The role of prototyping in mathematical design thinking

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The role of prototyping in mathematical design thinking Leah M. Simona, Dana C. Coxb, a b

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Dixie High School 300 South Fuls Rd. New Lebanon, OH 45345, United States Miami University, Department of Mathematics 301 S. Patterson Ave Oxford, OH 45056, United States

A R T IC LE I N F O

ABS TRA CT

Keywords: Mathematical design thinking Mathematical modeling Prototyping Narrative inquiry Technology

In this paper, we use the methodology of narrative inquiry to examine mathematical modeling through the lens of design and design thinking. Using Schön’s (1992) definition of design thinking as knowing in action, we wonder how might we articulate mathematical knowing in action and its role in mathematical modeling. Together, the authors tell the story of a mathematical design project where Leah develops an animation of a zip line rider, focused on the role of prototyping in and impact of design on the mathematical modeling process. The act of differentiating prototyping as an act of mathematization and broadening the current modeling theory to include more of what the design world refers to design thinking has implications for both teaching mathematics and classroom research.

1. Introduction Mathematical modeling has received increased attention in United States classrooms with the advent of the Common Core State Standards for Mathematics (CCSSI, 2010). But there is little consistency in definitions of what mathematical modeling is, how to do it (modeling cycle), or how to represent it diagrammatically (Cirillo, Pelesko, Felton-Koestler, & Rubel, 2016). Kaiser (2017) details six different perspectives on modeling that can be historically identified, each leading to different characterizations of the process and goals. Looking across these perspectives, there is some agreement that the process is cyclic, though some emphasize following a complete modeling cycle through to solving a real-life problem, while others focus intently on the relationship between real and mathematical models and conceptual development. In an effort to come to a fuller definition and situate the process, Cirillo et al. (2016) cite similarities among representations of the mathematical modeling cycle and the design cycles used in engineering design and software development. In each of these contexts there is use of an iterative process including moments of problem definition, product development, testing, and refining. These noted similarities between modeling and design cycles beg the question: how might principles of design thinking inform how classroom mathematical models are built and how students come to understand this process? Design thinking is a multifaceted term, typically referring to the ways in which designers work and think. The Interactive Design Foundation (Dam & Siang, 2019) describes it as encompassing iteration and problem solving, but also as a way to redefine problems or as a way to question assumptions and our initial understanding of a problem. Schön (1992) describes design knowledge as tacit and non-verbal and defines designing as “knowing in action” (p. 131), which can be difficult to articulate or pull into a metacognitive space like a mathematics classroom. As we consider what it means to view mathematical modeling through the lens of design, we take Schönös (1992) construct as our theoretical framework and ask How might principles of design thinking inform how classroom models are

Abbreviations: IGS, interactive geometry software; PLTW, Project Lead The Way ⁎ Corresponding author. E-mail addresses: [email protected] (L.M. Simon), [email protected] (D.C. Cox). https://doi.org/10.1016/j.jmathb.2019.100724 Received 1 February 2018; Received in revised form 1 July 2019; Accepted 26 July 2019 0732-3123/ © 2019 Elsevier Inc. All rights reserved.

Please cite this article as: Leah M. Simon and Dana C. Cox, Journal of Mathematical Behavior, https://doi.org/10.1016/j.jmathb.2019.100724

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built and how students come to understand the process? In order to examine student modeling practice using the framework of knowing in action, we also need to understand the classroom as a workplace and the influence of creativity and innovation within the modeling process. To get a better sense of the ways in which we can develop a culture that supports modeling, we can examine the ways in which other industries have taken up design thinking to encourage and develop innovation and creativity in the workplace. Kolko (2015) identifies five principles for design thinking. What designers know about how best to construct artificial worlds for examination and validation comes to bear on how we help students do the same. In this paper, we use the methodology of narrative inquiry to articulate mathematical knowing in action in the classroom. Our narrative takes place in the mathematics classroom and in a technological space supported by Interactive Geometry Software (IGS). Our story will show how constructing an artificial world (in which a zip line behaved in expected ways) influenced our mathematical thinking as well as our perception and awareness of how this thinking manifests in action. We will reflect within our narrative on how the five principles of design thinking (Kolko, 2015) supported our experience. 2. Background and literature It is important to go into more depth about establishing a design culture to support mathematical modeling instruction. In this section, we will explore five principles of design thinking that are laid out by Kolko (2015). These principles are: 1 2 3 4 5

Create models to examine complex problems. Focus on users’ experiences, especially their emotional ones. Tolerate failure. Exhibit thoughtful restraint. Use prototypes to explore potential solutions.

These principles, intended to support the integration of design thinking, is a framework through which to create or modify the instructional context in which mathematical modeling occurs. Before we begin, we wish to clarify that in the design world, models and prototypes are different types of products constructed for different purposes. Zbiek and Conner (2006) differentiate between those actions taken relative to a Real Word Situation versus a Mathematical Entity. The Design literature would associate modeling with those actions regarding the real world situation and prototypes with those associated with the mathematical entities. This is different from the way these terms are defined in the world of mathematics education, however, where models are seen as a product, modeling an overarching and cyclic process that includes prototyping as a subprocess. We want to make more careful distinctions. We will refer to mathematical knowing in action, as the knowledge that is brought to bear during the modeling cycle. Like design thinking, it occurs where there is a transaction taking place between the mathematician, the real life context, and the developing mathematical entity (model). These transactions are similar to what Zbiek and Conner (2006) refer to as modeling subprocesses, examples of which include Mathematizing, Aligning, or Communicating. To this, we will be adding prototyping as a broad category of transactions that support these subprocesses. Prototyping provides an experimental lens (Fountain, 1990; Kolko, 2015; Rothenberg, 1990) that is helpful in articulating how a designer or mathematician works in between the problem and solution, or real life context and mathematical entity. 2.1. Modeling versus model creation From an epistemological or theoretical perspective on modeling (Kaiser, 2017), the work of design is deeply aligned with that of mathematization (Freudenthal, 1968). Freudenthal’s intention was that modeling should be seen as “a complex interplay between mathematics and the real world, based on various kinds of mathematization processes,” (Kaiser, 2017, p. 259), and that mathematics should be framed as an action to be taken rather than as a created object or model. Zbiek and Conner (2006) echo this and describe mathematization as a subprocess of the modeling cycle as introducing mathematical ideas that link the context to the mathematical entity. Schönös (1992) construct of knowing in action is a way of referencing the ways in which design thinking is linked to and dependent upon the act of design. Mathematical knowing in action is, therefore, revealed in acts of mathematization, which continues the theme of presenting mathematics as action, and eludes to the interconnectedness of mathematical ways of knowing and the ways in which we, as mathematicians, act. These actions are separate from its products, namely, mathematical models. The distinction can be made by contrasting what Cirillo et al. (2016) refer to mathematical modeling with the construction of mathematical models through Model Eliciting Activities (MEAs; Lesh, Hoover, Hole, Kelly, & Post, 2000). In the classroom, both involve students creating models to examine complex problems. However, according to Cirillo et al. (2016), mathematical modeling refers to an iterative process including moments of problem definition, product development, testing and refining. The focus on the process is different from what Lesh et al. (2000) have written about relative to MEAs, or activities that “focus on the development of constructs (models or conceptual systems that are embedded in a variety of representational systems) that provide the conceptual foundations for deeper and higher order understandings,” (p. 592) of mathematics. The focus on the process or product (the model) is an important distinction. In the classroom, focusing on the process allows 2

