in grades 5 and 6 Specificallywe evaluated two classroom studies focusing on buildingpre-planned constructions cooperative building and scalingthree-dimensional constructions Based on our findings webelieve the appropriate classroom use of Minecraft to be anenriching experience regarding the concept of space at the endof primary education However we would like to emphasizethat we do not want to advocate for Minecraft replacing partsof the current education curricula Rather we see Minecraftas a useful tool to augment teaching in the curriculum ldquoTobe used when appropriate not to be applied when other toolsare more useful and not taught for its own sakerdquo [48]

The mentioned related work in Section II lists many morepossibilities for the further use of Minecraft We think thatespecially the usage of programmable agents in this context ispromising as it allows for visual debugging by observing theagentsrsquo actions in a virtual world ndash a topic we plan to explorefurther at undergraduate level [34]

Another future opportunity is the use of Virtual Reality (VR)for spatial geometry or concepts of space [49] [50] [51] withsome research ideas being already over 15 years old [52][53] Since recently Minecraft is also available in a VRversion [54] with Azmandian et al [55] already exploring thepossibilites of haptic targeting eg to build a castle [56] Oncethe VR hardware7 and appropriate software becomes widelyavailable to schools we anticipate exciting new directions foraugmented learning in the classroom

Lastly we would like to conclude with a statement of astudent in Figure 13 with which we wholeheartedly agree

Fig 13 ldquoThe whole game consists of mathrdquo ndash Response from a student ingrade 6 when asked about how Minecraft relates to mathematics

ACKNOWLEDGEMENTS

We would like to thank the anonymous reviewers of the8th IEEE Global Engineering Education Conference (IEEEEDUCON) for their helpful comments on our submission

REFERENCES

[1] G Kadunz and R Straesser Didaktik der Geometrie in der Sekun-darstufe I [Didactics of Geometry in Lower Secondary] HildesheimFranzbecker 2009

[2] M Ludwig and H-G Weigand Didaktik der Geometrie fuer die Sekun-darstufe I [Didactics of Geometry in Lower Secondary] SpektrumHeidelberg 2009 ch Konstruieren [Constructions] pp 55ndash80

[3] H-J Elschenbroich Mathematik-Didaktik [Didactics of Mathematics]Cornelsen Berlin 2003 ch Unterrichtsgestaltung mit Computerunter-stuetzung [Creating Teaching with Computer Support] pp 212ndash233

[4] M Ludwig ldquoVorlesungskapitel Propaedeutische Geometrie in Klasse5 und 6 [Lecture Chapter Propaedeutics of Geometry in Grades5 and 6]rdquo Available at httpwwwmathuni-frankfurtdesimludwigvorlesungenskriptendidgeoskript didgeohtml 2007

[5] B Andelfinger Geometrie [Geometry] Soest Landesinstitut fuerSchule und Weiterbildung 1988

[6] P H Maier Raeumliches Vorstellungsvermgen Ein theoretischer Abrissdes Phaenomens raeumliches Vorstellungsvermoegen [Spatial Imagina-tion A Theoretical Outline of the Phenomenon of Spatial Imagination]Auer Verlag 1999

[7] K Luig and R Straesser ldquoFoerderung ausgewaehlter Aspekte derRaumvorstellung mit dynamischer Geometriesoftware [Promotion ofSelected Aspects of Spatial Representation with Dynamic GeometrySoftware]rdquo in Beitraege zum Mathematikunterricht 2009

[8] H Schumann Raumgeometrie Unterricht mit Computerwerkzeu-gen[Space Geometry Teaching with Computer Tools] CornelsenVerlag 2001

[9] M Ludwig ldquoRaumgeometrie mit Kopf Herz Hand und Maus [SpaceGeometry with Head Heart Hand and Mouse]rdquo in Beitraege zumMathematikunterricht 2001

7Currently ranging from hundreds (Oculus Rift HTC Vive) to thousands(Microsoft HoloLens) of dollars for a single headset as of November 2016A less powerful alternative is to use modern smartphones as a display forvirtual reality eg Google CardboardDaydream or Samsung Gear VR

[10] H-G Weigand and T Weth Computer im Mathematikunterricht NeueWege zu alten Zielen [Computers in Mathematics Teaching New Waysto Old Goals] Spektrum Akademischer Verlag 2002

