Mining and Crafting Mathematics: Designing a Model for Embedding Educational Tasks in Video
Games
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Heather McCreery Kellert, M.A.
Graduate Program in Education Teaching & Learning
The Ohio State University
2018
Dissertation Committee:
Dr. Patricia A. Brosnan, Advisor
Dr. Theodore Chao
Dr. Sarah Gallo
Dr. Karen E. Irving
Copyright by Heather McCreery Kellert
2018
Abstract
The purpose of this study is to investigate the design and modification of a rich mathematical task embedded in an educational video game Minecraft through analysis of observations of student engagement in mathematical practices. Student engagement in both the game and the overarching task is framed using the application of Self-Determination Theory to video games (Przybylsky, Ryan & Rigby, 2010) while analysis of the researcher-teacher’s design, implementation, and modification is framed through an iterative design process.
Results of this qualitative study include multiple modifications of the researcher-teacher in attempt to enhance engagement in the Standards of Mathematical Practice (Common Core
Standards State Initiative, 2018). The design and modifications of the task were then fit into a model to depict the relationships between four categories of student engagement including (1) engagement with the game, (2) engagement with mathematics, (3) engagement possibly motivated by factors of Self-Determination Theory, and (4) the task’s ability to provide opportunities for students to engage in Standards for Mathematical Practice (Common Core
State Standards Initiative, 2018).
Key terms: educational video games, design cycle, design process, rich mathematics tasks, student engagement, mathematics task design
i Acknowledgements
First and foremost, I would like to thank my advisor, Dr. Patti Brosnan. You have been an inspiration from the time of our first meeting, and have spent countless hours with me through my doctoral journey. I am eternally grateful for my experiences in the Mathematics
Coaching Program, which not only solidified my educational philosophies, but also evolved my own practices and goals for influencing how children might learn and love mathematics. You have truly been a wonderful thinking partner throughout my messy research processes, and my researcher and educator identities would not be what they are without your guidance.
I am also emphatically grateful to Dr. Azita Manouchehri, who spent a myriad of her time providing me with thoughtful and critical feedback throughout my final writing process. It was in your final course that I reached many of my “a-ha” moments due to your careful questioning and thought-provoking challenges. I truly appreciate the time you spent with me reviewing my analysis, and instilling in me the passion you carry for mathematics education research.
I would also like to thank my committee members Dr. Teddy Chao, Dr. Sarah Gallo, and
Dr. Karen Irving. Your collective expertise has been overwhelmingly helpful throughout the refinement of my study from a “stadium of squirrels” to (just about) “one squirrel.” I am grateful for the time you set aside to meet with me to detail your insight throughout the multiple stages of my writing process and articulation of ideas, as well as the time and energy you invested in me in your classes and during conferences.
Dr. Annalies Corbin and Dr. Sheli O. Smith: Words cannot describe how grateful I am for the opportunity to work and thrive within the PAST family, but I will try. Thank you for bringing
ii me onto the team, listening to my ideas, and entertaining my well-intentioned mischief. The two of you exemplify strong, non-nonsense, female leaders who believe in what is right for children, and you know no boundaries.
To the PAST Team (in alphabetical order): Alyssa, Andy, Ashley, Diana, Jim, Kat, Kayla,
Lori, Maria, Monica, Pam, Ruth, and Sam. You saw and heard it all, uncut and unfiltered. Thank you for listening and supporting me through one of the most intellectually challenging times in my life. You are truly a thinking partner collective and I will forever be in your debt for the ongoing support you provided me during my face-to-face time with you.
To James Ross and the fine folks at Nodecraft: Thank you for the generous scholarship you provided to the Minecraft Mathematics program. Your server helped immensely with classroom management and task delivery to students. The students were so excited to have their “own” server that they could enjoy together, and the ability to backup our worlds was invaluable!
Finally, I would like to thank my life partner and love Brian: You and our sweet daughter
Finley Brooke are my light and inspiration for life-long learning. I would not be where I am without you and your support along every step of the way. We are the captains of our ship, and I am looking forward to many more voyages to come, including our second daughter on the way!
iii Vita
August 2005 ………………………………………………….……….B.A. Mathematics, University of Washington
Seattle, Washington
Winter 2008 - Summer 2008 ………………………………….Teacher of Mathematics, Nathan Hale High
School, Seattle, Washington
June 2008 ……………………………………………….……………..M.A. Education, University of Washington
Seattle, Washington
Autumn 2008 - Summer 2011 …………………………..……Teacher of Mathematics, Garfield High
School, Seattle, Washington
June 2010 – Winter 2012……………………………….…….…Textbook Co-Author and Production
Coordinator, Kinetic Books Company, Inc.
January 2013 – August 2014……………………………………Academic Advisor, Department of
Mathematics, The Ohio State University
August 2014 – May 2015………………………………………..Graduate Research Associate, The Ohio
State University
August 2014 – May 2015………………………………………..Mathematics Tutor Supervisor, The Ohio
State University
May 2015 – Present….………………………………………..….STEM Innovator and Hybrid Teacher, PAST
Foundation, Columbus, Ohio
Fields of Study
Major Field: Education: Teaching & Learning
iv Table of Contents
Abstract ...... i
Acknowledgements ...... ii
Vita ...... iv
Table of Contents ...... v
Table of Tables ...... viii
Table of Figures ...... x
Chapter 1: Introduction ...... 1
Minecraft as an Educational Video Game ...... 3
Background of Minecraft Mathematics Workshop ...... 7
Research Questions ...... 8
Definition of Terms ...... 9
Chapter 2: Literature Review ...... 11
Research on Educational Video Games ...... 11
Research on Engagement with Video Games ...... 21
Research on Engagement with Mathematical Practices ...... 31
Research on Design Cycles ...... 37
Theoretical Framework ...... 40
Chapter 3: Methodology ...... 51
Characteristics of Minecraft Mathematics Workshop Setting ...... 52
Procedure and Timeline of Study ...... 59
The Task ...... 61
v Participants ...... 64
Instrumentation ...... 65
Analysis ...... 74
Researcher-Teacher Beliefs and Biases ...... 80
Credibility ...... 81
Chapter 4: Results ...... 83
Workshop 1 ...... 83
Workshop 2 ...... 90
Mathematics Incorporated within the Build The Room Task ...... 96
Chapter 5: Analysis ...... 98
Theme 1: Intertwining the Game with Task Objectives ...... 101
Theme 2: Incorporating Opportunities for Student Interactions Inside and Outside the Game
...... 140
Chapter 6: Discussion ...... 181
The Engagement Amplification Model: Using Student Engagement to Inform Task Design .. 181
Implications for Practice ...... 189
Implications for Future Studies ...... 190
Limitations of the Study ...... 196
Conclusion ...... 197
References ...... 199
Appendix A: Build the Room Challenge #1 template from workshop 2, Build the Room’s
Dimensions ...... 210
vi Appendix B: Build the Room Challenge #2 template from workshop 1, Document a Craft ...... 211
Appendix C: Build the Room Challenge #5 template from workshop 1, Same Volume Different
Dimensions ...... 212
Appendix D: Build the Room Challenge #4 template from workshop 2, Same Volume Different
Dimensions ...... 213
Appendix E: Day-by-Day Rich Task Design Cycle template ...... 214
Appendix F: Three-Column Engagement Template ...... 215
Appendix G: Post-Workshop Student Interview Questions ...... 216
vii Table of Tables
Table 1: Timeline of Phase and Cycle Iterations……………………………………………………………………….48
Table 2: Students’ Self-Rated Levels of Game expertise…………………………………………………..………55
Table 3: Learning Ecology Elements of Cobb, et al. (2003) in Minecraft Mathematics……………..59
Table 4: Timeline of the Procedure of the Study…….………………………………………………………………..60
Table 5: Challenges of the Build the Room Task.………………..………………………………..…………………..62
Table 6: Student Demographics of Workshops 1 and 2…………………………………………….………………65
Table 7: Phases and Day-by-Day cycles of Instrumentation……………………………………………………..73
Table 8: Workshop 1 Day-by-Day Rich Task Design Cycle…………………………………………………………87
Table 9: Student Responses in First and Second Workshops ……………………………………………………91
Table 10: Challenges by Workshop…………………………………………………………………………………………..92
Table 11: Workshop 2 Day-by-Day Rich Task Design Cycle……………………………………………………….93
Table 12: Components of the Cognitive Rigor Matrix, Standards for Mathematical Practice and
Self-Determination Theory Observed During Task.………………..…………….……………….………96
Table 13: Key of Raw Data References……………………….………………………………………………………….100
Table 14: Modification and Anticipation of Build the Room’s Dimensions: Pilot Studies to
Workshop 1………………………………………………………………………………………………………………..114
Table 15: SMP Noticed in the Challenge Build the Room’s Dimensions.…………………………..……..120
Table 16: Design and Anticipation of Same Volume Different Dimensions: Workshop 1…………125
Table 17: SMP Noticed in the Same Volume Different Dimensions challenge, Workshop 1…….135
Table 18. Design and Anticipation of Communication in Same Volume Different Dimensions:
Workshop 1……….……………………………………………………….………………………………………………145
viii Table 19: Additional SMP Noticed in the Same Volume Different Dimensions Challenge………..157
Table 20: Modification and Anticipation of Same Volume Different Dimensions: Workshop
2……………………………………………………………………………………………………..………………………….167
Table 21: SMP Noticed in the Same Volume Different Dimensions Challenge, Workshop
2…………………………………………………………………………………………………………………………………172
Table 22: SMP and Their Corresponding SDT Motivational Factors of Focus………………………….186
ix Table of Figures
Figure 1: First-person view of a Minecraft player as another player plants flowers……………………5
Figure 2: Smith and Corbin’s (2014) design cycle for design of problem-based learning
modules...... 39
Figure 3: The Day-by-Day cycle of the Rich Task Design Cycle adapted from Smith and Corbin
(2014)...... 45
Figure 4: The Day-by-Day and Workshop-by-Workshop cycles of the Rich Task Design Cycle…..46
Figure 5: Theories within each component of the study’s process...... 50
Figure 6: Day-by-Day Rich Task Design Cycle template with jottings...... 67
Figure 7: Three-Column Engagement template with jottings...... 68
Figure 8: Resulting engagement when early access to the game, and little structure within the
task’s procedure was provided...... 106
Figure 9: Resulting engagement with late access to the game, and rigid structure within the
task’s procedure was provided...... 107
Figure 10: Resulting engagement when access to the game was provided mid-way through the
task with a medium amount of structure in the task’s procedure...... 108
Figure 11: Screenshot taken within the indoor farm of a student’s Build the Room product that
shows the door to the additional room containing the pet llamas……………………………..110
Figure 12: Resulting engagement before the task provided an opportunity for students to
connect the product to their fantasy worlds...... 111
Figure 13: Resulting engagement after the task provided an opportunity for students to connect
the product to their fantasy worlds...... 111
x Figure 14: Predicted changes in student engagement after modifications made to the task’s first
challenge, Build the Room’s Dimensions...... 116
Figure 15: A capture from the screencast of Adam as he views one of the rooms in his team’s
underground desert base...... 118
Figure 16: Students in Team Desert Base begin to carve out an underground tunnel and space
for a room in the Build the Room’s Dimensions challenge...... 120
Figure 17: Resulting student engagement after modifications made to the task’s first challenge,
Build the Room’s Dimensions...... 122
Figure 18: Relationships of SDT and SMP to engagement within the Engagement Amplification
Model...... 123
Figure 19: Predicted student engagement after consideration of SDT and SMP in the new
challenge Same Volume Different Dimensions...... 127
Figure 20: Resulting student engagement after implementation of Same Volume Different
Dimensions challenge...... 138
Figure 21: Modified relationships of SDT and SMP to engagement within the Engagement
Amplification Model...... 139
Figure 22: Predicted student engagement after consideration of SDT’s relatedness and
autonomy as well as the SMP in the new challenge Same Volume Different
Dimensions…...... 147
Figure 23: Allen’s 3D drawing of a rectangular prism, labeled with height, length, and width..153
Figure 24: Oscar moves his avatar to an open underground space near the team’s original room
to show Allen where the new room will be located...... 155
xi Figure 25: Oscar questions Allen’s position of the floor he is building in relation to the
ceiling...... 155
Figure 26: Resulting student engagement after implementation of Same Volume Different
Dimensions when considering SDT’s relatedness and the SMP Construct viable
arguments and critique the reasoning of others...... 159
Figure 27: Comparison of resulting student engagement of the Same Volume Different
Dimensions challenge before and after a focus on communication between
students……………………………………………………………………………………………………………………..160
Figure 28: Comparison of resulting engagement with varying levels of access to the game and
degrees of structure…………..………..………..………..………..………..………..………..………..……….161
Figure 29: Further modified relationships of SDT and SMP to engagement within the
Engagement Amplification Model...... 165
Figure 30: Predicted student engagement after modifications made to the challenge Same
Volume Different Dimensions for Workshop 2...... 168
Figure 31: Resulting student engagement after implementation of Same Volume Different
Dimensions when considering SDT’s relatedness and the SMP Construct viable
arguments and critique the reasoning of others...... 174
Figure 32: Modified relationship of SDT dial in comparison to SMP to dial within the
Engagement Amplification Model...... 175
Figure 33: The Engagement Amplification Model with two dials of SDT and SMP, and two
volume bars of game and mathematical engagement...... 182
xii Figure 34: Relationships within the Engagement Amplification Model between the four
components of student engagement...... 185
Figure 35: The Engagement Amplification Model with a dial for specific mathematics concepts
instead of specific SMP...... 192
Figure 36: The Engagement Amplification Model with a dial for an alternative discipline’s
concepts or practices, and an engagement volume bar for that discipline………………….193
Figure 37: The Engagement Amplification Model with a dial for transdisciplinary concepts or
practices, and volume bars for engagement in multiple disciplines…………………………….194
Figure 38: The Engagement Amplification Model with a dial for transdisciplinary concepts or
practices, and two volume bars for game and transdisciplinary engagement…………….195
xiii Chapter 1: Introduction
The purpose of this dissertation is to examine how a rich mathematical task embedded in an educational video game might be designed and modified through analysis of students’ game and mathematical engagement. Specifically, I used the popular game Minecraft in an after-school enrichment program to examine my iterative process of (1) anticipation, (2) implementation, (3) evaluation, and (4) modification to inform the addition of new challenges within the task as well as modifications to existing challenges during the two workshops of the study. Design and modifications were informed by careful documentation of how I noticed student engagement in mathematical practices before, during, and after each class day as well as before and after each workshop. Therefore, this dissertation aims to provide evidence of how an educational video game might be used to engage students in mathematical practices through rich task design, and develops a model to depict relationships of student engagement with the game, with the task, and with mathematics.
Multiple studies and publications have advocated for the use of Minecraft, or digital game-based learning in classroom settings (Cheng & Annetta, 2012; Devlin, 2011; Eichenbaum,
Bavelier, & Green, 2014; Gee, 2007; Hanghøj, Hautopp, Jessen, & Denning, 2014; Niemeyer &
Gerber, 2015; O’Connor, 2015; Rigby & Ryan, 2011). However, an argument worth addressing is specifically how games such as Minecraft may be used for learning. In their literature review regarding the use of Minecraft in the classroom, Nebel, Schneider, and Rey (2016) quote a teacher who believes in the importance of student collaboration, but does not believe in the
“diversion” of using a computer game when everyone sits “in front of this box” (p. 361). The
1 teacher’s statement provides an example for the need to conduct and disseminate research that not only explores why video games can aid student learning, but how they can be used to do so.
True, students working together in Minecraft are sitting in front of a screen. However, students can be positioned to sit next to one another or in groups facing each other (Bailey,
2016; Marcon & Faulkner, 2016), and given an activity that requires face-to-face communication both as part of the task as well as part of classroom norms set by the teacher
(Callaghan, 2016). The teacher can also pedagogically orchestrate the class so that he/she is interacting with students both within the game and face-to-face by asking questions about their constructions, probing for conceptual explanations, and demonstrations of learning. Indeed, some research has already found that meaningful use of Minecraft relies heavily upon the pedagogical approaches of teachers (Callaghan, 2016; Hanghøj, Hautopp, Jessen, & Denning,
2014; Meyer, 2015), a theory that this study hopes to confirm and expand upon through examination of my process as the researcher and teacher (researcher-teacher), of mathematical task design and modifications in Minecraft.
While particular case studies examining the use of Minecraft in a classroom environment exist, few, if any, studies delve into specific processes of the teacher in his/her task design embedded in an educational video game, and the modifications implemented due to interpretations of student behaviors and actions. Without empirical evidence of teacher processes of task design, the educational potential of the game has been questioned due to an already existing “swamp of gimmicks” within educational technology tools that claim to impact student learning (Zimmer, 2016). My design-based research approach (Cobb, Confrey, diSessa,
2 Lehrer, & Schauble, 2003; Design-Based Research Collective, 2003) combined with an ethnographic perspective (Green & Bloome, 1997; Heath & Street, 2008) utilizes systematic procedures of observation and documentation of empirical evidence that reflect task implementation and modification over a particular time and space. Thus, my research aims to contribute to the field by providing empirical evidence of a researcher-teacher’s task development while taking into account evidence of students’ engagement with the game and mathematical practices. Through detailing the task as well as how it was developed to cater to students’ intrinsic motivation during interactions with task objectives and game environment, I show that Minecraft may have the potential to be used as a tool to engage students in mathematical practices, an engagement that supports development of student mathematical thinking (Carpenter, Fennema, Franke, Levi, & Empson, 1999; Stein, Grover, & Henningsen,
1996). This engagement in mathematical thinking may possibly lead to mathematical achievement that can be measured and analyzed in future studies.
In this chapter, I will begin by providing a context for my research by first describing the background and features of Minecraft as well as its potential as an educational video game.
Next, I will provide a brief background of Minecraft Mathematics. Finally my research questions and definitions of key terms will be presented.
Minecraft as an Educational Video Game
Although Minecraft’s vast, open-ended, building-block world harbors opportunities for mathematical exploration, the game was not originally designed for educational purposes.
Swedish programmer Markus “Notch” Perrson was inspired to develop Minecraft after playing a similar block-building game, Infiniminer (Handy, 2010). Infiniminer was originally released in 3 2009 as a free download with the intention of “allowing players to build while they were playing the game, and have it be really natural and free form” (Nierenberg, 2015). Perrson happened upon Infiniminer and realized that it was a game he wanted to develop further. He had always wanted to create open-ended games in which a player had complete autonomy.
I wanted to make huge games. Games where you can do anything… no trees you can’t
cut down, and no made-up story being told to the player to motivate them. Instead, the
player would make their own story, and interact with the game world, decide for
themselves what they want to do. (Perrson, 2013)
Indeed Minecraft fits Perrson’s original vision. He implies that a major motivational factor of the game was intended to be autonomy, the presentation of opportunities for choice in a game world as defined by Self-Determination Theory applied to video games (Rigby & Ryan,
2011; Ryan, Rigby, & Przybylski, 2006). Educators and advocates for the game alike have commented on how the endless opportunities for creativity, and for players to create their own stories while engaging with the game is a large part of the game’s popularity among children and adults alike (Gilbert, 2017; Starkey, 2016; Tromba, 2013). Therefore, one might argue that an objective of Minecraft is to become immersed in one’s own story, rather than “win” a final level.
Upon launching the game, the player is situated in an open environment that includes trees, hills, mountains, rivers, deserts, and pretty much any other biome one can imagine. The game’s graphics are rudimentary; the environment is visibly pixelated (Figure 1), and the game’s main building components consist of large cubic blocks that have dimensions equivalent to about half the player’s avatar’s height, a relationship that has the potential for mathematical
4 exploration. If playing in “creative mode,” one can pull up a menu of multiple colored and textured blocks, switches, wiring, weapons, and decorations. From there, the player is free to decide what he/she desires to build.
For example, Figure 1 shows the view of a player as she watches another player in the game world plant flowers. The mountains in the background as well as the trees and grass blocks were all generated by the game to make up the environment of the virtual world. The student built the house with a pathway to the front door, and was starting to create the landscaping of the front yard, which included shrubs and flowers. Players and students often work together on builds such as these, communicating both inside and outside of the game, their ever-changing plans for creating their products and stories.
Figure 1. First-person view of a Minecraft player as another player plants flowers
5 When opening a Minecraft world, players can choose between game modes. When playing in creative mode, one can also choose to “turn off” hostile creatures that may attack one’s avatar or build, so that the player may concentrate solely on building and creating. In creative mode, one can “win” the game by conquering the Ender Dragon in The End if hostile enemies have been “turned on” when setting up one’s game world in the mode. After the dragon is slain, end credits appear on the screen before taking the player back to the game to continue building.
In survival mode, the player can “die” by failing to eat, falling a large distance, encountering hostile creatures, or falling victim to explosions, fire, or lava. In this mode, players cannot bring up the menu with all blocks and resources available in infinite amounts.
Instead, they must mine and craft materials to keep themselves safe and nourished. Winning survival mode also consists of building a portal to The End and conquering the Ender Dragon, although it is a much more challenging feat that may take hours, days, or months.
One may argue that another way of “winning” Minecraft, particularly in creative mode, could be accomplishing an impressive build, where the definition of impressive depends on one’s own opinions and the audience for whom the player is creating. Multiple online forums exist for the game that showcase everything from building instructions for replicating others’ creations (www.grabcraft.com), to narrative screencast videos that tour a detailed world that a player or group of players have created (Niemeyer & Gerber, 2015). The goal in this case, is to create something wonderful to garner excitement.
The motivation and engagement involved in building impressive environments in creative mode is one aspect of my study. Observations, documentation, and analysis of student
6 engagement in the game as well as in mathematical practices informed the initial design and modifications I made to my mathematics challenges implemented with students. The purpose of these challenges as part of an overarching rich mathematical task (a term to be defined later in the chapter) is to make transparent, the mathematical practices of students that may already occur as students create and build within the game.
Thus, Minecraft is not a “math game” in which traditional mathematics exercises pop up on the screen, but rather a game with an environment that can be utilized to engage students in mathematical practices. For example, large blocks and slabs (half blocks) used in the game for building require concepts of counting, measurement, and dimension during the building process. Conversions and scaling may be relevant if students are replicating an existing structure with pre-determined measurements. Thus, Standards for Mathematical Practices
(Common Core State Standards Initiative, 2017) such as Model with Mathematics, and Look for and make use of structure might be enacted as students work through rich mathematical tasks with objectives that intertwine game and mathematical components.
Background of Minecraft Mathematics Workshop
This study occurs at an educational non-profit facility within an after-school enrichment program, Minecraft Mathematics. Both Minecraft Mathematics workshops in the study consisted of four class days, and will be detailed further in the third chapter. Students enrolled in after-school programs at this particular organization are informed that because the facility is one of research and development, teachers such as myself research how students learn as we design activities that attempt to optimize their learning. Thus, from my experiences, students
7 seem to understand that while they are not in their daily classroom environment, many of the classroom norms related to school are still in place within the classrooms at the facility.
Minecraft is primarily a game of exploration, and a literature review by Nebel, et al.
(2016) found that students are already accustomed to engaging in a certain amount of non- academic freedom, or “free-play with self set goals using exploration and discovery” (p. 360) as opposed to teacher or task-guided objectives within the game. As I had experienced during pilot studies, adding mathematical constraints around the creative freedom students normally experience can interfere with the student-perceived norms already in place around gameplay.
Therefore, within my iterative process of task redesign, I was pushed as the researcher to make systematic and comprehensive details of constraints such as these, and consider ways to approach them with constructive intention to attempt to optimize student engagement in mathematical practices.
Research Questions
The research questions that guide the study explore how my interpretations of students’ engagement in mathematical practices inform task design and modifications.
1. Through rich task design, what are some ways that an educational video game can
be used to engage students in mathematical practices?
2. How does students’ engagement inform the process of rich task design in Minecraft?
3. How does students’ engagement inform the process of rich task modifications in
Minecraft?
8 Definition of Terms
Definitions are provided for each of the following terms to clarify their meanings.
Terms defined include (1) rich mathematical tasks, (2) mathematical practices, (3) student engagement with mathematical practices, and (4) student engagement with Minecraft.
Rich mathematical tasks: Rich tasks, also referred to as high cognitive demand tasks, are tasks that engage students in high-level mathematics thinking and reasoning (Stein, Grover, &
Henningsen, 1996) through a process that involves complex and non-algorithmic thinking. For example, rather than going through a well-rehearsed step-by-step procedure that has been provided to them by the teacher or another source, students must analyze constraints of a problem to explore processes, connections, and relationships of the content (Collins and NCTM,
2011). Students are encouraged to monitor and regulate their own thought processes (Pape,
Bell, & Yetkin, 2003), and then verbalize and write about these processes. Features of rich tasks include:
1. Multiple entry points (Gojak, 2013)
2. Multiple solution paths (Cohen, 1994; Gojak, 2013; Stein, et al., 1996)
3. Multiple answers (Cohen, 1994; Gojak, 2013; Stein, et al., 1996)
4. Multiple representations (Cohen, 1994)
5. Relevance for students in content and context (Cohen, 1995; Gojak, 2013)
6. A demand for communication in either oral or written display of mathematical
thinking (Stein, et al., 1996)
9 Mathematical practices: These practices are the eight Standards of Mathematical
Practice (SMP) that describe ways that students may engage with mathematics throughout their K-12 schooling (Common Core State Standards Initiative, 2018).
Student engagement with mathematical practices: Observable actions of student engagement with the SMP listed above may include (1) documentation of procedures or solution strategies as suggested by Stein, et al. (1996); (2) communication to peers or the teacher of mathematical ideas and representations as detailed in teaching philosophies of
Cognitively Guided Instruction (Carpenter, Fennema, Franke, Levi, & Empson, 1999) and
Complex Instruction (Cohen, 1994); and (3) demonstration of attention to solving a problem through writing mathematical thinking on paper, counting aloud, and using tape measures.
Student engagement with Minecraft: Observable actions of student engagement with the game may include (1) talking about events in the game (Bailey, 2016); (2) collaboration through discussion of planning and building, and changes made in the game world (Callaghan,
2016; Marcon & Faulkner, 2016; Meyer, 2015; Nebel, et al., 2016); (3) students teaching one another how to perform certain actions within the game (Marcon & Faulkner, 2016; Niemeyer
& Gerber, 2015); and (4) basic interaction with the game through exploring, viewing materials, or building, regardless of involvement with other students or players (Starkey, 2016).
10 Chapter 2: Literature Review
This chapter offers a summary of the existing literature central to my study: (1) educational video games, (2) engagement with video games, (3) engagement with mathematical practices, and (4) the use of a design cycle as an iterative process. First, literature on educational video games is reviewed to depict the multitude of terminology used to describe their varying functions as well as provide examples of current gaps in research surrounding video game usage for educational purposes. Literature that surrounds learning and cognition in the presence of video games as well as literature that call for greater empirical research in educational video game use is reviewed as well. Next, in relation to engagement with video games, the application of Self-Determination Theory to video games (Przybylsky,
Ryan & Rigby, 2010) is detailed to highlight a component of the theoretical grounding of my proposed study. Additional literature that both explores and provides reasoning for the use of video games such as Minecraft within an educational setting is also reviewed. Third, engagement with the Standards for Mathematical Practice (Common Core State Standards
Initiative, 2017), another component of my theoretical grounding, is reviewed through literature that describes how rich mathematical tasks might garner engagement, especially in regards to their design and facilitation. Fourth, because my theoretical frame and methodology rely upon the use of an iterative design cycle, research on design cycles in engineering as well as in education are reviewed.
Research on Educational Video Games
Many terms are used to describe and categorize video games used in education. In the following section, I review these terms to situate the original intention of Minecraft’s design 11 and its use in my study. Then, I cite recent calls for empirical evidence in educational video game research.
Educational Video Game Terms. Exploration of video games as an educational tool is an emergent area of research, and a variety of terms exist to describe games’ educational intention and delivery. These terms include, but are not limited to the following:
• serious educational games/ serious games/ digital serious games
• computer games for learning
• edugames
• game-based learning
• educational video games
• v-learning/ virtual learning environments
• e-learning/ edutainment
While a fair amount of overlap exists between some of these terms, others are antiquated and carry negative associations with failed educational technology movements (Susi, Johannesson,
& Backlund, 2007). Terms are listed above in an order pertaining to their educational purpose and relation to game play. Types of games near the top of the list are designed with educational intent, whereas terms in the middle incorporate game play for educational purposes regardless of the game designers’ original intentions. Terms at the bottom may encompass an educational purpose in design, but may not include a game element or successful implementation of academic content according to research. These terms will be described briefly to depict where Minecraft lies on the spectrum of educational intention and delivery in video games research.
12 Most literature agrees that serious educational games are developed with the primary purpose of educating their players in academic areas (Cheng & Annetta, 2012; Petridis,
Dunwell, Liarokapis, Constantinou, Arnab, de Freitas, & Hendrix, 2013; Zikas, Bachlitzanakis,
Papaefthymiou, Kateros, Georgiou, Lydatakis, & Papagiannakis, 2016). Synonymously abbreviated as serious games, digital serious games (Arnab, Lim, Carvalho, Bellotti, de Freitas,
Louchart, Suttee, Berta, & De Gloria, 2015; Mildner, Beck, Reinsch, & Effelsberg, 2016), or computer games for learning (Mayer, 2015) typically have the intention of raising awareness of academic topics, and supporting the acquisition of new knowledge and skills enabling learners to engage in situations that would otherwise be impossible to experience (Ypsilanti, Vivas,
Räisänen, Biitala, Ijäs, & Ropes, 2014). The term serious educational games has only recently evolved to include games more specifically in the K-12 educational realm since Corti (2006), and
Connolly, E. A. Boyle, MacArthur, Hainey, and Boyle (2012) initially defined them more broadly as games developed for behavior change in business, industry, healthcare, and education.
Edugames (Calaghan, 2016) relate to supporting collaboration and learning in K-12 education safely and meaningfully, and may thus fall within the realm of serious educational games depending on the designer’s intentions for the game.
Connolly, et al. (2012), Corti (2006), and Zikas, et al. (2016) also use game-based learning as a way to describe the implementation of serious educational games, although
Connolly, et al. propose that game-based learning is a stronger educational term compared to serious educational games, a suggestion that may already be anachronistic due to the number of more recent publications that use serious educational games in reference to education.
According to the EdTech Review editorial team (2013) and Farber (2016), game-based learning
13 is game play that has defined and intended learning outcomes that provide meaning for students. Farber argues that “the game is not the teacher… the game is just an activity” (p. 1) and recommends that play should not be graded, but rather teachers can assess the learning transfer from the game experience to the curriculum, a transfer facilitated by the teacher.