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students to use whatever mathematics is available, interesting or useful when developing a mathematical model. However, MEAs are designed around specific mathematical constructs that have been deemed by external authorities to be important, and while it is possible to also focus on the process, these tasks are developed with the product in mind. The emphasis placed by Lesh et al. (2000) is on evaluating how and how well students solve provided problems using processes approved by adult stakeholders. The focus on product, including the product of process, is reiterated by Doerr and English (2006) who define a MEA as an activity where “students’ thinking processes are explicitly revealed via their descriptions, explanations, justifications, and representations both as they engage with the task and as they present their end products,” (p. 9). The emphasis is still on the specific mathematics to be learned and the process is used as another avenue for assessment. Students of mathematics are rarely asked to design mathematical models for an audience outside of their immediate school context or for a more general purpose than knowledge acquisition or assessment. There are both implicit and explicit norms for model building conveyed by others in that context (Brophy, Klein, Portsmore, & Rogers, 2008). We need to better understand how these norms are conveyed as well as what emotional or social structures might also impact behavior. 2.2. Design culture Principles two, three, and four honor the human experience of engaging in design thinking. In this section we explore these principles in the context of mathematical modeling with emphasis on ways in which the classroom culture influences the ways in which student mathematical knowledge is put into action. 2.2.1. Focusing on user experiences Focusing on our students’ experiences could be interpreted as a plea for formative assessment and a careful documentation of action and intention. In design culture, however, focusing on a user’s experiences means understanding what the user is feeling and how the user engages with design. In mathematical design, this translates to understanding how an individual is engaging with a mathematical design either as a designer or a user. When we expect our students to engage in mathematical design, we must consider our students as well as the audience for whom students are designing mathematics. Due to the nature of the classroom setting, this audience is typically limited to themselves, classmates, and teachers. Brophy et al. (2008) documents a shift for engineering design projects in the classroom, explaining that students initially design for themselves, and subsequently design for others. In mathematics classrooms, we believe that this shift could be reversed. Students generally start working with the expectation that they will have something to demonstrate to others, the teacher in particular. When students are challenged to do mathematics (Smith & Stein, 1998), they make the jump to designing for themselves. It is at this point, when students are truly thinking mathematically and independently; the teacher moves into the role of an advisor during the design process, listening for students’ mathematical thinking. As there is a need to document, communicate, and verify mathematical design in the classroom, mathematical empathy (Araki, 2015) is important. Empathy entails the search to comprehend another person through their frame of reference. Mathematical empathy requires that sort of comprehension, but instead of emotions and feelings it is focused on mathematical ideas (mathema). Thus, mathematical empathy refers to the ability to comprehend another person's mathematics and the true meaning or purpose behind them, seeking to utilize the other person’s frame of reference (p. 118). 2.2.2. Tolerate failure Understanding student relationships with mathematics, including emotion, is key to developing a culture of design thinking. Asking students to engage in mathematical design requires that there is room within the classroom and school cultures for risk taking and ambiguity, two challenges that face any group that wants to adopt a design culture (Kolko, 2015). In the act of problem solving, student ambition and creativity may be hampered by feelings of risk, as many are conditioned to value a polished solution over the actual process of building one. Ambiguity about strategy and even purpose may feel uncomfortable in an academic setting where you are evaluated on the production of solutions. This works against the essence of mathematics (Dreyfus & Eisenberg, 1996; Ginsburg, 1996). The fear that a solution will not be found (and thus an assignment not completed) may impair our willingness to ask and pursue ambitious mathematical questions, to create ambitious models for data or natural phenomena, or to move beyond the boundaries of what we already know into an area where insight may be profound. In order to do this, we must foster a culture tolerant of intellectual risk taking (Jansen, Cooper, Vascellaro, & Wandless, 2017) and tolerate failure. Intellectual risks come in response to complex problems. Smith and Stein (1998) characterize the highest level of mathematical tasks as those in which students are “doing mathematics.” These tasks have no clear path to a solution, require students to conceptually explore and understand the relevant mathematics while monitoring their progress, and “may involve some level of anxiety for the student” (p. 348). It is only through complex situations that students are given opportunities to iteratively explore solution spaces, rather than find solutions. This is an important distinction in light of the other principles. While it is important to create viable designs, we must also acknowledge that in the classroom context, the designing is as important, if not more important, than the design. If the goal is to move with dexterity or fluency within a solution space, then we must provide students room to maneuver. We contend later that we should elevate prototyping as a method for that maneuvering. 2.2.3. Exhibit thoughtful restraint Lastly, for students to have agency and status they must be allowed to decide for themselves what their models cannot and should not do. Thus, they must be given the freedom to exhibit thoughtful restraint. In the context of this paper, thoughtful restraint is 3