[11] (2016) List of interactive geometry software 3D Software [Online]Available httpsenwikipediaorgwikiList of interactive geometrysoftware3D Software

[12] A Goebel Archimedes Geo3D [Online] Available httpwwwraumgeometriededrupalen

[13] mdashmdash Experimentieren im Geometrieunterricht [Experimentation inGeometry Teaching] Franzbecker 2006 ch Dynamische Raumge-ometriesoftware - ein ideales Werkzeug zum Experimentieren in derGeometrie [Dynamic Spatial Geometry Software - an ideal Tool forExperimentation in Geometry] pp 27ndash36

[14] M Hohenwarter et al GeoGebra [Online] Available httpswwwgeogebraorg

[15] M Hohenwarter and J Preiner ldquoDynamic mathematics with geogebrardquoAMC vol 10 p 12 2007

[16] POV-Ray [Online] Available httpwwwpovrayorg[17] POV Team and others ldquoPersistency of vision ray tracer (pov-ray)rdquo

Version 10 Tech Rep 1991[18] BAUWAS [Online] Available httpwwwbicsbeschuledeson

machmitswbauwasindexhtm[19] H Meschenmoser Computergestuetzte Konstruktion und Visualisierung

das Konstruktionsprogramm BAUWAS [Computer-Assisted Constructionand Visualization the Construction Program BAUWAS] BerlinMachmit-Verlag 1999

[20] M Persson and J Bergensten ldquoMinecraftrdquo Stockholm Sweden MojangAB 2011 [Online] Available httpminecraftnet

[21] Smurrayinchester (2007) Mori Uncanny Valley CC BY-SA 30 httpscreativecommonsorglicensesby-sa30legalcode [Online] AvailablehttpscommonswikimediaorgwikiFileMori Uncanny Valleysvg

[22] inCrysis (2009) Guide to the game Crysis - Screenshots and ToD art[Online] Available httpsakamsincrysis

[23] M Mori K F MacDorman and N Kageki ldquoThe uncannyvalley [from the field]rdquo IEEE Robotics Automation Magazinevol 19 no 2 pp 98ndash100 June 2012 [Online] Availablehttpdoiacmorg1101109MRA20122192811

[24] M Franke Didaktik der Geometrie In der Grundschule [Didactics ofGeometry in Primary School] Spektrum Akademischer Verlag 2007

[25] G D R Steve Nebel Sascha Schneider ldquoMining learning and craftingscientific experiments A literature review on the use of minecraft ineducation and researchrdquo Journal of Educational Technology amp Societyvol 19 no 2 pp 355ndash366 2016

[26] B Bos L Wilder M Cook and R OrsquoDonnell ldquoLearningmathematics through Minecraftrdquo Teaching Children Mathematicsvol 21 no 1 pp 56ndash59 2014 [Online] Available httpdxdoiorg105951teacchilmath2110056

[27] O Hill (2016 June) Wersquove sold Minecraft many many timesLook Mojang [Online] Available httpsmojangcom201606weve-sold-minecraft-many-many-times-look

[28] J Pagliery (2014 September) Microsoft buys Minecraft for $25billion CNN [Online] Available httpmoneycnncom20140915technologyminecraft-microsoft

[29] C Huang and M Huang (2012) The Scale of the Universe 2 [Online]Available httphtwinsnetscale2

[30] D Short ldquoTeaching scientific concepts using a virtual world -minecraftrdquo Teaching Science-the Journal of the Australian ScienceTeachers Association vol 58 no 3 p 55 2012

[31] C Zorn C A Wingrave E Charbonneau and J J L Jr ldquoExploringminecraft as a conduit for increasing interest in programmingrdquo inProceedings of the 8th International Conference on the Foundationsof Digital Games FDG 2013 Chania Crete Greece May 14-17 2013 G N Yannakakis E Aarseth K Joslashrgensen andJ C Lester Eds Society for the Advancement of the Scienceof Digital Games 2013 pp 352ndash359 [Online] Available httpwwwfdg2013orgprogrampaperspaper46 zorn etalpdf

[32] CodeBlocks (2012) [Online] Available httpsdevbukkitorgbukkit-pluginscodeblocks