According to these authors, game-based learning describes how games are used as a tool for educational purposes. The game may or may not have been designed with educational intentions, but the teacher can extract and connect learning opportunities from the students’ game experiences to academic content. While serious educational games and related terms are situated on one end of the spectrum regarding educational intent, educational video games and game-based learning are positioned in the middle, because they refer to using the games as a tool to teach regardless of the intention behind games’ design (Nebel, Schneider & Ray,
2016).
Next, v-learning and virtual learning environments, used interchangeably, refer to a type of instructional design that may or may not be game-based. As their names suggest, these terms refer to immersive, three-dimensional spaces where learners can interact in real time
(Annetta, Klesath, & Meyer, 2009; O’Connor, 2015; O’Connor & Domingo, 2017). While one might argue that Minecraft is a virtual learning environment since the player has an avatar and can interact in real time with classmates in a three-dimensional space, v-learning tends to incorporate environments that emphasize interactions between avatars in dialogues and teaching scenarios such as in the popular environment Second Life (http://secondlife.com).
Virtual learning environments have been used in classroom settings to mimic scientific field trip experiences such as insect collecting (Annetta, et al., 2009) and to facilitate virtual class
14 meetings and presentations (O’Connor & Domingo, 2017). Therefore, this study does not define Minecraft as a virtual learning environment. In the Minecraft virtual environment, students interact with one another, but largely focus on designing and building products together rather than on engaging in a discussion together (they do this face-to-face) or taking part in simulations of traditional learning experiences like lectures, presentations, and collection of insects.
E-learning and edutainment sit at the end of the list, because of their generality and antiquity. According to Susi, et al. (2007), e-learning applies to anything computer-enhanced, computer-based, or used in distance learning, while edutainment refers to video games from the 1990s designed for pre-school and early-elementary school students. They note that some researchers believe the edutainment movement failed due to the heavy weight on drill-and- practice in the games.
This study refers to Minecraft as an educational video game, a game that was not originally designed for educational purposes, but may be used as an educational tool.
According to Minecraft game designers Zachary Barth and Markus Persson, Minecraft was not intentionally developed for behavior change or educational intent, but rather as a building game meant to be fun and engaging creatively (Handy, 2010; Nebel, et al., 2016; Nierenberg,
2015; Persson, 2009; Persson, 2013). Furthermore, using Minecraft as an environment and application for problem solving within a mathematical task constitutes game-based learning since the game is a major component of instructional design. The next sub-section details how game-based learning that includes both serious educational games as well as educational video games like Minecraft might be used to examine student learning and cognition.
15 Learning and cognition in the presence of video games. Few studies in educational research have explored learning and cognition in presence of video games, which has resulted in calls for a greater number of empirical studies in student learning outcomes within the field
(Connolly, et al., 2012; Mayer, 2015). Some of these studies are reviewed below to provide examples of how student learning outcomes have been examined thus far in the field’s brief history. While this dissertation does not aim to examine student achievement or cognition, its focus on enhancing student engagement in mathematical practices through purposeful task design has the potential to inform future studies that measure student learning outcomes associated with these tasks. Through an engagement in mathematical practices, students may engage in mathematical thinking (Carpenter, Fennema, Franke, Levi, & Empson, 1999; Stein,
Grover, & Henningsen, 1996), which may possibly lead to mathematical achievement.
The development and implementation of Quest Atlantis, a serious educational game that utilized design-based research and ethnographic perspectives, has been a frequently cited project in educational video game research according to Google Scholar as well as individual journal websites of the project’s many publications (Arici, 2008; Barab, Gresalfi & Arici, 2009;
Barab, Gresalfi, & Ingram-Golbe, 2010; Barab, Thomas, Dodge, Carteaux, & Tuzun, 2005;
Thomas, Barab, & Tuzun, 2009). These publications outline the development of the game and its activities in great detail, and establish the term transformational play as a game-based learning in which the player “takes on the role of a protagonist who must employ conceptual understandings to transform a problem-based fictional context” (Barab, et al., 2010, p. 525).
Two studies that were part of this project found that students who used the game in the
16 classroom learned more than the comparison group who used the same curriculum without the game (Barab, et al., 2009).
One of these studies (Arici, 2008) used a quasi-experimental design that also incorporated the use of ethnographic and observational data techniques. The study examined both measures of engagement and learning in four sixth grade science classrooms with the same teacher. The teacher’s four classes were randomly assigned a traditional delivery mode without the use of the game, or a delivery mode that used Quest Atlantis. While pre- assessments showed no significant differences between classrooms, post-assessments showed significant learning gains by the groups that received the delivery mode that used the game.
The groups that used Quest Atlantis also displayed greater engagement, suggesting a possible relationship between engagement and student achievement in video game-based learning.
My study differs from the Quest Atlantis project in that it (1) uses an educational video game not originally designed to cover a specific curriculum, and (2) primarily focuses on engagement in standards for mathematical practices rather than on specific science curriculum.
However, these differences highlight the unique features and goals of my study. While I also apply design-based research, I use an engagement theory that involves intrinsic motivational factors of students (reviewed later in the chapter) to affect my task design in real time as students and I move through the timeline of two workshop iterations rather than to test a serious game’s ability to affect student achievement distinct from its development process.
A study that examines an educational video game rather than a serious educational one is that of Stanton (2017). Stanton examined the use of Minecraft as an educational tool with fifth grade geometry students in the course of his dissertation research. He argued that
17 because research on educational game use rarely encompasses commercial off-the-shelf games such as Minecraft, he aimed to provide evidence that student achievement can increase with the use of these types of games in the classroom. Thus, his quasi-experimental quantitative study centered on the collection of student learning outcomes after students participated in a series of interventions in geometry defined as (1) traditional with no digital gameplay, (2) lesson-based digital gameplay, and (3) play-based digital gameplay. Despite providing the interventions in varying orders to different student groups, he found that Minecraft-based interventions led to increased achievement compared to traditional interventions, and that there was little difference in student achievement associated with the two types of Minecraft- based interventions.
Because Stanton’s (2017) study was quantitative in nature, the focus was not on the mathematical tasks that were embedded within Minecraft. While outlines for the lesson-based interventions were provided in the appendix, outlines consisted only of a brief series of steps designed to assist students in moving forward in the game. They did not present or suggest opportunities for mathematical practices and student discussions that captured human and tool interactions towards learning specific concepts. Therefore, a goal of my study is to address this gap in task detail as I provide specific explanations behind the design and modification of embedding mathematical tasks within the game, decisions that are informed by my observations of student reactions and engagement with the tasks. Furthermore, my study aims to rely on learners’ dialogue in the course of their interactions with the game, a component that is absent from more quantitative studies.
18 Calls for empirical research. With the exception of the aforementioned studies on learning and cognition as well as those on engagement reviewed in following sections, many publications and research on educational video games fail to include either qualitative or quantitative empirical evidence surrounding student and teacher interactions with the game.
For example, Mayer (2015) advocates for a greater collection of empirical evidence in his call for moving away “from broad theoretical frameworks to more focused theories that spell out specific learning mechanisms in a testable way” (p. 350). He acknowledges that the field of educational games research is still young, but that speculating on broad perspectives for game design and offering “expert-inspired visions of how games might improve education” (p.350) is not enough.
While the establishment of broad theoretical frameworks might be necessary during beginning stages of a developing field, observations and critically designed experiments are necessary for progression in K-12 mathematics education (Begle, 1979) as well as in educational research in general to test and modify these theories. Eichenbaum et al. (2014) note the importance of systematic empirical research in video games education. They conclude that while educational games can be utilized for learning, games themselves are neither intrinsically good nor bad for learning; the nature of their impact will depend on how they are used with students, and systematic research is needed to explore specific outcomes. Furthermore,
Connolly, et al. (2012) performed a vast literature review on serious educational games and educational video games research. Their search found 7,392 papers, which confirmed a “surge in interest” in the field (p. 671). However, only 129 of these papers (less than 2%) provided empirical evidence concerning impacts and outcomes of playing games for learning.
19 For example, a recent study by Zikas, Bachlitzanakis, Papaefthymiou, Kateros, Georgiou,
Lydatakis, and Papagiannakis (2016) provides a thoughtful design framework for serious educational game creation in both augmented reality (AR) and virtual reality (VR), but their project does not yet apply empirical evidence to examine student engagement or learning outcomes with the games or to inform their theoretical model. The authors’ list their main research question as a focus on whether “games can, via novel Presence (feeling of ‘being there’ in a virtual or augmented world)…support and foster future learning and teaching” (p.
805) yet there is no mention of testing or measuring Presence with the game’s targeted audience of primary school students or any other group of participants. Rather, the study provides detailed information of how the game was designed, and predicts how students might react to certain activities and visual elements. Based on this article alone from their project, the study exemplifies Mayer’s (2015) aforementioned critique for a need to detail specific learning mechanisms within a model or theory, as well as apply empirical evidence to test a framework.
In contrast, my study aims to not only detail the design and modification of activities embedded within an educational video game, but illustrate how the design process connects to a theory and methodology informed by empirical evidence of student engagement in mathematical practices. My study’s documentation and analysis of engagement also may be used to inform future studies that connect engagement in mathematical practices to student achievement and learning outcomes of these mathematical practices. The next section details how engagement has been studied in relation to video games in a classroom environment, and compares recent studies that use empirical evidence to study engagement to my own research.
20 Research on Engagement with Video Games
Engagement in video games and in Minecraft specifically, has been explored in multiple studies that use a variety of theory, methodology, and instrumentation to examine engagement. First, I will review Self-Determination Theory as a model to study the intrinsic motivation and engagement within video games. Next, I will describe how student engagement in video games is conceptualized and related to the academic content that may be embedded in the games while relating the ideas to my own study. Then, I will review and synthesize a number of studies that examine student engagement in Minecraft within a classroom setting, while again, pointing to how my study may apply and strengthen some of the ideas within these studies.
Engagement with video games: Motivational factors of Self-Determination Theory.
Self-Determination Theory provides a model for intrinsic motivation, the natural tendency for human beings to explore activities of interest or enjoyment. As explained by Deci and Ryan
(1985), the creators of the theory, intrinsic motivation is different from extrinsic motivation, which is on-task behavior driven by external incentives such as money or approval of others.
While many studies have explored the use of Self-Determination Theory in sports and other leisure domains (Przybylski, Rigby & Ryan, 2010), Ryan, Rigby and Przybylski (2006) argued that
Self-Determination Theory (SDT) could also be used to measure motivation in video game participation, because “most players do not derive extra-game rewards or approval. Indeed, most computer game players pay to be involved, and some even face disapproval for participating” (p. 349). Thus, people engage with games because they are intrinsically satisfying.
21 A “mini theory” (Ryan, et al., 2006) of SDT, Cognitive Evaluation Theory (CET) explains specific psychological needs during intrinsic motivation, which include a need for (1) autonomy,
(2) competence, and (3) relatedness. Also referred to as motivational factors (Przybylski, et al.,
2010; Rigby & Ryan, 2011), these three psychological needs are applied to motivation in video game play. Autonomy is defined by the presentation of opportunities for choice, such as control over one’s environment or virtual world, and is the crucial element in intrinsically motivated activities (Deci & Ryan, 2000). Autonomy is not synonymous with chaotic, uncontrolled freedom. Rather, an individual can experience parameters within their choices, but still feel that he/she has a range of choices, and that the results of choices made are meaningful in game play (Rigby & Ryan, 2011, p. 47). The virtual world of Minecraft provides opportunities for user autonomy, and as discussed in later chapters, task design and modification in this study aim to incorporate autonomous elements such that students still experience meaningful choice in the game as well as in mathematical practices despite task parameters.
Competence encompasses a player’s need for challenge and feelings of effectiveness during the acquisition of new skills or abilities, and receipt of positive feedback within the game as opposed to that from a teacher, which might be extrinsic motivation in certain situations, especially if grading is involved such as in certain K-12 settings. Players experience competence when they are optimally challenged with a task that is neither too difficult nor too easy. When a player experiences the reward of the task, such as finishing a build in the virtual world of
Minecraft, the individual must connect the positive feedback to the competence applied
22 through autonomy such that the reward connects to his/her own choices (Deci & Ryan, 1985,
2000).
Finally, relatedness is when the player feels connected to others. While Ryan, et al.
(2006) are “intrigued by how needs for relatedness may be met by ‘computer generated’ personalities and artificial intelligence” (p. 350), they chose to focus studies of this motivational factor on interactions between real players, a focus of my study as well. Social connectedness or relatedness might differ from the extrinsic motivation of praise from a teacher if students connect teacher praise to the receipt of grades. However, in a study by Grolnick and Ryan, children in a classroom setting reported higher levels of intrinsic motivation when their teacher was warm and receptive (Deci & Ryan, 2000). While I attempted to encourage students as the researcher-teacher in my study, I focused primarily on interactions and social connectedness of students with one another both inside and outside of the game to document and analyze the motivational factor of relatedness as it related to my task design.
Whereas CET has been applied to other fields of research besides education and video game play, a second mini-theory of SDT, Player Experience of Need Satisfaction (PENS) specifically explores SDT in relation to video game play. PENS includes the motivational factors of autonomy, competence, and relatedness, but also introduces presence and intuitive controls as additional factors that might be assessed to measure motivation and engagement in a game.
Presence is described as “the sense that one is within the game world, as opposed to experiencing oneself as a person outside the game” (Ryan, et al., 2006, p. 350), and intuitive controls are game controls that are easily mastered, and do not interfere with one’s presence.
23 Because game controls are not manipulated by task design and modification, my study does not incorporate observation or assessment of intuitive controls as a measure or motivational factor.
Additionally, my study does not incorporate presence as a separate motivational factor to autonomy, competence, and presence, and instead places it within competence. Reasons for this decision are twofold. First, in their analyses of four studies, Ryan, et al. (2006) found that presence was associated with competence as players carried out effective actions related to their in-game interests. Through effectively carrying out actions that were meaningful to them, players experienced a greater feeling of being a part of the game’s world. Second, Witmer and
Singer (1998) used the Presence and Immersive Tendency Questionnaire to research and measure presence after participants’ game use to score multiple factors they defined to be related to presence including realism. Realism includes the concept of meaningfulness of experience for which presence increases as the in-game situation or environment becomes more meaningful to the person. Furthermore, meaningfulness is also related to motivation to learn or perform, and experiencing success after performing a learned skill may lead to feelings of effectiveness or competence. Thus, competence relates to presence through meaningfulness; when players feel present in the game world, they may experience competence when completing a goal or task in the game due to feeling that they are doing something meaningful in the world in which they are immersed.
An example of a study that used SDT’s three core motivational factors of autonomy, competence, and relatedness to explore student engagement in Minecraft within a K-12 classroom setting is that of Cipollone (2015). Her study showed that students were engaged in the game through SDT, and that the Minecraft video game environment affected students’
24 ability to problem solve, which was evidenced by pre- and post-assessments. However, one of her research questions regarding how a teacher or facilitator as a source of extrinsic motivation might foster intrinsic motivation to engage in the game was unable to be explored in depth due to the disengagement of the instructor in her study. Instead of leading the Minecraft components of the lesson plans as anticipated by the researcher, the instructor removed herself from the instructor role, and the researcher became the instructor. Thus, my study hopes to examine this missing piece in my exploration of how task design and modification as the researcher-teacher might influence student engagement, especially in regards to SDT.
Engagement with video games, in general. Video games can also be engaging, because they hold significance or relevance for the player. Devlin (2011) highlights the need for a connection to the real world of the student, if the student is to be engaged with the educational purposes of the video game. In relation to connecting mathematics to a game,
“the mathematics learned has to arise naturally in the environment, and have meaning in it, and the learner in that environment has to be motivated to carry out the tasks that involve that mathematics” (p. 25). Theories of problem-based learning also emphasize the importance of relevance to the learner to promote engagement (Smith & Corbin, 2014). Because my research was conducted in an after-school enrichment program within an organization that promotes and supports the use of problem-based learning, philosophies of problem-based learning influenced components of my design process. For example, a problem incorporated within the task was how students might design and build a room in the game with the same dimensions of the classroom. In my study, a natural and meaningful connection to the real world (Devlin,
2011) was connected to a relevant problem within the game (Smith & Corbin, 2014). I attempt
25 to depict how an intention to include relevance within the task’s problem influences the motivation and engagement of students while they worked through the task to solve the problem.
A quantitative study (Annetta, Klesath, & Meyer, 2009) also examined engagement in the form of presence and immersion as undergraduate students worked on an entomology bug collection project using a serious game designed by the authors. In their study, presence is defined by “the ability of people to be perceived as real, 3-D beings…despite not communicating face to face” (p. 30) and immersion is defined as “a psychological state is achieved where students perceive themselves to be enveloped by, included in, and interacting with the environment and other users” (p. 31). The Presence and Immersive Tendency
Questionnaire used by the U.S. military (Witmer & Singer, 1998) to measure presence was provided to students after using the game to score user presence on four factors of control, sensory, distraction, and realism on a five-point Likert scale.
Witmer and Singer define control as the degree and immediacy of autonomy, how well an individual can perceive what may happen next, how natural or well-practiced the interaction with one’s virtual environment feels, and the degree to which the individual can modify his/her environment. The definition of the sensory factor focuses on the individual’s visual perception of the virtual environment and movement. Distraction or the lack thereof, is defined by the degree to which an individual feels isolated from their actual physical environment during gameplay, as well as their willingness or ability to ignore their external surroundings. Finally, realism “refers to the connectedness and continuity of the stimuli being experienced” (p. 230) and is grouped with how meaningful the experience feels to the individual.
26 In Annetta, et al.’s (2009) study, Witmer and Singer’s (1998) questionnaire synthesized presence with immersion such that questions were asked to determine whether participants perceived themselves to be “enveloped by, included in, and interacting with the environment and other users” (p. 31) such that outside distractions were minimized. Findings suggested that students’ experienced high levels of presence and immersion, but the authors emphasize the importance of the instructor in synthesizing the content learned in the environment to ensure students connected the content to the experience. My study attempts to emphasize the authors’ suggestion of the importance of the instructor in connection with the content and gaming experience. Through intentional task design and modification of the researcher-teacher informed by observations of student engagement in mathematical practices, presence and immersion are incorporated through Self-Determination Theory’s motivational factor of competence. Presence and immersion were categorized in my study as part of SDT’s competence, because the degree to which students felt effective while working on a task related to the victories they experienced in Minecraft. A victorious experience could only occur if the students felt present and immersed in their gameplay through a connection between the task and game. Increasing the competence experienced by students during task implementation is a goal considered by the researcher-teacher as task modifications simultaneously attempt to enhance opportunities for mathematical practices within the task.
Furthermore, Annetta, et al. (2009) also cite their previous research (Annetta, Klesath, &
Holmes, 2008) in which they find virtual presence to be directly correlated with undergraduate students’ success and satisfaction in an online course. While my study does not examine student academic success, their research provides another example of the possible connection
27 between student engagement and student achievement, a connection that validates my own focus on student engagement.
Engagement with Minecraft. Similar to the virtual learning environment in the study of
Annetta, et al. (2009), the Minecraft world is a 3-D virtual environment, and recent research has noted some of the ways that it engages students. Bailey (2016) completed a year-long ethnographic study of 10 and 11 year old students who played Minecraft in an after-school club setting. He observed that students were simultaneously engaged in spaces both inside and outside of the game. Their singing, discussion, and body movement outside of the game related to in-game play, while also affecting the in-game play as it occurred. Bailey’s rich descriptions of student interactions suggest that Minecraft can be highly engaging for students, especially when positioned to sit with one another while playing in their virtual worlds. His findings also provide evidence of a complex and interwoven relationship between students’ in- game and face-to-face play when sitting in close proximity during game play.
Unlike the after-school enrichment program Minecraft Mathematics of this study, his club did not seem to incorporate academic tasks or goals. However, his illuminations of the relationship between student and game interactions relate to my study’s observational goals in utilizing these interactions to inform task design. My careful documentation of student interactions aimed to capture and play upon the students’ existing engagement in order to promote even greater engagement, especially in relation to the task and mathematical practices.
Bailey’s study did not aim to embed academic tasks or goals within Minecraft, but additional researchers and advocates for the game’s educational use (Callaghan, 2016; Marcon
28 & Faulkner, 2016; Meyer, 2015; Nebel, et al., 2016; Niemeyer and Gerber, 2015) have observed engagement through various forms of student collaboration. This collaboration includes students’ discussion of what is planned, built, and modified in the game, as well as instances of students teaching one another game mechanics, recipes for crafting materials, and methods for building. Starkey (2016), and Tromba (2013) define high levels of student engagement through constant attention and involvement with the game, even among students with varying interests due to the game’s ability to cater to these multiple interests via its open-world setting and spectrum of building materials and tools (an example of autonomy in Self-Determination
Theory). Callaghan (2016) notes that the teacher had to wait to start the game for the students after directions were delivered, due to their eagerness to engage with the game before receiving their assignments. Thus, the ability of the game to engage students in creative endeavors surrounding the building of products has been widely noted. My study aims to capitalize on this existing engagement, and show how purposeful task design might incorporate the communication and creativity to inspire mathematical practices.
Bailey (2016), Callaghan (2016), Hanghoj, et al. (2015), Meyer (2015), and Nebel, et al.
(2016) also note the importance of meaningful facilitation by the teacher to foster engagement, especially in relation to synthesizing a connection between Minecraft and the academic content. Indeed, proponents of game-based learning advocate for the necessity of the teacher to facilitate and assess the connection between the game and learning (Annetta, et al, 2009;
Barab, et al., 2010; Farber, 2016), and the same facilitation applies when using Minecraft as an educational tool. Bailey and Callaghan described the instructor or teacher interacting in the game alongside students to establish a presence in the game, engage students in tasks and
29 objectives, and provide opportunities for students to “show off” their builds to the teacher.
While students worked through tasks in the Minecraft Mathematics workshops, I often moved around the classroom to interact face-to-face with students while carrying and opening my laptop to interact with the students simultaneously in the game. In doing this, I also attempted to synthesize mathematics content and Standards for Mathematical Practice with game and task engagement.
A final study worth noting that surrounds the use of student engagement in Minecraft is that of Cipollone (2015) who studied seventh and eighth grade students’ use of the game in their mathematics classroom for learning purposes. As previously mentioned, Cipollone applied
Self-Determination Theory to video games, and used survey questions that were framed with the Player Experience Needs Satisfaction scale (Ryan, Rigby and Przybylski; 2006), a measurement tool of the theory. Cipollone’s Mixed-Methods study is rich in quantitative data regarding students’ ratings of themselves in their degree of motivation, a seven-point Likert scale, while playing the game in class. Qualitative measurements through field notes and focus groups were also documented and analyzed to provide examples and summaries of how students were and were not engaged with the game during class time. However, because the focus of the study was to examine the intrinsic motivation of students while integrating an educational video game into a classroom environment, detailed information regarding the mathematical tasks in the game as well as the mathematical practices in which the students were engaged was not present. Furthermore, due to complications within the study, she was unable to explore the teacher’s interactions with students during game implementation.
30 Therefore, my study may contribute to this gap in the research, because students’ engagement in the game is similarly examined using Self-Determination Theory, but applies the theory towards game-embedded task design and modification of the researcher-teacher to enhance students’ engagement in mathematical practices. The next section examines literature on student engagement in mathematical practices, especially in relation to task design and implementation. Engagement in mathematical practices through student work on a task will eventually be connected to student engagement in Minecraft with Self-Determination
Theory within my theoretical framework presented in the culminating section of the chapter.
Research on Engagement with Mathematical Practices
The Common Core State Standards for Mathematics outline eight Standards for
Mathematical Practice (SMP) that “describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years” (Common Core State Standards Initiative, 2017). The eight standards are as follows:
• Make sense of problems and persevere in solving them
• Reason abstractly and quantitatively
• Construct viable arguments and critique the reasoning of others
• Model with mathematics
• Use appropriate tools strategically
• Attend to precision
• Look for and make use of structure 31 • Look for and express regularity in repeated reasoning
Because my study aimed to garner engagement in these practices through the design and modification of tasks embedded within Minecraft, I will synthesize literature that addresses mathematical tasks in two ways. First, I will define particular features of rich tasks and how these features might be designed within a task to foster student engagement in mathematical practices. Second, I draw upon literature to present suggested facilitation of rich tasks that might optimize student engagement in mathematical practices.
Rich mathematical tasks as a vehicle for engagement. Rich tasks, also referred to as high cognitive demand tasks, are tasks that engage students in high level thinking and reasoning (Stein, Grover, & Henningsen, 1996). When students successfully engage in rich tasks, they are doing mathematics (Collins & NCTM, 2011; Stein, et al., 1996), a process that involves complex and non-algorithmic thinking. This description of doing mathematics differs from other low cognitive demand definitions that describe simply following predetermined rules laid out by the teacher (Lampert, 1990). For example, rather than going through a well- rehearsed step-by-step procedure that has been provided to them by the teacher or another source, students must analyze constraints of a problem to explore processes, connections, and relationships of the content (Collins & NCTM, 2011). Students are encouraged to monitor and regulate their own thought processes (Pape, Bell, & Yetkin, 2003), and then verbalize and write about these processes.
The structure or task features (Stein, et al., 1996) of rich tasks contain specific characteristics. In describing rigorous learning experiences, Gojak (2013) notes that entry points and extensions must be included for all students, suggesting that there should be
32 multiple ways to start the task, and multiple ways to arrive at a variety of solutions. Stein et al.
(1996) also list (1) multiple solution strategies, (2) offering opportunities for multiple representations, and (3) requiring communication between students and to the teacher of mathematical thinking. In the context of group work, Cohen (1994) discusses rich multiple ability tasks (p. 68) that aim to elicit high cognitive demand from students while problem solving as a team. Among her features, she lists that the tasks should (1) have more than one answer and solution path, (2) be intrinsically interesting and rewarding, (3) allow different students to make different contributions, (4) require a variety of skills and behaviors, (5) require reading and writing, and (6) be challenging. Rich mathematical tasks engage students as they experience these features through observable actions such as documenting solution strategies, communicating their representations, and demonstrating their attention to solving a problem.
Therefore, rich mathematical tasks in my study incorporate a combination of the features discussed above: (1) Multiple entry points (Gojak, 2013), (2) multiple solution paths
(Cohen, 1994; Gojak, 2013; Stein, et al., 1996), (3) multiple answers (Cohen, 1994; Gojak, 2013;
Stein, et al., 1996), (4) multiple representations (Cohen, 1994); (5) relevance for students in content and context (Cohen, 1994; Gojak, 2013); and (6) a demand for communication in either oral or written display of mathematical thinking (Stein, et al., 1996). These task features support the Standards for Mathematical Practice (SMP), because they set expectations for a task to provide opportunities for more cognitively demanding engagement with mathematics.
For example, when a task has multiple solution paths and multiple solution possibilities, students might employ the SMP of reason abstractly and quantitatively along with construct
33 viable arguments and critique the reasoning of others in order to convince themselves and others that their solutions make sense, rather than rely upon a single, given answer by the teacher. Rich tasks embedded within Minecraft must also include the task feature of relevance for students in content and context by linking SMP such as model with mathematics, and look for and make use of structure from real life contexts to the virtual game world, one of many connections that my research aims to portray.
Facilitation of student engagement with rich tasks. Similar to the argument of linking game play to academic content cited earlier, one of the many roles of the teacher in task delivery is to ensure a connection for students between the task and its academic content.
Enacting tasks with students is part of curriculum development and design work done by teachers (Remillard, 2005). Simply giving a rich task to students without an appropriate introduction and relation to material, or without proper facilitation during and after the task is likely to diminish the engagement of the task, and detract from its high cognitive demand
(Stein, et al., 1996). Because rich tasks require such high cognitive effort, students may experience anxiety regarding the unknowns, and give up during a problem that does not have an easily attainable solution (Collins & NCTM, 2011; Stein et al., 1996). Building student expectations, perseverance, and a dependable routine instills a productive and engaged problem-solving community within classroom culture.
Cohen (1994) describes such a dependable routine for structuring and timing a rich task.
First, students are pre-trained and receive orientation, which may be a teacher-driven warm-up activity, or demonstration of basic knowledge needed to solve the task. The purpose of the orientation is to provide substantial guidance to the students, without showing a procedure for
34 solving a version of the task itself. After the orientation, the task is introduced to the students.
Students read the task and design a process for moving towards the solution. While students may not yet know exactly how they will arrive at the solution, they are encouraged to come up with a preliminary plan for getting started. When the allotted time for the task has been provided, students share their solutions. During the wrap-up students have an opportunity to discuss, reflect and connect mathematics concepts to the application they have just experienced. The wrap-up can occur completely during the sharing of solutions, or can extend to a teacher-led discussion afterwards. Each class day in my study’s workshops followed a similar dependable routine in attempt to establish a productive and engaged classroom culture.
During time that students work with their groups on the task is often when reduction in task complexity may take place. Henningsen and Stein (1997) found that an otherwise well- designed rich task recedes when (1) there is a shift in focus from student understanding to correctness and completeness of an answer, (2) students pressure the teacher into providing the procedure or completing the more challenging parts of the task, and (3) the answer itself is emphasized instead of the solution path for achieving the answer. Therefore, as the instructor of Minecraft Mathematics classes, I attempted to be mindful of holding a strong focus on student understanding while underscoring students’ processes when formatively assessing to foster engagement in mathematical practices. I documented these actions as part of my field notes, reflection and analysis for further task design and modification.
Anderson (1989) also speaks to the importance of maintaining a level of high cognitive demand during student work on a task to ensure that the task remains rich in the mathematical opportunities provided to students. For example, if students are struggling with the open-
35 endedness of a task, a teacher can scaffold the lesson to provide support without decreasing the cognitive demand, or have a student model a thinking strategy or process. Colburn (2004) suggests finding a blend of open-ended and structured task design that caters to the needs and experiences of students in a particular space and time, so that they may still engage in high- cognitive demand work without becoming disengaged due to a perceived lack of structure.
Colburn’s blend of open-endedness and structure was one I often reflected upon in my study.
As detailed further in my analysis chapter, providing students with the appropriate balance of constraints and freedom during one of the first challenges in the Build the Room task required multiple iterations of design and modification during pilot studies as well as during the workshops of this study to establish a task format and delivery that seemed to produce greater levels of engagement in SMP.
Henningsen and Stein (1997) also found ways that instructors supported rich tasks that included (1) building off what students already know and understand, (2) scaffolding the task,
(3) allowing the appropriate amount of time for tasks, (4) modeling high-level thinking processes and strategies to students, and (5) sustaining press for explanation and meaning during a task. Cahnmann and Remillard (2002) reaffirmed many of these student supports with the addition of maintaining high expectations for students, which overlaps with not giving into student pressure for answers. The role of the teacher in facilitating rich tasks is not to be the beholder of all knowledge, but to provoke thinking and problem solving (Cohen, 1994;
Carpenter, et al., 1999) and to give students ownership of the problem. When students have ownership of the problem, they will be more engaged, leading to better understanding
(Cahnmann & Remillard, 2002; Smith & Corbin, 2014).