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interpreted as giving students agency over both ambition and risk, allowing them to decide when and how far to push personal mathematical boundaries. When students are provided the opportunity to take full ownership of the design process, they simultaneously take ownership of the outcome. By giving students agency and fully adopting these principles as central to establishing a design culture in our classrooms, we create a classroom where students want to be and one that empowers them to contribute and challenge themselves. They are able to express authority as well as autonomy. This can have a positive impact on a student when they find meaning and purpose in their work. 2.3. Prototypes to explore the solution space Prototyping is one method of exploring, both individually and collaboratively, a solution space (mathematical entity) for a problem. In this section we will more carefully define prototyping. By focusing on prototyping as a transaction rather than a physical object or product, we position it as a form of mathematization (Freudenthal, 1968), where it is a creative act of epistemological generation as one makes mathematical sense of a phenomena. In order to do so, we make a distinction between the world of mass production and that of software design (Floyd, 1984). Creating a summative prototype in mass production is the process of running a test case or sampling a consumer good to prove that a production line is sound. Geometry students might be asked to create geometric prototypes of specific quadrilaterals to determine the validity of emerging definitions. In contrast, prototyping in software design represents less of the final product and more the act of thinking and working within the solution space. In this sense, and within the mathematics classroom, prototyping serves as far more than a final summative verification. Formative prototypes can help expose flaws sooner, improving the final result. Rather than waiting until the entire system is created, we can explore it on a smaller scale or experiment with just a part (Rothenberg, 1990). Described both as serious play and pragmatic action (Schrage, 1999), prototyping is an act of transforming an idea into something valuable. From this perspective, prototyping can be described as a way of communicating ideas and bringing solutions into a place for consideration and evaluation. While prototyping, designers can push and change the parameters of a problem to create solutions. This looks very different depending on the environment where it is used. We will detail a few different ways this action has been described in the design literature. For a given problem, Kolko (2015) suggests that we engage in prototyping only after we have come to an understanding of the problem space and as a means to explore the solution space of the problem. Prototyping is the act of exploring and experimenting with possible solutions and results and is an example of a transaction that requires we move flexibly between context and mathematical entity to explore and discover solutions that meet some or all of the constraints established. While the modeling cycle would eventually bring a final proposed model up for verification and testing, prototyping does not require that a model be complete, or even entirely applicable to the situation. In the spirit of mathematical questions that begin with under what conditions might… or what would happen if…, we also posit that prototyping is initially possible in the absence of a posed question and might play a role in problematizing a situation. Fountain (1990) describes prototyping as a cycle: determining what requirements the prototype will meet, constructing the prototype, validating the prototype, and reassessing the specified requirements or implementing and using the prototype. This cycle leads to two different ways of classifying a prototype, throwaway and evolutionary. A prototype that is used as temporary test case and is meant to be left behind as the project learns and moves on is a throwaway prototype. Alternatively, a prototype that is created to be retained and lead to further development is an evolutionary prototype. In the second case, the prototype is thought to be part of the result. Another way to characterize prototyping action is to consider the type of ideation generated by the prototype, which can be classified as evaluative or generative. Evaluative ideation is a way of determining when a work in progress or a result is sufficient, while generative ideation is a way of expanding what we might come to expect from a result. Rothenberg (1990) describes evaluative prototyping as experimentation to determine whether an aspect of a proposed system will work. In contrast, strawman prototyping is a form of generative ideation, which answers open ended questions about the expected behavior of a system; these questions often reveal gaps and inconsistencies between the prototypes and expected results early in the design process. 3. Methods In this section we will describe the methods of our inquiry. After a brief introduction to narrative inquiry as a methodology, we will delineate the perspectives from which we are writing our narrative. This will enable to reader to better situate the narrative within the academic, intellectual, and emotional context in which it takes place. Further, it will serve as a personal introduction of both of the authors to the reader. The goal is not to identify bias or target it for removal, rather, to state unequivocally that who we are as people is an indelible piece of what we experienced and the story loses meaning in anonymity. 3.1. Narrative inquiry Narrative Inquiry (Clandinin & Connelly, 2000) is used to capture personal and human dimensions of experience. It is especially powerful when experiences are recorded as they occur, and later are layered with reflection upon those experiences. While the experiences are temporal and situated, their meaning expands through reflection. In this sense, the emergent narrative is both product (data) and process (analysis) (Clandinin & Connelly, 2000; Connelly & Clandinin, 1990). 4

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In this paper, the temporal experiences of designing mathematics were documented in a series of files, images, notes, emails, and sketches. While we also had memories of the events depicted in these data, those did not serve as temporal data, but rather elasticized the other forms of data, helping us uncover meaning that was either available to us earlier and forgotten, or only available after the fact. These data support the development of a reflective narrative that includes two perspectives, the perspective of Leah, the prototyper, and the perspective of Dana, the instructor; neither of us could tell the story alone as it requires knowledge of both of our intentions and actions. The narrative originated from an animation task situated in IGS in which students were to create a design of a real world phenomena. During this task, Leah retained files and sketches that she created, and websites that she utilized while developing an animation of the zip line rider. These data served as a foundation for our review of the project and our analysis of the various episodes of design thinking. We later reviewed the data from the two-week project, stepping ourselves back through the design process used in creating the animations, relying on our memories to uncover meaning and intentions in creating an opportunity for and working within the design process. Then we reconstructed the mathematical design episodes within the larger context of the course and assignment to create the narrative that we presented in this paper. We wrote this joint narrative to emphasize mathematical knowing in action and the resulting mathematical design process, merging the temporal data with our memories of the project, and then layering our reflections upon both. We have tried to keep the narrative close to the original experience, supplementing it with our memories and referencing, but not including, the mathematical calculations that were necessary to create the various equations and relations in IGS. We set about answering the question, how might principles of design thinking inform how classroom models are built and how students come to understand the process? This narrative articulates mathematical knowing in action and shows that Leah’s understanding of and solutions to the design challenge were constantly evolving as she engaged in prototyping and various prototyping actions (Fountain, 1990; Kolko, 2015; Rothenberg, 1990). Upon analysis, the event broke into natural sections as we considered moments that were major design catalysts through the perspective of mathematical knowing in action. From there, we classified each section of the narrative based on the predominant prototyping actions, Throwaway vs. Evolutionary and Evaluative vs. Generative, while considering Leah’s intentions, actions, and the outcome of her intentions and actions (Fountain, 1990; Rothenberg, 1990). Reflecting on Leah’s experiences in the larger context of the course as a whole creates a rich text that helps understand the nuance of prototype development and its role in articulating mathematical knowing in action. In the course of telling this story, we share what we have learned about mathematical knowing in action and ourselves; narrative gives us a way to share our experience with mathematical design and our relationship to it (Richardson, 1994). Prior to sharing the narrative of the zip line creation we will first provide you with some background into our perspectives and the context in which we were working. 3.2. Dana’s Perspectives perspective Leah and I met in a course called Mathematical Problem Solving with Technology. The course is designed for sophomores who are interested in teaching secondary mathematics. In this course, students revisit their own learning of secondary mathematics and investigate mathematical concepts by way of problem solving with various technological tools such as IGS. Prior to this study, I have grounded my description of the course in the language of problem solving and problem posing. In other publications, I highlighted the activity of exploration, strategizing, and communicating with others mathematically (Cox & Harper, 2016). After this study, I include my intention in the course to create a culture of design where creativity and innovation are evident and to give explicit attention to mathematical knowing in action. Within that course, I assign an animation task. In the animation task, I ask students to use IGS to design an animated version of a dynamic phenomenon. Ambitious phenomena that students have chosen to model in the past include the behavior of physical objects such as pendulums, hanging springs, or a bouncing ball. They have also included periodic motion in the form of Ferris wheels, working analog clock displays, or the rise and set of the sun. Leah chose to work on a model of a zip line, strung between trees. Whether this task is a modeling task is questionable, though it requires mathematical design. Thompson and Yoon (2007) invoke replication as one of many motives for modeling. As a response to the desire to replicate a dynamic phenomenon, what is developed is, indeed, a model. However, modeling, as a mathematical activity, has traditionally been thought of as activity in response to an asked question. Similarly, from a design perspective, modeling is conducted in the problem space and in response to complex problems. In this sense, mathematical modeling is about generating a model that helps understand a problem or motivate a particular solution or strategy. This task is about replicating a known phenomenon in an artificial world. Wagner (1993), puts forward a second metaphor for inquiry: the blind spot. We can use inquiry as a way to expose blind spots and to ask questions that we had not previously thought to ask. In this sense, we start with something we think we understand and find (in the end) that there is more to learn. The animation task is designed as this sort of inquiry. I direct students to take a mathematical lens on something they otherwise may not be motivated to see. Rather than pursue an answer to a grand question, I ask them to problematize something that had either previously been taken for granted or not even noticed. Another important facet that makes this an opportunity for mathematical design is exposure to risk and the development of mathematical ambition. Asking students to engage in a task that is ill-defined (does not begin with a specific question to be answered) gives students a chance to set their own working goals and pose their own problems to solve. Building a model for something familiar, for which a mathematical model is previously known and understood, requires little ambition and exposes students to little risk. It is likely that a student will be able to complete the assignment quickly, be satisfied with the behavior of the animation, and the mathematics behind the animation will be sound. However, the mathematics uncovered is not new, and there is little potential for 5