[33] D Saito H Washizaki and Y Fukazawa ldquoInfluence of the programmingenvironment on programming educationrdquo in Proceedings of the 2016ACM Conference on Innovation and Technology in Computer ScienceEducation ser ITiCSE rsquo16 New York NY USA ACM 2016 pp 354ndash354 [Online] Available httpdoiacmorg10114528994152925477

(a) Never (4 CG5) (b) A little (1 CG5) (c) Few hrs (1 CG2) (d) Often (6 CG10)

Fig 11 Results in the second classroom experiment from the geometry questions in the questionnaire (prepost with 4 + 1 + 1 + 6 = 12 students) andthe control group (CG with 5 + 5 + 2 + 10 = 22 students) The captions denote how much the students played Minecraft before also showing how manystudents are in each group Responses in blue are correct incorrect ones are in orange The results before the lectures are roughly similar to the control group(except for question 1 in Figure 11a) but have improved to mostly correct results after the two lectures

(a) Sketch of the crane in Fig 12b (b) Construction of the crane fromFig 12a properly scaled

(c) Properly scaled house but toolarge to complete in time

(d) Properly scaled house in twodimensions but not in height

(e) Sketch of the construction inFig 12f each block outline shown (f) Properly scaled construction (g) Windows not properly scalable (h) Another take on scaling

Fig 12 Examples of constructions from the second classroom experiment The students all chose to build houses in the first lecture (top row) except for onestudent building a crane in Fig 12b In the second lecture all the students built constructions related to buildings ie merlons (battlements) in Fig 12f

1) Students in grade 12 that did not play Minecraft (orother first-person games) before had more problemsorienting themselves in the virtual world of Minecraftthan the corresponding group of students in grade 56They especially lacked the playful behavior of youngerchildren who accustomized themselves to the new en-vironment quickly Furthermore these students in grade12 also continued to consult the instructions in Figure 5from time to time unlike the students in grade 5 and 6

2) The students in grade 12 logged in at roughly similarlevels from home as the younger children but their col-laborative behavior was different Cooperative buildingwas limited most students built on their own Further-more some students actively damaged other buildingswhen nobody else was logged on ndash which did not happenin grade 56 Two female students noted that they weredisappointed by this behavior of their fellow classmateswhich they deemed inappropriate for their age

We did not study these occurrences in depth as we did notexpect them to happen beforehand However we believe thatfurther research in this area would definitely be interesting

VI SUMMARY AND OUTLOOK

We studied the usage of Minecraft as a tool to teach spatialgeometry in virtual worlds in grades 5 and 6 Specificallywe evaluated two classroom studies focusing on buildingpre-planned constructions cooperative building and scalingthree-dimensional constructions Based on our findings webelieve the appropriate classroom use of Minecraft to be anenriching experience regarding the concept of space at the endof primary education However we would like to emphasizethat we do not want to advocate for Minecraft replacing partsof the current education curricula Rather we see Minecraftas a useful tool to augment teaching in the curriculum ldquoTobe used when appropriate not to be applied when other toolsare more useful and not taught for its own sakerdquo [48]

The mentioned related work in Section II lists many morepossibilities for the further use of Minecraft We think thatespecially the usage of programmable agents in this context ispromising as it allows for visual debugging by observing theagentsrsquo actions in a virtual world ndash a topic we plan to explorefurther at undergraduate level [34]

Another future opportunity is the use of Virtual Reality (VR)for spatial geometry or concepts of space [49] [50] [51] withsome research ideas being already over 15 years old [52][53] Since recently Minecraft is also available in a VRversion [54] with Azmandian et al [55] already exploring thepossibilites of haptic targeting eg to build a castle [56] Oncethe VR hardware7 and appropriate software becomes widelyavailable to schools we anticipate exciting new directions foraugmented learning in the classroom

Lastly we would like to conclude with a statement of astudent in Figure 13 with which we wholeheartedly agree

Fig 13 ldquoThe whole game consists of mathrdquo ndash Response from a student ingrade 6 when asked about how Minecraft relates to mathematics

ACKNOWLEDGEMENTS

We would like to thank the anonymous reviewers of the8th IEEE Global Engineering Education Conference (IEEEEDUCON) for their helpful comments on our submission

