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Engineers frequently utilize computer simulation as part of their design processes to model and understand the behavior of complex systems. Simulation is also an important tool for developing students’ understanding of modeling and strengthening their intuition for problem solving in complex domains. This project uses a two-joint robot arm problem and accompanying computer simulation to demonstrate to AP BC Calculus students how and why we would use calculus concepts simultaneously in Cartesian and polar coordinate systems. We developed the simulation in a way that allows students to experience mathematical modeling in an applications-based engineering context. A small cohort of students in AP BC Calculus completed an open-response survey of their perceptions on mathematical modeling before and after completing our simulation. Analyzing these data using direct content analysis showed that students seemed to increase their understanding of mathematical modeling as an iterative process, although some students narrowed their description to focus on computer simulation. This study supports the role of simulation in developing students’ understanding of mathematical modeling and developing specific content knowledge, and how engineering can provide a valuable context for the application of mathematical modeling. Introduction Mathematical modeling is a critical component of math, science, and engineering education [1]–[7] . Both the Common Core State Standards for Mathematics (CCSSM) and the Next Generation Science Standards (NGSS) emphasize the importance of mathematical modeling [1] . Mathematical modeling in the classroom helps to develop the critical thinking and math skills required for engineering [2] . It allows students to “revise their preconceptions and... understand the underlying principle[s] of mathematics” [8] and integrate topics similar to professionals in the field [1] . Students are expected to engage in modeling throughout engineering, math, and science curricula [3] . One way to bring mathematical modeling into the classroom is to use a simulation task with engineering applications. In this study, researchers investigated how completing such a task influences student perceptions of mathematical modeling. Using a simulation provides quick and efficient feedback in a cost-effective manner [4] . Simulations also allow students to explore cause and effect relationships between variables [5] , test a large number of different models [6] , and develop intuition about difficult concepts [7] . Researchers selected the two-joint robot arm as the simulation task for students enrolled in AP BC Calculus (BC Calculus) classes at a public high school in the intermountain west region of the United States. BC Calculus is an Advanced Placement course that is roughly equivalent to the second semester of college calculus. The two-joint robot arm is often used in upper-level engineering courses or modules with a prerequisite of differential equations and linear algebra (e.g., [9] , [10] , [11] ), but it has also been used earlier in the curriculum as a real-world application of trigonometry and calculus-based physics [12] , [13] . Sultan designed an activity for students in a precalculus class who have the geometric understanding to determine the position of the end effector, but do not yet understand the concepts of differentiation and rates of change necessary to calculate the velocity [12] . Berkove & Marchand [13] described a robot arm in space, and students calculated forces that the robot arm exerted. Their activity required students to have a strong background in physics, and high school students have varying levels of physics understanding. After more than a year of introductory calculus, though, high school students in BC Calculus are well versed in the differential relationship between position and velocity. This knowledge allows students to explore the two-joint robot arm from the perspective of the motion of the end effector without the need to introduce new physics concepts, differential equations, or complex matrices. The two-joint robot arm simulation addresses the exploration of parametric equations and working between polar and Cartesian coordinate spaces. These skills are part of the BC Calculus curriculum, and tested on the exam [14] . To address student perceptions of mathematical modeling, researchers designed the task to maximize student engagement in as many aspects of the GAIMME modeling process [15] as possible within the time constraints of the course. Literature review There are many definitions for mathematical modeling and across the STEM spectrum [15] . In this study, mathematical modeling was defined using the GAIMME modeling process [15] which includes six interrelated steps: ● Identify and specify the problem to be solved ● Make assumptions and define essential variables ● Do the math: get a solution ● Implement the model and report the results ● Iterate as needed to refine and extend the model ● Analyze and assess the model and the solutions Many studies investigate how students engage in mathematical modeling or simulation (e.g. [1], [7], [16] ), but not how students define the mathematical modeling process. McKenna and Carberry [3] focus on a broader definition of modeling in the engineering design process that includes “any representation of some physical phenomena”. They assessed modeling because it is prolific across math, science, and engineering courses. They found that while students consistently responded about physical aspects of the design process (e.g. prototypes, drawings, charts), students mentioned mathematical modeling significantly less frequently than professors. The category of mathematical modeling included “ideas represented by mathematical equations and calculations” [3] . Students were also less likely to mention other abstract models, specifically theoretical/conceptual and verbal models. McKenna and Carberry [3] concluded that, while students engage in mathematical and other abstract modeling activities throughout the engineering curriculum, they do not necessarily recognize the importance of these tools in the design process. Instructors have introduced engineering tasks in calculus classes as a way to increase students’ problem solving skills [17] and improve critical thinking [2] . When students engage in mathematical modeling in the engineering curriculum, it is often in the form of simulation [5] . Simulation may help students develop modeling skills while also deepening their intuition of complicated math topics [5]–[7] . Dickerson and Clark [7] researched the role of SPICE (an electronics circuit simulation computer program) in university microelectronics courses. They explored the difference between teaching a course using an interactive simulation in-class versus teaching the course without. Students reported that engaging in the simulation helped them with test and quiz problems, and that they felt they understood something from the simulation that they would not have learned without it. These students scored higher on the final exam than students who did not take the course with interactive simulation [7] . Modeling in the classroom, including the mathematical modeling task for this project, is often different from modeling in a professional context [4] . Develaki [4] points out the difference between modeling by scientists and modeling in an educational context. Scientists use modeling and simulation in conjunction to develop new, unproven theories that they then test and modify [4] . Similarly, engineers use modeling and simulation in the design process to develop new and innovative solutions to problems [3] . Students, on the other hand, engage in “educational modeling”, where they change specific parameters and initial conditions to develop their understanding of a system that is already well-understood [4] . In this study, researchers designed a simulation to engage BC Calculus students in educational modeling [4] of an engineering problem that illustrates how parametric functions and their derivatives in a polar reference frame (angular joint motion and arm length) to describe straight-line horizontal and vertical motion. Development of the simulation included careful attention to the steps of the GAIMME modeling process [15] , particularly assessment and analysis. The researchers used preand post-survey data to compare student perceptions of mathematical modeling before and after completing this simulation activity to address the research question: how do student perceptions of mathematical modeling change before and after completing an engineering simulation activity? Methods Simulation development and implementation The two-joint robot simulation was created using Unity [18] . The structure of Unity as a gaming engine allowed for simplification of the code and easier implementation of graphics. Unity also includes the ability to compile and run in WebGL, which allows students to access the program using Chromebooks. Equations for horizontal and vertical motion were pre-programmed in the simulation, and students could vary coefficients within these constraints. Students could also vary angle values, the rate of change of θ 1 , and the maximum value of θ 1 (Figure 1). As the end effector (end of the robot arm) moves, it drops a series of game objects to trace the path for student observation. Figure 1. Two-joint robot arm simulation. Classroom implementation of the simulation occurred over three consecutive days. Students had a total of 120 minutes to work on the simulation activity. Prior to completing the activity, students received introductory instruction on polar and parametric equations and vector calculus. They also watched a video of the Harris T7 Explosive Ordnance Detection (EOD) robot investigating “suspicious” packages in a car [19] . Students discussed the motion of the EOD robot and how the extension of its arm in a linear pa

Student Perception of Mathematical Modeling Before and After Completing a Two Joint Robot Computer Simulation Task