36 My study largely focuses on how I as the researcher-teacher intended to find ways of supporting and enhancing student engagement in rich mathematical tasks through facilitation strategies delineated above as well as through integrating some of these strategies into the design of the expectations and objectives of the task itself. I reflected upon and analyzed these teacher moves and rich task design strategies through a design cycle process. The next section attends to literature on design cycles to begin to frame my own iterative process of task design within my study.
Research on Design Cycles
In this section, I will review the concept of the design cycle as an iterative process for design work, and some of the ways it has been applied in research. First, I will discuss the design cycle as a set of stages used in engineering collaboration. Then, I will outline the use of design cycles within STEM education and problem-based learning to guide students through a design process.
Design cycles in engineering. Ostergaard and Summers (2009) review the collaborative design process used in engineering, and propose a taxonomy for the classification of collaborative design situations. One of their top-level attributes is design approach, which includes a review of research on the structure and process of design work. They cite a study by
Austin, Steele, Macmillan, Kirby, and Spence (2001) in which designers believed that they performed better as a team when they agreed upon and followed a design cycle, referred to here as a design process. Additionally, a positive relationship was shown between ratings of a team’s processes and designs when a design process was agreed upon and followed within the team (Bursseri & Palmer, 2000). While a number of authors have proposed names and 37 descriptions for design process stages, Ostergaard and Summers (2009) found that the stages defined by Pah and Beitz (1996) were among the most popular in their reviewed research.
These stages are: (1) clarification of the task, (2) conceptual design, (3) embodiment design, and
(4) detail design.
IDEO Technologies, an engineering firm located in Silicon Valley, also demonstrated the use of a design cycle around the design of an improved shopping cart (Koppel, 1999). Their acronym IDEO stands for Interaction Design Engineering Organization, in which interaction design refers to a design process that examines how users interact with a product
(https://www.ideo.com/about). The team worked together to move through a recognizable design cycle that included a series of steps. First, they defined the problem: How can they make a better shopping cart? Second, the team conducted research of existing shopping carts, prices, needs of stores, and problems of current designs. Third, the team built various prototypes. Fourth, the team evaluated each prototype, and ideas were synthesized to fifth, build and share a final product.
Design cycles in education. IDEO’s design cycle closely parallels the design cycle detailed by Smith and Corbin (2014) that theorizes a design process that teachers can follow with students as they implement problem-based learning (Figure 2). Smith and Corbin’s (2014) design cycle is used frequently at the educational non-profit PAST Foundation to frame a process for student learning as well as for teacher planning (Deaner & McCreery-Kellert, 2018).
This design cycle is described as “a universal process of human thinking that helps us organize our attempts at solving problems” (p. 21) and contains the four elements of (1) planning, (2) implementation, (3) analysis, and (4) dissemination. Because the design cycle is presented to
38 teachers through professional development and classroom support, teachers are encouraged to choose language that resonates with their students in order to establish a lexicon used continuously throughout design work.
Figure 2. Smith and Corbin’s (2014) design cycle for design of problem-based learning modules
Aydin, Bakirci, Artun, and Ceptni (2011) also applied a design cycle with teachers in a professional development context in a qualitative case study examining the planning and implementation of a design activity in the elementary classroom. Teachers experienced a professional development session during which they were presented with the technological design cycle, and worked through an activity before implementing the same activity with their students. The steps of this design cycle are: (1) identifying and defining a problem, (2)
39 investigating and researching the problem, (3) generating and considering a design, (4) modeling the selected design, (5) testing and evaluating the selected design, and (6) reporting and presenting results. The study found that while teachers put the design cycle into practice, their teacher content knowledge was lacking around certain concepts addressed in the activity that prohibited the authors’ originally envisioned execution of the activity.
There have yet to be any studies that apply a design cycle to a teacher’s iterative process in using rich mathematical tasks with students, particularly within a game-based environment. Thus, my study aims to contribute to the field through this application. In the next section, I will detail my theoretical framework that uses a design cycle in an innovative way to analyze the evolution of a rich mathematics task through iterative cycles. Additionally, my observations and analysis of student interactions with the task, game, and with one another are framed within the Self-Determination Theory model of Deci, Przybylski, Rigby, and Ryan (2000,
2006, 2010, 2011) as I attempt to enhance student engagement in Standards for Mathematical
Practice.
Theoretical Framework
This study is framed by (1) the application of Self-Determination Theory to video games
(Przybylski, et al., 2010; Rigby & Ryan, 2011; Ryan, et al., 2006), (2) the application of Standards for Mathematical Practice to rich mathematical tasks, (3) design-based research theories and a design cycle to examine my iterative process of task design and modification, and (4) an ethnographic perspective. In this section, I first outline how Self-Determination Theory is used to situate student engagement in the game as well as in the task of the study. Next, I discuss the connection between Standards of Mathematical Practice and Self-Determination Theory in 40 how student engagement might be examined to inform rich task design and modification.
Then, through a presentation of design-based research, a design cycle is introduced that pertains specifically to my iterative process of rich task design and modification. Finally, I coalesce these concepts with philosophies of ethnography that inform my observations of student engagement throughout the study.
Self-Determination Theory. As discussed earlier in the chapter, Self-Determination
Theory contains three motivational factors of autonomy, competence, and relatedness, which can be connected to human needs for satisfaction during video game play (Przybylski, et al.,
2010; Rigby & Ryan, 2011; Ryan, et al., 2006). This study uses the motivational factors to frame observations of student interactions with the game, task, and with each other. For example, students who were observed to be communicating about their actions in the game during a task may be driven by a motivational factor of relatedness, since they may feel connected to one another as they build within the virtual environment of Minecraft while discussing their actions face-to-face outside of the game. Additionally, if these students were debating decisions for a specific type of Minecraft building block for use in the task’s objective, the students may be driven by a motivational factor of autonomy, because they are engaged in a decision-making process that involves their own choices to affect an outcome.
These motivational factors are also fused with task design to support analysis of how engagement in the task might be enhanced, both in relation to Minecraft and Standards for
Mathematical Practice. For example, the students in the hypothetical example above may become more engaged in a task that allows them freedom to choose their own block types for a creation despite other mathematical constraints of the creation’s objectives. Furthermore, if
41 the students are free to choose other mathematical parameters such as length, width and height of a structure that still results in a particular volume constraint of the problem, autonomy is applied to the task through this freedom of choice while opportunities for
Standards of Mathematical Practice such as model with mathematics and attend to precision are incorporated through the students’ creation of a model that represents otherwise abstract dimensions that must equate to a precise volume parameter.
Standards for Mathematical Practice. While the example above aims to show how
Standards for Mathematical Practice (SMP) might be directly intertwined with Self-
Determination Theory (SDT), the two concepts are also connected through features of rich tasks discussed earlier in the chapter. For example, a task may include the rich task feature of multiple solutions, because its objective involves the creation of a product built within the game that can be represented with varying dimensions of length, width, and height to achieve a volume constraint. Because multiple solutions are possible, students must engage with the
SMP of look for and express regularity in reasoning as they build their representation and
“maintain oversight of the process, while attending to the details” (Common Core State
Initiative, 2018, p.1) by connecting how different combinations of length, width and height might create the same volume, regardless of how the values are assigned to each dimension.
Thus, autonomy in SDT is also incorporated, because students have choice in how they attend to the details of the process, choice, organization, and modeling of their dimensions.
Furthermore, the task feature of multiple solutions may also engage students in the same SMP of look for and express regularity in reasoning as they “continually evaluate the reasonableness of their intermediate results” (Common Core State Initiative, 2018, p.1) to
42 make sure their dimensions produce the specified volume constraint. This type of student engagement in SMP might be observed through student dialogue and communication that would also take into account the SDT of relatedness through their interactions within and outside of the game. My study aims to provide evidence for how student engagement in SMP might be enhanced within a rich task embedded in Minecraft, and how SDT can be used to propel the engagement. Thus, task design and modifications implemented and analyzed throughout my study aim to incorporate rich task features that enhance SMP through their attention to SDT.
Design-based research. Connections between SMP and SDT in rich task design involve philosophies of design-based research that were applied to study my evolution of a mathematics task in a complex system. According to the Design-Based Research Collective
(2003), good design-based research: (1) intertwines design of a learning environment with theories; (2) uses an iterative process to design, test, and modify; (3) leads to shareable theories with practitioners; (4) accounts for how designs function in authentic settings through documentation of successes and failures with a focus on interactions that refine understanding; and (5) relies on solid methods of documentation to connect processes of enactment to outcomes of interest. My research embodies the first two characteristics, because my design of rich mathematical tasks in the Minecraft environment will take into account theories of engagement with mathematical practices and educational video games, as well as theories of task design through an iterative design cycle. The third characteristic connects to the goal of my research to contribute to the mathematics education field in examining how an educational video game such as Minecraft might be used to foster engagement as the teacher’s iterative
43 task modification process is explored. Finally, the fourth and fifth characteristics pertain to my research through my systematic documentation methods and procedures that involve an ethnographic focus on how I interpret student engagement in both the game and mathematical practices, that in turn affect my modification of a rich mathematical task that I designed for students within Minecraft.
The design cycle. The study is theorized through the Rich Task Design Cycle, a modified version of the Smith and Corbin (2014) design cycle (Figure 2). This particular design cycle frames my iterative process for developing a rich mathematical task to engage students; the cycle is specific to the teacher’s task design and focuses on the teacher’s analytical process.
The framework includes a Day-by-Day cycle (Figure 3) as well as a Workshop-by-Workshop cycle
(Figure 4) within its process.
44 Day-by-Day Cycle
Figure 3. The Day-by-Day cycle of the Rich Task Design Cycle adapted from Smith and Corbin
(2014)
45
Figure 4. The Day-by-Day and Workshop-by-Workshop cycles of the Rich Task Design Cycle adapted from Smith and Corbin (2014).
In the model, Anticipate replaces the upper-right quadrant of Smith and Corbin’s
Planning. Additionally, Smith and Corbin’s Analysis quadrant is stretched into two quadrants of
Evaluate and Modify, the latter of which replaces Smith and Corbin’s Dissemination quadrant.
In task design, dissemination is encompassed when the task is shared with students during the next iteration, and associated issues are anticipated, implemented, evaluated and modified
46 again. An entry point is added between the Modify and Anticipate that represents the new or revised version of the task, Version X.
This model includes three specific time periods or phases of my task design analysis that make up two cycle types of the process: (a) Day-by-day, and (b) Workshop-by-workshop. Phase
1 is the first half of the day-by-day cycle and consists of the observations and analysis I conduct just as class is beginning (Anticipate) as well as during class (Implement). Here, I systematically documented student engagement with the task in relation to the game and to mathematical practices. Phase 2, the second half of the day-by-day cycle is the analysis I synthesized directly after class as I reviewed my observations from class, interpreted (Evaluate), and documented the modifications I would apply before the next class day (Modify). Because the overarching task in my study occurred over four class days, I moved through the day-by-day cycle four times in one Minecraft Mathematics workshop.
The timeline of the phases and cycles is outlined in Table 1 as the task is implemented over four days. As the task is implemented on the first day, Phase 1 occurs. At the culmination of the first day, Phase 2 analysis encompasses an overview of the entire first day lesson. On the second day of the task, a new iteration of the day-by-day cycle begins in which Phases 1 and 2 occur a second time. A third and fourth iteration of the day-by-day cycle occur on those days of the task as well. Phase 3 occurs after completion of the entire task over the four-day time period when I analyze all day-by-day cycles to modify the task for the iteration in the next workshop with a different group of students.
47 Table 1
Timeline of Phase and Cycle Iterations
Day of Task During/After Phase Day-by-Day Workshop Class Cycle Cycle 1 During 1 1 1 After 2 2 During 1 2 After 2 3 During 1 3 After 2 4 During 1 4 After 2, 3 1 (new During 1 1 2 iteration) After 2
Ethnographic perspective. During the three phases delineated above, an ethnographic perspective drove the systematic documentation and reflection of my research in addressing the complexity of a learning environment. Cobb, et al. (2003, p. 9) use the metaphor of an ecology to emphasize the complex nature of interacting systems of the classroom environment which include (1) tasks or problems for students, (2) kinds of discourse encouraged, (3) norms of participation that are established, (4) tools and related material means provided, and (5) the practical means for teachers to connect these intertwined and overlapping elements. Analysis of a classroom ecology can be supported by an ethnographic perspective through recommendations to the researcher to take into account his/her own experiences and biases, and use them to inform and analyze the same observations (Emerson, Fretz, & Shaw, 2011).
48 Emerson, Fretz, and Shaw (2011) suggest attention to how routine actions within the environment of study are organized, and how they take place, because “asking how also focuses the ethnographer’s attention on the social and interactional processes through which members construct, maintain, and alter their social worlds” (p. 27). In this study, my attention focused on three components of social and interactional processes. First, I attended to students’ engagement in how they interact and interpret a task within the game. Second, I focused on how I interpreted engagement based on my own beliefs. Third, as the researcher, I focused on how my interpretations of engagement affect modification of a rich mathematical task. An ethnographic perspective assisted me in removing expectations of what is “normal” or
“right,” and motivated me to take into consideration “what one knows, how such knowledge has been acquired, and the degree of certainty of such knowledge, and what further lines of inquiry are implied” (Hammersly & Atkinson, 2007, p. 192). This perspective also provided opportunity for taking into account my perspectives and biases as the researcher-teacher.
Thus, I responded to my observations of students’ engagement with the game and mathematical practices, analyzing these observations to inform my task modification.
My procedures for systematic documentation, detailed in the next chapter, also draw from ethnography. My in-depth documentation supported design-based research by focusing on how I modified the task in certain contexts, since one of my research goals is to produce
“robust explanations of innovative practice, and provide principles that can be localized for others to apply to new settings” (Design-Based Research Collective, 2003, p. 4). As recommended by Barab and Squire (2004) as well as Briggs (1986), another underlying goal of my research was to detail as many contextual factors as possible, as well as my own biases, in
49 order to present my analysis as transparently as possible. I aim to do this so that future studies might compare and contrast contexts, and modify or strengthen my attached theories related to task design and modification in an educational video game environment.
Figure 5 shows how each of the four theories discussed are enveloped within the components of my research process. While Standards for Mathematical Practice (SMP) and
Self-Determination Theory (SDT) are also applied during the observational and analysis components of the study, they were the theories of focus during the rich task design and modification components. Similarly, an ethnographic perspective was adapted throughout the study, but was largely considered during the documentation and analysis of task implementation. Finally, Design-Based Research and the Design Cycle were present within the overall process of the study, as the research was conceptualized in both day-by-day iterations as well as workshop-by-workshop iterations.
Figure 5. Theories within each component of the study’s process
50 Chapter 3: Methodology
In this qualitative study, I used an iterative design-based research process, a design cycle, to systematically document and analyze my observations of student engagement both in mathematical practices, and in the game. My observations of student engagement were informed from an ethnographic perspective, which emphasized my focus on how participants interacted with one another and the game. An ethnographic perspective also contributed to the procedures and instrumentation utilized for documentation and analysis.
Research surrounding the application of an iterative design cycle to modify a task from the teacher’s perspective is currently limited, especially in relation to the use of the teacher’s observations of student engagement to drive the task modifications. Current research has shown that when a teacher uses Minecraft with intentional pedagogical approaches, the game is more meaningful for students in creating opportunities for engagement with content
(Hanghøj, et al., 2015; Nebel, et al., 2016). Furthermore, a teacher’s implementation of activities over time that involve the use of a video game, seem to point to a positive affect on student engagement (Bell & Gresalfi, 2017). These arguments highlight the importance for the exploration of my iterative process of intentional task evolution over time based on my analysis of student engagement.
Because the focus of my study is to gain an improved understanding of the “how” of task modification, a qualitative inquiry that adapts design-based research and an ethnographic perspective is most suitable in order to examine the development a task that is driven by emergent behaviors of students while taking into account my own beliefs as a researcher and teacher (DBRC, 2003; Hammersley & Atkinson, 2007) that also inform iterative task
51 modification. This chapter will first attempt to support this claim through descriptions of (1) the setting of the study including affordances and constraints, (2) procedure and timeline of the study, (3) the rich mathematical task of focus, and (4) the participants involved in the study.
Next, data collection procedures and instrumentation will be outlined to provide detail of how student engagement was documented, and how the task design and modification process was executed. Then, the analysis process of the study will be delineated before researcher-teacher beliefs and biases are summarized, and credibility of the study is presented.
Characteristics of Minecraft Mathematics Workshop Setting
My study occurred in an after-school enrichment program, Minecraft Mathematics located in a classroom space at an educational research and development non-profit organization. While enrolled in a workshop, each student received his/her own laptop for use during class along with a mouse and graph-paper notebook to document work. Students were positioned in groups around tables, and were encouraged to work together.
Timing of workshops and timing constraints. Originally, the timeline for the study’s workshops was much longer, consisting of one 120-minute class per week for eight weeks, for a total of eight classes per workshop. A student in one of these workshops might have spent a rough total of sixteen hours in the workshop. In addition, three workshops of eight classes each were planned in order to create an opportunity for the study to move through three iterations of the task of focus. However, due to financial constraints of the organization as well as other factors out of the researcher’s control, both the number of workshops and classes per workshop were greatly reduced. Only four students registered for the first initially scheduled
52 Minecraft Mathematics workshop. The organization determined that holding the workshop was not financially feasible, and the workshop was cancelled.
The organization then reduced the duration of the remaining two workshops from eight to four days, and reduced the class time from 120 minutes to 90 minutes. Reasons provided to the researcher for this decision were additional financial constraints as well as an understaffing issue in a separate division of the organization in which the researcher was employed. One may note that this was a decision articulated by the organization, and not able to be controlled by the researcher.
The required age group for the students of the remaining workshops was also changed.
Originally, the workshops were to be offered to middle school students in grades five through eight. However, due to concerns of achieving financially feasible registration numbers for the workshop, the age group was changed to an elementary school level, requiring students to be in grades three, four, or five. Due to her greater experience with middle school students in
Minecraft Mathematics, the researcher had originally planned for the study’s workshops to involve this same age group. However, the researcher also had experience with elementary aged students, although in fewer Minecraft Mathematics workshops, so this decision was not as monumental as the reduction in workshop duration.
Each of the two Minecraft Mathematics workshops occurred over four class days that were spread between multiple weeks. The first workshop occurred once per week for four weeks, and the second workshop occurred twice per week for two weeks. Each class day was
90 minutes long, not including the time that students arrived early and stayed late to play in the
53 game or continue to work on a task’s product in the game. Thus, a student might have spent a rough total of six hours in a workshop.
Affordances of the Minecraft Mathematics environment. Despite the unforeseen changes detailed above, there were many affordances of using the Minecraft Mathematics workshops as the environment and setting of my study. One affordance was the strong probability that students attended the workshop due to their existing interests in Minecraft.
Evidence of students’ existing interests in the game was supported through an introductory student questionnaire that included a question asking the students to rate their current level of expertise with the game (Appendix H, Question 2). This questionnaire was not an instrument designed or intended for the study; I had provided the questionnaire to the 14 workshops prior to this study as a teaching practice to learn about my student population. However, information from one of its questions may support a more detailed illustration of how students in the Minecraft Mathematics environment in general perceived their own game expertise.
Therefore, the questionnaire is included in the appendix to provide additional information and context about the Minecraft Mathematics environment as suggested for design-based research (Design-Based Research Collective (DBRC), 2003). Table 2 shows the number of students who selected each answer choice for the Minecraft expertise question.
More than 93% of the 126 students rated themselves at a skill level of intermediate or higher, which may portray a strong interest in Minecraft of this particular student population.
54 Table 2
Students’ Self-Rated Levels of Game Expertise
Answer choice Total no. students who selected answer a) Expert 34 b) Advanced 71 c) Intermediate 13 d) Novice 7 e) Not familiar with game 1 Total 126
A second affordance the Minecraft Mathematics workshops was my experience in instruction of a total of 17 workshops before the two workshops of this study. Unlike the workshops of this study, each of these previous workshops was six to eight weeks in length occurring once per week for two hours after school, or five days per week during the summer for three to four hours each day. As a result of this experience, technology issues (internet and server network connectivity issues, computer security, server interface) were limited due to having been addressed in former workshops.
A third affordance was my two and a half year experience with over 140 students using my own pedagogical and content knowledge, and integrating that knowledge with the emergent technology of the game (Mishra & Koehler, 2006). Throughout this time, my existing knowledge of mathematics content, and the variety of instructional strategies in which I had found success in my former teaching years were combined and further developed with my knowledge of Minecraft. I observed what I perceived to be student understanding and application of targeted mathematical learning objectives, as well as the gaps in that understanding, and used these to leverage task design and modification. This previous
55 experience in the setting has assisted me in conceptualizing and connecting my work to the vast literature I have read during this time, and deepened by technological pedagogical content knowledge in the context of the Minecraft Mathematics learning ecology (Cobb, et al., 2003).
For example, I had always considered Self-Determination Theory, but I had not aimed to analyze and articulate the connection of theoretical construct to targeted mathematical objectives of a task embedded within an educational video game. This dissertation aims to clarify that connection and present a model based on the context of this study’s two workshops of focus.
Beginning with the third Minecraft Mathematics workshop (of the 17 previous workshops prior to this study), I became the director of the Minecraft Mathematics program, a final affordance. Through my position as program director, I had the freedom of designing and modifying my own activities that incorporated the mathematics content of my choice. The first few workshops consisted largely of exploratory teaching, which involved emphasis on the experiential aspects rather than hypothesis testing or retrospective analysis (Steffe &
Thompson, 2000). My ethnographic methodology courses at the university presented opportunities for me to begin to test and experience certain methodological components related to ethnography. As a result, my research questions evolved as I took into account broad patterns I had noticed throughout the multiple workshops, and the workshops evolved into pilot studies.
At this time, the Minecraft Mathematics workshops moved from exploratory teaching, to teaching experiments (Steffe & Thompson, 2000), during which I practiced data collection and tested various hypotheses. This process continued into my dissertation proposal phase
56 during which I attempted to narrow research questions. Thus, despite the constraints of time in this study’s workshops, I have had the fortune of obtaining experience in multiple Minecraft
Mathematics workshops, experience that has greatly informed my dissertational research methodology and focus. For example, I designed and modified multiple tasks during the pilot studies to cater to student interests I noticed as I observed and interacted with students. The
Build the Room task originally asked students to create a model of the classroom as accurately as possible which students interpreted as an objective of replicating not only the dimensions of the room, but the room’s furniture, color of the walls and carpet, and even a representation of the teacher in the classroom. I adjusted the wording of the task to ask students to model the room’s dimensions as accurately as possible after I noticed how engaged students were when the task’s objective and product included fewer constraints around decorations and aesthetics in the game, something which will be detailed further in Chapter 5. Thus, the version of the
Build the Room task examined in this study was not the first, and had been informed by previous iterations and pilot studies over three years and 17 workshops.
Additional constraints of the Minecraft Mathematics environment. In addition to the timing constraints of the two workshops in this study, constraints of the Minecraft Mathematics environment included the possibility that students did not participate in the workshops with the objective of learning mathematics, although this objective may have differed for their parents or other family members who enrolled them in the program. Some students, such as two particular students in the first workshop of my study to be detailed later, voiced their displeasure for mathematics and their hope that the workshop would focus only on the game.
Motivation for students that may have existed in school to perform well or earn specific grades
57 may have not been present in Minecraft Mathematics due to the program’s extracurricular nature.
Daily routine. Similar to Cohen’s (1994) dependable routine for structuring and timing a rich task, students were first introduced to a new task, or reminded of their work and objectives from the previous class day. During the allotted time that students spent designing solutions and products of the task, I moved around the room to check in with students, asked them questions about their thinking and processes, and formatively assessed engagement and progress while systematically documenting my observations to be analyzed after class. As both teacher and researcher, I moved between two modes described by Emerson, et al. (2011, pp.
21-22) of (1) attending to events in full participation as the teacher in the environment, to (2) documenting events that I find informative to my research. I moved more explicitly into the second mode during the wrap-up of the class days on which students shared or “showed off” their progress in building and extending upon task products through a presentation to the class, as well as in between classes as I documented and analyzed field notes, memos, and recordings.
Minecraft Mathematics as a learning ecology. According to Cobb, et al. (2003), “Design experiments ideally result in greater understanding of a learning ecology—a complex, interacting system involving multiple elements of different types and levels—by designing its elements and by anticipating how these elements function together to support learning” (p. 9).
Additionally, the DBRC argues, “the value of attending to context… produces a better understanding of an intervention… it can lead to improved theoretical account of teaching and learning” (2003, p. 7). Therefore, Table 3 outlines the interacting elements of the Minecraft
58 Mathematics learning ecology to provide specific contexts for each of Cobb’s elements.
Furthermore, an ethnographic perspective is incorporated through a strong consideration of these contexts within a complex system, especially in regards to the researcher-teacher’s role in her affect on the environment through task design and interpretations of student engagement during task implementation.
Table 3
Learning Ecology Elements of Cobb, et al. (2003) in Minecraft Mathematics
Learning Ecology Element Context of Element in Minecraft Mathematics
1 Tasks and problems students solve Task of focus (Build the Room)
2 Types of communication encouraged Thinking and explaining out loud; students teaching strategies to classmates
3 Norms of participation established Safe play; collaboration; sharing of ideas
4 Tools and materials provided Laptops; secure online shared Minecraft world
5 Practical means by which the teacher can Researcher-teacher’s ability to observe, relate these elements document, and analyze for task modification
Procedure and Timeline of Study
As detailed earlier in the chapter, constraints placed upon the researcher by the organization hosting the Minecraft Mathematics program resulted in a shorter timeline for this study’s workshops than anticipated. However, because the study largely focuses on my actions and design process as the researcher-teacher, the time spent before, in between, and after the eight class days of the workshop as I moved through the multiple phases of the design cycle are considered in addition to the eight class days of the Minecraft Mathematics workshops. For
59 example, the study began with preparation and anticipation of the first class, two weeks before its implementation. The process continued with, and consisted of the time spent after each class, and between classes and workshops analyzing in multiple iterations of the design cycle through field notes, memos, transcriptions, and analysis of audio, video and screencast footage to be detailed later in the chapter. Thus, regardless of time and experience in previous studies, this data collection began two weeks before the first class day of the first workshop, and culminated at the end of analysis and writing.
Table 4
Timeline of the Procedure of the Study
Timeline Activity Summary of the Activity Jan 2018 Preparation for Phase 1 began two weeks prior to first Workshop 1 class day: Reviewed field notes from previous two pilot studies; completed anticipatory field note to assist planning and organization of rich task to be used in study
Jan – Feb 2018 Workshop 1 Phases 1-3 executed: Jottings, fieldnotes, conceptual memos written; logged audio/video recordings, screencasts, and post-workshop student interviews
Feb 2018 Post-workshop Phase 3 continued: Transcribed Analysis and audio/video recordings, screencasts, and Preparation for student interviews; completed analytic Workshop 2 memo; completed initial round of coding;
60 completed anticipatory field note to assist planning and organization of rich task modifications Feb – Mar 2018 Workshop 2 Phases 1-3 executed: Jottings, fieldnotes, conceptual memos written; logged audio/video recordings, screencasts, and post-workshop student interviews Mar 2018 Post-workshop Phase 3 continued: Transcribed Analysis audio/video recordings, screencasts, and student interviews; completed analytic memo Mar – May 2018 Further Coding Completed second round of coding and and Analysis additional tabling of data; reviewed coding methods with expert team of researchers to increase validity and reliability of coding and analysis; model of engagement was conceptualized
The Task
The rich mathematical task of focus was the Build the Room task (Appendices A-D).
During my pilot studies, this task was the first task of the workshop. During this study, Build the
Room was presented to students as the overarching task, which included multiple sub-tasks or challenges. The task, as its title entails, challenges students to build a model of the classroom in
Minecraft as accurately as possible given the measurements of the building blocks in the game.
Some perceived mathematical content areas include number sense, measurement, geometry, spatial visualization, conversions, and problem solving. Multiple perceived Standards for
Mathematical Practice of specific challenges will be detailed within the analysis of this
61 dissertation’s chapter five. The task occurred over the entirety of each workshop’s four days of class, culminating with student presentations of their work throughout the workshop. Table 5 outlines the challenges of Build the Room. Challenges that include more than one challenge number such as Same Volume Different Dimensions were implemented in a different order between workshops. Explanations for the differing orders will be provided in the following chapters.
Table 5
Challenges of the Build the Room Task
Title of Challenge
Challenge 1 Build the Room’s Dimensions
Challenge 2 Document a Craft
Challenge 3 Present Your Builds
Challenge 2 or 4 Label Dimensions with Signs
Challenge 4 or 5 Same Volume Different Dimensions
Challenge 5 or 6 Present Your Work
Challenge 6 or 7 Final Presentation
Build the Room aims to arrive at the intersection of the highest level Create of Bloom’s
Revised Taxonomy (Anderson & Krathwohl, 2001) and Webb’s Depth of Knowledge (DOK) Level
4 in which students “design a model to inform and solve real-world, complex, or abstract situations” (Hess, Jones, Carlock, & Walkup, 2009, Appendix). Throughout the task, students create a model of the room both in dimension and volume through a process of steps, many of 62 which they define and organize themselves. Students first create the dimensions of length, width and height of the classroom, which require measurement of the actual classroom and conversion from feet to meters. Within the game, students have a certain degree of autonomy in Self-Determination Theory (SDT), defined as the presentation of opportunities for choice
(Ryan, et al., 2006), because they are free to choose any type of room to build at any location, with any materials and decorations, as long as the dimensions match those of the classroom.
Therefore, I attempted to design and implement the open-endedness and encouragement given to students to monitor and regulate their own thought processes during a rich mathematics task (Gojak, 2013; Pape, Bell, & Yetkin, 2003; Stein, et al., 1996) through the frame of autonomy in SDT to encourage mathematical engagement.
Students are expected to work together to measure and model the dimensions of the classroom in the game through their creative builds. Students are able to see one another’s avatars within the game world, and often engage in fluid communication both inside and outside of the game, an observation supported by the entirety of my pilot studies, and detailed by Bailey (2016) in his study of student interactions in a Minecraft afterschool club. Teamwork and communication were encouraged in order to provide opportunities for mathematical discussion during a rich task (Cohen, 1994; Stein, et al., 1996) as well as opportunities for collaboration regarding creative choices and the theme of their fantasy world. The latter opportunity is framed and supported by the motivational factor of relatedness in SDT, which is when the player experiences a feeling of connection to others through gameplay (Ryan, et al.,
2006). Thus, by requiring students to work in a team, I aimed to intertwine conversational
63 game engagement with conversational mathematics engagement as students worked towards their task and game objectives.