REFERENCES

[1] G Kadunz and R Straesser Didaktik der Geometrie in der Sekun-darstufe I [Didactics of Geometry in Lower Secondary] HildesheimFranzbecker 2009

[2] M Ludwig and H-G Weigand Didaktik der Geometrie fuer die Sekun-darstufe I [Didactics of Geometry in Lower Secondary] SpektrumHeidelberg 2009 ch Konstruieren [Constructions] pp 55ndash80

[3] H-J Elschenbroich Mathematik-Didaktik [Didactics of Mathematics]Cornelsen Berlin 2003 ch Unterrichtsgestaltung mit Computerunter-stuetzung [Creating Teaching with Computer Support] pp 212ndash233

[4] M Ludwig ldquoVorlesungskapitel Propaedeutische Geometrie in Klasse5 und 6 [Lecture Chapter Propaedeutics of Geometry in Grades5 and 6]rdquo Available at httpwwwmathuni-frankfurtdesimludwigvorlesungenskriptendidgeoskript didgeohtml 2007

[5] B Andelfinger Geometrie [Geometry] Soest Landesinstitut fuerSchule und Weiterbildung 1988

[6] P H Maier Raeumliches Vorstellungsvermgen Ein theoretischer Abrissdes Phaenomens raeumliches Vorstellungsvermoegen [Spatial Imagina-tion A Theoretical Outline of the Phenomenon of Spatial Imagination]Auer Verlag 1999

[7] K Luig and R Straesser ldquoFoerderung ausgewaehlter Aspekte derRaumvorstellung mit dynamischer Geometriesoftware [Promotion ofSelected Aspects of Spatial Representation with Dynamic GeometrySoftware]rdquo in Beitraege zum Mathematikunterricht 2009

[8] H Schumann Raumgeometrie Unterricht mit Computerwerkzeu-gen[Space Geometry Teaching with Computer Tools] CornelsenVerlag 2001

[9] M Ludwig ldquoRaumgeometrie mit Kopf Herz Hand und Maus [SpaceGeometry with Head Heart Hand and Mouse]rdquo in Beitraege zumMathematikunterricht 2001

7Currently ranging from hundreds (Oculus Rift HTC Vive) to thousands(Microsoft HoloLens) of dollars for a single headset as of November 2016A less powerful alternative is to use modern smartphones as a display forvirtual reality eg Google CardboardDaydream or Samsung Gear VR

[10] H-G Weigand and T Weth Computer im Mathematikunterricht NeueWege zu alten Zielen [Computers in Mathematics Teaching New Waysto Old Goals] Spektrum Akademischer Verlag 2002

[11] (2016) List of interactive geometry software 3D Software [Online]Available httpsenwikipediaorgwikiList of interactive geometrysoftware3D Software

[12] A Goebel Archimedes Geo3D [Online] Available httpwwwraumgeometriededrupalen

[13] mdashmdash Experimentieren im Geometrieunterricht [Experimentation inGeometry Teaching] Franzbecker 2006 ch Dynamische Raumge-ometriesoftware - ein ideales Werkzeug zum Experimentieren in derGeometrie [Dynamic Spatial Geometry Software - an ideal Tool forExperimentation in Geometry] pp 27ndash36

[14] M Hohenwarter et al GeoGebra [Online] Available httpswwwgeogebraorg

[15] M Hohenwarter and J Preiner ldquoDynamic mathematics with geogebrardquoAMC vol 10 p 12 2007

[16] POV-Ray [Online] Available httpwwwpovrayorg[17] POV Team and others ldquoPersistency of vision ray tracer (pov-ray)rdquo

Version 10 Tech Rep 1991[18] BAUWAS [Online] Available httpwwwbicsbeschuledeson

machmitswbauwasindexhtm[19] H Meschenmoser Computergestuetzte Konstruktion und Visualisierung

das Konstruktionsprogramm BAUWAS [Computer-Assisted Constructionand Visualization the Construction Program BAUWAS] BerlinMachmit-Verlag 1999

[20] M Persson and J Bergensten ldquoMinecraftrdquo Stockholm Sweden MojangAB 2011 [Online] Available httpminecraftnet

[21] Smurrayinchester (2007) Mori Uncanny Valley CC BY-SA 30 httpscreativecommonsorglicensesby-sa30legalcode [Online] AvailablehttpscommonswikimediaorgwikiFileMori Uncanny Valleysvg