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uncovering a blind spot. Selecting a phenomenon for which a mathematical model is not previously known is more ambitious and introduces significant risk. Students may not be fully satisfied with the animation that they create and the mathematics behind their design may be flawed. Students are not conditioned to recognize this as worthy of submission or as a “final product”. It can be difficult for students to recognize the value in the process over the product. I must admit that it can be difficult for me, at times, to trust that the time spent on broken or flawed products was useful or that learning occurred on the basis of self-report alone. I caution students to find a balance between ambition and risk. My evaluation process leaves room for broken or inaccurate animations to exist simultaneously as “complete”. I impose time limits to communicate to students that a boundless search for perfection is not needed or warranted. Lastly, I encourage students to share not only the final animation, but to present to one another some of the challenges they overcame and obstacles that still linger. 3.3. Leah’s two perspectives I took Problem Solving with Technology in the fall of my sophomore year. The majority of my classmates, including myself, were pre-service teachers studying to teach mathematics to students in grades 7 through 12. In this course, we learned about and used many different technologies, including IGS, to solve open-ended problems. I have now graduated and, after being exposed to different technologies in this course, I continue to use IGS to help my current students develop an understanding of geometry concepts. The classroom environment greatly influenced how I approached my assignments. The first graded assignment set the tone of the class. For homework, Dana asked, “How many different tessellations can you make using squares and equilateral triangles?” We used IGS to investigate the question and had to explain how we determined our answer. After we turned our homework in, we discussed the question in depth during class. Our answers ranged from 2 to 27 different forms of tessellations depending on how we classified the tessellations we made. We all shared our different approaches and thoughts, but looked to Dana to tell us who was “right”. To our initial disappointment, Dana prioritized the diversity of answers and processes over the elevation of “one best” solution. She turned the question back on us and asked us “under what conditions is each of these ideas right?” This became a theme throughout the course. Dana was okay if we did not have polished assignments to turn in at the deadline as long as we showed ambition, problem solving, and clearly explained our thought process. During class we exchanged ideas, discussed open-ended problems, and worked in small groups. As a result, I often thought about mathematics from new perspectives. Typically, I left class with more questions than I had walking in. Whenever I could, I would return to my dorm after class and continue working to answer these questions. This is exactly what happened while I was developing my animation. When Dana assigned this project, I had to decide what to build. I really wanted to engage with and enjoy the project so I considered phenomena that also related to my hobbies (camping and outdoors activities). I quickly settled on creating an animation of a person moving down a zip line. Dana gave us two weeks to complete the project and she advised us to spend only one of the two weekends before the due date working on the animation. From the start I had a feeling that this animation would end up taking more time than a single weekend, but I was okay with that. I knew developing this was a risk because I was not sure how I was going to create it even though I chose the phenomenon I wanted to do. I love zip lining, but that is completely different than looking at the mathematics, and sometimes physics, behind it. For two weeks I prioritized this assignment over all of my other school work. There was a ton of work ahead of me because I wanted to make the zip line as accurate as possible. I researched how to model the speed of a rider, the shape of the zip line wire, and the equations I needed to make the animation. In the process I taught myself the mathematics that I wanted to use. At times I did not even know if I was on track at all. As the two weeks progressed, I realized creating this animation was really ambitious. (Enough so that I created two animations instead of one.) Additionally, I had to accept that my second animation was not going to be perfect when I turned it in; what helped is that I knew Dana did not expect it to be. I knew she would consider both of my animations and the amount of effort I put into them when she gave me a grade. 4. Modeling and mathematical knowing in action In this section, we will work together to describe in great detail an episode of modeling that occurred in our classroom. Convention is to use one perspective as data and the other as analysis, however we are taking a different stance. Narrative inquiry can be used as a way to learn about others and to study participants. It is also a powerful tool for self-learning, as we are using it here. In this sense, we are both participants and researchers in this study and narrative inquiry, focused on lived experiences, exists simultaneously as product (data) and process (analysis) (Clandinin & Connelly, 2000; Connelly & Clandinin, 1990). “Designers, I shall argue, are in transaction with a design situation; they respond to the demands and possibilities of a design situation, which, in turn they help create,” (Schön, 1992, pg. 4). This is the story of how Leah and Dana acted together in transaction with a design situation. We begin with Leah’s first-hand account of the design process with the purpose of examining more closely the experience of riding a zip line. Together we will then tell the story about how we engaged with one another to work around an obstacle. Then, we will work together to assess what we learned from the task as well as writing this narrative. 4.1. Defining the problem Leah: My goal was to create an animation that realistically showed a person moving down a zip line while taking into account three key variables: the weight of the rider, the shape of the zip line, and the speed and acceleration of the rider. To model a person 6