Participants
Participants in the study were the students who were registered by parents or other adult family members, and attended one of the two Minecraft Mathematics workshops. I was also a participant of the study as the researcher and teacher of the workshops who interacted with students and documented students’ interactions with one another as well as with the game and mathematical practices. This section will outline the selection process of the students, as well as student demographics.
Selection process through marketing. Minecraft Mathematics workshops were advertised on the website of the educational non-profit organization that hosted the program, in the organization’s monthly newsletters, and through its email listserv, which includes business partners, parents and family members of students who have participated in other programs, and local community members. Website information described the workshop as one in which students mine and craft mathematics skills using the game (PAST Foundation, 2018), and included multiple photos of girls and boys presenting their builds, interacting with one another, and playing the game. Students were enrolled in each workshop on a first-come, first- served basis before the limit of 10 students was reached, due to the number of laptops available.
Student demographics. Demographics of the students in the study’s two workshops are outlined in Table 6. Sixteen students attended the workshops. Seven of these students were enrolled in third grade and nine were enrolled in fourth grade. All 16 students identified as 64 male. All but one student attended public school from the same school district (District A) within a mostly white, upper-middle class suburban neighborhood outside a major university in the Midwest. The student from District B attended a public charter school in an urban environment adjacent to District A.
Table 6
Student Demographics of Workshops 1 and 2
Number of Students Attended 16 Grade 3 7 Grade 4 9 Identify as Male 16 Identify as Female 0 District A 15 District B 1
Instrumentation
During each workshop, I applied a robust list of instrumentation to collect and analyze data. Because I anticipated the narrowing of my original research foci, dependent upon the emergent patterns during my analysis phase, I collected a larger and more diverse corpus of data than would ultimately be used to support the analysis as recommended by Lemke (2003).
Also recommended by Lemke (2003), I collected data from students exhibiting diverse levels of mathematical engagement for comparison purposes. In this section, I will first outline the instrumentation that supported my systematic documentation of my ideas for task design and modification, which defined my iterative process for analysis. Second, I will describe the
65 instrumentation that supports the documentation of my observations of student engagement that informed my task modifications.
Documentation of researcher-teacher process. A variety of instruments were utilized to systematically document my task modification process over the Day-by-Day cycle and
Workshop-by-Workshop cycle. These instruments included (1) ethnographic jottings (Emerson, et al., 2011) on two organizational templates, (2) my ethnographic fields notes, and (3) two types of ethnographic memos that aided my synthesis of task modification over the Workshop- by-Workshop cycle as well as supported my analysis of how I noticed student engagement.
Ethnographic jottings. Emerson, et al. (2011) define jottings as “a brief written record of events and impressions captured in key words and phrases” (p. 29). Although often in the form of quickly rendered words or phrases, jottings aided my recall of conversations and student actions during class, and are later translated into more concise descriptions in field notes. I used two organizational templates in class to direct and structure my jottings, the Day- by-Day Rich Task Design Cycle template and the Three-Column Engagement template.
Templates were attached to a clipboard as I moved around the classroom and interacted with students.
Day-by-Day Rich Task Design Cycle template. This template (Appendix E) is a simple graphic of the Rich Task Design Cycle that focuses on a single Day-by-Day cycle. Jottings were written outside each of the four areas of the cycle (Anticipate, Implement, Evaluate, Modify) as they pertained to each part of the current iteration (Figure 6). Jottings recorded near the
Anticipate quadrant pertained to how I imagined students might interact with the task depending on what I had observed before class began. The Implement quadrant included
66 actions, events, and dialogue of students during class. Jottings recorded near Evaluate related to thoughts that I had during class about the successes and barriers of the task, while jottings recorded outside of Modify focused on changes I envisioned during class based on observations of students at that time. Evaluate and Modify were explored more deeply after class as I synthesized jottings in fieldnotes and memos, to be described in later sections.
Figure 6. Day-by-Day Rich Task Design Cycle template with jottings
Three-Column Engagement template. This template (Appendix F) contains three columns: (1) Engagement in the game, (2) engagement with mathematical practices, and (3) task modifications. Observations jotted in a single row connected to one another, often 67 containing jottings in either column one or two, and then in column three (Figure 7).
Observations of engagement with the game, and with mathematical practices were driven by definitions outlined in Chapter 1 of each type of engagement. Jottings within the task modifications column were immediate thoughts I had during class regarding future task modification as a result of the engagement I was observing and documenting in the previous columns.
Figure 7. Three-Column Engagement template with jottings
Ethnographic field notes. Fieldnotes are detailed entries of daily experiences and observations in the field that include accounts of events in real time, notable short phrases of participants, and significant features of the environment and occurrences that may inform research (Emerson, et al., 2011; Heath & Street, 2008). First, fieldnotes detailed my decision- making process as the teacher, student dialogue excerpts that might have informed the study, and my observations through a teacher’s lens of classroom events as they occurred with students. Fieldnotes aided documentation of my task modification in the Day-by-Day as well as the Workshop-by-Workshop cycle. 68 Second, fieldnotes documented my observations of student engagement in both mathematical practices as well as with the game as they occurred in class. Goals of fieldnotes were to harness as much detail on paper as quickly as possible after class to capture recent thoughts and reactions; evaluation and editing of word choice were saved for memos and later writing (Emerson, et al., 2011). Therefore, immediately after each day of Minecraft
Mathematics, I retreated to my desk in the same building, but away from the classroom to type fieldnotes with the goal of producing detailed recollections that captured involvement and excitement about the day’s events (Emerson, et al., 2011; Heath & Street, 2008). I adapted multiple organizational strategies in my fieldnotes, which included (1) tracing activities and observations in chronological order, (2) opening the writing session with an incident or event that stood out, or (3) moving through incidents related to specific topics of interest (Emerson, et al., 2011) such as students’ reactions to a particular challenge, or engagement in Minecraft such as when they expressed interest in learning how to craft fireworks and beacons in the game.
Because fieldnotes were written immediately after each class day, they are situated in
Phase 2 of the Rich Task Design Cycle, whereas jottings are situated in Phase 1. Fieldnotes written after the last day of a workshop include observations and reflections of the task across the entire workshop, thus providing a bridge to Phase 3. However, conceptual and analytic memos more precisely detailed synthesis of the Workshop-by-Workshop cycle while fieldnotes focused on daily events.
Conceptual memos. In the midst of collecting data, conceptual memos assist a fieldworker in better understanding what his/her fieldnotes may mean (Heath & Street, 2008).
69 My conceptual memos reflected upon how I interpreted student engagement, both with the game, and with mathematical practices. As previously mentioned, I hold certain beliefs as an educator and researcher about teaching practices and student behaviors. Therefore, documentation and analysis of how I was noticing student engagement, and how it was affecting my task modification in field notes was considered.
Two conceptual memos were written during each workshop. Fieldnotes from the first two days of a workshop were reread and reflected upon to inform the first conceptual memo.
A second conceptual memo was completed at the start of Phase 3 after the fourth day of class before my analytic memo was written to synthesize all data. Similar to the conceptual memo structure recommended by Heath & Street (2008, p. 80), I noted (1) unexpected occurrences related to engagement; (2) primary sources of data that informed my observations of engagement; and (3) patterns, insights, and breakthroughs surrounding my interpretations of engagement. The conceptual memos, along with my fieldnotes were then used to inform each workshop’s analytic memo.
Analytic memos. Unlike fieldnotes, but similar to conceptual memos, analytic memos contain evaluation and attention to word choice, but are not fully developed working papers.
According to Hammersley & Atkinson (2007) analytic memos are described as follows:
These are… occasional written notes whereby progress is assessed, emergent ideas are
identified, research strategy is sketched out, and so on. It is all too easy to let one’s
fieldnotes and other types of data pile up... it is a grave error to let this work accumulate
without regular reflection and review. (p. 191)
70 In my study, analytic memos were written during Phase 3, after the conclusion of each workshop as I synthesized my fieldnotes, conceptual memos, and data surrounding student engagement to modify the task for the next iteration of a workshop. Through these memos, I examined my research progress and noted significant “aha” moments I experienced in task design as well as modifications that had been made in relation to my findings. Analytic memos also summarized task modifications made throughout the workshop, and anticipated the redesign for the next workshop.
Documentation of student engagement. In this section, I will describe the instrumentation that I used to systematically document my observations of student engagement to inform task modifications. A variety of instruments were utilized over the Day- by-Day and Workshop-by-Workshop cycles to capture how students engaged with the mathematical practices, with the game, and with each other. These instruments included (1) audio and video recordings of my own conversations with students as they worked on a challenge, (2) formal interviews with students at the conclusion of each workshop, and (3) students’ screencasts during work on certain challenges.
Audio/video recordings of student explanative verbal reports. During class as I moved around the room and interacted with students, I noticed engagement in mathematical practices, as defined earlier in Chapter 1 by students’ (1) documentation of procedures or solution strategies, (2) communication of mathematical ideas to peers or to me, and (3) attention to solving a problem through writing on paper, counting aloud, and using tape measures. These observations inspired me to record certain conversations with students as they worked on a challenge. The purpose of recording these conversations was to document
71 verbal reports of students’ explanations as I generated questions, as described by Chi (2006) in attempt to understand how students may have been engaging in Standards for Mathematical
Practice, and whether the task was designed in a manner to effectively provoke this engagement. Recordings provided additional data to be reviewed and analyzed after class, and corroborated particular field notes and memos related to student engagement. To clarify students’ actions, and what I interpreted to be student engagement, I asked questions that pertained to students’ (1) excitement about something they were building or doing in the game, (2) plans for continued building or play, and (3) procedures they were enacting to solve the challenge.
Student-recorded screencasts. During work on specific challenges such as Build the
Room’s Dimensions and Same Volume Different Dimensions, students were instructed to record screencasts of their gameplay. Students then saved their recordings to their laptops, so that I was able to download and view them for analysis after class during Phase 2 after the first and third days of class. Screencasts were between eight and 50 minutes in duration. While I viewed and logged most screencasts, I only transcribed screencasts that seemed to point to breakthrough ideas and significant events that informed task modification, a process documented in my fieldnotes and memos. Furthermore, I used screencasts to verify events and conversations that were documented in my fieldnotes.
Post-workshop interviews with students. At the culmination of each workshop, students were interviewed in order to draw additional insight of engagement with the task as well as with the class in general. My interviews took into account sensitivities to my position as the teacher interviewing students as recommended by Briggs (1986) to optimize reflexivity and
72 authenticity while taking into account my power as an adult and perceived authority figure. As
Spradley (1979) suggests, I framed my research purpose to the students, but in language that intended to make sense to them (Briggs, 1986) regarding engagement with the game and mathematics of the overarching task. I used a combination of descriptive, structural, and contrast questions (Spradley, 1979) listed in Appendix G. Students were selected based on my observations of their engagement throughout the workshop with an objective of choosing one student who seemed emphatically engaged, and another student who did not seem as engaged. These interviews informed my Phase 3 analysis and analytic memo.
Summary and timeline of instrumentation. Due to the short amount of time that I spent with each group of students, I used a variety of instrumentation to capture data. This data pertained to my own ideas and processes that I applied, as well as to the student engagement that informed task modification. Table 7 summarizes the instrumentation in relation to the phases and day-by-day cycles of the Rich Task Design Cycle iterative process.
Table 7
Phases and Day-by-Day Cycles of Instrumentation
Instrumentation Phase Day-by-Day Cycle(s) Documentation of Jottings 1 1, 2, 3, 4 researcher-teacher Field notes 2 1, 2, 3, 4 process Conceptual memo 2 2, 4 Analytic memo 3 4 Documentation of Jottings 1 1, 2, 3, 4 student engagement Field notes 2 1, 2, 3, 4 Conceptual memo 2 2, 4 Analytic memo 3 4
73 Audio/video 1 1, 2, 3, 4 Recordings Student screencasts 1 1, 3 Post-workshop 3 4 student interviews
Analysis
As previously detailed, the analysis process was incorporated largely within the second and third phases of my documentation, which included both evaluation and analysis. In addition to writing field notes, memos, and reviewing the three types of recordings, multiple rounds of coding and tabling of data were also implemented to establish themes that assisted in answering research questions. This section will (1) expand upon the data collection methods and instrumentation outlined in the previous section in order to emphasize their analysis components, (2) delineate the coding methods applied when reviewing and synthesizing data, and (3) describe the additional coding process involving two additional experienced researchers to strengthen the validity of the study’s coding procedure.
Analysis during documentation. Analysis occurred continuously throughout the phases of the Rich Task Design Cycle, but was largely concentrated in Phase 2 (after each class day) of the Evaluate and Modify quadrants of the Day-by-Day cycle, and Phase 3 (after the workshop) during reflection and analysis of the task’s implementation of the Workshop-by-Workshop cycle. Fieldnotes were constructed from jottings, and contained immediate evaluation and analysis directly after implementation of task components with students. This analysis was reflected upon and evolved as conceptual memos were constructed from the re-visitation of field notes, and review of audio/video recordings and screencasts. Analytic memos were written as further analysis that aimed to synthesize emergent ideas and themes after the 74 conclusion of the first workshop, and then again after the second workshop. After the first workshop, the analytic memo relied upon a second review of field notes, but was also influenced by the coding of these field notes, transcriptions of recordings, and the two conceptual memos. The second analytic memo was written at the culmination of the second workshop and was informed by (1) the first round of coding after the first workshop, (2) an additional review of all field notes, (3) further review of recording transcriptions, now including those from the second workshop, (4) a synthesis of four conceptual memos, and (5) a second round of coding that included these documents.
Audio/video footage, and screencasts that had been recorded during class were immediately downloaded to a private computer, and de-identified. These recordings were then logged either directly after class in the evening, or during the following day. While some analysis occurred during initial logging during these times, the majority of analysis took place as specific recordings were transcribed and reviewed again during the coding process. As recommended by Steffe and Thompson (2000) when conducting teaching experiments, “Careful analysis of the videotapes offers the researchers the opportunity to activate the records of their past experiences with the students and bring them to conscious awareness” (p. 292). Thus, when the recordings were revisited during later analyses phases, mental records were enhanced by attention to a specific theme or idea inspired by the coding process. Recordings and their transcripts were revisited additional times during second rounds of coding and afterward as examples of evidence were selected to support and depict themes.
75 Coding applied after workshops. At the culmination of the first workshop, a first round of coding was applied to the corpus of data at that time, which included field notes, conceptual memos, and transcriptions of class audio/video, screencast, and student interview recordings.
Because a research goal of this study was to develop a new theory about a researcher-teacher’s process of task design, grounded theory (Corbin & Strauss, 2015) and accompanying coding methods of Process and Concept Coding were applied (Saldaña, 2016). Process Coding was used to highlight the student actions observed by the researcher-teacher, as well as the actions of the teacher-researcher in response to the students. Examples of process codes applied to students were measuring, communicating, complaining and exploring, while process codes applied to my own actions included structuring, reminding, and modifying. Concept Coding was simultaneously applied to attach suggested meanings to many of the actions and responses of both the students and researcher-teacher. Examples of concept codes applied were autonomy, competence, relatedness, fantasy worlds, and safe environment.
Through the use of these process and concept codes, the data began to yield a focus on the researcher-teacher’s multiple goals of promoting mathematical conversation among students, creating opportunities for the students to experience targeted learning goals both inside and outside of the game, and attempting to incorporate student interests in the tasks to encourage on-task behavior. These goals influenced further development of the concept codes, which evolved into themes. The major themes were then articulated further in summary tables and an analytic memo written after the first round of coding in order to integrate and synthesize ideas and patterns found thus far in the data (Saldaña, 2016). This
76 analytic memo became part of the corpus of data to be coded after the class of the second workshop, as the summary tables were combined, and modified.
The corpus of data at the culmination of the second workshop included field notes, conceptual memos, one analytic memo from the first workshop, and all transcriptions of audio/video, screencast, and student interview recordings. A second round of coding was applied at this time, taking into account the patterns established during the first round of coding within the summary tables to notice differences between the workshops, and themes that were prominent in both. Thus, Theoretical Coding was applied in attempt to find the core categories and themes that provide a theoretical explanation for the phenomenon (Corbin &
Strauss, 2015, p. 13) of how task design embedded within an educational video game might be conceptualized when taking into account targeted learning objectives and motivational factors.
As recommended by Saldaña (2016), final sets of codes were elaborated upon in a second analytic memo. The second analytic memo propelled the focus of multiple summary tables from the first round of coding to a single table that denoted six major themes. These six themes were eventually narrowed to two themes: (1) intertwining the game with mathematics, and (2) noticing student interactions to inform task development.
Reliability and validity of coding procedure. After the workshops had concluded, and I had analyzed and coded the majority of my data, I met with two experienced researchers in my field who were familiar with my study to increase the reliability and validity of my analysis procedures. I provided the researchers with random samples of transcriptions and the contents of field notes, and asked them to offer their analysis through using specific codes and themes I had developed during my second round of coding and tabling of data. Some issues
77 and new observations were discussed including (1) students’ intended meanings of specific words or phrases, (2) differing interpretations of the students’ involvement in particular
Standards for Mathematical Practice, and (3) attention to particular behaviors and reactions that may be “norms” of Minecraft players.
During the review of one transcript in particular, one researcher noticed the recurrence of a specific word used by a student that I had interpreted to be negative and authoritative towards the other student. However, upon re-examination of the data, the word “no” when used in this particular dialogue seemed to insinuate both agreement and disagreement with the other student, and did not carry a negative connotation. For example, “no” seemed to sometimes precede a phrase in which the student was referring to himself as incorrect, now agreeing with the other student rather than disagreeing. He also seemed to used “no” when addressing himself as incorrect or altering his own ideas in mid-dialogue. This new interpretation allowed me to consider a different relationship between the two students, connecting some of their dialogue to a specific Standard for Mathematical Practice (SMP) in a different way.
My interpretations of how students were engaging in specific SMP were also discussed in some of the data examined by the group. For example, some documented student actions and transcriptions may have overlapped multiple SMP. However, the group came to the conclusion that choosing one or two on which to focus might allow the analysis to be more focused, especially if particular instances within the dialogue were cited to show examples of these particular SMP. Furthermore, the researchers believed different parts of a particular transcript incorporated a certain SMP in more depth than I had originally analyzed. This led to a
78 discussion on the definition of the particular SMP and my intended modifications during the second workshop. An agreement was reached that considered the researcher’s observations and definition of the two SMP, but took into account the shortcomings of students’ engagement in relation to my intended goals for engagement described within my field notes.
Finally, because one of the researchers was an experienced Minecraft player and had experience in research examining how students interacted with mathematics in the game, he was able to provide additional insight about decisions that seemed to be made by a pair of students in a particular transcript. While my field notes had observed that the students had avoided discussion of a particular decision, his interpretation was that the decision had been made, but was not formalized and articulated in a mathematical way due to “norms” of
Minecraft play. These “norms” included an ability to quickly build and destroy blocks such that if a mistake or miscalculation was made, students did not need to discuss their modifications, and could quickly correct the error. Also, he suggested that decisions around a particular build within the transcript were made due to a number sense already established by routine players of the game. Depending on their experience playing Minecraft, some students may recognize that a room with a larger length and width may be more enticing aesthetically compared to a tall room with a small floor area. Attention to these “norms” modified my developing analysis surrounding possible reasons for the students’ decisions to use particular dimensions in their build.
Including the expertise of two mathematics education researchers with experience in
Minecraft research and the use of technology in the classroom allowed me to deepen analysis in particular areas through consideration of additional interpretations. While their
79 interpretations did not affect task modifications made during the second workshop (due to our meetings occurring after the workshops), their insight shed new light on student interactions that I had believed to be static or non-conclusive. Additionally, they were able to verify some of my analyses, and pointed to additional evidence that I had not yet considered.
Researcher-Teacher Beliefs and Biases
As recommended by Hammersley and Atkinson (2007), as well as by the Design-Based
Research Collective (2003), I have considered my perspectives as both a researcher and teacher, and how they might have affected my interpretations of student engagement during analysis as well as my facilitation of the task during implementation. Throughout the study, I was influenced by my knowledge of theory, mathematical content, pedagogical practices that I believed to be effective, and the prospective use of Minecraft as an educational tool. As a former high school mathematics teacher and current doctoral candidate, my beliefs have been affected by my experiences (1) working with students in a K-12 public school setting, (2) working with teachers as a colleague within the same department as well as one who is primarily providing support externally, and (3) working with students in the Minecraft
Mathematics environment in workshops prior to this study.
Through these experiences and my knowledge of theory developed throughout my education in teaching and learning, I believe that rich mathematical tasks foster learning, that students may learn more when working together to communicate their ideas, and that video games can be used as an educational tool when the tool is applied thoughtfully, carefully, and with intention to address specific learning targets. As a former high school mathematics teacher, my job was to support students in their mathematical exploration and thinking, often 80 through application of some of my beliefs described above as well as knowledge of theories such as Cognitively Guided Instruction (Carpenter, et al., 1999). As the teacher of the Minecraft
Mathematics workshops, I further attempted to encourage this mathematical thinking and exploration, and extract it for documentation and analysis purposes. I pushed students to engage in Standards for Mathematical Practice (SMP), and to behave as mathematicians while engaging with an educational video game. Thus, during this study I was biased through my beliefs, knowledge, and intentions to engage students in SMP.
My biases are also influenced by emotional aspects that stem from my experiences and identity as a mathematics educator, and educator in general. I experience negative emotions when students voice their hatred or dislike of mathematics, and positive ones when students display strong engagement in mathematical practices. I also enjoy observing what I interpret to be students’ “a-ha” moments when grappling with a challenging concept. Finally, I admit a strong interest in Self-Determination Theory (SDT) applied to video games, because I am excited at the notion of leveraging students’ existing enjoyment and motivation applied in video games for educational purposes, and I believe that SDT might support additional theories and ideas of how this may be possible. These biases and beliefs were considered throughout my study, particularly during my data collection and analysis processes as I interpreted my own actions and responses to student engagement.
Credibility
Credibility of the study is supported by measures taken to enhance the reliability and validity of coding procedures referred to earlier in the chapter, data triangulation, and consideration of the researcher-teacher’s beliefs and biases. Data was triangulated using 81 multiple data sources outlined previously in describing my instrumentation. The Design-Based
Research Collective (DBRC) (2003) describes how “design-based research typically triangulates multiple sources… to connect intended and unintended outcomes to processes of enactment”
(p. 7). Here, the process of enactment envelops the rich mathematics task implementation and modification in Minecraft, and the outcomes are my modifications made to the Build the Room task that were guided by my observations of student engagement. Here, student engagement was based on my interpretations that were influenced by my own beliefs and biases. The DBRC
(2003) also advocates for the questioning of the designer-researcher’s beliefs during the process of enactment, a questioning that I incorporated during conceptual memos, coding, and tabling.
82 Chapter 4: Results
This chapter provides results of the study by first summarizing each workshop with (1) a brief overview of student demographics, (2) students’ perceptions of themselves as Minecraft players, and (3) examples of some of the broader modifications applied in the workshop. A table at the end of each workshop’s summary provides the results of all modifications documented during movement through the Day-by-Day Rich Task Design Cycle of anticipation, implementation, evaluation, and modification. Next, each challenge of the Build a Room task is connected to the Cognitive Rigor Matrix, Standards for Mathematical Practice, and Self-
Determination Theory components.
Workshop 1
The demographics of this workshop were much different in grade, gender, and school district, compared to those of previous workshops prior to this study. In this first workshop, eight of the ten students were in fourth grade, and two were in third grade. This elementary school-aged population was different from the majority of the 15 previous workshops of which
11 consisted of middle school students. Thus, the majority of my previous experience in
Minecraft Mathematics workshops was with middle school students, although two of the three most recent workshops consisted of elementary school-aged students. This was also the first workshop in which all students identified as male.
An additional first occurrence in a Minecraft Mathematics workshop at this particular organization was that all students attended school within the same school district, but were not familiar with everyone in the class due to attending different elementary schools within the district. A parent of a former student from this school district described children from this 83 particular district and neighborhood as “used to having their opinions matter” and “speaking loudly at home” (Field Note (FN) 1/25/2018). This generalization correlated with my own observations during the first class day, an observation evidenced by the volume of voices and continuous interruption of one another and me as I attempted to lead a class discussion and provide instructions.
Furthermore, responses to a particular question regarding Minecraft expertise on the introductory questionnaire provided at the beginning of each workshop (Appendix H) also differed from the majority of students in previous workshops. One may note that this questionnaire was not an instrument of the study, but rather used as an indicator and possible reason for lack of engagement of this particular group of students on the first day of the first workshop. The questionnaire was a teaching tool that had also been used in the last fourteen workshops to inform my knowledge of how student played the game at home so that I could more easily anticipate how they might interact with the game (i.e., would they struggle with controls? What types of materials and builds were most familiar to them?).
Seven out of ten students in this workshop selected the answer choice of this particular question that described themselves as experts in Minecraft. This was the highest percentage of students claiming to be experts in any workshop. Out of 126 student responses encompassing sixteen workshops (including those in this first workshop), only 26.7% of students have circled the answer describing themselves as experts. Therefore, the number of self-perceived experts in the first workshop may have contributed to behavioral issues on the first day of class.
Despite a detailed outline and lesson plan that aimed to optimize student engagement in mathematical practices, the first class day contained the least amount of mathematical
84 engagement in comparison to the second, third, and fourth classes. Throughout the first class, students interrupted introductions to activities often to ask when they would play Minecraft; one student in particular voiced his dislike of math, calling for the class to only focus on the game only, a call which was echoed repeatedly by multiple students and seemed to negatively affect structured, on-task behavior for the remainder of class (FN 1/25/2018; Conceptual Memo
(CM) 2/6/2018).
As a result of the first class, multiple modifications were anticipated and implemented in the second class. Modifications included changes in the delivery of instruction by the researcher-teacher as well as changes and additions to the original Build a Room task with the design and insertion of Challenge 2: Document a Craft, discussed further in the next chapter.
Additional behavior-inspired modifications included (1) a checklist of planned activities for the day written on the whiteboard, (2) time constraints written next to each activity, (3) the use of a timer to time each activity with verbal instructions provided to students to meet the time constraint as an additional challenge, (4) shorter and more concise verbal instructions to students, (5) instructional templates for each of the task’s challenges (Appendices A-D) with structure of answer spaces provided in comparison to a blank notebook of graph paper, and (6) the reward of free play at the end of the whiteboard checklist as the reward for meeting time constraints and completion of challenges.
I observed a significant change regarding students’ on-task behavior, students’ verbal interruptions of one another as well as myself, and general students’ attitude on the second, and remaining days of class that may have been a result of the modifications listed above as well as those related to the design of the task. Additional examples of my anticipations,
85 implementations, evaluations, and modifications are summarized in Table 8, which includes these components as the Day-by-Day Cycles of four classes. These components not only take into account student behavior, but also the engagement I noticed and documented as the researcher-teacher surrounding students’ mathematical practices and attention to the game.
86 Table 8
Workshop 1 Day-by-Day Rich Task Design Cycle
Anticipate Implement Evaluate Modify Workshop Students may have weak Some students quickly Students correctly answered Less time will be spent on conversions than (W) 1, Class proportional reasoning converted with correct questions on PreTest related to anticipated during group discussions and Day (C) 1 skills division and shared conversions with exception of J check-ins; will provide attention to J and R methods or just answers and R with others; J, AS, R seemed to struggle. Students may have weak Students measured All students answered measuring Less time will be spent on measurements than measuring skills quickly and accurately question on PreTest correctly anticipated Students may struggle In two groups: Only one Students need more structure to For W2, I will provide a template with with documentation of member documented guide their documentation; structured spaces for measurements and measurements measurements. Other Wrote in strange areas in graph conversions; Use templates for all future two groups: No one paper notebooks (inside cover activities in W1 and W2 documented more than and around border) one measurement. Interest in recording Added question to This PreTest question prompted Adjusted wording of this question to ask for an students’ number sense in PreTest about estimation many verbal questions during estimation in both feet and MC blocks instead estimating height of room room’s height in MC PreTest of just MC blocks in Minecraft (MC) blocks blocks given that one MC before after workshop block is about 3 feet
Use class outline to Boys asked many Students were constrained by the • Checklist of all activities on board manage time spent on questions about when task, seemed to perceive not • Time constraints added to each with timer activities and guide and how they would get enough time for fun, perhaps • Short, concise instructions students with verbal to play the game because weren’t aware of the • Carrot of free play! instructions planed activities despite my • Speak with boys complaining about having verbal delivery at start of class to do math in the workshop N/A General disrespect for Almost all boys expert answer on • Make bulleted changes above teacher and one another questionnaire; these results are • Establishment of my bias brought out different than all previous pilot importance of context: This workshop is study workshops different because of demographics and questionnaire answers that imply self- 87 proclaimed experts • Create Challenge 2 to encourage thinking aloud through explanations to others, collaboration, and harness interest in building items in which students expressed interest
Anticipate Implement Evaluate Modify W1, C2 Apply delivery Implemented all bullets Class was more organized and Continue to implement these modifications modifications from attitudes improved bulleted list in row 5 above (checklist, time constraints, etc.) Create Challenge 2 to Students engaged in the Questioned whether there is To increase engagement and deepen SMP for encourage classroom norm challenge, but lost enough SMP in Challenge 2 last component, have students sit in different of positive communication, interest during last pairs and craft items following their partner’s but also to combine game component of explaining written directions interest of building certain builds to team or partner items Created Challenge 3 to Some groups did not use Presentations were short, but Students need more presentation experience provide first opportunity the checklist while other confidence was an important and greater accountability to attend to for practice presentations groups used checklist in focus provided checklist detail W1, C3 Boys do not have Students were engaged • Requiring students to show me • Continue to use computers as a second measurements from first and on-task despite their documented dimensions carrot near beginning of class to encourage challenge documented; having access to the before receiving game access teamwork and completion of smaller class Challenge 4 aims to game may have motivated them to objectives engage them in collaborate and share • Class discussion about purpose of signs: a organization, conversions, information within teams way to help presentations and organize and building • Students struggled with sign information placement; perhaps did not • Provide another opportunity for see relevance to class presentations. presentations Create Challenge 5 for Short class discussion This went very well compared to • Repeat Challenge 5 in W2, but include time Build the Room to about volume while Challenge 4; multiple SMP for presentations, so students might introduce or review server was stopped. observed; Possible reasoning: observe utility of signs 88 concepts of volume while Students then had to find • Students had experience • Include more opportunities for students to incorporating student different dimensions with building a room with given relate dimensions they have calculated by interests in building the same volume of dimensions factoring, to the actual arrangement of additional rooms original room and build it • I emphasized the challenge’s length, width and height of their new room in the game. creative components Anticipate Implement Evaluate Modify W1, C4 Time will be a major All goals accomplished, I prefer to give students feedback In W2, students will take post-test first, and constraint. Goals: but less time spent on on their delivery of the then practice presentations with less time students (1) organize their practicing presentations mathematics by asking them spent on organizing the presentation presentations, (2) take and modifications than mathematics questions during PostTest, (3) practice desired practice presentations; I was presentations, (4) modify unable to do this to the degree presentations and builds, that I desired (5) final presentations for parents
89 Workshop 2
The demographics of this workshop were surprisingly similar to those of Workshop 1, although the initial behavior issues of the first workshop were not present perhaps due to implementation of the behavior-related modifications listed previously that included the whiteboard checklist of activities, time constraints, and free play among others. Additionally, class size may have been a variable as there were only six students in the second workshop, as compared to ten in the first workshop. Four fewer students in the class meant that there were two fewer teams of students, as team sizes ranged from two to four students. Another difference was that the students in this workshop were predominantly third graders with one fourth-grader in comparison to the first workshop that was predominantly fourth-graders with two third-graders. Similarities between the two workshops included all students’ identifying as male, and five out of six students attending elementary schools in the same upper-middle class suburban school district as the students attended in the first workshop.