[22] inCrysis (2009) Guide to the game Crysis - Screenshots and ToD art[Online] Available httpsakamsincrysis

[23] M Mori K F MacDorman and N Kageki ldquoThe uncannyvalley [from the field]rdquo IEEE Robotics Automation Magazinevol 19 no 2 pp 98ndash100 June 2012 [Online] Availablehttpdoiacmorg1101109MRA20122192811

[24] M Franke Didaktik der Geometrie In der Grundschule [Didactics ofGeometry in Primary School] Spektrum Akademischer Verlag 2007

[25] G D R Steve Nebel Sascha Schneider ldquoMining learning and craftingscientific experiments A literature review on the use of minecraft ineducation and researchrdquo Journal of Educational Technology amp Societyvol 19 no 2 pp 355ndash366 2016

[26] B Bos L Wilder M Cook and R OrsquoDonnell ldquoLearningmathematics through Minecraftrdquo Teaching Children Mathematicsvol 21 no 1 pp 56ndash59 2014 [Online] Available httpdxdoiorg105951teacchilmath2110056

[27] O Hill (2016 June) Wersquove sold Minecraft many many timesLook Mojang [Online] Available httpsmojangcom201606weve-sold-minecraft-many-many-times-look

[28] J Pagliery (2014 September) Microsoft buys Minecraft for $25billion CNN [Online] Available httpmoneycnncom20140915technologyminecraft-microsoft

[29] C Huang and M Huang (2012) The Scale of the Universe 2 [Online]Available httphtwinsnetscale2

[30] D Short ldquoTeaching scientific concepts using a virtual world -minecraftrdquo Teaching Science-the Journal of the Australian ScienceTeachers Association vol 58 no 3 p 55 2012

[31] C Zorn C A Wingrave E Charbonneau and J J L Jr ldquoExploringminecraft as a conduit for increasing interest in programmingrdquo inProceedings of the 8th International Conference on the Foundationsof Digital Games FDG 2013 Chania Crete Greece May 14-17 2013 G N Yannakakis E Aarseth K Joslashrgensen andJ C Lester Eds Society for the Advancement of the Scienceof Digital Games 2013 pp 352ndash359 [Online] Available httpwwwfdg2013orgprogrampaperspaper46 zorn etalpdf

[32] CodeBlocks (2012) [Online] Available httpsdevbukkitorgbukkit-pluginscodeblocks

[33] D Saito H Washizaki and Y Fukazawa ldquoInfluence of the programmingenvironment on programming educationrdquo in Proceedings of the 2016ACM Conference on Innovation and Technology in Computer ScienceEducation ser ITiCSE rsquo16 New York NY USA ACM 2016 pp 354ndash354 [Online] Available httpdoiacmorg10114528994152925477

The mentioned related work in Section II lists many morepossibilities for the further use of Minecraft We think thatespecially the usage of programmable agents in this context ispromising as it allows for visual debugging by observing theagentsrsquo actions in a virtual world ndash a topic we plan to explorefurther at undergraduate level [34]

Another future opportunity is the use of Virtual Reality (VR)for spatial geometry or concepts of space [49] [50] [51] withsome research ideas being already over 15 years old [52][53] Since recently Minecraft is also available in a VRversion [54] with Azmandian et al [55] already exploring thepossibilites of haptic targeting eg to build a castle [56] Oncethe VR hardware7 and appropriate software becomes widelyavailable to schools we anticipate exciting new directions foraugmented learning in the classroom

Lastly we would like to conclude with a statement of astudent in Figure 13 with which we wholeheartedly agree

Fig 13 ldquoThe whole game consists of mathrdquo ndash Response from a student ingrade 6 when asked about how Minecraft relates to mathematics

ACKNOWLEDGEMENTS

We would like to thank the anonymous reviewers of the8th IEEE Global Engineering Education Conference (IEEEEDUCON) for their helpful comments on our submission

REFERENCES

[1] G Kadunz and R Straesser Didaktik der Geometrie in der Sekun-darstufe I [Didactics of Geometry in Lower Secondary] HildesheimFranzbecker 2009