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zip lining, I needed to understand precisely how a person moves along a zip line from a mathematical perspective and how this movement related to each of my variables. The rider’s weight remains constant as he or she zips down the wire. This weight is known before a rider even clips into the zip line, but not every rider has the exact same weight. The rider’s weight affects velocity, acceleration, and the shape of the curve of the wire that forms the zip line. A rider who weighs more will accelerate faster and have a higher velocity than a rider who weighs less. Additionally, a rider will move faster along a zip line with a large vertical displacement between the starting point and the lowest point because the zip line starts out with a steeper slope. At the same time, the shape of the wire is dependent on a rider’s weight and speed, but changes shape as the rider moves along it. My challenge was to create an animation that simultaneously accounted for all three variables. Riding a zip line is entirely different than creating an animation of a zip line rider in IGS. In the physical world, gravity automatically acts upon objects and we do not think twice about it. When zip lining, I have noticed how I accelerate along a zip line and come to a stop at the right time. When I watch other people zip line, I can see the sag of the wire as they zip down. In the virtual world of IGS, gravity must be created mathematically. Starting out, I had little knowledge of how to do this. That was how I problematized the task. 4.2. From evaluative to evolutionary prototyping Leah: To establish criteria and amass resources for building my design, I began by researching zip lines to find answers to all my questions. One of the first and most useful resources that I came across was the website, “The Science of Riding a Zip Line,” (Lang, Stocking, & Hoover, 2017). The website introduces relevant terminology and explains the mathematics involved in calculating a rider’s maximum speed on a zip line. I learned that a free hanging wire, without the weight of a person, creates a catenary curve. (I had never ever heard of that curve before, requiring further research on my part.) Additionally, the website triangulates a zip line for the purpose of calculating a rider’s maximum speed. The zip line is simplified into two right triangles where the hypotenuse of each spanned the distance from the endpoint where the zip line was secured to a tree or pole to the lowest point of the wire. I referenced the equations on this website to assist me in making my own. I had an insight when I came across the graph in Fig. 1. It displays the loss of acceleration, due to friction, air resistance, and other factors, based on a rider’s weight as he or she moves along a wire. I developed equations directly from this graph to enter into my IGS file. I continued my research to learn more about catenary curves, a curve formed by a rope or wire that is hanging freely from two anchors, as this is the shape of a zip line when there is no added weight on the wire. Finding the equation of the curve was straightforward. However, I wanted to find or create an equation that modeled the sag in the wire as a rider zipped down it. Additionally, there are two ways for the rider to slow to a stop depending on whether they are on a zip line course or a single zip line. On a zip line courses, the rider stops using only manual braking. I chose to create a single zip line where the rider slows naturally through oscillations before they are manually braked. To investigate the forces that were acting on a rider, I sketched a free-body diagram in Fig. 2, something I learned from my high school AP Physics class. I hoped that I could make some equations that modeled the speed of the rider and the shape of the wire from fully understanding the forces that acted upon the rider. This sketch helped me to understand the forces as an object, in this case a person, was hanging from a wire at a specific point. The forces I had to consider included gravity, friction, tension, and air resistance. These forces changed at every point because the angle between the rider and the cable changed. I could not figure out how to create an equation that would allow me to model the sag in the wire from my sketch. Due to time constraints, I decided to not take into account the sag in the zip line when I created my animation. This fixed one of

Fig. 1. Determining loss based on a rider’s weight (Lang et al., 2017). 7

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Fig. 2. Free-body diagram of rider and related equations.

my variables, the shape of the zip line, to be a catenary curve. I also viewed the weight as fixed because it did not change throughout a single ride. I created my first prototype to take into account the direct relationship between a rider’s weight and the corresponding loss in acceleration. I linked these together using a slider within IGS, leaving little mathematical ambiguity because I developed equations directly from Fig. 1 to explain this relationship. All I had left was to develop a way to determine the rider’s speed along the wire so that I could animate the rider in IGS. (While the rider’s velocity is dependent on their acceleration, I was limited to animating a point based on velocity along the x-axis within IGS.) This relates back to the challenge of creating gravity mathematically which involved an enormous amount of experimentation and validation within IGS while I played with many different iterations. First, I referenced the equations in “The Science of Riding a Zip Line” to develop and manipulate similar kinematic equations that could represent a rider’s velocity at any point along a zip line (Lang et al., 2017). That did not work how I wanted it to. Then I found the derivative of the catenary curve, hoping that would show the velocity of the rider at any and all points along the curve. Next, I found the velocity in the vertical direction and used that and the line tangent to the catenary curve to find velocity in the horizontal direction. In the end, I decided to combine two of these methods together because, when I experimented with animating the point, it accelerated in a motion similar to a zip line rider. The prototype that I generated within IGS is pictured in Fig. 3 and shows the numerous functions I used to try to animate the rider, represented by point C, based on speed along the x-axis. Once the initial prototype was built, I analyzed the accuracy and appropriateness of it by considering. a few serious flaws. First, it appeared that the rider appeared to never move across the lowest point on the zip line; mathematically, this is because the rider’s speed approached infinity at that point. This is impossible in real life, but I could not figure out how to fix it within the animation. Next, the rider never slowed to a stop but instead would oscillate, always starting and ending at the same position on the wire. This contradicts how a zip line works so I tried to fix this flaw. I played around with some different equations and some calculus, hoping that somehow I would stumble across something that would change this. I wanted the point to slow to a stop at the lowest point along the curve, but that never happened in this sketch. I was completely stuck and frustrated, but I was also determined to create a more accurate animation than what I currently had. Looking back on this prototype now, I realize that at least part of my velocity equation was incorrect and my scaling may have caused everything from the equation of the catenary curve to the loss functions to be partially inaccurate. Despite this, I had learned about catenary curves, loss, weight, and how to animate a point based on a variable speed while creating my first animation. I also learned more about the kinematic equations of velocity and acceleration. However, this prototype was self-limiting and I needed to