In contrast to the students of the first workshop, five out of six students in the second workshop circled the answer choice on the introductory questionnaire that described themselves as advanced rather than as experts. Students in this workshop circled answer choice (b) “I have played Minecraft quite a bit, and I know most things about the game,” in comparison to (a) “I know almost everything about Minecraft. I’m totally an expert.” Table 9 compares the number of students in each workshop who selected each answer choice for this question in which answer choices are ordered according to a declining level of expertise with
(a) denoting expert status, and (e) equating to complete unfamiliarity with the game. One might argue that the lack of self-proclaimed experts might have affected the behavior observed 90 during the second workshop. Students may have believed that they still had some things to learn about the game, and that this learning could occur within the workshop.
Table 9
Student Responses in First and Second Workshops
Answer No. students who No. students who Total no. students choice selected answer in selected answer in who selected Workshop 1 Workshop 2 answer (a) 7 1 8 (b) 2 5 7 (c) 1 0 1 (d) 0 0 0 (e) 0 0 0
During the first class day of the second workshop, I noticed that students helped one another craft particular items, and immediately began to collaborate in brainstorming ideas for the theme of their fantasy world in the game upon entering the classroom (FN 2/26/2018).
Students also collaborated during the measurement activity and checked one another’s work.
Thus, I decided not to use the second challenge from the first workshop, Document a Craft to encourage additional teamwork, and instead focused on comparing measurements and exploring conversions in the second class. After evaluating my observations of implementation of the second day, I also made a modification to set aside more time for student presentations.
I had noticed in the first workshop that students might have benefitted mathematically from explaining their thinking to the class if they had more opportunities for peer and researcher- teacher feedback during a presentation format.
Differences between the two workshops in the order and implementation of sub-tasks or challenges of the overarching Build the Room task are outlined in Table 10. Reasoning for some
91 of these changes are explored in the next chapter, and also present in Table 11, which shows additional examples of my anticipations, implementations, evaluations, and modifications through the Day-by-Day Cycles of the four classes in the second workshop.
Table 10
Challenges by Workshop
Workshop 1 Workshop 2 Challenge 1 Build the Room’s Dimensions Build the Room’s Dimensions Days 1-2 Days 1-2 Challenge 2 Document a Craft Label Dimensions with Signs Day 2 Day 2 Challenge 3 Present Your Builds Present Your Builds Day 2 Day 2 Challenge 4 Label Dimensions with Signs Same Volume, Different Day 3 Dimensions Day 3 Challenge 5 Same Volume Different Dimensions Present Your Work Day 3 Day 4 Challenge 6 Present Your Work Final Presentation Day 4 Day 4 Challenge 7 Final Presentation N/A Day 4
92 Table 11
Workshop 2 Day-by-Day Rich Task Design Cycle
Anticipate Implement Evaluate Modify W2, C1 Students may have weak Students did not begin their Four out of six students knew to divide Plans to provide extra attention to RR proportional reasoning conversion procedures yet, on the proportional PreTest question and W to observe and provide support skills since we spent time measuring during conversions Students may have weak Students measured accurately All students answered measuring Similar to W1, less time will be spent measuring skills and with great organization question on PreTest correctly, and found on measurements than anticipated especially Team Cookie (E, M, strategic procedures to measure as and compared to pilot studies R); measurement took longer accurately as possible than during W1 Students may struggle All six students wrote down This may have been a result from the use Still some confusion around feet versus with documentation of measurements! of the instructional template in Minecraft blocks despite label of feet measurements comparison to W1 at end of lines; will emphasize this more during verbal delivery next time Interest in recording No verbal questions from I had adjusted wording of this question No changes students’ number sense in students when they answered to ask for an estimation in both feet and estimating height of room this question on the PreTest MC blocks that may have helped in Minecraft (MC) blocks before after workshop Plan to notice degree of Many positive conversations Students taught one another how to We will not use Document a Craft but communication and within groups, and cooperative build items without my facilitation focus on conversions, placement of collaboration between teamwork in measuring signs, and practice presentations to students to inform use of classroom and building together encourage explanations of Document a Craft in game mathematical thinking challenge on day 2 W2, C2 Anticipate Team Memes Team Memes, especially G Team Cookie seems to take challenges Encourage Team Cookie to use more of (RR, W, G) will need help needed support with seriously and do not incorporate as much their fantasy world through small with conversions due to conversions; Team Cookie creativity; Team Memes is the opposite: verbal prompts asking about plans; PreTest scores quickly completed task eventually finished conversions but did encourage Team Memes to meet the not build room in game challenge objectives which may involve coming up with a plan for who will build what
93 Plan to compare Measurements were very close Team Memes struggled but drew Desire to give Team Memes additional measurements as a class between groups; discussed as a intricate diagrams on the board as a practice with conversions; will provide to notice differences and class how to convert feet to team that included tables and horizontal a warm-up question next class day accuracy meters in order to model in lines to work through conversion process game W2 Challenge 2 will be to Team Cookie quickly met this Team Memes was distracted with Perhaps they did not see relevance to label dimensions in feet challenge and provided creative elements of game and did not presentation until the presentation and meters with signs in evidence during presentations stay on task to create signs occurred; will provide opportunity for the game another presentation next class day W2 Challenge 3 will be a Team Cookie completed all Team Memes may have really noticed Provide adequate time for checklist for presentations items on checklist, Team Memes how lacking their presentation was; presentations and discussion next class did not and I provided feedback Team Cookie may have realized that they day had opportunities to be more creative Anticipate Implement Evaluate Modify W2, C3 Plan for warm-up question All students participated and Provided closure to previous class day I hoped that warm-up would support on conversions to give Team Cookie checked answers with conversions; also prompted their end-of-day presentation, but we additional practice with each other discussion of looking for a larger or ran out of time for presentations! I will smaller number in the answer depending have them refer to this warm-up next on the conversion units class while prepping presentations W2 Challenge 4 will be Students needed more guidance 5 out of 6 students are in 3rd grade, However, discussion required some Same Room Different compared to W1, but eventually which may have been why they had prompting from me; next time include Dimensions with additional took ownership in finding more questions; addition of arrangement a question that has them decide on questions on arrangement factors. questions brought out discussion of their final arrangement that they will of factors; students may choosing heights! actually build struggle with multiplication compared to W1 W2 Challenge 5 will be We did not have time for I aimed not to err on the side of While the 15 min at start of class another checklist for presentations after 15 min of providing too much structure with the began with me addressing technical presentations time in the beginning and 45 mathematics and provide opportunities difficulties, it evolved into free play min of discussion and work on for students to build their products and which may have increased focus when the template immerse themselves in the game we began the warm-up and challenge 4; may include free play at beginning of next class day W2, C4 Students will need time for Provided a 30 min chunk of time Presentations were more organized, Practice presentations seem to support practice presentations and for practice presentations and contained more discussion about the SMP of Constructing viable feedback from both me feedback mathematics and students seemed more arguments; include same amount or 94 and peers comfortable than in W1 more time for practice in future workshops Students will also need Provided students with 20 Free play time allowed them to build A balance between free play and time to continue building minutes of free play at more of their fantasy world to show off structured work on the task seems to their products; balance beginning to continue task during final presentations; may have vary between workshops, but may help this with the above product development and helped increase engagement during engagement in SMP decorations; also 15 minutes presentations between practice and final presentations Anticipate Implement Evaluate Modify
95 Mathematics Incorporated within the Build The Room Task
Design and modifications applied to the task and its challenges aimed to address particular Standards for Mathematical Practices (SMP) (Common Core State Standards
Initiative, 2018) as well as meet the rigorous critical thinking standards present in the Cognitive
Rigor Matrix (Hess, Jones, Carlock, & Walkup, 2009) that overlies Boom’s taxonomy with
Webb’s Depth of Knowledge. Multiple studies have shown that student engagement is peaked at the higher levels of Webb’s Depth of Knowledge and Bloom’s Taxonomy (Anderson &
Krathwohl, 2001; Hess, et al., 2009; Paige, Sizemore, & Niece, 2013). Thus, I aimed to incorporate the higher levels of the revised Bloom’s Taxonomy of Anderson and Krathwohl
(2001) Analyze, Evaluate, and Create to provide opportunities for greater critical thinking during task challenges. While I evaluated one of the smaller challenges to be level one, the remaining challenges seemed to fit Depth of Knowledge levels three and four. I also used the motivational factors of Self-Determination Theory (SDT) as applied in video games (Przybylsky, et al., 2010;
Rigby & Ryan, 2011; Ryan, et al., 2006) to frame my observations of students’ engagement in
SMP and the game that in turn, informed my design and modification of the task.
Table 12
Components of the Cognitive Rigor Matrix, Standards for Mathematical Practice and Self-
Determination Theory Observed During Task
Challenge Cognitive Rigor Matrix Standards for Self-Determination Theory Mathematical Practices components Build the Room’s Bloom’s Create and Webb’s • Making sense of • Autonomy Dimensions DOK Level 4 Extended problems and persevere • Competence Thinking: Design a model to solving them • Relatedness inform and solve real-world, • Reason abstractly and complex, or abstract quantitatively situations • Model with mathematics 96 • Use appropriate tools strategically • Attend to precision Document a Bloom’s Evaluate and Webb’s • Attend to precision • Relatedness Craft DOK Level 3 Strategic Thinking • Look for and make use of • Autonomy and Reasoning: Verify structure • Competence reasonableness of results • Look for and express regularity in reasoning Label Dimensions Bloom’s Analyze and Webb’s • Use appropriate tools • Competence with Signs DOK Level 1 Recall and strategically Reproduction: Retrieve • Attend to precision information from a table or graph to answer a question Same Volume Bloom’s Create and Webb’s • Making sense of • Relatedness Different DOK Level 4 Extended problems and persevere • Autonomy Dimensions Thinking: Design a model to solving them • Competence inform and solve real-world, • Reason abstractly and complex, or abstract quantitatively situations • Construct viable arguments and critique the reasoning of others • Model with mathematics • Attend to precision • Look for and express regularity in repeated reasoning Presentations Bloom’s Evaluate and Webb’s • Look for and express • Relatedness DOK Level 4 Extended regularity in repeated • Competence Thinking: Draw and justify reasoning conclusions
Table 12 matches elements from the Cognitive Rigor Matrix with SMP and SDT components for each Challenge of Build the Room. The relationship between SMP and motivational factors of SDT is examined further in the next chapter through specific episodes within the two challenges of focus, Build the Room’s Dimensions and Same Volume Different
Dimensions. These episodes provide evidence of student engagement in the task or lack thereof, and the modifications that resulted in attempt to increase engagement in targeted mathematical learning objectives.
97 Chapter 5: Analysis
Through analysis within the iterative progression of the Rich Task Design Cycle, this chapter begins to conceptualize a model for mathematical task design embedded in an educational video game. During the final coding and tabling phase of this study, two themes were established in relation to the goals of task design: (1) intertwining of the game with task objectives, and (2) incorporating opportunities for student interactions inside and outside of the game. While the overarching goal of the task was to engage students in targeted mathematical learning objectives, the two themes detailed in this chapter support specific task design developments that aimed to meet this overarching goal by leveraging existing student engagement and enjoyment in the game.
The research questions of the study are restated below, and are addressed within the examination and analysis of each of the two themes listed above.
1. Through rich task design, what are some ways that an educational video game can be
used to engage students in mathematical practices?
2. How does students’ engagement inform the process of rich task design in Minecraft?
3. How does students’ engagement inform the process of rich task modifications in
Minecraft?
Both themes provide answers to the first research question, because they suggest (1) ways that task objectives might be combined with game objectives, and (2) ways that communication between students might be enhanced to foster greater engagement in the intended mathematical practices of the task. Within analysis of each theme, examples of student engagement perceived by the researcher-teacher are provided, and task-related actions of the
98 researcher-teacher are discussed. Thus, the second and third research questions are answered through the analysis of the additional task design and modifications applied as a result of these engagement examples, particularly in relation to Self-Determination Theory (SDT) and intended
Standards for Mathematical Practice (SMP).
This chapter is organized by the two themes presented above, starting with the first theme and ending with the second. Each of the two themes is examined through the components of the Rich Task Design Cycle, moving first through modification and anticipation, then through implementation and evaluation before repeating the cycle a second time. During the modification/anticipation section, how I think the students will respond to the task, and design of a task is discussed. A table is provided that summarizes the problems of engagement noticed during previous iterations of the same task or during similar experiences that relate to the task or challenge of focus. Unique to the first modification/anticipation section, the
Engagement Amplification Model is introduced, and simple examples are depicted that draw from pilot study evidence collected previous to the workshops of this study. Other modification/anticipation sections do not include this introduction, and simply provide the model to predict the implementation of the task or challenge of focus.
An implementation/evaluation section follows each modification/anticipation section.
Key examples were chosen from my corpus of data that illustrated my observations and best provided evidence for my analysis. Table 13 provides a key for raw data referenced when describing evidence of engagement.
99 Table 13
Key of Raw Data References
Abbreviation Raw Data FN Field Note CM Conceptual Memo AM Analytic Memo AR Audio Recording VR Video Recording SC Screencast
After examples are provided and analyzed, a table that summarizes all observed evidence of students’ engagement in SMP from my data collection is provided and raw data is referenced.
Then, the Engagement Amplification Model is applied again, but used as an evaluation tool to depict the results and evidence discussed in the section as opposed to a predictive tool to aid in the design or modification of the task. Finally, the implementation/evaluation section concludes with an analysis of the model itself to draw connections between the relationships within it, based on the results of the section’s iteration.
Each theme includes two modification/anticipation sections, and two implementation/evaluation sections for a total of four sections or two iterations through the design cycle. At the conclusion of the chapter, research questions are revisited after the examination of the themes. The models provided throughout the chapter are purely qualitative; future studies may assign numerical values to the model. However, the purpose of this study is to establish the model and discuss the relationships within it based on the evidence collected during the workshops.
100 Theme 1: Intertwining the Game with Task Objectives
To intertwine the game and task objectives, students seemed to move through a process that involves the use of SMP while simultaneously moving towards the completion of a game objective such as finishing one’s own creation, build, or product within the Minecraft world to show others, or to enjoy individually. Creating a finished product speaks to the SDT motivational factor of competence, a player’s need for challenge and feelings of effectiveness when the challenge is met (Rigby & Ryan, 2011). When intertwining this in-game reward of
Minecraft with a mathematical task, the challenge met that might inspire feelings of effectiveness is the completion of a product, such as a room, structure, or statue, outlined by the mathematical task. Students may step back to admire the product they have created, a product that was designed with mathematical learning objectives in mind, and required the use of SMP to create.
Another feature of the game that may inspire motivation and engagement is the ability for one to become immersed in the Minecraft world such that it becomes one’s fantasy or pretend world (Bailey, 2016; Starkey, 2016). This immersion speaks to presence in SDT, such that the game world has become the temporary reality of the player (Ryan, et al., 2006).
Designing and extending a task that is situated within students’ fantasy worlds, but that also incorporates SMP and intended mathematical learning goals through the completion of its objectives may offer opportunities to enhance mathematical engagement through leveraging this motivational factor of the game. For example, the first challenge in the Build the Room task asked students to create the finished product of a room within their fantasy worlds. However, the mathematical constraints of the problem required the room’s dimensions to model those of
101 the physical Minecraft Mathematics classroom. Therefore, to build the room, students had to measure the length, width, and height of the classroom, and then use proportional reasoning to recreate the measurements within Minecraft’s game world.
The SDT motivational factor of autonomy, defined as the presentation of opportunities for choice (Rigby & Ryan, 2011), was also considered in intertwining the game with task objectives. My design attempted to incorporate autonomy by providing opportunities for exploration of relationships of the mathematical content. Rather than lead the students through a step-by-step procedure, the structure of the challenge aimed to motivate students to find one of many solution paths on their own, a feature of rich mathematical tasks (Collins &
NCTM, 2011). Autonomy was also incorporated through allowing students freedom of choice in materials and building location, freedoms that simultaneously intended to cater to presence with a suggestion given to students that they create a room that fits within their fantasy worlds.
In conclusion, by intertwining SDT motivational factors of competence through feelings of effectiveness and presence, as well as autonomy in students’ gameplay with task objectives, I aimed to design a task that engaged students mathematically in targeted learning objectives that involved the use of SMP.
This section provides evidence of my modification process using the Rich Task Design
Cycle with a goal of intertwining the game with task objectives. I begin by describing my pilot study experiences and modification decisions in anticipation of the study’s workshops. Then, I move through two iterations of the design cycle to delineate my implementation and evaluation of the modifications as well as the modifications and anticipations I made for a second implementation and evaluation. Analysis of the data is conceptualized through a
102 relationship between SMP and SDT within a developing model, the Engagement Amplification
Model.
Modification and Anticipation: Choosing Build the Room for further design. The task
Build the Room was originally a one- or two-day activity that asked the students to build a room within the game that had dimensions equivalent to those of the physical classroom. I selected the task for the study, because I had noticed connections between student engagement and the design of the task during previous implementations, but I wanted to apply a more organized and systematic methodology to test my hypotheses. I intended to frame my observations using
SDT to examine how the theory might relate to task design and SMP. Thus, I began the workshops with Build the Room, and evolved it into a four-day task that consisted of multiple challenges or sub-tasks. The original room product of the task became the product for the first challenge, now titled Build the Room’s Dimensions in this study. Build the Room became the name for the overarching task that included multiple challenges such as Build the Room’s
Dimensions, noted in earlier chapters. Below, I review the modifications that I made to Build the Room prior to the first workshop to exemplify how I intertwined the game with task objectives, the first theme in connecting task design and engagement. These modifications informed my anticipations for the first class of the first workshop.
Intertwining game access and task structure. The degree of autonomy I applied to the
Build the Room task varied during my pilot studies and former teaching experiments. I explored different ways of facilitating the structure of the activity including the steps I encouraged students to take first, and when students were given access to the game. I arrived at a modified structure in which students were instructed to measure first before having access to a
103 computer to begin their builds. Students had autonomy regarding how they measured, as long as they documented their measurements. After their measurements were documented and shown to the researcher-teacher, students were given their laptops to begin building rather than instructed to perform proportional reasoning calculations to convert their measurements in feet and inches to meters for use in the game (each Minecraft block in the game is equivalent to one cubic meter). Thus, students were provided autonomy in how they modeled their measurements within the game. They were provided choices in determining their own solution paths towards finding a way to represent their measurements of a physical space in feet and inches, to the virtual game space that operated with the metric system, an application of the
SMP Model with Mathematics.
The task structure regarding computer access and objectives described above was a result of implementing and evaluating multiple iterations of the task with students that included degrees of autonomy on opposite ends of the spectrum. For example, if I provided access to the game before students measured, students often became engaged in the game and failed to engage with measurement and conversion, two targeted learning goals I aimed to encompass in the task. Figure 8 depicts the students’ engagement in the game and mathematics in this scenario. Engagement in the game is high, because students actively played the game in this scenario, and even attempted to replicate the classroom in some cases.
However, they did not focus on mathematical objectives or the task, and replicated the classroom by estimating what they thought the size might be, so mathematics engagement is low. Autonomy is defined as the opportunity for choice, not as unbridled freedom such that there is little meaning or connection for the students (Rigby & Ryan, 2011, p. 47). Thus, the
104 “autonomy dial” is set to low, because while the students were granted a large amount of freedom in how they could solve the task, the freedom was chaotic and uncontrolled.
The “SMP dial” is set to low in Figure 8, because the task was not structured such that students were motivated enough to use the SMP of Make sense of problems and persevere in solving them, or Use appropriate tools strategically. While students laid blocks in the game to approximate the walls and ceiling of the classroom, there was little to no mathematical measurement involved in their processes. For example, when I asked one student how she knew to use a certain number of blocks for one of the walls, she replied that she “eye-balled” or guessed based what she saw from sitting at her desk. During my task design, my objectives had been for the students to measure the lengths of the walls with the providing yardsticks and measuring tape. Then, I had intended for the students to convert their measurements in feet and inches to meters, in order to match their measurements to the given dimensions of building blocks in Minecraft, in which one Minecraft block is equivalent to one cubic meter
(Curse Inc., 2018). Therefore, students did not Make sense of the problem in the way that had intended, nor did they Use appropriate tools strategically to approximate the size of the room with measurement tools, documentation, and conversions.
105 Figure 8. Resulting engagement when early access to the game, and little structure within the task’s procedure was provided
On the opposite end of the spectrum, providing too much structure by requiring students to complete conversions before having game access created disengagement with the entire task, possibly due to lack of emphasis placed in game enjoyment. Similar to Figure 8,
Figure 9 shows the students’ engagement in the game and mathematics in this second scenario.
Engagement in the game is now low, because students were not granted access to their computers until late into the task, and then struggled to build their room, unsure of which of their documented measurements (feet or meters) to use. Engagement in the mathematics is medium-low, because while they engaged in more mathematics than in the first scenario, the process was forced due to a more rigid task structure. The rigid task structure constrained the autonomy of the task, so the “autonomy dial” is set to low. The “SMP dial” is set to medium, because while the task encouraged the SMP of Make sense of problems and persevere in solving
106 them, or Use appropriate tools strategically, students eventually disengaged. After measuring and converting, they no longer persevered in their problem solving and strategic use of the numbers and game to model the classroom.
Figure 9. Resulting engagement when late access to the game, and rigid structure within the task’s procedure was provided
In attempt to achieve greater engagement in both mathematics and with the game, I adjusted the structure of the task so that students received computer access after measurement documentation, but before proportional reasoning exploration. In this way, the task did not provide structure around measurement conversions, but rather opened the problem to the students to discover and address after starting to build with their measurements in feet and inches in the game. As a result, students generally converted their measurements upon recognizing that the room seemed too tall or large, with little instructional prompting. Figure 10 portrays this scenario with game engagement as high as in the first
107 scenario, and mathematics engagement higher than both scenarios. The autonomy dial is set to high, because the students were provided much more freedom within the task than in the second scenario. Also the freedom was more controlled than during the first scenario, since there was still a significant amount of structure around measurement, computer access, and task objectives. The SMP dial is set to high, because the task design now seemed to provide opportunities for the students to persevere in their problem-solving and strategically use measurement tools, conversion procedures, and the game to model the dimensions of the physical classroom.
Figure 10. Resulting engagement when access to the game was provided mid-way through the task with a medium amount of structure in the task’s procedure
Intertwining fantasy worlds and task products. In earlier iterations of Build the Room before this study, students were instructed to model the classroom itself within the game, which included the carpet, color of the paint on the wall, desks, and other furniture. However, I
108 modified the task to incorporate an opportunity for the students to choose their own theme and function of the room in contrast to specifically building a classroom, after I noticed strong engagement and presence during times when students had greater creative freedom in their task products. For example, during one pilot study, two girls worked together on a task that included building a 48-block structure in which they had the freedom to determine the shape and dimensions, as long as they could prove that they used exactly 48 blocks. The girls described to me with enthusiasm, not only how the structures incorporated 48 blocks, but their purpose in the girls’ world, a harmonious village of half-dinosaur-half-unicorn characters. The
48-block challenge had created a product that fit within their world rather than a more abstract one that merely met an instructional objective.
Therefore, I attempted to incorporate this same degree of creativity in Build the Room by modifying the task to require students to only replicate the dimensions of the room, rather than the entirety of the room’s physical properties. The students could choose the theme and function of the room, which provided an increase in SDT autonomy with the purpose of increasing ownership, and thus mathematical engagement in SMP such as Modeling with mathematics, and Attend to precision. Similar to the girls who were immersed in their fantasy world of half-dinosaur-half-unicorns, students in a separate pilot study also became immersed in their fantasy worlds as well as in the SMP when students replicated the dimensions of the room rather than the room itself in Build a Room. One student built an indoor farm with an adjoining room for his pet llamas (FN 11/15/16). He carefully measured the barn doors present on one of the physical classroom’s walls so that he could create the same-sized space for his
109 llamas to commute between the two rooms (Figure 11), a demonstration of the SMP Attend to
Precision.
Figure 11. Screenshot taken within the indoor farm of a student’s Build the Room product that shows the door to the additional room containing the pet llamas
Before this increase in the Build the Room task’s autonomy, students may have not been as engaged with the game, because they were not as immersed applying creative freedom towards their fantasy worlds (Figure 12). After the change in product expectations, the students may have been more engaged in the game, and possibly with the mathematics as well due to their increased ownership over the task’s product (Figure 13).
110
Figure 12. Resulting engagement before the task provided an opportunity for students to connect the product to their fantasy worlds
Figure 13. Resulting engagement after the task provided an opportunity for students to connect the product to their fantasy worlds
111 Because presence relates to competence in SDT through feelings of effectiveness (Ryan, et al., 2006), the “autonomy dial” has now changed to an “SDT dial” to include any and all SDT motivational factors of autonomy, competence, and relatedness applied during task modifications. The SMP dial was also turned up, because through experiencing greater engagement in the creation of their product, students may have also experienced greater engagement in the aforementioned SMP necessary to complete the product’s objectives. Thus, because students were passionate about their product, they may have been more engaged in moving through the SMP processes to complete the objective associated with their product.
As previously mentioned, measures depicted within the developing Engagement
Amplification Model are qualitative. For example, in figures 12 and 13, I observed a significant increase in both mathematical and game engagement after the task provided a connection of the task product to students’ fantasy worlds. Future studies may examine more quantitative measurements for engagement, but one of the primary purposes of this study is to explore the nature of the relationship between SDT and SMP, and develop the model itself before quantitative measurements are explored.
Relationship of SDT and SMP through the Engagement Amplification Model.
Relationships between SDT, SMP, and both types of engagement were findings that emerged as
I reviewed my designs, implementations, and modifications of the Build the Room task throughout the study’s workshops, beginning with a review and reflection of the task during pilot studies. After reviewing and tabling my data, I began to notice how adjusting the ability of the task to engage students through SDT in SMP resulted in the success of the task in fostering mathematical engagement as well as task-related engagement in the game. Complex
112 relationships emerged, and I developed the Engagement Amplification Model to pictorially represent the findings in my data.
Based on experiences prior to the study’s first workshop, the Engagement Amplification
Model (Figures 9-10 and 12-13) shows a relationship between the following three components:
(1) student engagement in both mathematics and the game, (2) the degree to which motivational factors of SDT were considered and applied in the task design through modifications, and (3) the degree to which opportunities of the intended SMP of the task were realized. In this model, the “dials” of SDT and SMP can be turned to the left or right to qualitatively/pictorially represent an observed decrease or increase in SDT and SMP application within task design and modification. The graph bars or “volume levels” of game engagement and mathematics engagement represent an observed increase or decrease as a result of the dial movement. In this study, the volume levels as well as the degree to which the dials are turned are not yet based on quantitative measurements, but rather used for demonstration purposes as a qualitative comparison based on data of what occurred before and after specific task modifications.
In addition to a relationship between engagement in the game and mathematics, the model also aims to show how factors of SDT and SMP, when taken into consideration during task development, might relate to one another as well as to the observed increase and decrease of student engagement. The turning of the SDT and SMP dials represents what occurs during modification to address problems of engagement observed during task implementation.
For example, Table 14 summarizes the problems of engagement noticed during prior implementations of Build the Room previous to this study, and connects each problem to a task
113 modification that applied a specific motivational factor of SDT with the intention to create greater opportunities for specific SMP.
Table 14
Modification and Anticipation of Build the Room’s Dimensions: Pilot Studies to Workshop 1
Problem of Engagement Modification Applied SDT Intended SMP Overemphasis on measuring Intertwined game with task • Autonomy • Make sense of and not enough gameplay by embedding conversions problems and OR as a stumbling block to persevere in solving unstructured opportunities for create task’s product in them gaming due to lack of game • Modeling with emphasis on measuring mathematics • Use appropriate tools strategically Task did not incorporate Intertwined game with task • Autonomy • Modeling with enough student choice; by adjusting expectations • Competence mathematics expectations required a for product so students’ through • Attend to precision replication of the entire room had freedom to choose presence including furniture, carpet, function and theme of among other aesthetics room
These former problems of engagement were considered in preparation for the first workshop in the study, and affected final modifications to the task before implementation on the first day. The first problem of engagement noticed was related to an overemphasis on either measuring or gameplay without a balance established between the two. To address this problem, I intertwined the task with the game so that the students might discover one of the task’s mathematical problems on their own, and then solve the problem as part of the game.
This modification involved an increase in the SDT motivational factor of autonomy, because students had the opportunity to choose when they would perform conversions in order to complete the task. The modification aimed to increase opportunities for the SMP of Modeling with mathematics, because students might recognize that building the room’s dimensions in feet may not depict an accurate representation or model of the classroom. The modification 114 also aimed to increase opportunities for the SMP of Use appropriate tools strategically, because students might recognize that conversions were needed in order to complete the task’s objective. Finally, the modification attempted to help students Make sense of problems and persevere in solving them by finding that balance between game access and autonomy within the task.
The other problem noticed was that the task did not include enough student choice or autonomy in regards to the task’s product. By modifying the requirements of the challenge’s product to only include the constraint of the dimensions, opportunities for student choice increased such that students could determine the function and theme of the room. This modification involved an intentional incorporation or increase in the SDT motivational factors of both autonomy and competence, the latter of which involved students successfully finishing a product that they could make a part of their fantasy world or in-game story, as well as presence through establishing a connection between their product and their fantasy world.
This modification aimed to increase the SMP of Modeling with mathematics and Attend to precision such that students might compare their fantasy room’s size to the size of the classroom in which they were sitting, recognizing that both rooms incorporated roughly the same amount of space with minor adjustments made due to the rounding necessary within the game’s constraints.
