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Linear algebra instruction is an essential competency that is necessary for success in multiple engineering disciplines. Research in realistic mathematics education and the development of an empirically tested curriculum in inquiry-oriented practices for teaching linear algebra helps improve the ability of instructors to teach the content via multiple lenses and modes. While there have been good instructional materials and strategies developed to apply inquiry-oriented instruction for linear algebra, students struggle to apply and connect the different modes. Game-based learning provides a platform to creatively include multiple modes and strategies in a fun and engaging manner. In this paper, we present we discuss the addition of game-based learning elements into an existing curriculum that teaches undergraduate linear algebra via an inquiry-oriented pedagogy. The aim of this paper is to discuss the game design strategies used in connecting game based learning to inquiry oriented methods. 

We present preliminary results of students’ strategies playing Vector Unknown: Echelon Seas [VUES], a 3D videogame intended to support student reasoning about vectors. Our team designed VUES by drawing on theories from Inquiry-Oriented Instruction (IOI), Game-Based Learning [GBL] and Realistic Mathematics Education [RME]. VUES builds from a prior 2D game by giving players vectors with 1, 2, or 3 components, depending on the level. We use codes from our team’s prior analysis (Mauntel et al, 2020) to analyze strategies in the 3D game. Early results show that students develop similar strategies during 3D gameplay as other students developed while playing the 2D game. However, we have also found new strategies that we did not witness with 2D gameplay, requiring us to extend our coding scheme. Further, early results emphasized the need for design changes to the 3D game to better support players’ progress. 

Developing a rich understanding of linear combinations is key to understanding linear algebra. In this paper, I explore the rich connections students make between the geometric and numeric representations of linear combinations through playing and analyzing a video game. I look at population of students who have never taken linear algebra before and analyze how they structure space using the video game, Vector Unknown, as a realistic starting point. I detail and analyze this activity including the activities that transition them from 2D to 3D space. 

We present findings from a study analyzing and comparing the strategies participants deployed in playing the game Vector Unknown and completing the Magic Carpet Ride task. Both the game and task are designed to give students an introduction to basic concepts about vectors needed for success in linear algebra. We found that participants used a diverse array of strategies, tending to favor algebraic approaches to the Magic Carpet Ride task. We also found that participants tended to try the same strategies in both tasks, but did not usually follow through with the same strategy in both contexts. These findings have implications for instructors considering using one or both tasks in their linear algebra class. 

In this report, we characterize seven of twenty-five students’ responses to a single written homework assignment from the Spring 2021 academic semester. The homework was designed to incorporate the Vector Unknown 2D digital game to investigate how students answered questions about span and linear independence after playing various levels of the game. We present our modification of the roles and characteristics framework of Zandieh et al. (2019), our identification of students’ grammatical use of game language and math language, as well as the results of analyzing students’ homework responses using our framework. 

Understanding linear combinations is at the core of linear algebra and impacts their understanding of basis and linear transformations. This research will focus on how students understand linear combinations after playing a video game created to help students link the algebraic and geometric representations of linear combinations. I found that having students reflect upon the game and create their own 3D version of the game illustrated which elements of 2D understanding could be translated into 3D. Also, students' creation of easy, medium, and hard levels provided insight into how students progressively structure space. 

We present results of a grounded analysis of individual interviews in which students play Vector Unknown - a video game designed to support students who are taking their first semester of linear algebra. We categorized strategies students employed while playing the game. These strategies range from less-anticipatory button-pushing to more sophisticated strategies based on approximating solutions and choosing vectors based on their direction. We also found that students focus on numeric and geometric aspects of the game interface, which provides additional insight into their strategies. These results have informed revisions to the game and also inform our team's plan for incorporating the game into classroom instruction. 

The results we report are a product of the first iteration of a design-based study that uses a game, Vector Unknown, to support students in learning about vector equations in both algebraic and geometric contexts. While playing the game, students employed various numeric and geometric strategies that reflect differing levels of mathematical sophistication. Additionally, results indicate that students developed connections between the algebraic and geometric contexts during gameplay. The game’s design was a collaborative effort between mathematics educators and computer scientists and was based on a framework that integrates inquiry-oriented instruction and inquiry-based learning (IO/IBL), game-based learning (GBL), realistic mathematics education (RME). 

Demands in undergraduate education are shifting to reach larger student populations - especially learners beyond the brick-and-mortar classroom - which has led to more pressing demands to incorporate technologies that afford such learners access to high-quality, research- based, digital instructional materials. In this article, we explore three theoretical perspectives that inform the development of such instructional materials. In our team’s efforts to develop a game-based learning applet for an existing inquiry-oriented curriculum, we have sought to theoretically frame our approach so that we can draw on the corpus of researcher knowledge from multiple disciplines. Accordingly, we will discuss three bodies of literature – realistic mathematics education’s [RME’s] approach to curriculum development, inquiry-oriented instruction and inquiry-based learning [IO/IBL], and game-based learning [GBL] - and draw on parallels across the three in order to form a coherent approach to developing digital games that draw on expertise in each field.