Fig. 3. My first animation of the rider. 8

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look at modeling a rider from a new perspective. 4.3. Creating a “throwaway” prototype Leah: Over the course of the two weeks that I had to complete the project, I always had thoughts running around in my head about the accuracy of the animations I was building. I asked myself question after question: How will the scale of my zip line affect the speed of the rider? How fast does the rider accelerate? How do I modify a catenary curve to show the sag? If I assumed the rider’s weight did not change the shape of the zip line, how would this affect the speed of the rider? Does the rider slow down faster if there is a sag in the wire (caused from the rider’s weight)? Asking these questions served two purposes. First, they helped me to use my conceptual understanding of zip lines to continuously develop my mathematical understanding of them. Second, these questions helped me consider to what extent my prototypes both realistically and unrealistically showed a person moving down a zip line. In an actual zip line, the weight of a person changes the shape of the wire. I was not able to find or develop a mathematical equation that represented the position of the rider or the shape of the zip line in terms of my three variables, the weight of the rider, the shape of the zip line, and the speed of the rider. Early on while developing my first prototype, I made an assumption that the weight of the rider was negligible, so it did not change shape of the wire. This reduced the complexity of the mathematics to something I could manage within my time constraints, consequently reducing the overall risk I took and the accuracy of my animation. This decision remained in my mind throughout the project. It drove me to create mathematics to determine the speed of a rider at any point along the catenary curve and create a representation of this in an animation; this proved to be more than a twoweek task in itself. Every decision I made in this project impacted both subsequent decisions and the final outcome. I wondered, what if I did not hold the shape of the wire constant; would the speed of the rider have been more accurate and easier to determine and represent? The scale of my first prototype of the animation likely factored into the speed of the rider. I tried to scale the catenary curve of my first prototype so that one square unit equaled one square foot. In my second prototype, I sacrificed this aspect to focus on the physical appearance of my ‘polished’ animation due to time constraints. During the research phase of creating my second prototype, I found what was a gold mine of equations, so much so that the document is only physical papers I kept from this project. This document contained many pages of equations, diagrams, explanations, and graphs dating back to 1927 in which it detailed the physics and mathematics of how tramways work. This find caused me to reconsider how a rider’s weight affects the shape of the zip line from both physical and mathematical perspectives (Advanced Cableway Equations). A portion of the document is in Fig. 4. I analyzed this document with the goal of developing an equation that would more closely model the sag in the wire. The equations within the document detailed the physical impact of a weight hanging from a wire and took into account wire tension, length, important angle measures, and more. I wanted to transform these equations into a single equation that would show the shape

Fig. 4. Excerpt from Tramway Engineering, LTD., 2017 Advanced cableway equations. Retrieved from http://www.tramway.net/Advanced %20Equations.pdf. 9

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of the wire when a weight was hanging from any point along the wire. I struggled to make enough sense of the tramway document to be able to do this. From what I could tell, many equations in the document assumed that there was a known distance from where the cable was anchored to the location of the weight. None of the equations modeled the shape of the curve, but I thought there was a chance that I could combine multiple equations to get the equation I wanted. I kept going back and forth in my head trying to figure out a way to consolidate the mathematics into an equation of the shape of the wire. The challenge I kept running into was how to relate all the variables. The shape of the curve affected the rider’s speed and vice versa so I would need to figure out how to separate these variables to make a position equation. Also, I was unsure of how to model the shape of the wire with an equation, even if the person was moving at a constant speed. I wanted to dive into these equations more, but it may have taken weeks or longer. If I had more time to analyze and understand the equations in depth I could probably have figured out a way to use them. However, I had under a week left to create my final animation. I decided to abandon or “throwaway” these ideas since I did not have enough time to develop them into something useable. 4.4. Return to evaluative prototyping Dana: Leah was stuck and came to see me after class. I was overwhelmed by the project at first. When Leah showed me her sketch, she expressed frustrations with being unable to make the rider move naturally. Being dropped into the project with very little introduction, it was difficult for me to figure out how to help. For me, it was not so much the complexity of the mathematics or technology, but the complexity of Leah’s inquiry that caused me to struggle. Leah’s visit to my office was a bit like someone opening a tightly packed trunk only to have the contents explode all over the room. There was an obstacle, but to engage me with overcoming that obstacle, she also had to unpack all that she had learned and struggled with along the way. It was important to me to walk beside her in this struggle and to do that, it was necessary for me to understand both the obstacle she was seeking to resolve and also how she had arrived at that obstacle in the first place. At that point, I was attending to her mathematically, but I needed to attend to her emotionally as well. I could tell that the project was the source of some anxiety by the urgency with which she talked about wanting to resolve it. I could also tell that this was a source of great pride. After listening to her talk about the mechanics of her design and ways it was both satisfying and frustrating to her, I asked her to show me her prototype. It was clear that what she had developed thus far was more than she had expected of herself and that she was delighted by the animation in the way that I might be after completing half of a jigsaw puzzle before standing up for a good stretch. The prototypes she had created became discussion tools. We talked about her expectations compared to her designs. I was careful in my questioning to ensure that she remained the authority on this project. I offered some mathematical tools that I thought would be helpful, but left their usage up to her. Leah: Dana suggested that I research damping functions, sine functions, and parametric equations and then think about the mathematics of animating a rider with these functions in mind. I was really resistant to the idea of rebuilding the animation at first because it meant I would need to start with a new blank file, new equations, and essentially have to repeat the time consuming process it took to build my other animation. (I really was enjoying the project and all the challenges of it, but there was only so much time I could spend on this assignment while keeping up with all my other classes.) I was completely stuck though. I also had a problem to solve. The more I thought about Dana’s suggestions, I realized I was going to have to make a new prototype, as frustrating as it was to throw out my first one. I kept a few parts of my first animation. I decided to continue to describe the shape of the zip line wire with a catenary curve; as much as I wanted to find a curve that modeled the sag in the wire, there was just not enough time in the next seven days to do so. I did transform the catenary curve into a parametric equation so that I could animate the rider based on time. I also kept the equations that represented the loss in acceleration based on the rider’s weight (from Fig. 1) since these were tried and true values from an actual zip line. I knew this was not causing issues with the speed of the rider, it was just how I used this value that could have an impact on the rider’s speed. One aspect of the first animation that I did not keep was the scale of the zip line; the first I scaled to be representative of the length of an actual zip line, the second was not. Then it was on to Dana’s suggestion to consider damping functions. I researched them some before I created one of my own. I animated my rider based on this function so it would show the rider slowing down during every oscillation along the zip line. I wanted the rider to complete one full oscillation while slowing down due to the damping function, which represents the outside forces that cause the rider to naturally slow down. After that, I wanted the rider moving slow enough to be manually stopped at the lowest point on the zip line. The function I created behaved partially how I wanted it to. During each oscillation, the rider did not go as high as the previous one. The rider did not slow to a stop after the second or third oscillation, but I got much further on the second animation than I did on the first using by animating the rider based on the damping function. The new challenge was how to quickly slow the oscillation to a stop. I experimented with modifying the damping function and the different variables to make this happen, but it did not seem like any of these adjustments made noticeable progress to my goal. Instead I considered my overall progress on the project and decided to settle for what it was. I was able to make a rider move down a zip line and eventually slow down. The loss functions were correct and I think my damping function was somewhat accurate too. I was likely limited by the shape of the wire and the scaling of the second zip line. To finalize the appearance of my animation, I edited the aesthetics of it, as shown in Fig. 5, which addressed the issue of stopping the rider. To made the rider stop, I created an image of a stationary rider on a ladder that would appear at a specific time, precisely the same time and location when the image of the moving rider disappeared. This imitated a rider stopping and getting off a zip line. The image of the rider is rectangular and so it contains three points at the vertices that I can use to resize the image. The image is 10

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Fig. 5. My final animation.

actually attached to three catenary curves, all vertical translations of each other and two of which are hidden. All the points are animated at the same speed to make the image of the rider appear move while staying the same size. This caused two more issues. Over time the rider would change sizes due to the distance between curves. Also, the rider’s starting point changed depending on where the rider ended the during the last animation. This could cause the rider to not oscillate over the lowest point. I did not want this to occur when Dana graded my project so I wrote the instructions in the top right corner so that she would know how to restart the animation to ensure that the rider would not appear to change size or position along the curve.