By studying problems of engagement and making modifications that seemed to solve the problems observed, I intended to give students access more readily to the mathematics that was previously “hidden” within the game. My goal in task design was to intertwine the game and task objectives such that the game provided a context and reason for making the
115 mathematical conversions, and a means for exploration in number sense within size, measurement, and rounding. Figure 14 shows the Engagement Amplification Model for the focused modifications for the Build the Room’s Dimensions challenge in anticipation of the first workshop.
Figure 14. Predicted changes in student engagement after modifications made to the task’s first challenge, Build the Room’s Dimensions
Implementation and Evaluation: An underground room. As a result of my modifications discussed above, students seemed highly engaged in both the game and with the mathematics in relation to SMP as well as the targeted learning objectives related to measurement and conversions. Student engagement was observed through (1) SDT’s motivational factor of competence by students’ presence in their fantasy worlds; and (2) SDT’s autonomy of the task that catered to students’ choices to work towards the task objective, independent of instructor facilitation.
116 Competence and autonomy to persevere. All students in both workshops determined a common building theme with their team, and were engulfed in their resulting Minecraft fantasy worlds. For example, one group of three students incorporated their room into their
“underground desert base,” which highlighted their interests in armor, weapons, hidden chests, and mechanical traps constructed from redstone, a material in the game that allows players to create electrical connections between buttons, levers, lights and mechanical movements (FN
1/24/2018; FN 1/31/2018). This group, Team Desert Base, integrated their game-related interests such that their room was no longer just a room, but a room with a purpose in their shared fantasy world that may have increased the competence and presence experienced by the students during the task. Figure 15 shows a room from the team’s underground base that is reinforced with the Minecraft block types Chiseled Stone Bricks (grey) and Chiseled Sandstone
(yellow) to correlate with their desert base theme. Notice the player Adam, whose screen we are viewing, has an inventory displayed at the bottom of the screen that includes lava, stone bricks, armor, and a pick axe among other materials that correlate with building and playing in the team’s fantasy underground desert base.
117
Figure 15. A capture from the screencast of Adam as he views one of the rooms in his team’s underground desert base
Common among all groups in the workshops, Team Desert Base was motivated to complete their measurements so that they could receive access to the game, evidenced by their on-task behavior and cooperation in physically measuring the room with the provided tape measures. As observed in each of their screencasts, the group began carving out space underground immediately upon establishing their space in the game world. They developed their underground base theme, and agreed to make the room of the task underground as well.
As they carved out space underground, they talked about how big the room should be and how they might use their measurements of length, width, and height.
During this time, evidence of the SMP Making sense of problems and persevere in solving them was observed when due to one group member’s absence in the second class, the other two group members realized that they had recorded only one of the three dimensions 118 and needed their absent friend’s documentation. Upon receiving the information, they also realized that they had not yet converted the length and width to meters, and would need to do so since Minecraft blocks were equivalent to cubic meters. They then took the necessary steps so they could build the classroom’s dimensions in meters within their underground base to model the dimensions of the classroom as accurately as possible.
These students also engaged in the SMP of Reason abstractly and quantitatively, because they compared the length to the width of their room to determine the space that needed to be carved out in rectangular areas. They compared the use of feet to meters with
Minecraft blocks, realizing that using their measurements in feet would produce a much larger area in the game world than if they first converted the measurements to meters, and then used the numbers to create their dimensions. Thus, because the task incorporated autonomy by avoiding direct instruction and allowing productive struggle to occur, the students pursued the challenge and engaged in multiple SMP. Figure 16 shows a first-person view of one student who is looking at his teammate in front of him as they build their room. The image from a screencast shows the degree of measurement and problem solving in which the students must have engaged to get to this point, because they are building underground and must carve out the dimensions calculated through their conversions to model the dimensions required by the task.
119
Figure 16. Students in Team Desert Base begin to carve out an underground tunnel and space for a room in the Build the Room’s Dimensions challenge
Relationship of SDT and SMP. A variety of evidence listed in Table 15, shows that students engaged in five SMP, engagement that may have resulted from the modifications made to the task before implementation that aimed to increase SDT motivational factors of autonomy and competence.
Table 15
SMP Noticed in the Challenge Build the Room’s Dimensions
Standard for Observed Evidence Mathematical Practice Making sense of Students seemed to understand the objective, and took the appropriate steps to problems and persevere measure and convert despite some initial difficulty in most groups (VR_HI2 solving them 1/31/2018, VR_OAA 1/31/2018, 2 FN 2/26/2018, FN 2/28/2018)
Reason abstractly and To acquire the height of the room, students measured the height of the quantitatively classroom’s back wall cinder blocks or the windows of the classroom’s garage door, multiplying the measurement by the number of cinder blocks or windows (VR_OAA 1/31/2018, FN_2/28/2018) 120
Model with Mathematics Students replicated the dimensions of the classroom within the game; some groups built the room with measurements in feet first before realizing they needed to convert to meters (VR_HI2 1/31/2018, VR_HI3 1/31/2018, FN_2/28/2018)
Use appropriate tools Students used their measurement and proportional reasoning skills to replicate the strategically dimensions of the room from feet to meters (VR_HI2 1/31/2018, VR_OAA 1/31/2018, FN 2/28/2018)
Attend to precision Students checked their conversions within their groups to make sure their results were consistent, and made sense numerically (i.e., the conversion in meters should be less than the original measurement in feet) (FN 1/31/2018, VR HI2 1/31/2018)
By increasing autonomy and competence when embedding a task in Minecraft, the game was intertwined with task objectives in a way that engaged students in building the task’s product.
As students worked towards designing and building the task’s product, they engaged in multiple
SMP through their motivation and engagement with the game and task. Thus, Figure 17 shows the resulting balance of engagement, SDT, and SMP according to these observations as predicted during the modification and anticipation stage in the previous section.
121 Figure 17. Resulting student engagement after modifications made to the task’s first challenge,
Build the Room’s Dimensions
The consideration of both SDT motivational factors and opportunities for students to engage in SMP seem to be dependent on one another through the game. First, the task design aimed to incorporate greater autonomy and competence in SDT, and as a result, students seemed to engage more with both the game and the mathematics. Thus, the consideration and application of SDT motivational factors in a task may result in greater engagement in the game as well as in mathematical practices as seen by the arrows connecting the SDT dial to both game and mathematics engagement volume bars in Figure 18. SMP seemed to act as a motivational factor as well when students realized they had to use specific mathematical strategies in order to meet the task objective for their product in order to experience the intrinsic rewards of ownership and effectiveness through autonomy and competence in SDT.
Thus, SDT influenced SMP, and vice versa as denoted by the double-sided arrow connecting the two dials. “Turning up” one dial may result in the same directional movement of the other dial, 122 although the degree to which the other dial turns as a result is not yet known. Finally, an increase in SMP seems to denote an increase in mathematics engagement, but whether this increase affects game engagement is unknown at this time, since increasing opportunities for
SMP in the scenario shown previously in Figure 9 with greater structure seemed to decrease game engagement. Therefore, the arrow stemming from SMP to the engagement volume bars only connects to mathematics engagement for now.
Figure 18. Relationships of SDT and SMP to engagement within the Engagement Amplification
Model
Modification and Anticipation: Creating an additional room and challenge. Due to the students’ engagement in this first challenge during the first workshop as evidenced in the previous section, I was encouraged to modify the overarching task by designing an additional challenge for the Build the Room task, titled Same Volume Different Dimensions (Appendix C).
The design of this challenge also aimed to intertwine the game with its task objective through having the students create a product that would serve a purpose in their fantasy worlds. 123 Because I noticed students starting to create additional rooms attached to their original room, the objective of Same Volume Different Dimensions was to build another room with the same volume as the original, but with a different length, width, and height.
Targeting specific SMP with attention to SDT. Still a part of the overarching Build the
Room task, my instructional goals were for the students to explore dimensions in greater depth through examining their relationship to volume while hopefully incorporating related SMP such as Reason abstractly and quantitatively and Look for and express regularity in repeated reasoning. Through engaging in Reason abstractly and quantitatively, I hypothesized that students might connect abstract values of length, width, and height with a relationship to volume by experiencing how three new dimensions affect a new room’s size in comparison to their original room. The task’s challenge might incorporate opportunities for students to engage in Look for and express regularity in repeated reasoning as they worked through a procedure of using the volume from their original room to calculate new dimensions. For example, students might think about whether certain calculations could be repeated when factoring their volume, and whether there was a general method or shortcut that could be used to do so (Common Core State Standards Initiative, 2018).
These instructional goals and incorporation of SMP were considered as I simultaneously applied the SDT motivational factors of autonomy and competence in an attempt to enhance engagement with the task’s new challenge. Autonomy was incorporated through the creative freedom allowed not only in the functionality of the room in their world, but also mathematically in the variety of answers (combinations of dimensions) possible to keep the task open-ended by design (Gojak, 2013; Stein, et al., 1996). Competence in game play was also
124 applied to the task, because of its similarity to building the first room with certain dimensions in a specific space, a process that all groups had completed at this point. My hope was that students would experience greater confidence while calculating the dimensions of, and building the second room since they had completed a similar activity earlier. Table 16 summarizes the problem of engagement in relation to the design choices made that aimed to increase student’s motivational factors in SDT and engagement in SMP.
Table 16
Design and Anticipation of Same Volume Different Dimensions: Workshop 1
Problem of Design Applied SDT Intended SMP Engagement Another challenge Intertwined game with • Autonomy • Reason abstractly and needed to continue task by embedding a • Competence quantitatively momentum of similar product within the through presence • Look for and express mathematical task's objective while and feelings of regularity in repeated engagement considering opportunities effectiveness reasoning experienced through for additional learning the room product of a goals and SMP in task previous challenge structure
Predictions through the Engagement Amplification Model. During the first workshop, the problem of continuing to propel the students’ momentum of mathematical engagement in connection with their fantasy worlds arose as they completed the Build the Room’s Dimensions challenge and continued to add to their creations. To address this problem, I designed an additional challenge with a product similar to that of the first challenge to continue to intertwine the game with the task. This modification also involved an increase in SDT motivational factors of autonomy and competence, because students again had the freedom to choose the theme and function of their additional room. Because students already had
125 experience building a room with dimension constraints, slightly adjusting the constraints of the challenge to those surrounding volume and dimensions aimed to enhance their feelings of effectiveness, a by-product of competence as they moved through a familiar process.
SDT motivational factors were also considered in incorporating the SMP of Reason abstractly and quantitatively and Look for and express regularity in repeated reasoning. The task structure included rich task features of open-endedness, multiple solution paths, and multiple answers within the expectation that students would calculate their own dimensions to produce the specific volume of their original room. These opportunities for choice in solution path and answer provided autonomy within the task, and competence, because students might use their ideas about dimensions from the first challenge to reason through a procedure of calculating a volume and using the volume to find new dimensions.
Figure 19 shows the anticipated relationship of student engagement, SDT, and SMP within the Engagement Amplification Model. A similar strategy of intertwining a balance of task structure with student fantasy worlds and task products was used to design this new task in comparison to the modifications applied to Build the Room’s Dimensions. Therefore, the predicted student engagement outcome of the new challenge mirrors that of figures 14 and 17, predicted and resulting outcomes of the first challenge.
126 Figure 19. Predicted student engagement after consideration of SDT and SMP in the new challenge Same Volume Different Dimensions
An additional similarity to Build the Room’s Dimensions is the interdependence of SMP and SDT in the design and anticipation of this new challenge as well. Both dials are turned to high, because both were strongly considered, and seem to rely on one another regardless of order of consideration. For example, students may engage in the SMP of Reason abstractly and quantitatively by “attending to the meaning of quantities” (Common Core State Standards
Initiative, 2018, p. 1), relating their combinations of measurement calculations to in-game dimensions as they design their additional room with new dimensions. As a result, they may experience SDT’s competence as they experience “a-ha” moments while moving through the problem-solving process and comparing their new room with their original room. Reversing the order of consideration such that SDT is considered first, students might experience the SDT motivational factor of autonomy when choosing their combination of dimensions for their new room (dependent on the constraint of their first room’s volume). As a result, they may be 127 engaged while looking for and expressing regularity in repeated reasoning to deduce a process for successful factoring, the other intended SMP of this challenge.
Implementation and Evaluation: Autonomy and shortcuts. Immediately after receiving an introduction to Same Volume, Different Dimensions, most students in both workshops began to write on the challenge’s instructional template (Appendices C-D) and punch numbers into their calculators. Most students worked together in their groups to take apart their volume by repeatedly dividing it by an assortment of whole numbers until they found one that yielded another whole number (VR_OAA 2/7/2018, VR_RJ 2/7/2018, VR_IH 2/7/2018, VR_JM 2/7/2018,
AR_W 3/5/2018, FN 3/5/2018). Then, some students experienced a barrier of not knowing what to do until they were reminded by either the researcher-teacher, but more often by their group members that they needed three dimensions. They then divided their resulting whole number from the first calculation by another assortment of numbers until they found a final whole number. Finally, many students multiplied the three whole numbers from their calculations together to check if the product resulted in their volume calculated at the top from their original dimensions. Students did not seem to be familiar with the term factoring, and when asked, many stated that they had never been taught volume before, but that it made sense, because “you need a way to measure the space of a room with blocks in 3D” (FN
3/5/2018). High levels of engagement in the task by almost all students, and a connection of the numbers to geometric dimensions within the game seemed to evidence a movement through the SMP Reason abstractly and quantitatively.
During the introduction to the challenge in which I led a brief class discussion and overview of volume in relation to dimensions of length and area, I turned off access to the
128 server, the in-game Minecraft world. I turned off server access in both workshops often when making class announcements or leading class discussions to minimize distractions for the students. However, after the first group had established their new dimensions and checked in with me, I turned on access to the server so that the group could begin to build their new room.
Because the first groups in both workshops audibly and emphatically expressed competence and feelings of effectiveness about having found new dimensions, other students were aware that access to the server was now available. Despite this recognition, students continued to persevere through finding their factors and did not become distracted by the game. Such a strong focus and high attention to their factoring processes may have been a result of the autonomy provided within the task, especially in relation to the SMP Look for and expressing regularity in repeated reasoning as they worked to find short cuts within their division process.
One student, Isaiah (all student names are pseudonyms to protect the anonymity of the students) even went to another group when he and his group member Han became stuck in their process. Despite he and Han having a different number to factor due to a different volume calculation, Isaiah applied a step from the other group’s process to his own process to find the three factors (FN 2/7/2018, VR_I 2/7/2018).
Flat rooms and disengagement. Despite strong engagement observed of most students during Same Room Different Dimensions, one group in the first workshop did not engage as strongly with the task. Manny and Jay attempted to simply rearrange the factors of length, width and height to produce the same product rather than find new factors, my original instructional objective. As the teacher, I interpreted their actions as a shortcut in order to gain access to the game more quickly (the server was not yet accessible at this point). In attempt to
129 extract some mathematical thinking, I prompted a conversation about whether the room would have different dimensions by definition if the numbers were rearranged. While Jay seemed to take time to think, Manny may have believed that I would not grant them access to the game if they were to simply rearrange the numbers (despite my plans to do otherwise if they were to provide adequate reasoning about dimension rearrangement), and decided to use the measurement of one meter as one of his dimensions, another action which as the teacher, I perceived to be a shortcut.
Regardless, I believed that there was still an opportunity to engage Manny in the SMP of
Mathematical Modeling by connecting his calculations to a build in the game. The following excerpt from a video recording during student work time in class (VR_M 2/7/2018) begins with
Manny (M) typing an estimation of length, width, and height into the calculator that would result in his goal volume of 440 cubic meters as I, the researcher-teacher (RT) ask him to transfer the mathematics to the game.
1. M: [Types into calculator] 11 times 40 times 1? …440! Yay!
2. RT: So if you had a height of 1 in your room, what would that look like?
3. M: …It would just be flat.
4. RT: You would have to duck down and walk around in it—
5. M: But it’s still a room! [Laughs]
Had Manny continued to engage in conversation with me regarding the possible utility or functionality of a “flat” room, as the teacher I would have been satisfied with his mathematical
130 engagement surrounding dimensions and space, using my power as the teacher to then grant him access to the game. However, Manny, who I deemed as strong in certain mathematical skills due to his perfect score on the Pre-Test completed on the first day of class, may have realized he had the capability to explore further. He began to enter combinations of dimensions into the calculator before finding a combination of 11, eight and five to produce a product of 440 through a guess and check method dividing 440 by 11 and then other whole numbers.
Manny’s original lack of engagement in the mathematical process can be noted to inform future modification. Perhaps building a second room was not as interesting to him, and he perceived the task to be an abstract mathematics problem that acted as a barrier to his game access rather than as an opportunity to explore and compare room sizes that could be added to one’s fantasy world. While the SMP Model with Mathematics was somewhat present in my conversation with Manny, our conversation allowed me to consider whether this standard could be expanded upon in modification to the challenge, or with the addition of another challenge. For example, how might the arrangement of the length, width, and height dimensions affect the “flatness” of a room? Students might engage in the SMP Model with
Mathematics as they replicate measurements in the game. However, if students were to think about how specific numbers assigned to each dimension affects the shape of the room, they may be more deeply engaging in the SMP of Reason abstractly and quantitatively, because each dimension would be attended to in its meaning, not just in its computation (Common Core
State Standards Initiative, 2018, p. 1).
131 Feelings of effectiveness observed in mathematically engaged and non-engaged students. While I observed less mathematical engagement or interest from Manny during the
Same Volume Different Dimensions challenge, students such as Han, Isaiah, and Adam engaged in the task, with Han and Isaiah in particular taking time to compare the sizes of their rooms and engage in a short discussion with one another (FN 2/7/2018). In their cases, evidence of their engagement may be noted within transcription excerpts of their interviews on the last day of class regarding their interests in the game and perception of the tasks. Han and Isaiah worked in a group of two during one of the workshops, and were interviewed together. I selected Han and Isaiah to be interviewed because they seemed highly engaged in the mathematics of the challenges. In contrast, I selected Adam from the same workshop to interview, because he did not seem as mathematically engaged compared to other students.
In the episode below from the interview with Han and Isaiah on the last day of class, I asked Han (H) and Isaiah (I) about the differences between playing Minecraft recreationally
(known as free play time during class), and playing Minecraft with the challenges I provided in the workshop.
1. RT: What would be the difference between working on something I give you, and free play?
2. H: Well free play is a bit more fun because you just get to go and do whatever—
3. I: But—
4. H: And there’s no worksheets—
5. I: Well what I like, what I like about your thing is usually I say I’m going to build something,
6. then I never get around to doing it—
132 7. H: Well yeah—
8. I: But for you I have to do it—
9. H: Yeah that’s true.
The boys seemed to convey their appreciation of having focused objectives within the game, despite these objectives incorporating the use of the instructional templates (Appendices A-D), which Han refers to as worksheets. The focus on a particular build to its completion may cater to SDT’s factor of competence through feelings of effectiveness.
During my interview with Adam (A), I observed competence through presence and his immersion in the game during the first challenge Build the Room’s Dimensions. Adam was slow to answer many of the interview questions, but quickly and enthusiastically spoke about building multiple rooms with his team when asked about measuring and building.
1. RT: Tell me about some of your experiences with the challenge where you guys had to do
2. the measuring, and then build it…
3. A: Well…we kind of, like, measured the classroom and put it into Minecraft for our
4. underground desert base, I guess.
5. RT: Yeah—
6. A: And like…we made, well we technically made three rooms. One was the main room, and
7. two other ones were for, well one was a secret room, and then the other one was for…like…
8. just like, a freedom, like an escape-plan room.
9. RT: What was your favorite room to build out of all those?
133 10. A: Probably the room that we had like, a secret lava thing, that if we took down a secret
11. lever then lava would come down and when we pulled it back up the lava would stop so we
12. could get in the room when things were trying to get us.
When prompted to share his experiences working with his team, Adam admitted to receiving
“quite a bit of help” from his team in filling out the templates, because “there were a lot of things that I didn’t really know what to do and they would just tell me what to write and I would write it” (AR_A 2/14/2018). Some of my video recordings and field notes corroborated this information as well. However, the transcription above depicts Adam’s attention to the multiple rooms and his immersion in their underground desert base theme. While Adam was not engaged with the mathematics, he seemed interested in the functions of the multiple rooms and the contraptions built within them for protection purposes, and still participated in the task by filling out the accompanying instructional template and helping build the new room based on instructions and leadership from his other two group members. Therefore, while
Adam seemed to experience presence in SDT’s competence, he may have also experienced feelings of effectiveness through his involvement in helping his team build multiple rooms.
Based on these excerpts from Han, Isaiah, and Adam, I argue that the success of the
Same Volume Different Dimension challenge’s ability to foster engagement for certain students was based on the intertwining of the game with the task objective. This success was achieved through building a specific product (Han and Isaiah) that was an additional room in their fantasy world (Adam). Through the creation of this product, students may have experienced presence and feelings of effectiveness within SDT’s competence, regardless of their perceived differing
134 levels of mathematical engagement by the researcher-teacher. These students seemed to enjoy building multiple rooms and having a finished product to show off to their peers in class, or to enjoy viewing and experiencing among themselves.
Results through the Engagement Amplification Model. Compared to Manny and Adam, most other students including Adam’s group members as well as Han and Isaiah were observed to have stronger mathematical engagement during the Same Volume Different Dimensions challenge. Table 17 lists the SMP that were observed in mathematically engaged students through additional video recordings and field notes.
Table 17
SMP Noticed in the Same Volume Different Dimensions Challenge
Standard for Observed Evidence Mathematical Practice Making sense of problems Students seemed to understand the objective, and took the appropriate and persevere in solving steps to factor and check their work, despite some difficulties (FN 2/7/2018, them VR_M 2/7/2018, VR_I 2/7/2018, SC_O 2/7/2018)
Reason abstractly and Students connected abstract numbers factored to dimensions of a room quantitatively within the game (VR_I 2/7/2018, SC_O 2/7/2018)
Model with Mathematics Students related the factors of a product to dimensions of a room’s volume and created a model in the game world (SC_O 2/7/2018, VR_JM 2/7/2018, VR_IH 2/14/2018, AR_W 3/5/2018)
Attend to precision Students used multiplication to check the accuracy of the factors they had found by using division (SC_O 2/7/2018, VR_I 2/7/2018)
Look for and express Students moved back and forth between using multiplication and division to regularity in repeated find various dimensions of a room’s volume; some students invented a reasoning general procedure for factoring their volume; all students used multiplication and division to find factors for a product (SC_O 2/7/2018, VR_I 2/7/2018)
A variety of evidence shows that students engaged in these five SMP during the challenge, engagement that may have paralleled the intention to increase SDT motivational
135 factors of autonomy and competence. After noticing that students were engaged in building particular products in the game (additional rooms), the design of the challenge aimed to incorporate an objective that would revolve around a similar product (a new room with new dimensions). Through allowing structured freedom such that students were to determine the dimensions for the room as long as the dimensions created an equivalent volume to the room from the task’s first challenge, autonomy and competence were infused within the task in an attempt to enhance engagement in targeted learning objectives of factoring and volume.
Intended SMP of Reason abstractly and quantitatively and Look for and express regularity in repeated reasoning were also incorporated with the autonomy and rich task features of open- endedness and multiple answers.
Figure 19 shows the resulting student engagement of the Same Volume Different
Dimensions challenge in comparison to the predicted engagement described in the previous section. While game engagement remained high, mathematical engagement was not as high as
I had originally expected due to the experiences described with Manny, and Adam. Manny in particular demonstrated an ability to engage in the mathematics content, but his motivation to do so was not as strong as that of other students, either because he was not as interested in creating the product of the task, or possibly because the SMP of Reason abstractly and quantitatively was not as strongly connected to the task.
For example, in a previous dialogue excerpt discussed, Manny was able to articulate his visualization of a “flat room” when he rearranged his dimensions so that the room had a height of one block or meter (VR_M 2/7/2018), but this notion did not explicitly help him attain competence within the task (FN_2/7/2018). If the structure of the task had provided an
136 opportunity for him to take his teacher-perceived shortcut, Manny may have experienced greater competence in his ability to demonstrate his existing knowledge, and perhaps would have felt rewarded for finding a more efficient way to solve the problem. Instead, Manny may have recognized that he was taking a shortcut, but due to the structure of the task, did not believe his solution path to be acceptable, resulting in lack of competence on the task.
Furthermore, if Manny had felt validated in his choice to take the shortcut, autonomy may have increased as well due to opportunity for an additional choice meaningful to him. These modifications to Same Volume Different Dimensions will be explored later in the chapter.
However, the present shortcomings of the SDT motivational factors of autonomy and competence, as well as their relation to the SMP of Reason abstractly and quantitatively are denoted in Figure 20 by a slight “turning down” of the SDT and SMP dials. The dials are still turned to the higher side of the spectrum, because the task seemed to provide opportunities for most of the other students to experience the intended SMP and SDT motivational factors.
137
Figure 20. Resulting student engagement after implementation of Same Volume Different
Dimensions challenge
The slight turning of the SDT and SMP dials did not seem to affect game engagement, even for
Manny, who returned to the game upon completing the instructional template and built his new room, connecting it to the Nintendo theme that he and Jay had been developing (FN
2/7/2018, FN 2/14/208). Therefore, game engagement remained high, while mathematical engagement was affected negatively by a decrease in SDT and SMP compared to their intended dial settings prior to implementation.
The lack of the dials’ effect on game engagement may insinuate that a much stronger dial turning, or greater difference between anticipated and resulting dial settings must occur for a change in game engagement to occur. While students already seem to have an existing game engagement regardless of their work on tasks, design of a task can still affect students’ game engagement as previously observed when (1) students received late access to the game during work on the task (autonomy), (2) when there was too much structure and not enough choice 138 (autonomy), or (3) when the task’s product did not incorporate as much creative freedom in connection to students’ fantasy worlds (competence)(Figures 12-13). However, these three cases denote larger differences in consideration and application of SDT motivational factors, and do not consider SMP. Therefore, Figure 21 presents a modified version of the relationships between amplification of engagement, SDT, and SMP, such that the arrow connecting the SDT dial and game engagement is not as pronounced. The arrow is not as pronounced as the others, because the SDT dial seems to have less of an effect on game engagement than on mathematical engagement. The SDT and SMP dials still seem dependent upon one another, because they both turned in the negative direction as a result of the task’s shortcomings in autonomy and competence around the SMP of Reason abstractly and quantitatively. Thus, the arrow connecting the two dials is still present in the model.
Figure 21. Modified relationships of SDT and SMP to engagement within the Engagement
Amplification Model
139 In conclusion, the first theme of intertwining the game with task objectives was encompassed in multiple design and modification elements of the task. Designing the task’s products such that they included the students’ existing game interests incorporated the SDT motivational factor of competence to propel task-related mathematical engagement.
Considering SDT factors of autonomy and competence as specific SMP were designed into the task also seemed to enhance mathematics engagement, especially if the SDT factors simultaneously provided opportunities for choices during in-game play. While this first theme of intertwining the game with task objectives largely applied autonomy and competence in SDT, the second theme applies the third motivational factor relatedness through its focus on student interactions.
Theme 2: Incorporating Opportunities for Student Interactions Inside and Outside the Game
As challenges of Build the Room were implemented, great attention was paid to how students interacted with one another to inform further task design and modification. In an attempt to enhance mathematical engagement in targeted learning objectives as well as in
SMP, I documented when and how students communicated and worked together through the
SDT motivational factor of relatedness, defined as a player feeling connected to others in the game (Rigby & Ryan, 2011). Before moving through the design cycle to describe the task design process as in the previous section, an overview of how relatedness might apply in Minecraft will be provided.
Opportunities for relatedness in Minecraft. Minecraft involves a degree of relatedness through its inclusion of Artificial Intelligence (AI) entities such as villagers, animals, and monsters. Although a player cannot interact conversationally with the AI, they are highly visible 140 as other life forms present in the virtual world. During both workshops as well as in most previous Minecraft Mathematics classes, I turned off the existence of the villagers and monsters in the world with the purpose of creating fewer distractions from task objectives.
However, non-hostile animals were popular, because many students liked to have “pets” in their fantasy worlds, and I wanted to provide opportunities for presence with animals in tasks, especially since they acted more as decorative rather than interactive entities.
In addition to AI, players can also view other human players depending on the accessibility of their selected game world through a server, an online or local area network that allows players to connect to a shared virtual world within the game. A player may also open an exclusive virtual Minecraft world inaccessible to other players and play alone. In the Minecraft community, many players create and open servers and virtual worlds for community play or play with friends. During the workshops of this study as well as most of the pilot studies, we used a shared game world that was accessible only to students in class, and controlled solely by me as the researcher-teacher. In addition to control of access, I also possessed control of certain game settings such as the ability for players to deplete health from one another’s avatars. I turned off this feature to limit distractions and hostile interactions, since I wanted to promote a classroom norm of students helping one another build rather than destroying one another’s health and builds.
A clean, unused game world without any previous students’ builds was provided to each workshop on the first day of class. Students’ avatars spawned, or entered the world at the same location, and were thus able to see one another at this time. I encouraged teams to find separate locations to build within the world in order to give as much space as possible for
141 creative freedom. Also, I intended to discourage opportunities for accidental destruction of another team’s progress. Students were granted access to commands, which allowed them to teleport to one another by entering a simple algorithm into the chat box. Students could visit other groups outside of their own, but a classroom norm established on the first day of both workshops outlined how teams were not allowed to build or destroy items in another team’s space.
Minecraft involves multiple conversational opportunities with other players both within and outside of the game world. For example, players can use a chat feature by typing messages into a space near the bottom of the screen so that they will appear at the bottom of the screens for everyone within the same virtual world. Signs with messages of the player’s choice can also be displayed within the world such that when other players approach a specific location, they may read the sign posted by another player. A player may also have an in-person conversation with other players about events happening in real time within the game world as they build alongside one another, which was the most common conversational occurrence in the
Minecraft workshops compared to chatting and signs as evidenced by multiple screencasts.
Bailey (2016) also notes the fluidity of communication in and out of the virtual Minecraft world in his yearlong ethnographic study that investigated a group of ten and eleven year old children’s engagement with Minecraft as they collaborated to build a virtual community.
I attempted to incorporate SDT’s relatedness into task design by not only requiring students to work in teams, but by orchestrating the task such that its objective entailed a product that the students would create together in attempt to harbor mutual feelings of ownership as well as SDT’s competence through experiencing the reward of completion. This
142 section explores movement through the rich task design cycle (anticipation, implementation, evaluation, and modification) of Build the Room’s challenge Same Volume Different Dimensions between the first and second workshops (Appendices C and D). As described previously, the challenge required students to build an additional room in their fantasy world that had an equivalent volume, but an alternative length, width and height to the original room from the first challenge Build the Dimensions of the Room. My intention was for students to work together to find new dimensions, and discuss how the dimensions related to the shape and size of their room. Through these discussions, I aimed to connect specific SMP and targeted mathematical learning objectives surrounding volume and factoring to students’ existing engagement with the game and with each other.
