Hello Mark, thank you for the question. We've collected qualitative data about students strategies for completing the game and examined some data from homework assignments relating to the game as well. We are currently working on designing a pre/post-test and will be comparing it with student scores in the game, but have not completed this research yet. 

In terms of creating their own games. I can approach this from two different angles. The first angle is that we had computer science students design both games (2D and 3D) for a capstone project. This required the students to think deeply about the mathematics being presented in the game and how best to represent it in the game. It also required some linear algebra developing tilts, rotations, grid development, and placement of goals.  I would say the project was an overall success in not only designing the games, but also having these students be exposed to the mathematics behind the game.

Second, for my dissertation, I am had students play the 2D game, analyze the game, and create a mock 3D game that they could play in GeoGebra. These students had not taken linear algebra before and had a mixture of experience with gaming. Some of them had not even plotted points in 3D before. I found the intersection between thinking about the game and designing a game to be overall very fruitful. For example, the students had to create easy, medium, and hard levels and this allowed the students structure their thinking and communicate which concepts they found most difficult. Concepts borrowed from the original playthrough, such as the limitation of the coordinate axis to +/-20 allowed the students to explore linear combinations that would work in a limited 2D environment. While exploring this limited environment, they discovered aspects such as linear independence and dependence. Also, I had one group that tried to make a "bluff" vector as a prospective answer. This gave rise to the question of what vectors would not work for a problem. For example, one student proposed switching the goal coordinates as a "bluff." Then there was an interesting discussion about if this was actually a bluff vector. 

All in all, I found having students design their own game to be very fruitful. The linear algebra students were not experts in game design, but from the playing the game they could obtain structures that they could try to mimic and could be used to organize their thoughts for productive mathematical discovery. This is my first pass at this type of research, and I think more time would have to utilized for developing such materials towards targeted goals for a course (but it could be very productive as a class project).

Nathan, thank you for your question. I would say this was a collaborative effort. The team consisted of an expert game designer, two math ed researchers, and a few graduate students in addition to the ASU capstone students. The math ed researchers met and discussed what topics they wanted to develop in the game. I will say that one was an expert in mathematical metaphors which students use while taking linear algebra. The capstone students met with the math ed researcher and the game design expert who was teaching the class in order to discuss the material covered in the game and several ways the researchers envisioned the game (in particular there is a travel metaphor that is popular for students to use while discussing linear combinations and both researchers wanted this metaphor incorporated into the game in some way). I would say the overall process was very collaborative.

Students were given material from the Inquiry Oriented Linear Algebra curriculum (https://iola.math.vt.edu/) to help form the basis for their understanding in addition to the discussions in the meetings. They would often present their example programs and there was a back and forth that balanced gameplay with incorporation of the concepts. For example, one aspect of the 2D game that was student proposed involved presenting students with four vectors with two pairs of linear dependent vectors which helped the player get a feel for the linear dependence. Another game that we are refining is a cannonball game that is fun (you blow up ships), but the gameplay initially was just calculating a lot of matrix multiplication. Through a discussion and experience with a task from the IOLA curriculum we are changing the mechanics to better resemble a task that explores change of basis. 

In summary, I would say that the design process was balanced by having students need to design an interesting game and the mathematics educators having knowledge from both teaching and researching the topics providing insight guiding this process. In particular thinking about contexts inspired by mathematical metaphor research and and in-game decisions that mimic student thinking from researched curricular tasks. 

Joshua, thank you for the question. I would say that when we use the word real, we do not necessarily mean real-world. Instead, we mean something that is experientially real for the students. I would say that a video game is an experientially real scenario that they can draw upon. I will say though that our team has worked to design materials for the classroom that connect to the game to bridge the students' gaming experiences to formal mathematics. In the class, they will do real-world problems, including a traffic flow problem that could use some concepts developed in the game. 

On a side note, for my dissertation I had students design a 3D version of the game (this was before the 3D version of the game above was completed). Looking at the data, I found that students would use certain views in GeoGebra to do certain activities in 3D space. So for example, they would occupy one viewpoint to see if two vectors were linearly dependent. They would also create 2D projections to see if the projection of the vector (from their view) spanned that 2D space and then altered the view to see the tilt of a plane spanned by 2 vectors. These activities provide a context for students breaking down and orienting themselves to 3D space. They also provide important experiences that students can mimic when they are for example designing objects in CAD or doing some engineering projects. All in all I see the games as providing another resources that students can draw upon to solve problems and the more resources we provide to and connect mathematical ideas to the better.