Mathematical Modeling of Minecraft – Using Mathematics to Model the Gameplay of Video Games Thesis Presented in Partial Fulfillment of the Requirements for the Degree Masters of Mathematical Sciences in the Graduate School of The Ohio State University By Raymond T Cox Graduate Program in Mathematical Sciences. The Ohio State University 2015 Committee: Herb Clemens, Advisor Azita Manouchehri Copyright by Raymond T Cox 2015 Abstract The use of video games to teach mathematics is decades old, but has been predominated by designing video games that directly teach basic skills to young children. The rise of internet culture and modern gaming has created a population of older students that are emotionally and intellectually invested in digital entertainment. Simultaneously, changing educational standards favor problem solving and modeling behavior over basic skills. This paper explores the possibility of using commercial, popular video games as a “real world problem” that can be modeled and solved using mathematics commonly taught to high school students. A specific example of this possibility is drawn from the extremely popular game Minecraft. ii Dedication This document is dedicated to my kitten, who was born after and died before. iii Vita 2002....................................................Bridgewater-Raritan High School 2007....................................................B.S. Mathematics, York College of Pennsylvania 2011 to present...................................M.M.S., The Ohio State University Fields of Study Major Field: Graduate Program in Mathematical Sciences. iv Table of Contents Abstract ................................................................................................................................ ii Dedication ........................................................................................................................... iii Vita ...................................................................................................................................... iv List of Tables ..................................................................................................................... vii List of Figures ................................................................................................................... viii Chapter 1: Modeling Behavior in Mathematics .................................................................. 1 What are models? ............................................................................................................. 1 What are the phases involved in mathematical modeling? .............................................. 1 How do models develop? ................................................................................................. 7 How are models used in teaching and learning? ............................................................ 10 What is a good model? ................................................................................................... 12 Why use video games to model mathematics? ............................................................... 13 Why use Minecraft, specifically? ................................................................................... 14 Chapter 2: Modeling Activity: Working on the Railroad ................................................... 17 The World of Minecraft .................................................................................................. 17 The Story ........................................................................................................................ 18 v The Assignment .............................................................................................................. 22 The Transition ................................................................................................................ 31 Conclusion ...................................................................................................................... 32 References ...................................................................................................................... 35 Appendix A: Handouts ....................................................................................................... 36 Minecraft: Tunneling ...................................................................................................... 37 Minecraft-Minecart: Elevated Rail ................................................................................. 39 Minecraft-Minecart: Transition from Tunnel to Elevated Platform ............................... 40 Minecraft-Minecart: Terrain Composition ..................................................................... 41 Minecraft-Minecart: Crafting Recipes ........................................................................... 42 At the Crafting Table .................................................................................................. 42 At the Furnace ............................................................................................................ 42 Minecraft-Minecart: Tools ............................................................................................. 43 vi List of Tables Table 1: A Student's Progression Through the Modeling Activity.....................................30 Table 2: Crafting Recipes...................................................................................................42 Table 3: Tool Recipes.........................................................................................................43 vii List of Figures Figure 1: Modeling Cycle....................................................................................................5 Figure 2: Jack's House.......................................................................................................18 Figure 3: Forest and Extreme Hills Biomes.......................................................................19 Figure 4: Rails don't work well in a mountain...................................................................20 Figure 5: Rails laid directly on the ground........................................................................20 Figure 6: Creepers obstruct the railway.............................................................................21 Figure 7: A creeper explodes, destroying the rails.............................................................21 Figure 8: Rail built above the ground, with creepers safely below it................................21 Figure 9: A player standing at (-23, 40, 317) as shown in-game.......................................22 viii Chapter 1: Modeling Behavior in Mathematics What are models? Modeling can be used to describe how people interact with and learn from the world. It is a natural process wherein an individual forms conceptual objects, models, that describe his real-life experiences from his own perspective (Lesh & Doerr, 2003). This school of thought is derived from the constructivist that an individual's perspective of “reality” is unique to that person (Lesh & Doerr, 2003). The modeling thought process is that models are used to interpret and organize the reality of their lives. Reality is objective to this school of thought, but the models are individual and thus every individual person interprets the same experience differently (Lesh & Doerr, 2003). From this perspective, a “model” can be seen as the smallest complete unit of knowledge (Lesh & Doerr, 2003). Although it has many pieces, the entire model is needed for those pieces to have context, meaning, and understanding. A model will have internal and external components, with the external components consisting of representations that allow the individual to communicate the model to other people (Lesh & Doerr, 2003). What are the phases involved in mathematical modeling? Mathematical modeling consists of multiple subprocesses that describe how the modeler defines the problem at hand, turns that definition into a mathematical construct, and interprets that mathematical construct to solve the original problem (Zbiek & Conner, 2006). As with any description of a method of thinking, the modeling cycle is an 1 abstraction of a continuous process into discrete pieces. As such, different researchers divide the cycle into similar but different cycles depending on their own needs. For this work, I have chosen to use the modeling cycle described by R.M. Zbiek. Refer to Figure 1. A working mathematical model is a gestalt of four distinct elements. These are a mathematizable situation, a mathematical object, a purpose or problem that spurred the modeling process, and the relationships between these three things (Zbiek & Conner, 2006). In simpler terms, these are the actual thing that has called the student's attention, the math that the student has used to describe the situation, the reason the student is thinking about the situation in the first place, and the way the student both turns raw information into math and then math into a solution. In the following activity, the mathematizable situation is the game Minecraft, the mathematical object is the collection of equations and formulas the student creates during the activity, the purpose is the scenario of the activity itself, and the relationships are the rules of Minecraft. In this paper I will often refer to the “rules” of a game, particularly Minecraft. When I speak of “rules” I refer not only to the allowed actions of the player but also the way objects within the game behave. For example, to create