4.5. Articulating knowing in action Dana: The role of IGS in this task is to enrich the validation phase of the design process and to ensure that the mathematics is made explicit. When animating a sketch, observing, and then evaluating its behavior, we have a method by which to validate the design or to suggest direction for a subsequent iteration. Furthermore, utilization of the software requires students to adopt a mathematical lens and structure for their design. Taking the zip line as an example, we can see the role IGS plays in making the math explicit. Creating a physical model with string and a weight would enable the maker to take gravity for granted, as it would automatically act on the weight. Modeling the zip line with IGS requires the maker to attend to gravity in an explicit way. Asking how to get the technology to do something requires that students simultaneously ask, what is the mathematics behind what I am trying to do? In this sense, the mathematics behind the sketch is explicit both in terms of whether the model works, but also because it is on display within the sketch. The product is very much available for analysis, but the process is a different story. While I ask students to submit their designs as IGS files at the end of the project, I also give them time in class to present their design to others. Furthermore, while a demonstration of each design is expected (and often delights), it is just as important to prompt each student to share more about the process by which the design emerged. Just as with Leah’s zip line, the students are in the best position to assess the meaning behind their design and to tell me whether and what they have learned from the mathematical design process. Leah: The animation that I turned in was actually only a small portion of what the final product would be. Though I turned in my project, I still had many unanswered questions because my animation did not fully or accurately show a zip line and rider that behaves naturally. I started the project with just one question: how do I create a person riding down a zip line? I ended with many more unanswered ones. Some of these include: How do I make the person fully stop after passing by the lowest point in the wire twice? How do I scale the length of the zip line appropriately? How could I make a zip line so that one end of the wire is lower than the other? If I had more time, the next step would be to tackle the equation of the catenary curve so that I could calculate an equation of a curve that is affected by the rider’s weight at any given position. I would have to start over again, this time modifying the equation of the zip line, while keeping the loss equations and the damping equations the same. I would also correct the scale. This subsequent iteration would get closer to creating a realistic and accurate animation. Dana: In many cases, students cannot articulate knowing in action; just because it is difficult to verbalize design thinking does not mean that it did not occur or that it was not profound. It is difficult to assess or analyze the process of mathematical design. Here, Leah has preserved artifacts and has remembered for us not only her design, but the intention and knowing in action. I have learned from writing and reading this emerging narrative to trust that even if it cannot be expressed so clearly as here, the act of designing 11

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mathematics is an act of knowledge generation that defies accountability measures that require proof or evidence of it. Resnik (1994) challenges educators to create environments that “engage learners in construction, invention and experimentation,” (p. 24). In my attempt to do this in our classroom, I had to “design things that allow students to design things,” (p. 24). In that way, my work was to establish a decentralized design culture in my work with students. By decentralized, I mean that I adopted a participatory role, much as I have done in this narrative. My role in the classroom and in this paper is not to evaluate, lead or command, but to provide an environment there where mathematical knowing in action might occur and be made explicit. Doerr and English (2006) found that MEAs (Section 2.1.1) enable teachers to learn alongside their students, a claim that my experience echoes. I believe that my learning, and I suspect the learning of these teachers, is directly tied to the culture of design. They note that teachers learned to engage in forms of interpretative listening and observing. This is the consideration of student experience (Section 2.1.2) and honoring students mathematical authority and autonomy (Section 2.1.4). During her office visit, I practiced this with intention and then encouraged Leah and her classmates to do so with one another during presentations. 5. Discussion To conclude, we will discuss the ways in which principles of design were influential in creating the environment for this experience to unfold. We will pay close attention to the ways in which prototyping facilitated the modeling subprocesses of aligning and communication. Finally, we will parse specific prototyping actions evident in our narrative. 5.1. Creating a culture of design The five design principles were integral in creating a design-centric culture that supported mathematical modeling. The assignment was purposefully ambiguous and required imagination. Leah, having selected the context of a zip line, worked in a nonlinear and iterative way back and forth between that real context and the mathematics behind it. Using Geogebra, she created a mathematical version of a real life event and held that mathematical version accountable to reality, but also learning more about the reality of a zipline as she worked. This non-linear process was necessitated by the complexity of the task at hand, which also provided a unique opportunity to articulate mathematical knowledge in action. Zbiek and Conner (2006) hypothesize that given a more structured task, the modeler follows a more linear path. This example is one of the opposite and tracks the meandering and iterative path supported by a complex task. In order to manage that complexity in the context of a mathematics classroom, a careful balance was struck between Leah’s ambition and Dana’s tolerance for failure. With a lower tolerance for failure, Leah would have taken on a less ambitious project. This power differential is evident in a classroom where the instructor takes on most of the responsibility for assessment and evaluation. Here, it was the instructor who bore the responsibility for establishing acceptance of failure as an instructional principle and to communicate that to Leah. Creating a safe environment for risk and failure required that both of us recognize that mathematical modeling was a process rife with emotion. We both made room for emotions such as frustration, pride, embarrassment, and anxiety as natural byproducts of the modeling experience. Furthermore, the responsibility of exhibiting thoughtful restraint needed to be placed solely in the hands of Leah, the student. As the instructor, Dana could not stand in judgment of Leah’s determination of what aspects were important to pursue and which to leave behind. 5.2. Prototyping as alignment and communication Zbiek and Conner, (2006) position aligning and communicating as subprocesses that permeate the modeling cycle. Aligning is the work of determining whether a mathematical model is appropriate for a given context or purpose and reconciling inconsistent results. Communicating is putting forth ideas, information, or details. Prototyping, for us, served as a way of articulating mathematical knowledge in action because it brought mathematics into a place for aligning and communication. In this case, prototyping was a way to bring ideation and research about the mathematics behind a zip line to a technological place for consideration and validation. It was also a structure that enabled both iteration and evolution. Lastly, it was a record of practice that enabled the two of us to have ongoing conversations about the sufficiency of the developing model. Under the umbrella of rough draft thinking (Jansen et al., 2017), prototyping may be one way to offload the emotional burden of talking about work that is incomplete. While there is some conceptual overlap between prototypes and solutions, prototypes can have a temporal quality and lack the permanence and sense of completion of a solution. Labeling work as incomplete, or identifying thinking as prototypical makes students more comfortable making their work public (Thanheiser & Jansen, 2016). 5.3. Distinct protoyping actions Prototyping is a way that students can move fluidly in a non-linear way within the solution space for a given problem. Within the solution space, we have identified four distinct prototyping transactions along two distinct spectra: Throwaway vs. Evolutionary and Evaluative vs. Generative. In the first episode, Leah engaged in a full prototyping cycle (Fountain, 1990). As Leah created her initial design, she did so with an evaluative intent. The ultimate goal was to build a prototype that could be tested. This evaluative prototype was intended to be a fully operational test case. Within this initial prototyping cycle, criteria were developed, a prototype was constructed using IGS, and 12