Workshop 1 Modification and Anticipation: “It’s easy, bro!” During the first class of the first workshop, I noticed multiple occurrences of a student asking a team member for assistance, and the team member denying or ignoring the student’s request. These observations were corroborated in both field notes and screencasts (FN 1/25/2018, SC_R
1/24/2018, SC_I 1/24/2018, SC_AD 1/24/2018, SC_AS 1/24/2018). For example, while searching for a place to build, Isaiah asked Han how to build a beacon, to which Han simply replied, “It’s easy bro!” and continued to build and explore in the game (FN 1/25/2018).
Additionally, Adam often asked both of his group members where they were in the game world, or about what they had built so far (SC_AS 1/24/2018), questions that went unanswered while his group members asked non-task related questions such as “who is invisible?” and counted blocks independently without talking with one another (FN 1/25/2018, SC_AD 1/24/2018).
143 These instances led to a decision to add to the overarching Build the Room task, a challenge entitled Document a Craft (Appendix B) in attempt to encourage students to work together to complete a series of objectives. Document a Craft involved students documenting both visually and verbally, a procedure for building a particular item that would be taught to the rest of the team, and then created around the original room of the first challenge, Build the
Room’s Dimensions. I had observed student interests in particular items such as beacons, fireworks, and redstone (FN 1/25/2018). In an attempt to enhance SDT’s factor of relatedness, I designed the task to connect players to one another through having to explain and teach a procedure related to their existing in-game interests.
Targeting communication. While Document a Craft seemed to encourage positive communication within groups (Interview with Han and Isaiah 2/14/2018, VR_R 2/14/2018), I wanted to continue to provide opportunities for positive communication and behavior within the design of remaining challenges. Therefore, as I designed the new challenge Same Volume
Different Dimensions for the third class of the first workshop, I not only aimed to intertwine the game with the task objective as mentioned in the previous section, but I designed the structure of the task such that the students would need to discuss and plan the task’s product together.
For example, near the beginning of the challenge, students find factors of a number that is their original room’s volume. Because I had noticed students checking their answers with their group during Document a Craft, I hoped that students would find multiple answers (autonomy) and then work together to find an agreed-upon answer in the present challenge (relatedness).
After finding factors, the third question of the original Same Volume Different Dimensions instructional template (Appendix C) asked students to draw different areas of the room
144 including the floor and a wall in attempt to inspire a conversation about which dimension should be used for the height as compared to the length or width, and how the resulting room might be shaped (e.g. short and flat, tall and skinny, etc.).
Thus, this challenge aimed to incorporate both of SDT’s relatedness and autonomy for the group by providing freedom to choose the factors (with some constraints), and then the arrangement of the new room’s dimensions. The challenge also aimed to integrate deeper engagement in the SMP of Reason abstractly and quantitatively as well as Construct viable arguments and critique reasoning of others by providing opportunities for students to discuss how choosing different dimensions might affect the room’s shape and perceived size. Table 18 summarizes the problem of engagement in communication during the first workshop, and the modifications made that aimed to increase student’s motivational factors in SDT as well as their engagement in SMP. This table is similar to that of Table 16 from the previous theme’s exploration, but Table 18 focuses on the problem of lack of student communication mathematically or in general rather than the problem of continuing the momentum of mathematical engagement.
Table 18
Design and Anticipation of Communication in Same Volume Different Dimensions: Workshop 1
Problem of Engagement Modification Applied SDT Intended SMP Students were not Added Same Volume • Relatedness • Reason abstractly and communicating or Different Dimensions to • Autonomy quantitatively collaborating in the game continue to encourage • Construct viable mathematically or socially collaboration and arguments and critique discussion; attempted to reasoning of others include opportunities and expectations for student communication within task structure
145 Predictions through the Engagement Amplification Model. As SDT motivational factors were increased during task development, SMP were also increased. Again, the relationship between SDT and SMP was interdependent; they affected one another simultaneously without one being independent from the other. For example, as I aimed to engage students in the SMP of Reason abstractly and quantitatively, I applied SDT’s relatedness and autonomy so that students might decide with their team, which dimensions would create an additional room that fit their fantasy world, both in shape and in function. Furthermore, upon considering student engagement in SDT’s relatedness, the SMP of Construct viable arguments and critique the reasoning of others surfaced as a way to engage students in conversations with one another so that they might feel connected within the game around an agreed-upon decision.
Therefore, in predicting student engagement in communication for the Same Volume
Different Dimensions challenge, Figure 22 shows that both SDT and SMP dials are turned up to high, because relatedness, autonomy, and the aforementioned SMP were strongly considered to optimize the task structure for mathematics communication. The hope of the researcher- teacher was that implementation would yield high engagement levels in both the game and mathematics due to communication surrounding appropriate dimensions for a new room within the teams’ fantasy worlds.
146
Figure 22. Predicted student engagement after consideration of SDT’s relatedness and autonomy as well as the SMP in the new challenge Same Volume Different Dimensions
Workshop 1 Implementation and Evaluation: A height of five blocks. During the first iteration of Same Volume Different Dimensions, students worked backwards to guess and check factors using combinations of multiplication and division. Three out of four groups worked closely together (VR_I 2/7/2018, SC_O 2/7/2018, FN_2/7/2018), with the exception of Manny and Jay as described previously. While all students completed the challenge’s instructional template, collaborative effort was not as strong as I had hoped in observing and discussing how the selected dimensions might affect the shape of the room. However, all groups were able to establish new dimensions and show me as the teacher, how their dimensions multiplied to the volume product calculated from their original dimensions. Groups then built their additional room with their newfound dimensions, often assigning specific group members to build each dimension and area of the room (e.g., walls, ceiling) such that everyone had a job. As groups began to replicate their new dimensions in the game, they often discussed materials that they 147 would use to build their new room as well as the room’s function in their fantasy world (FN
2/7/2018).
Communication between Oscar and Allen. Communication exemplified between Oscar and Allen in the following sub-section was fairly typical between group members engaged in the challenge. The students worked together to calculate their factors by dividing their volume by different numbers. Then, they continued to work through the challenge’s instructional template by diagraming what the shape of the room might look like before collaboratively building it together in the game. Similar to other students, Oscar and Allen did not engage in the SMP Reason abstractly and quantitatively to the extent that I had envisioned, as evident in their short or non-existent discussion of choosing a height of five meters. However, they engaged in the SMP of Construct viable arguments and critique the reasoning of others through checking each other’s work and vocalizing issues they encountered while factoring and building.
The three excerpts below were taken from a screencast that recorded student voices
(SC_AD 2/7/2018) as well as the computer screen of a student Oscar (SC_O 2/7/2018) while he and a group member Allen worked on the challenge during the third day of the first workshop.
In the first excerpt, the two students work through the challenge to find their factors. The students then imagine how the factors might represent height, length and width in the second excerpt before returning to the game and beginning to build a new room with these dimensions during the third excerpt. The first excerpt begins shortly after I had provided students with their instructional templates for the challenge and reviewed expectations. Oscar and Allen’s original dimensions were four, ten, and 11 meters, factors that produced a product or room volume of 440 cubic meters.
148
1. O: The same volume, but different dimensions. So it has to be different than all of these,
2. and it has to equal 440. So we could do… 11…
3. A: 12… even if you do 440 divided by…
4. O: Oh yeah.
5. A: 440--
6. O: 440--
7. A: 440 divided by… 9 equals….
8. O: [Laughs] point point point point
9. A: just a bunch of points
10. O: 440 divided by seven…
11. A: Seven. Okay.
12. O: No! Why?
13. A: Okay, 440 [gasps] 440 divided by 10!
14. O: No, cuz that’s—that’ll get 11. Wait, divided by…
15. A: 10?
16. O: No, divided by five, sorry.
17. A: We have to divide it by five…wait, what does that make--
18. O: No for this one, erase that—no don’t erase the end. Ya get 5, 11, and 8. [Repeats]
19. 5, 11, 8.
20. A: 5 height…
21. O: 5 height.
149 22. A: 11…8…
23. O: That equals it, watch: 8 times 11 equals that times 5, see?
To find factors of 440, the students divide the number with various whole number guesses other than their original factors of 10 and 11 until they no longer receive an answer on the calculator with decimals, which may be interpreted from “point point point” and “just a bunch of points” (lines 8-9). While their calculators were not visible during this recording, I hypothesize that Oscar was showing Allen that the factors created a product of 440 on the calculator in line 23. Although how Oscar calculated the factors of 11 and 8 after finding a factor of five in line 18 is unclear, a possibility may be that dividing 440 by five resulted in 88, which to him may have been clearly divisible by 11, leaving a remaining factor of eight.
The students also begin to refer to five as the height in lines 20 and 21. While five was the first number stated in Oscar’s sequence (line 18), height is the last of the three dimensions provided on the instructional template. The students may have picked five for the height, because it was the closest of their new factors to four, the height of their original room. Also, because the boys were experienced Minecraft players, they may have recognized that using the smallest factor for the height would provide them with a larger floor area (CM 2/17/2018).
Regardless, there was limited discussion related to how a height of five was decided upon, as evident in the next excerpt, which occurs as the boys begin to work through the third question of the challenge.
Because Oscar and Allen quickly moved from articulating the numbers as part of the room’s volume to more abstract factors (lines 1-3), they were engaging in the SMP Reason
150 abstractly and quantitatively. The displayed the ability to “decontextualize—to abstract a given situation and represent it symbolically” (Common Core State Standards Initiative, 2018, p. 1).
They somewhat moved back to contextualizing when they assigned the factor of five as the height, but there was limited discussion around the connection, and an opportunity for the task to include additional structure that might initiate more robust conversations about the assignment of each factor to a dimension.
A greater emphasis on the assignment of each factor to a dimension may also enhance the SMP Construct viable arguments and critique the reasoning of others, because students would have to “compare the effectiveness of two [or more] plausible arguments” (Common
Core State Standards Initiative, 2018, p. 1) for why a height of five, for example, would be best for the new room in relation to the desired shape. While Oscar and Allen slightly engage in this
SMP as they work through the calculations together through vocally guessing and disagreeing with one another’s guesses (lines 12-16), they seem to be mostly in agreement rather than constructing arguments to convince one another of conflicting choices.
The third question asked them to draw area diagrams of a floor and wall in the new room, and label the dimensions.
1. A: So I already drawed it. Do the same thing exact thing as me… How many walls are there?
2. O: 4. 1, 2, 3, 4, yeah 4.
3. A: So on the bottom this way, it’s an 8.
4. O: No 5 height, 8 long, 11—
5. A: Oh so just like this.
151 6. O: No wait length is here, height is here…
7. A: Yeah so it’s sideways.
The students were now documenting how a floor and wall might look when they built them in the game, and Allen checked his work with Oscar as he sketched the wall’s height and length. The students still do not discuss how or why the height is five, and Allen seems to only modify the orientation of his drawing (Figure 23) upon disagreement with Oscar (lines 3-7), suggesting that he was in agreement about dimension assignments, but in disagreement about his representation as “sideways” in comparison to Oscar’s. As evidence, erasure marks in
Allen’s drawing seem to show an initial rectangle with “5H” labeled at the top, and “11L” labeled on the side above and to the left of rectangular prism. While this original representation was correct for a two-dimensional drawing of a wall as the challenge’s question asked, the drawing was oriented “sideways” (line 7) because it displayed the height as a horizontal line compared to vertical lines.
Allen’s drawing is not two-dimensional as the instructional template prompted.
However, the purpose of the task’s question was to support students in the contextualizing of their factors as dimensions. Thus, one may argue that the prompt achieved its purpose in this matter, but failed to inspire conversation of the SMP Construct viable arguments and critique the reasoning of others about why the height should be five. Despite the autonomy of choices provided for students to choose the factors that can be assigned to each dimension, the students did not engage in the opportunity to do so. However, the boys still slightly engaged in
Construct viable arguments and critique the reasoning of others as they examined and critiqued
152 each other’s work. Furthermore, the student’s ability to quickly draw and label a 3D object freehand shows that he was able to visualize the three-dimensional space of the game and represent it on paper, possibly a result of his engagement in building rooms for the task.
Figure 23. Allen’s 3D drawing of a rectangular prism, labeled with height, length, and width
Regardless of the students’ lack of engagement to the depth anticipated for the SMP
Construct viable arguments and critique the reasoning of others, the consideration of SDT’s relatedness may have been effective, because the students continued to work together throughout the entire task. After the boys completed the instructional template, they began to discuss plans for building the room in the game. At this point, the screencast showed their movement in the game, which is referred to between brackets in the third excerpt below.
153
1. O: Hey what color should we make our walls in our first house? Should we make it
2. cobblestone?
3. A: Yeah, yeah!
4. O: Yes, I’m in, I’m in! [He has just entered the virtual Minecraft world]
5. O: [Moves into an open underground space (Figure 24)] Allen! It’s all of this. We have to
6. fill in the bottom first.
7. A: So build that, Oscar. [Moves avatar towards ceiling] Wait, Oscar, are you using
8. Sandstone?
9. O: Yeah.
10. A: Okay. What should be the floor?
11. O: It’s up to you.
12. A: Oh, my dog! My little fluffy, very tiny dog… Oh there’s my dog!
13. O: Aww, he’s just like standing there, he’s standing there.
14. A: Are you going to get your own dog?
15. O: Maybe. [After building the roof, O returns to the inside of the new room]
16. A: Do you want to make the wall, Oscar?
17. O: [Examining A while he builds floor (Figure 25)] How do you know it’s five blocks tall,
18. Allen?
19. A: Don’t you see? See, it’s five blocks.
20. O: Wait, are you sure? [O measures the height in the game by building blocks vertically from
A’s floor to the ceiling]
154 21. O: One…two…three… it’s only three blocks tall! [O starts to break the “floor” blocks that A
has built. Then he places 5 blocks vertically from the ceiling]
Figure 24. Oscar moves his avatar to an open underground space near the team’s original room to show Allen where the new room will be located
Figure 25. Oscar questions Allen’s position of the floor he is building in relation to the ceiling
155 At this point in the dialogue, the students seemed set upon a height of five, and Allen’s first build of the floor that is only three blocks from Oscar’s ceiling does not seem to entail a misconception surrounding the use of one of their other factors of eight or 11 as a height.
Rather, Allen may have miscounted or perhaps counted a distance of five blocks from below his floor to where he was currently building. Another possibility is that Allen may have been counting both the ceiling block and the floor block as part of the height, when Owen as well as the researcher-teacher considered the height to be measured inside of the floor and ceiling.
Both boys seem comfortable with five blocks as the height (lines 17-18), as well as with the SMP
Reason abstractly and quantitatively as they moved back to contextualizing their abstract factors as dimensions within the game.
While the excerpt does not detail further the decision to make the height five meters, it shows the students’ integration of their fantasy world with the task as well as the SDT’s autonomy permitted in the freedom for material choices as they discussed via SDT’s relatedness, the “color” or material of the walls in the new room or “house” as well as a pet dog
(lines 1-2, 8, and 12-15). While Oscar made sure the height constraint was met, the students did not discuss the positioning of the room’s dimensions, or its size in comparison to the original room. Throughout the three excerpts, the students did not address the possibilities of using the factors of eight or 11 meters for the height, or more generally, the option of arranging the factors in a different sequence. These are modifications that will be explored later in the chapter.
Results through the Engagement Amplification Model. During the first iteration of
Same Volume Different Dimensions, students engaged in the SMP of (1) Make sense of
156 problems and persevere in solving them, (2) Reason abstractly and quantitatively, (3) Model with Mathematics, (4) Attend to Precision, and (5) Look for and express regularity in reasoning listed previously in Table 17. Table 19 shows the addition of the SMP Construct viable arguments and critique the reasoning of others with a brief description of how students engaged in this SMP.
Table 19
Additional SMP Noticed in the Same Volume Different Dimensions Challenge
Standard for Observed Evidence Mathematical Practice Construct viable Students worked together throughout the entire task to determine factors, arguments and critique the checking one another’s work, and correcting misconceptions when building reasoning of others the determined dimensions within the game (FN 2/7/2018, VR_I 2/7/2018, VR_OAA 2/7/2018, SC_O 2/7/2018)
Students’ engagement in these SMP might have been enhanced by the discussion within their groups, discussion that I aimed to encourage through the questions listed on the challenge’s instructional template. Furthermore, because the objective of the challenge resulted in a single product that could be intertwined with the students’ fantasy world, there might have been greater motivation to discuss the planning and properties of the product among group members, both aesthetically as well as mathematically.
While students engaged in a number of SMP, the depth in which they engaged in both
Reason abstractly and quantitatively and Construct viable arguments and critique the reasoning of others was not what I had intended during task design. Students had the freedom to choose their own dimensions (autonomy) and worked as a group to calculate these dimensions
(relatedness), but discussion regarding how and why particular factors would be selected for
157 each dimension was lacking. Students did not seem to consider their choices for the shape of the room, and how the shape might be altered dependent on the factor representing height.
Thus, engagement in Reason abstractly and quantitatively was deemed to be lower than intended due to the lack of articulation surrounding the application of factors as specific dimensions. Engagement in Construct viable arguments and critique the reasoning of others was also lower, because students did not articulate their choices for selecting a particular height, or debate the affects of the choices on the shape of the room.
Figure 26 depicts these results through the Engagement Amplification Model in which both game and mathematics engagement are high, but not as high as originally intended.
Students were still engaged with the game, but not to the degree anticipated because they did not immerse themselves in the dimensional aspects of the new room, especially in relation to how their new room would fit and look within their fantasy world. Students were also quite engaged with the mathematics, but did not discuss shape and size of the new room in relation to their factors. Consideration of SDT’s motivational factors autonomy and relatedness seemed to have their desired affect with the exception of the students’ lack of articulated engagement in choosing the height of their room, and discussing the choice. Therefore, the SDT dial is almost as high as anticipated, and still turned higher than that of the SMP dial that is turned slightly lower due to fewer opportunities for students to deeply engage in Reason abstractly and quantitatively and Construct viable arguments and critique the reasoning of others than anticipated.
158
Figure 26. Resulting student engagement after implementation of Same Volume Different
Dimensions when considering SDT’s relatedness and the SMP Construct viable arguments and critique the reasoning of others
This scenario provides evidence that a relationship between the SMP dial and game engagement may exist, contrary to previous speculation. Earlier in the chapter, a relationship between the SDT dial and game engagement was discussed such that the effect of the SDT dial on game engagement was less than its effect on mathematical engagement (Figure 20). This conclusion was drawn from the observation of how a slight turning of the SDT dial in the negative direction did not seem to have an affect on game engagement when Manny still engaged in building his room despite seeming to experience less motivation surrounding autonomy and competence than intended during engagement with the task.
However, in the most recent example of implementation of Same Volume Different
Dimensions, now taking into consideration relatedness and the additional SMP, game engagement has decreased. Figure 27 displays the changes observed between intentions of
159 Same Volume Different Dimensions to enhance SDT and SMP both with and without a focus on communication that involved SDT’s relatedness and the SMP of Construct viable arguments and critique the reasoning of others.
Figure 27. Comparison of resulting student engagement of the Same Volume Different
Dimensions challenge before and after a focus on communication between students
The SDT motivational factors seemed to have been reduced by a similar amount compared to the previous examination of the challenge in which the task did not seem to provide as much choice (autonomy) or feelings of effectiveness (competence) as intended in its design. Lack of student motivation in these regards was only deemed to be slight, similar to the slight lack of
160 autonomy and relatedness observed in this section’s examination of the same challenge with the same group of students. In contrast, the challenge’s opportunities for student engagement in SMP were evaluated to be much less present than intended compared to the original examination. A decrease in SMP was observed through lack of engagement in attending to the room’s shape as affected by its factored dimensions. Did the greater decrease in the SMP of focus cause a decrease in game engagement?
In an example discussed earlier in the chapter (Figure 28), a greater increase in SMP led to a decrease in game engagement, so the answer to this question is likely negative.
Figure 28. Comparison of resulting engagement with varying levels of access to the game and degrees of structure 161
Furthermore, reviewing an excerpt from the previously examined interview transcription with
Han and Isaiah may also support the likelihood that decreasing SMP within gameplay may not necessarily decrease game engagement.
1. RT: What would be the difference between working on something I give you, and free play?
2. H: Well free play is a bit more fun because you just get to go and do whatever—
3. I: But—
4. H: And there’s no worksheets—
According to Han, free play, which is when students have time after completion of a challenge to explore and build on their own without constraints of a structured activity, is “more fun” because of the freedom involved and because there are “no worksheets” (lines 2-4). Because there are no SMP involved in free play, one may conclude that a complete lack of SMP hardly decreases game engagement, but rather optimizes it. So what were the variables that occurred during implementation of Same Volume Different Dimensions that caused mathematics engagement to decrease when opportunities for SMP were less present in the task than intended?
Rather than conclude that the SMP dial directly negatively or positively affects game engagement, a more complex relationship may be explored through examining the connection between the SMP of focus and a description of game engagement. During previous evaluation of Same Volume Different Dimensions in which higher engagement in the game was observed,
162 observations and analysis focused on the SDT of autonomy and competence and the SMP of
Reason abstractly and quantitatively and Look for and express regularity in repeated reasoning.
Although opportunities for engaging in these SMP were not perfectly designed into the task
(the dial in the top display of Figure 27 is still turned slightly in the negative direction), game engagement was not affected, because students still engaged in these SMP as they built their product in the game, integrating them into engaged gameplay. For example, students moved between abilities to decontextualize and contextualize their factors (Reason abstractly…) and looked for shortcuts in their factoring process when dividing their volume by a variety of numbers (Look for and express regularity...). During their game engagement, students applied their newfound factors (Look for and express regularity…) as dimensions (Reason abstractly…), experiencing both autonomy in the selection of materials and placement of the new room as well as competence in its accuracy and completion.
When the analysis focus of Same Volume Different Dimensions encompassed communication through the SMP of Construct viable arguments and critique the reasoning of others, there seemed to be a slight decrease in game engagement in that students did not apply the task’s objectives that were affiliated with this additional SMP to the build of the new room in the game. While the students successfully created the new room, the product’s intended qualities of focus such as the effect of the dimensions on the room’s shape (Reason abstractly…) were not conjectured, justified, or communicated to others (Construct viable arguments…). Students still engaged with building their product in the game, but did not integrate team decisions regarding the shape and size of the new room within their fantasy world. Therefore, the SMP of Reason abstractly and quantitatively and Construct viable
163 arguments and critique the reasoning of others were not integrated into gameplay, because students did not articulate arguments or decisions to choose specific factors derived from their volume for each dimension before or during the build. Students still may have made decisions about factor assignment to specific dimensions before and during building (Reason abstractly…), which is why game engagement was not affected in the first scenario. However, their lack of articulation regarding these decisions (Construct viable arguments…) affects their game engagement in the second scenario, because the articulation was absent during gameplay.
One may also note that different SDT motivational factors were paired with the combinations of SMP described above. In the first scenario, autonomy and competence were the SDT aspects of focus through students’ attention to choices in factors and feelings of effectiveness when finding their factors. In the second scenario, SDT’s relatedness was connected to the SMP of Construct viable arguments and critique the reasoning of others, because teams were expected to interact and communicate with one another about their factor-dimension decisions incorporated by the SMP of Reason abstractly and quantitatively.
Thus, changes to the evolving Engagement Amplification Model are recognitions that (1) specific SMP of focus are encompassed by the SMP dial, (2) these SMP may be paired with a particular SDT such as relatedness to examine key motivational factors, and (3) a combination of the SDT and SMP dials may affect game engagement rather than the SDT dial alone. Figure
29 incorporates these changes into the relationships within the model.
164
Figure 29. Further modified relationships of SDT and SMP to engagement within the
Engagement Amplification Model
The arrow connecting the SDT dial, now specific to motivational factors that parallel the SMP of focus, has been removed and replaced with the solid arrow connecting the interdependent relationship between the SDT and SMP dials to game engagement. The SDT and SMP dial connection is now denoted with a dotted double-sided arrow to differentiate it from the arrows connecting specifically to engagement. Each dial is still connected to the mathematics engagement bar, because increasing the SDT and SMP present in a task still seems to positively affect mathematics engagement.
Workshop 2 Modification and Anticipation: Arranging dimensions. While Same
Volume Different Dimensions seemed to provoke a certain degree of mathematical conversation among groups as evidenced in the previous excerpts of Oscar and Allen’s work together, there was potential for the students to engage in deeper conversations pertaining to
165 the SMP of Reason abstractly and quantitatively and Construct viable arguments and critique the reasoning of others. Most students including Oscar and Allen did not seem to deliberate the possibility of rearranging the factors they had found to consider a variety of heights for their new room.
Targeting deeper communication. In an attempt to provide an opportunity for this deliberation, I added a question to the instructional template that asked students to list other arrangements of the newfound dimensions such that the factor originally listed for height might be listed for length or width instead (Question 3, Appendix D). Additionally, I added comparison questions to encourage students to think about the height and shape of their new room as a result of the new dimensions (Questions 5 and 6). Finally, as a result of a modification related to behavior from the first workshop, students in the second workshop were not provided graph paper workbooks, so the former question on the template regarding drawing areas on a grid remained on the template, but with the addition of grids (Question 4).
These modifications aimed to incorporate SDT’s motivational factors of relatedness and autonomy a second time, but strengthen their presence in the task as compared to the first iteration. Relatedness was considered in the additional questions to provide an opportunity for students to discuss mathematical relationships between dimensions and volume. For example, the question, “How does the rest of the room’s shape compare to your original room?” might inspire a conversation about how one room is flatter or narrower than the other due to the factor assigned to height. Through these additional questions, students might recognize a variety of options for choosing the final shape of their room, thus experiencing autonomy when selecting a final arrangement of dimensions for their room. These additional questions also
166 aimed to strengthen the incorporation of the SMP Reason abstractly and quantitatively through students having to consider how abstract numbers might affect the shape of their room, and possibly engage in a conversation about choosing the best arrangement for the shape of the room they wanted in their fantasy world through Construct viable arguments and critique reasoning of others. Table 20 outlines the problems of engagement observed in the first workshop, and the modifications implemented in consideration of specific SDT motivational factors and SMP for the second workshop.
Table 20
Modification and Anticipation of Same Volume Different Dimensions: Workshop 2
Problem of Engagement Modification Applied SDT Intended SMP Lack of conversation about Added question asking • Relatedness • Reason abstractly and possible arrangements of students to show multiple • Autonomy quantitatively factors as dimensions factor arrangements to encourage discussion of • Construct viable possibilities arguments and critique reasoning of others Lack of conversation about Added two questions to • Relatedness the affect of the dimension encourage discussion of arrangement on shape of comparing height and the room shape of room with original room
Predictions through the Engagement Amplification Model. Problems of engagement noticed during the first iteration of Same Volume Different Dimensions revolved around lack of communication and mathematical conversation about choosing how factors could be assigned to specific dimensions of the new room. To address the problems, I modified the challenge for the second workshop to include questions that encouraged discussion of comparing height and the shape of the room. Figure 30 shows anticipated engagement as a result of intending to
167 increase the SDT and SMP dials. In accordance with the updated model discussed previously, there are now SDT motivational factors of focus, autonomy and relatedness that are paired through their interdependency with the SMP of Reason abstractly and quantitatively and
Construct viable arguments and critique reasoning of others. Expected engagement is higher than that of the first iteration in anticipation of the modifications having their intended affect on increasing students’ mathematical discussion and decision-making processes in relation to their build within their fantasy world.
Figure 30. Predicted student engagement after modifications made to the challenge Same
Volume Different Dimensions for Workshop 2
Workshop 2 Implementation and Evaluation: “We’re not building a freakin’ skyscraper!” In general, students successfully calculated their factors by using a similar process to that of the students in the first workshop. However, compared to the first workshop, students in the second workshop seemed to require more guidance and support from me, as
168 evidenced by questions about having to do the third question (FN 3/5/2018) or more generally,
“what do we do on the back of the paper?” (AR_WG 3/5/2018) in reference to the last three questions. However, once I verbally explained the directions for each of these questions, the students were able to work together to find the answers they needed, and successfully build their new rooms in the game.
Wesley and Graham’s non-skyscraper. Wesley and Graham had started with a volume of 385 cubic meters and calculated new factors of five, seven, and 11. While working on the question regarding the arrangement of factors and their assignments to specific dimensions
(Question 3, Appendix D), Wesley (W) began to explain to his group member Graham (G) why he wanted the height of their new room to be five while Graham interjected ideas as well. As the researcher, I saw an opportunity to record evidence of a possible mathematics discussion, and as the teacher I saw an opportunity to extract and push for articulation of their mathematical thinking.
1. RT: So tell me again why you want the height to be five?
2. W: Umm… well because you don’t want it so high but then the floor is like skinny because
3. it’ll just look like a, like a skyscraper.
4. RT: Which height would make it look like a skyscraper?
5. G: Um 11, 11, this one [points to 11 on his instructional template]
6. RT: Mmm-hmm.
7. G: Just imagine 11 blocks, 5 and 7: that’d be a really tight fit.
8. RT: So—
169 9. G: Unless you, but, it could fit a billion stories so I'm not sure if that could also work—
10. W: A billion?
11. G: No. It won’t work. Because that’s not the room, we’re not building a freakin’ skyscraper!
Wesley showed evidence of imagining the dimensions’ affects on the size of a room or rectangular prism when he described a tall height and “skinny” floor (lines 2-3). Skinny might have referred to a smaller floor area due to the use of smaller numbers for the length and width, thus causing the shape of the room to be tall and skinny rather than short and wide.
Graham seemed able to recognize that of their three factors, the largest factor of 11 would encompass Wesley’s skyscraper description. He asked that one “imagine” the orientation of the blocks (lines 5 and 7), suggesting that he may have been visualizing and contextualizing the numbers as well, considering the affects of the dimensions on the size of the room. A “tight fit”
(line 7) may also parallel Wesley’s description of “skinny” in that the area of the floor would be much smaller or tighter than a floor area that included 11 as its length. A smaller area would allow less space for furniture, decorations, or other items placed within the room such that to fit all items may insinuate a “tight fit.”
Then, Graham then seemed to infer that “a billion stories” (line 9) could fit within a room that had this taller height of 11, perhaps visualizing how a single room could become multiple rooms with the inclusion of stories. While “a billion” stories would not fit within a height of 11 blocks due to the constraints of the Minecraft world, Graham’s consideration of breaking up the room into multiple stories to increase available floor area shows that he may have been thinking of a way to increase the space and utility of an otherwise small floor area
170 with a high ceiling. However, he seemed to articulate uncertainty (“I’m not sure”) regarding this idea at the end of line 9, and Wesley responded by questioning his use of “billion” (line 10).
Graham then quickly dismisses his idea in a way that seems to imply that his idea is incorrect and perhaps unimportant, because “we’re not building a freakin’ skyscraper” (line 11). Another possibility is that Graham may have been embarrassed that Wesley seemed to disapprove of his answer, and was attempting to revert back to the topic of what the students would be creating instead of what they would not be creating.