[2] M Ludwig and H-G Weigand Didaktik der Geometrie fuer die Sekun-darstufe I [Didactics of Geometry in Lower Secondary] SpektrumHeidelberg 2009 ch Konstruieren [Constructions] pp 55ndash80

[3] H-J Elschenbroich Mathematik-Didaktik [Didactics of Mathematics]Cornelsen Berlin 2003 ch Unterrichtsgestaltung mit Computerunter-stuetzung [Creating Teaching with Computer Support] pp 212ndash233

[4] M Ludwig ldquoVorlesungskapitel Propaedeutische Geometrie in Klasse5 und 6 [Lecture Chapter Propaedeutics of Geometry in Grades5 and 6]rdquo Available at httpwwwmathuni-frankfurtdesimludwigvorlesungenskriptendidgeoskript didgeohtml 2007

[5] B Andelfinger Geometrie [Geometry] Soest Landesinstitut fuerSchule und Weiterbildung 1988

[6] P H Maier Raeumliches Vorstellungsvermgen Ein theoretischer Abrissdes Phaenomens raeumliches Vorstellungsvermoegen [Spatial Imagina-tion A Theoretical Outline of the Phenomenon of Spatial Imagination]Auer Verlag 1999

[7] K Luig and R Straesser ldquoFoerderung ausgewaehlter Aspekte derRaumvorstellung mit dynamischer Geometriesoftware [Promotion ofSelected Aspects of Spatial Representation with Dynamic GeometrySoftware]rdquo in Beitraege zum Mathematikunterricht 2009

[8] H Schumann Raumgeometrie Unterricht mit Computerwerkzeu-gen[Space Geometry Teaching with Computer Tools] CornelsenVerlag 2001

[9] M Ludwig ldquoRaumgeometrie mit Kopf Herz Hand und Maus [SpaceGeometry with Head Heart Hand and Mouse]rdquo in Beitraege zumMathematikunterricht 2001

7Currently ranging from hundreds (Oculus Rift HTC Vive) to thousands(Microsoft HoloLens) of dollars for a single headset as of November 2016A less powerful alternative is to use modern smartphones as a display forvirtual reality eg Google CardboardDaydream or Samsung Gear VR

[10] H-G Weigand and T Weth Computer im Mathematikunterricht NeueWege zu alten Zielen [Computers in Mathematics Teaching New Waysto Old Goals] Spektrum Akademischer Verlag 2002

[11] (2016) List of interactive geometry software 3D Software [Online]Available httpsenwikipediaorgwikiList of interactive geometrysoftware3D Software

[12] A Goebel Archimedes Geo3D [Online] Available httpwwwraumgeometriededrupalen

[13] mdashmdash Experimentieren im Geometrieunterricht [Experimentation inGeometry Teaching] Franzbecker 2006 ch Dynamische Raumge-ometriesoftware - ein ideales Werkzeug zum Experimentieren in derGeometrie [Dynamic Spatial Geometry Software - an ideal Tool forExperimentation in Geometry] pp 27ndash36

[14] M Hohenwarter et al GeoGebra [Online] Available httpswwwgeogebraorg

[15] M Hohenwarter and J Preiner ldquoDynamic mathematics with geogebrardquoAMC vol 10 p 12 2007

[16] POV-Ray [Online] Available httpwwwpovrayorg[17] POV Team and others ldquoPersistency of vision ray tracer (pov-ray)rdquo

Version 10 Tech Rep 1991[18] BAUWAS [Online] Available httpwwwbicsbeschuledeson

machmitswbauwasindexhtm[19] H Meschenmoser Computergestuetzte Konstruktion und Visualisierung

das Konstruktionsprogramm BAUWAS [Computer-Assisted Constructionand Visualization the Construction Program BAUWAS] BerlinMachmit-Verlag 1999

[20] M Persson and J Bergensten ldquoMinecraftrdquo Stockholm Sweden MojangAB 2011 [Online] Available httpminecraftnet

[21] Smurrayinchester (2007) Mori Uncanny Valley CC BY-SA 30 httpscreativecommonsorglicensesby-sa30legalcode [Online] AvailablehttpscommonswikimediaorgwikiFileMori Uncanny Valleysvg

[22] inCrysis (2009) Guide to the game Crysis - Screenshots and ToD art[Online] Available httpsakamsincrysis