Results through the Engagement Amplification Model. My modifications of this challenge had aimed to provide an opportunity for students to think about the height and shape of their new room in consideration of their new dimensions, hopefully through discussing ideas with one another to decide how their factors would be assigned to each dimension.
While the students in this workshop needed further explanation for some questions, reasoning evidenced in the dialogue with Wesley and Graham may suggest that the modifications may have helped initiate discussions about the different heights possible for the room, and how the heights would affect the rooms’ shapes. Therefore, opportunities for the two SMP of focus
(Reason abstractly… and Construct viable arguments…) may have been enhanced, as students seemed to consider the affects of different heights, and in Graham’s case, even consider how floor area might be increased within a room by adding multiple stories to create additional rooms within a room. Table 21 summarizes SMP observed during the second implementation of Same Volume Different Dimensions.
171
Table 21
SMP Noticed in the Same Volume Different Dimensions Challenge, Workshop 2
Standard for Observed Evidence Mathematical Practice Making sense of problems While students initially struggled in understanding certain objectives of the and persevere in solving challenge, they took the appropriate steps to factor and check their work them within their teams (FN 3/5/2018, AR_TC 3/5/2018)
Reason abstractly and Students decontextualized the volume to find three factors for the number, quantitatively then contextualized the factors as dimensions, considering how choosing specific factors to assign to the room’s height might affect the room’s shape (FN 3/5/2018, VR_LEM 3/7/2018)
Construct viable Students worked together throughout the entire task to determine factors, arguments and critique checking one another’s work, and correcting misconceptions when building reasoning of others the determined dimensions within the game. Students also discussed within their teams which factor they should use as the height for their new room based on the building’s shape in the game (AR_WG 3/5/2018, AR_TC 3/5/2018, FN 3/5/2018)
Model with Mathematics Students related the factors of a product to dimensions of a room’s volume and created a model in the game world (AR_WG 3/5/2018, AR_TC 3/5/2018, FN_3/5/2018, FN 3/7/2018)
Attend to precision Students used multiplication to check the accuracy of the factors they had found by using division (FN 3/5/2018)
Look for and express Students moved back and forth between using multiplication and division to regularity in repeated find various dimensions of a room’s volume; some students invented a reasoning general procedure for factoring their volume; all students used multiplication and division to find factors for a product (FN 3/5/2018)
In this second iteration of Same Volume Different Dimensions, engagement in Reason abstractly and quantitatively was evidenced not only by the students’ attention to the factors as representative of varying dimensions, but also by how they articulated that the different heights would create differently shaped rooms with more or less floor space. This engagement was connected to the SDT factor of autonomy, because students seemed to recognize through
172 their articulation, that they had a choice to make about the shape of their room. While Wesley and Graham chose the smallest factor of five blocks for their height, another group chose their largest factor for their new room’s height, because they wanted to add a tall tower to their fantasy world (FN 3/5/2018, VR_LEM 3/7/2018). Therefore, as shown in Figure 31, engagement in both the game and mathematics was present, because students related their reasoning and mathematical findings from the task back to the game as they selected a specific height for a room imagined within their fantasy world. Both SDT factors of autonomy and relatedness were evaluated as high, because the students seemed motivated to communicate, and make choices as a group about the height of their rooms in relation to how the room might be situated within their fantasy world. Thus, the SDT dial is turned to the right on a setting of high as originally anticipated.
However, student engagement in Construct viable arguments and critique reasoning of others, while greater than in the first workshop, was still not as high as anticipated. As depicted in the interaction between Wesley and Graham, the students articulated the affect of selecting particular heights for their room, but did not construct arguments or critique one another’s reasoning beyond Wesley’s questioning of Graham’s “billion.” Perhaps a question asking the students to state why they had decided upon a specific height, or argue in favor of a particular height as compared to another, may have elicited a more robust interaction between students.
Thus, the SMP dial is turned farther to the right as an improvement from the first workshop, but is not turned as far as expected. Similarly, the mathematics engagement volume bar also increased, but is not as high as originally anticipated.
173
Figure 31. Resulting student engagement after implementation of Same Volume Different
Dimensions when considering SDT’s relatedness and the SMP Construct viable arguments and critique the reasoning of others
Although my modification goals for the challenge had been to increase both game and mathematical engagement such that they were at equally high levels, I perceived mathematics engagement to fall short of my aspirations due to lack of perceived evidence of students mathematically critiquing one another’s arguments. Through modifying the task in certain ways, I had anticipated that I was turning the SDT and SMP dials by the same amount, when the
SMP dial was not turned to the positive direction as much as the SDT dial. Therefore, despite intentions, the SMP dial may be less “sensitive,” requiring greater evidence to show whether the opportunities for students to engage in specific SMP were actually present in the task. In contrast, the SDT dial is affected by evidence showing whether students were engaged by certain motivational factors designed into the task, and this evidence may be easier to observe
174 or collect due to the variety of student actions that may constitute a depiction of autonomy, competence, or relatedness. For example, in order for students to engage in the SMP of
Construct viable arguments and critique the reasoning of others, I expected students to debate their choices and argue why other choices may not work. However, by simply articulating a choice about the dimensions, or communicating within the team about the choice, I considered students to be engaging in the task’s opportunities for autonomy and relatedness. Figure 32 shows a final addition to the relationships within the Engagement Amplification Model in which the SDT dial is “more easily turned” than the SMP dial, represented by its smaller size. The SMP dial requires greater evidence to turn, so it is larger, metaphorically requiring more strength for its affects to be observed.
Figure 32. Modified relationship of SDT dial in comparison to SMP to dial within the
Engagement Amplification Model
175 Finally, one may note that when the Engagement Amplification Models were shown to predict and evaluate student engagement before and after implementation of a challenge in this chapter, the dials as well as the bars of student engagement were never set to maximum.
This lack of maximization is due to my beliefs during this study that my tasks and challenges can always be improved upon to include even greater opportunities and displays of SDT and SMP.
For example, even though I considered my implementations of Build the Room’s Dimensions to be successful, I would continue to consider ways of improving the task to engage students in future studies, especially because the demographics and personalities of the new group of students may be different. Perhaps as further research is conducted, and quantitative measures surrounding engagement are conceptualized in relation to the dials and engagement bars, a more concrete idea of what a task optimized for engagement may look like will be theorized. These ideas may be implications for future studies and will be discussed in the next chapter. However, before implications for future studies and practice are presented, the research questions of the study are revisited.
Research Questions
In summary, the study’s research questions were examined through two themes: (1)
Intertwining the game with task objectives, and (2) incorporating opportunities for student interactions inside and outside of the game. A design cycle was used to trace the process of the researcher-teacher as challenges of the overarching Build the Room task were designed and modified through two workshops. Through examination of the process, the Engagement
Amplification Model was developed to depict the relationships between student engagement in
176 the game and mathematics with motivational factors framed by SDT and intended opportunities for students to engage in SMP.
Question 1: Through rich task design, what are some ways that an educational video game can be used to engage students in mathematical practices? An instructor must use educational video games thoughtfully and with intention if the goal is to engage students in targeted mathematical learning objectives and practices. This has been shown in previous research (Hanghøj, et al., 2014) and was further exemplified in this study through the various iterations of the task in how it was designed and modified in relation to the game. Through the use of the Engagement Amplification Model, a researcher and/or teacher can gauge the engagement that his or her task design may elicit by considering the interdependency of SDT motivational factors and SMP.
One way that an educational video game can be used to engage students in mathematical practices is for the task to intertwine its objectives with existing student enjoyment in the game (the first theme). Because a major appeal of Minecraft is its capacity for exploration (Nebel, et al., 2016; Starkey, 2016), integrating opportunities for student choice or autonomy within a task may parallel existing student enjoyment in the game. Establishing a balance of autonomy such that students may make choices within their mathematical exploration as well as within the presence of their fantasy world combines motivational factors of SDT as well as SMP. For example, during Build the Room’s Dimensions, students realized on their own, that they needed to convert their measurements from feet to meters, as they began to model their measurements in the game to achieve the task objective. The task was structured such that students had access to the game shorty after completing their
177 measurements, which allowed them the freedom to choose how and when to replicate their measurements in the game. In this way, autonomy and the SMP of Model with mathematics were intertwined to leverage one another, such that students may have experienced competence through feelings of effectiveness after observing that their converted measurements helped them achieve the challenge’s objective.
Another way that an educational video game can be used to engage students in mathematical practices is for the task to provide opportunities for student interactions both inside and outside of the game (the second theme). Not only does this strategy apply the SDT motivational factor of relatedness, but also it incorporates theories that students may learn more when they communicate their ideas, and engage in conversations with their peers
(Carpenter, et al., 1999; Cohen, 1994; Henningsen & Stein, 1997). Thus, a strategy for embedding a rich mathematical task within Minecraft may be to provide opportunities for discussion within groups through question prompts that aim to address gaps noticed in previous iterations of the task, or those anticipated based on assessed current student knowledge.
As shown earlier within the Engagement Amplification Model, providing opportunities for discussion within groups while working on a rich task in the game combines SDT motivational factors and SMP. For example, the SDT motivational factor of autonomy is incorporated in the SMP Construct viable arguments and critique reasoning of others, because students decide together through mathematical reasoning, which combinations of dimensions might give them a “skyscraper” if that is their goal. Thus, the SMP Reason abstractly and quantitatively is simultaneously included, because students are contextualizing and
178 decontextualizing room’s dimensions in the game while thinking about how those dimensions affect the shape and size of the room. As they discuss their modeling, they are engaged through the motivational factor or relatedness, because they are reasoning through a decision of what to add to their fantasy world in the game. By combining SDT motivational factors with opportunities for students to engage in SMP through peer interactions, an educational video game can be used to engage students in mathematical practices.
Question 2: How does students’ engagement inform the process of rich task design in
Minecraft? The Engagement Amplification Model can be used to inform the process of rich task design in Minecraft, and perhaps in other educational video games as well, through examining and predicting student engagement. The model divides student engagement into four components: (1) Engagement propelled by SDT motivational factors, (2) opportunities for engagement in SMP, (3) observed engagement in the game, and (4) observed engagement in mathematics. Targeted goals surrounding SMP (2) are paired with complimentary SDT motivational factors (1) in order to predict resulting student engagement in both the game (3) as well as in mathematics (4).
For example, during the design of the challenge Same Volume Different Dimensions, specific SMP of Reason abstractly and quantitatively and Construct viable arguments and critique the reasoning of others were considered as part of the targeted learning of the task.
The SMP were paired with SDT motivational factors of autonomy and relatedness that seemed to complement their application as they were incorporated into the task. As a result, the structure of the task aimed to provide opportunities for student choice and interactions both
179 inside and outside of the game while addressing mathematical ideas of contextualization, de- contextualization, and discussions around mathematical decisions.
Question 3: How does students’ engagement inform the process of rich task modification in Minecraft? The Engagement Amplification Model was not only used to design and predict student engagement of a task, but also to evaluate how well a task might have engaged students during its implementation in order to inform task modification for future implementation. After implementation, observations of engagement in both the game and mathematics were evaluated and rated within the “volume bars” of the model. Then, assessments of how well the task incorporated the SDT and targeted SMP were analyzed to determine the positioning of each of the SDT and SMP “dials” of the model. Problems of engagement were determined based on these evaluations, and modifications were considered with a goal of further increasing the engagement bars as well as continuing to turn each dial further in the positive direction. Therefore, through categorizing observations of student engagement into the four interrelated components within the Engagement Amplification
Model, modifications were made with the intention of optimizing engagement through careful consideration of the categories’ relationships within the model.
180 Chapter 6: Discussion
This study aimed to investigate the design and modification of a rich mathematical task,
Build the Room, which was embedded in the educational video game, Minecraft. Through systematic analysis of my observations of students’ engagement, I further developed and modified Build a Room with targeting mathematical learning goals encompassing the Standards of Mathematical Practice (SMP) (Common Core State Standards Initiative, 2018). Self-
Determination Theory (STP) in video games (Przyblyski, et al., 2010; Rigby and Ryan, 2011; Ryan et al., 2006) supported my framing of student engagement through motivational factors of autonomy, competence, and relatedness. Results included multiple modifications with emergent themes of intertwining the game with task objectives as well as incorporating opportunities for student interactions with one another inside and outside the game. This final chapter will first discuss the Engagement Amplification Model, a model for embedding educational tasks in video games such as Minecraft based on student engagement. Then, implications for practice and future studies will be presented before addressing limitations of the study.
The Engagement Amplification Model: Using Student Engagement to Inform Task Design
The Engagement Amplification Model frames the relationship between four interrelated components of student engagement to inform rich mathematics task design and modification within an educational video game. The model presents ideas for how the process of task design can be conceptualized, drawing attention to both the consideration and evaluation of student engagement in motivation, mathematics, and the game. While studies have explored the use of an educational video game such as Minecraft in relation to student learning outcomes 181 (Stanton, 2017) or student engagement (Cipollone, 2015), none have focused upon the process of the educator’s task design evidenced by student engagement, and how that design process may be framed to depict relationships between motivational factors and targeted learning goals. This section describes the model and its relationships in detail.
Defining the four engagement components. The model (Figure 33) categorizes student engagement into two major groups of (1) two dials that can be adjusted and manipulated by task design and modification, and (2) two volume bars depicting levels of student engagement, and that are affected by the turning of the dials.
Figure 33. The Engagement Amplification Model with two dials of SDT and SMP, and two volume bars of game and mathematical engagement
The dial labeled “SDT” represents the degree to which engagement is propelled by SDT motivational factors of autonomy, competence, and relatedness within the design of the task.
In anticipation of a task’s implementation, the SDT dial may be turned on a spectrum where the 182 left-most setting represents a lack of SDT motivational factors considered during task design and embedded into the task, and the right-most setting represents a complete consideration of
SDT factors. After a task’s implementation, the SDT dial may be adjusted in comparison to its anticipated setting to reflect the observed affect of the motivational factors that were intended in the task’s design. For example, autonomy may have been considered during task design, but during implementation, students may have not engaged in recognizing their choices and making decisions as expected. Evaluation of the degree to which SDT was present during task implementation can be used to inform modifications of the task for future implementation.
The dial labeled “SMP of focus” represents the degree to which opportunities for specific SMP are considered and present in the design of the task. The SMP dial does not represent all eight of the SMP that may be incorporated in a problem, but rather focuses on one or two for analyses. Before a task’s implementation, the dial may be turned within a similar spectrum to that of the SDT dial to anticipate the opportunities integrated into the task for students to engage in intended SMP. After a task’s implementation, the dial may be adjusted to evaluate observed student engagement in these particular SMP, reliant upon specific evidence of student communication and action that parallel the definition of each SMP.
Similar to the SDT dial, the evaluation of the SMP dial can also be used to inform modifications of the task to increase or refine opportunities for student engagement in the SMP of focus.
Additionally, the SMP dial is slightly larger than the SDT dial due to its reliance upon specific evidence in accordance with definitions of focused SMP. Thus, the dial is larger, because it requires a greater strength in evidence to turn it compared to the evidence required to turn the
SDT dial.
183 While the dials may be manipulated by the task designer as a prediction of the task’s inclusion of SDT motivational factors and SMP, the levels of the volume bars are not manipulated by the task designer. Rather, they are an anticipated or evaluated result of the positioning of the dials. For example, the game engagement volume bar moves upward or downward, denoting how engaged students are observed to be with the game as a result of the positioning of both the SDT and SMP dials. As developed in the previous chapter, the game engagement bar is also a reflection not only of how engaged students are in the game itself, but the degree to which they apply the task to the game, embedding the task’s product into their fantasy worlds.
The mathematical engagement volume bar has a similar relationship to the dials compared to that of the game engagement bar, but instead shows the degree of students’ observed mathematical engagement, which includes SMP as well as more focused mathematics concepts such as measurement, proportional reasoning, and volume. For example, observations may entail that students are highly engaged in mathematics due to their engagement in measurement activities, but not engaged in a specific SMP of focus due to the structure of the task. In this case, the SMP dial may be adjusted to a lower setting after implementation while the mathematics engagement volume bar may still reflect a medium degree of mathematics engagement. However, because the mathematical engagement bar incorporates SMP in its display of amplification, the bar would not be fully optimized without evidence of student engagement in the SMP of focus for that particular evaluation.
Relationships within the model. As explored throughout the previous chapter, the
Engagement Amplification Model includes a series of relationships within its four components
184 of engagement. Figure 34 depicts the relationship between the SDT and SMP dials as well as the relationships between each engagement bar with the dials.
Figure 34. Relationships within the Engagement Amplification Model between the four components of student engagement
The SDT and SMP dials are connected with a dotted arrow, because they are interdependent upon one another. The arrow is dotted to differentiate it from the arrows that connect the dials to the engagement volume bars. The arrow is double-sided to insinuate that one dial does not directly affect the other, but rather show that both dials have an ability to affect the position or adjustment of the other. Furthermore, because the SMP dial only represents one or two SMP of focus, one or two specific SDT motivational factors are selected that correspond with the SMP and vice versa. Table 22 suggests SDT motivational factors that may correlate to specific SMP. However, corresponding motivational factors are merely suggestions; depending
185 on the details of the mathematics and task involved, certain SDT factors may correspond more directly to the selected SMP than others.
Table 22
SMP and Their Corresponding SDT Motivational Factors of Focus
Standard for Mathematical Practice SDT Motivational Factors 1. Make sense of problems and persevere Autonomy in solving them Competence 2. Reason abstractly and quantitatively Autonomy Competence Relatedness 3. Construct viable arguments and Competence critique the reasoning of others Relatedness 4. Model with mathematics Autonomy Competence Relatedness 5. Use appropriate tools strategically Autonomy Competence Relatedness 6. Attend to precision Competence Relatedness 7. Look for and make use of structure Autonomy Competence 8. Look for and express regularity in Autonomy repeated reasoning Competence
For example, if an SMP of focus was Make sense of problems and persevere in solving them, then the motivational factors of autonomy and competence may be paired with this SMP to focus on opportunities for providing student choice (autonomy) within problem-solving procedures and feelings of effectiveness (competence) throughout the procedure to motivate perseverance. In turn, these SDT factors would also drive the structure of the task in incorporating that particular SMP. Autonomy might be encompassed in the provision of student choices in relation to the selection of mathematical strategies and representations 186 provided throughout the task’s structure. Student choices could also be incorporated within the design of the task’s product in the game, a product that would support the demonstration of the SMP and other targeted mathematical objectives. Thus, by increasing the autonomy involved within the task’s problem-solving opportunities, the SMP of Make sense of problems and persevere in solving them may also be increased to allow students to better understand the problem they are solving and choose the appropriate strategies to solve the problem.
This interdependency between SDT and SMP seems to have an ability to affect game engagement, especially since game engagement includes within its definition, the cohesion of the task’s product with the game. If (1) a task had successfully engaged students in the targeted SMP through the consideration and application of SDT motivational factors, and (2) the students had used their immersion in the SMP of focus to make decisions about their in- game product as a result of the task, then (3) game engagement would be high as a result of the SDT and SMP dial optimization. However, if students had not applied their engagement in
SMP to their in-game product as in the first iteration of Same Volume Different Dimensions examined in the previous chapter, then engagement in that SMP would (1) be evaluated as lacking due to a disconnection between the SMP and task product, and (2) result in a lower setting of the game engagement volume bar, since students would not be applying the task to the game, but rather interacting separately with the game such that the mathematical targets of the task were not incorporated.
The SDT and SMP dials also affect mathematical engagement, but in contrast to game engagement, the turning of one dial without the turning of the other may still increase or decrease mathematical engagement, hence the separate arrows. For example, turning the SMP
187 dial alone in the positive direction by increasing specific SMP may lead to a task that engages students in that particular SMP or mathematical concepts, despite a lack of autonomy, competence, or relatedness considered during task design. However, this type of task may fail to engage students in the game due to (1) lack of student choices able to be made within the game that connect to the task, (2) lack of feelings of effectiveness experienced within the game that are related to task completion, or (3) lack of feelings of connectedness to others within the game while working through the task due to its disconnection with the game.
Turning the SDT dial without turning the SMP dial may also affect mathematical engagement without affecting game engagement. For example, autonomy, competence, and relatedness might be considered in task design while opportunities for engaging in SMP of focus may be lacking. In this case, students may be motivated to engage in specific mathematics practices, although they may not necessarily encompass SMP, but rather focus on mathematics concepts such as measurement or proportional reasoning. Thus, students would still be mathematically engaged, perhaps because the task provided opportunities for choice, feelings of effectiveness, and a connection to others despite these motivational factors’ disconnection to a particular SMP. Game engagement may not be affected if again, the mathematical task objective is not connected to the in-game product. Therefore, the SDT and SMP dials can each individually affect mathematical engagement without reliance upon the other, but both dials must act in unison to affect game engagement due to the interdependent nature of intertwining a task objective with the game.
188 Implications for Practice
Implications for use in the educational field include a variety of applications. The model can provide a framework for educators and practitioners who are interested in designing rich mathematics tasks embedded in educational video games. The model provides a starting point for considering student engagement in four interrelated components during task design.
Beyond initial design, the model promotes an ability to evaluate a task and inform modifications, which encourage a practitioner not only to reflect on implementation, but also to modify the task based on documented evidence and analysis.
The model shows strong interdependence between SDT motivational factors and SMP, a relationship that has the potential to strengthen engagement in mathematics practices as well as application of the practices to an educational video game. Without the use of motivational factors to frame engagement, SMP might not connect as readily and noticeably to the game for students and vice versa. For example, a task that asks students to Model with mathematics by creating a scaled model of the Statue of Liberty may only be engaging for students who would be excited to model this particular monument in the game. By applying the SDT motivational factor of autonomy, students might be more motivated and thus more engaged if they choose their own monument to build, or simply use the dimensions of an existing monument to create their own monument with their team.
Furthermore, SDT motivational factors may rely on SMP as students work through a task. For example, if an objective of the aforementioned monument task is to model the monument in the game, the students must take the game’s height constraints into account before building the monument; students may need to scale their monument so that it will fit in
189 a particular location of the game, which may lead to them discovering that they need to use a constant scale factor and divide all dimensions before building the monument in the game.
Therefore, the model provides a helpful guide for practitioners in its ability to encourage the autonomy of student choice within a task (as well as other motivational factors) and link them to SMP so that students are engaged in both the mathematics as well as the game.
Finally, the model encourages practitioners to take into account students’ interactions with one another both inside and outside the game to inform task design. Providing students with opportunities to collaborate around a shared objective that combines student interest in the game with mathematical practices may provoke mathematical conversation as a by- product. Students deepen their mathematical thinking when they discuss solution paths of a problem with others (Cohen, 1994; Stein, et al. 1996). Thus, providing opportunities for mathematical discussion that relate to a meaningful product in students’ in-game fantasy worlds may deepen students’ mathematical thinking and increase ownership of the task objective. Then, as students have ownership of the problem, they will be more engaged, leading to better understanding (Carpenter, et al., 1999; Cahnmann & Remillard, 2002; Smith &
Corbin, 2014; Stein, et al., 1996).
Implications for Future Studies
In this study, I attempted to address a current gap in mathematics education research that pertains to the use of educational video games. This gap includes lack of empirical evidence (Connolly, et al., 2012; Eichenbaum, et al., 2014; Mayer, 2015; Zimmer, 2016) as well as a lack of studies that show specifically how a game can be designed for learning through the observation and analysis of student engagement. Without empirically-based theories for how 190 educational video games may be used with thoughtful intention to address specific learning targets, the potential of educational video game use will continue to be questioned due to an already existing saturation of educational technology tools that claim to impact student learning (Zimmer, 2016). While my research aims to address this problem, additional research is needed to provide a greater collection of evidence to validate and evolve the proposed model of this study.
For example, studies that apply the Engagement Amplification Model in alternative contexts may be helpful for testing the validity and reliability of the model. The design and modification of the overarching Build the Room task were imagined and implemented to provide greater opportunities for students to engage in particular SMP while taking into account qualities of rich mathematical tasks proposed by the research of Stein et al. (1996),
Pape, Bell, and Yetkin (2003), and Henningsen and Stein (1997) among others. A rich mathematical task aims to achieve the higher levels of the Cognitive Rigor Matrix, which have also been shown to engage students mathematically (Hess, et al., 2009). However, the
Cognitive Rigor Matrix is not applicable only to SMP or mathematics in general. Studies may be conducted that replace SMP with particular mathematics concepts such a measurement, volume, or specific Common Core Mathematics Standards, which may change the SMP dial to the mathematics concept dial (Figure 35) without detracting from the definition of a rich mathematics task.
191
Figure 35. The Engagement Amplification Model with a dial for specific mathematics concepts instead of specific SMP
Stepping away from mathematics as the sole focus, a rich task that aims to address higher levels of the Cognitive Rigor Matrix may also explore concepts and practices within the core disciplines of social studies, language arts, and science. In these contexts, the mathematical engagement bar would be changed to engagement for discipline of focus, and the SMP dial would become a dial adjusted according to the opportunities available within the task for students to engage in particular learning targets, such as standards, concepts, or practices of that discipline (Figure 36).
192 Figure 36. The Engagement Amplification Model with a dial for an alternative discipline’s concepts or practices, and an engagement volume bar for that discipline
Furthermore, one may wish to consider multiple disciplines through a task that revolves around a transdisciplinary problem. As shown in Figure 37, this application may increase the number of engagement bars or dials if one was to separate their evaluations and analysis of each discipline. Figure 38 shows a case in which the transdisciplinary content is merged into a single engagement bar and dial. Regardless, the model must consider targeted learning objectives as part of the task; content is inserted into the task thoughtfully, with intention, and in parallel to
SDT motivational factors.
193
Figure 37. The Engagement Amplification Model with a dial for transdisciplinary concepts or practices, and volume bars for engagement in multiple disciplines
194 Figure 38. The Engagement Amplification Model with a dial for transdisciplinary concepts or practices, and two volume bars for game and transdisciplinary engagement
Future studies may also incorporate the use of the model with learning outcomes that can be measured through the use of pre- and post-assessments. A measurement of learning outcomes may then lead to more quantifiable degrees of measurement within the model such that the dials may be turned on a scale of one to ten. Similarly in future studies, quantifiable measures may be assigned to engagement levels through the addition of student surveys and a numerical rating system that corroborates these surveys with researcher observations of engagement. Eventually, larger-scaled studies such as that of Stanton (2017) might test student learning outcomes as well as engagement ratings based on the implementations of rich mathematical tasks designed with the Engagement Amplification Model.
In addition, research regarding the use of the model outside of gaming environments may be useful to mathematics educators. Since mathematics teachers sometimes struggle with engaging students in tasks, perhaps the implementation and analysis of the interdependence
195 between SMP and SDT can support evaluation of task implementation. The interdependence between SMP and SDT may also inform modifications to that task.
Finally, this study depicts a model that was derived from a particular group of students within a specific after-school program. However, the classroom environment is one of an ecology that contains a complex nature of interacting systems (Cobb, et al., 2003). Therefore, the importance of exploring and examining a wide variety of contexts, not only in academic content, but also in relation to varying groups of students both socially and demographically must be underscored to test the reliability of the model. Thus, further studies are needed that examine groups of students in other geographic areas, that contain both similar and different demographic characteristics, and that both include and extend beyond the after-school program of Minecraft Mathematics.
Limitations of the Study
Limitations of this study include the homogeneity of the student population of the workshops, the extracurricular setting of Minecraft Mathematics, and the amount of time that the researcher was able to dedicate to documentation during implementation. As previously mentioned, further studies are needed to test the model on less homogeneous groups, or homogeneous groups with student demographic characteristics that are different from those in this study. Because my hope is that the Engagement Amplification Model might be used by K-
12 educators as well as educators involved in extra-curricular enrichment programs, an additional limitation of the study is its setting outside of a school classroom. Finally, while methodology and instrumentation were carefully considered for the dual role of the researcher-teacher, a limitation of the study was that the researcher-teacher might have been 196 unable to spend as much time collecting and documenting data during implementation compared to one with the sole role of researcher. However, the assistance of the technology of audio/video recordings and screencasts assisted with much of this data collection, as it allowed additional observations to be collected after implementation.
Conclusion
Exploration of the use of educational video games as a tool for learning may help educators make learning more accessible to a greater population of students. Video games that include a variety of problem-solving and collaborative opportunities may not only utilize
SDT motivational factors, but increase engagement in academic subject matter. Renowned educator and literacy expert James Paul Gee has observed the potential of video games for educational purposes as well in noting “wouldn’t it be great if kids were willing to put in this much time on task on challenging material in school and enjoy it so much?” (Gee, 2007, p. 2).
Gaming experts as well have questioned how motivation experienced during video game play might be harnessed to better society in asking, “What drives us to collect coins, snipe aliens, or scale the walls of ancient tombs until three in the morning? …Unlocking the mystery behind this desire may do more than help us understand our obsession – it could reshape and improve our society in powerful ways.” (Reeves, 2012, p. 34).
This study aims to capture the engagement involved in an educational video game, and argue how this engagement might be leveraged to enhance engagement in mathematical practices. An increase in relevance and engagement for students during learning has the potential to lead to a greater understanding of targeted learning objectives (Carpenter, et al.,
197 1999; Smith & Corbin, 2014). A greater understanding and appreciation for learning may cultivate a population of life-long learners and inspired problem solvers.
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209 Appendix A: Build the Room Challenge #1 template from workshop 2, Build the Room’s Dimensions
210 Appendix B: Build the Room Challenge #2 template from workshop 1, Document a Craft
211 Appendix C: Build the Room Challenge #5 template from workshop 1, Same Volume Different
Dimensions
212 Appendix D: Build the Room Challenge #4 template from workshop 2, Same Volume Different Dimensions
213 Appendix E: Day-by-Day Rich Task Design Cycle template
214 Appendix F: Three-Column Engagement Template
Engagement
with Minecraft
Engagement with Math
Practices
Ideas for Modification
215 Appendix G: Post-Workshop Student Interview Questions
Descriptive
1. Describe some of your favorite things to do while you play Minecraft
2. Can you tell me about your experience of the Build A Room Challenge?
3. Can you walk me through how you worked with ______[student’s team or partner]?
Structural
4. We’ve talked a little bit about how you play Minecraft and what you like to do in the
game… now I want to ask you a slightly different kind of question. I’m interested in
getting a list of all the different things you did during the Build a Room task. This might
take a little time, but I’d like to know about all the different things you did, and how
they either helped or didn’t help you during the activity.
Contrast
5. What’s the difference between “trolling” and “griefing”?
a. Did you have any experiences with either in class?
6. What’s the difference between playing Minecraft and completing a challenge or task in
class with Minecraft?
216 Appendix H: Introductory Student Questionnaire
217