[23] M Mori K F MacDorman and N Kageki ldquoThe uncannyvalley [from the field]rdquo IEEE Robotics Automation Magazinevol 19 no 2 pp 98ndash100 June 2012 [Online] Availablehttpdoiacmorg1101109MRA20122192811

[24] M Franke Didaktik der Geometrie In der Grundschule [Didactics ofGeometry in Primary School] Spektrum Akademischer Verlag 2007

[25] G D R Steve Nebel Sascha Schneider ldquoMining learning and craftingscientific experiments A literature review on the use of minecraft ineducation and researchrdquo Journal of Educational Technology amp Societyvol 19 no 2 pp 355ndash366 2016

[26] B Bos L Wilder M Cook and R OrsquoDonnell ldquoLearningmathematics through Minecraftrdquo Teaching Children Mathematicsvol 21 no 1 pp 56ndash59 2014 [Online] Available httpdxdoiorg105951teacchilmath2110056

[27] O Hill (2016 June) Wersquove sold Minecraft many many timesLook Mojang [Online] Available httpsmojangcom201606weve-sold-minecraft-many-many-times-look

[28] J Pagliery (2014 September) Microsoft buys Minecraft for $25billion CNN [Online] Available httpmoneycnncom20140915technologyminecraft-microsoft

[29] C Huang and M Huang (2012) The Scale of the Universe 2 [Online]Available httphtwinsnetscale2

[30] D Short ldquoTeaching scientific concepts using a virtual world -minecraftrdquo Teaching Science-the Journal of the Australian ScienceTeachers Association vol 58 no 3 p 55 2012

[31] C Zorn C A Wingrave E Charbonneau and J J L Jr ldquoExploringminecraft as a conduit for increasing interest in programmingrdquo inProceedings of the 8th International Conference on the Foundationsof Digital Games FDG 2013 Chania Crete Greece May 14-17 2013 G N Yannakakis E Aarseth K Joslashrgensen andJ C Lester Eds Society for the Advancement of the Scienceof Digital Games 2013 pp 352ndash359 [Online] Available httpwwwfdg2013orgprogrampaperspaper46 zorn etalpdf

[32] CodeBlocks (2012) [Online] Available httpsdevbukkitorgbukkit-pluginscodeblocks

[33] D Saito H Washizaki and Y Fukazawa ldquoInfluence of the programmingenvironment on programming educationrdquo in Proceedings of the 2016ACM Conference on Innovation and Technology in Computer ScienceEducation ser ITiCSE rsquo16 New York NY USA ACM 2016 pp 354ndash354 [Online] Available httpdoiacmorg10114528994152925477

[34] K-T Foerster M Koenig and R Wattenhofer ldquoA Concept foran Introduction to Parallelization in Java Multithreading withProgrammable Robots in Minecraftrdquo in Proceedings of the 17th AnnualConference on Information Technology Education ser SIGITE rsquo16D Boisvert and S J Zilora Eds New York NY USA ACM 2016 p169 [Online] Available httpdoiacmorg10114529781922978243

[35] ComputerCraftEdu (2015) [Online] Available httpcomputercrafteducom

[36] A Reem J Bana I Knight E Benson O Afolabi A Kerr P Blanch-field and G Hopkins ldquoDesign of a math learning game using aminecraft modrdquo in Proceedings of the European Conference on GamesBased Learning 2014

[37] K L Koslashrhsen and M Misfeldt ldquoAn ethnomathematical study of playin minecraftrdquo in Nordic research in mathematics education Norma 14H Silfverberg T Kaerki and M Hannula Eds 2014 pp 205ndash214

[38] A J Bishop Mathematical Enculturation A Cultural Perspectiveon Mathematics Education Springer Netherlands 1991 [Online]Available httpsdoiorg101007978-94-009-2657-8

[39] A Nguyen and S Rank ldquoStudying the impact of spatial involvementon training mental rotation with minecraftrdquo in Proceedings of the 2016CHI Conference Extended Abstracts on Human Factors in ComputingSystems ser CHI EA rsquo16 New York NY USA ACM 2016 pp 1966ndash1972 [Online] Available httpdoiacmorg10114528515812892423

[40] L Floyd (2016) Unleashing Math Thinking Power through Minecraft Hour of Curiosity [Online] Available httpwwwhourofcuriositycomminecraft

[41] Minecraft Wiki (2016 November) Minecraft classic [Online]Available httpminecraftgamepediacomClassic

[42] ClassiCube (2016) [Online] Available httpwwwclassicubenet[43] MCGalaxy (2016) [Online] Available httpsgithubcomHetal728

MCGalaxyreleases[44] K-T Forster ldquoRaumgeometrie mit minecraftrdquo Beitrage zum Mathe-

matikunterricht 2012 46 Jahrestagung der Gesellschaft fur Didaktikder Mathematik vom 53 2012 bis 93 2012 in Weingarten 2012

[45] B Schmidt-Thieme and H-G Weigand Didaktik der Geometrie fuerdie Sekundarstufe I [Didactics of Geometry in Lower Secondary]Spektrum Heidelberg 2009 ch Symmetrie und Kongruenz [Symmetryand Congruence] pp 187ndash214

[46] E Eggeling ldquoFoerderung ausgewaehlter Aspekte der Raumvostellungmithilfe von Geometrie-Software am Beispiel Minecraft [Promotingselected aspects of spatial reasoning with the help of geometry softwareusing the example of Minecraft]rdquo BA thesis Humboldt University ofBerlin Germany December 15 2016

[47] B Schmidt-Thieme Strukturieren - Modellieren - Kommunizieren Leit-bilder mathematischer und informatorischer Aktivitten [Structuring -

Modeling - Communicating Models of Mathematical and InformaticalActivities] Franzbecker 2005 ch Algorithmen faecheruebergreifendund alltagsrelevant [Algorithms - Interdisciplinary and relevant Every-day]

[48] K-T Foerster ldquoIntegrating Programming into the MathematicsCurriculum Combining Scratch and Geometry in Grades 6 and 7rdquo inProceedings of the 17th Annual Conference on Information TechnologyEducation ser SIGITE rsquo16 D Boisvert and S J Zilora EdsNew York NY USA ACM 2016 pp 91ndash96 [Online] Availablehttpdoiacmorg10114529781922978222

[49] C Lai R P McMahan M Kitagawa and I Connolly GeometryExplorer Facilitating Geometry Education with Virtual RealityCham Springer International Publishing 2016 pp 702ndash713 [Online]Available httpdxdoiorg101007978-3-319-39907-2 67

[50] A Kovrov and M Sokolsk ldquoUsing virtual reality for teaching solidgeometry A case study for a cube sectionrdquo in Interactive CollaborativeLearning (ICL) 2011 14th International Conference on Sept 2011pp 428ndash433 [Online] Available httpdxdoiorg101109ICL20116059620

[51] D Lai and A Sourin ldquoVisual immersive mathematics in 3d webrdquo inProceedings of the 10th International Conference on Virtual RealityContinuum and Its Applications in Industry ser VRCAI rsquo11 NewYork NY USA ACM 2011 pp 519ndash526 [Online] Availablehttpdxdoiorg10114520877562087856

[52] P Wilson Handbook of spatial research paradigms and methodologies1 Psychology Press 1997 ch Use of virtual reality computing inspatial learning research pp 181ndash206

[53] H Kaufmann D Schmalstieg and M Wagner ldquoConstruct3D AVirtual Reality Application for Mathematics and Geometry EducationrdquoEducation and Information Technologies vol 5 no 4 pp 263ndash2762000 [Online] Available httpdxdoiorg101023A1012049406877

[54] M Davies (2016 August) Minecraft VR coming OculusRift today [Online] Available httpsmojangcom201608minecraft-vr-coming-to-oculus-rift-today

[55] M Azmandian M Hancock H Benko E Ofek and A D WilsonldquoHaptic retargeting Dynamic repurposing of passive haptics forenhanced virtual reality experiencesrdquo in Proceedings of the 2016 CHIConference on Human Factors in Computing Systems ser CHI rsquo16New York NY USA ACM 2016 pp 1968ndash1979 [Online] Availablehttpdoiacmorg10114528580362858226

[56] mdashmdash (2016 May) Haptic retargeting Dynamic repurposing ofpassive haptics for enhanced virtual reality experiences Extendedvideo of the best video award at CHI rsquo16 [Online] Availablehttpswwwyoutubecomwatchv=SiH3IHEdmR0