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then the prototype was evaluated and found lacking. We can also think of this prototype as evolutionary in that some of what was created was absorbed into the final solution. Within that initial cycle, Leah created sketches and equations based on free-body diagrams she had learned to construct in her high school physics class. These sketches also served in a prototyping capacity, serving as mini strawmen aimed at generating a more rigorous idea of how forces interacted within the mechanics of the zip line. The second episode features a throwaway prototype that was generative in nature. We characterized it as a throwaway prototype because what she developed in this intermediate cycle was eventually tossed out because it was not feasible given the time constraints. In this sense, though the mathematics would have supported further development, the constraints of the assignment prevented it from being pursued. We consider it to be generative because it provoked new thinking about what she could expect from her solution. It exposed more of the complexity in the phenomenon and exposed mathematics that was new to Leah. In deciding to start over, we see Leah’s agency to show thoughtful restraint; she is empowered to impose limits on her design. The third episode culminates in a solution to the initial problem: to create an animation that simultaneously accounted for weight, speed, and the shape of the wire involved in a zip line. While Leah had thrown away her previous work and was starting fresh, she was not back at the starting line. Her initial prototypes had exposed a great deal of hidden mathematics in the mechanics of a zip line, and also exposed additional expectations that Leah held for the animation. These were maintained along with some of the constructed mathematical objects within IGS, and her final design was new, yet supported by her prior attempts. Within the final cycle, Leah identified new challenges, incorporated additional mathematics into her design, and learned how to become satisfied with the final product. In this way, the final prototype was a return to an evaluative purpose. 6. Conclusion The Common Core State Standards for Mathematical Practice (Common Core State Standards Initiative, 2010) emphasize modeling within the curriculum at all grade levels. Within these standards, modeling is viewed as a creative process defined as “the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions,” (p. 72). As a component of the design process, we see students as having more agency than simply choosing and applying appropriate mathematics. We look at the work of mathematical modeling as encompassing a wide range of knowing in action as they move within the problem to understand the complexity of the problem at hand and then mathematize it. In this paper, we posit that prototyping is one way to both observe and articulate the mathematics within those actions. We have two recommendations for teachers who want to support students as they engage in modeling. First, for students to fully engage in the modeling process, they must not be limited to exploring the problem space; they should have opportunities to explore mathematical models in flexible and non-linear ways. We have come to recognize the role of iteration in this process, but not every action moves us forward. We have portrayed how moving within the solution space can also lead to dead ends; however upon reaching a dead end, students do not always reject everything that led them there. Often, reaching a place and deciding that it is a dead end is, in and of itself, productive activity. Helping students to recognize when to use different prototyping actions may help them move more flexibly within a mathematical frame and recognize conceptual gains along the way. This exploration may not be customary in the classroom as students are often conditioned to expect efficient and linear progress towards a solution and often may experience frustration or low self-efficacy when struggling to make “forward progress.” Prototyping requires that students at all levels find a balance between risk and ambition. In this process, the teacher must support students to expand their mathematical knowledge while also learning to tolerate failure in (and as a final result of) the process. Second, in a mathematics classroom, students should be asked to engage with complex phenomena including those that can be explored geometrically. Dynamic Geometry Software (DGS) can be instrumental in this. Examples, in addition to the animation project described here, might include creating a dynamic geometry sketch that embodies a kaleidoscope, or another to examine, understand, and describe the mechanical impacts of changing gear ratios. Perhaps the task is to use mathematical tools to do traditional design tasks such as logo creation. DGS need not be involved, and we can report some anecdotal success with programming Ozobots to do a choreographed dance routine. In all of these examples, the act of prototyping encourages students to act with autonomy while they work within the modeling cycle. Activities where prototyping occurs must have a broad solution space for students to explore. Students should have access to complex problems where there is not a single “correct” solution or strategy, nor should the same level of sophistication or mathematical design be expected from every solution. Currently in schools, prototyping is often included in projects that incorporate engineering concepts into mathematics courses (c.f. Project Lead the Way, 2017). Inquiry is a central tenet among programs that seek to expose students to prototyping or design thinking and one of the key aspects of the PLTW curriculum is that students will engage in finding unique solutions to an open-ended design problems. 6.1. Implications for research There are three implications of this study for future research. First, as a field, we need more opportunities to observe students working flexibly with mathematical entities and between those entities and the realities they represent. Lesh and Zawojewski (2007) call for researching into how to document and assess understandings and abilities related to problem solving, which relies on data that consists of “auditable trails of documentation that problem solvers generate as they go through multiple cycles of expressing, testing, and revising their ways of thinking,” (p. 795). In terms of modeling, we need ways to document and articulate mathematical knowledge in action. 13

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Second, more can be done to develop a full theory about the creation of design culture in mathematics classrooms. This might have some overlap with existing theories about using technology to teach mathematics. Third, we urge the use of more empathetic forms of research (DöAmbrosio & Cox, 2015) such as narrative inquiry. Methods that include the voices of students and teachers will enable us to observe mathematical design from a more personal perspective, which has the potential to illuminate more of our blind spots with regard to student experience. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References Araki, M. E. (2015). Polymathic leadership: Theoretical foundation and construct development (Doctoral dissertation, PUC-Rio). Brophy, S., Klein, S., Portsmore, M., & Rogers, C. (2008). Advancing engineering education in P‐12 classrooms. Journal of Engineering Education, 97(3), 369–387. Cirillo, M., Pelesko, J., Felton-Koestler, M., & Rubel, L. (2016). Perspectives on modeling in school mathematics. In C. R. Hirsch, & A. R. McDuffie (Eds.). Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics. Reston, VA: National Council of Teachers of Mathematics. Clandinin, D. J., & Connelly, F. M. (2000). Narrative inquiry: Experience and story in qualitative research. CA: Jossey-Bass. Common Core State Standards Initiative (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Connelly, F. M., & Clandinin, D. J. (1990). Stories of experience and narrative inquiry. Educational Researcher, 19(5), 2–14. Cox, D. C., & Harper, S. R. (2016). Documenting a developing vision of teaching mathematics with technology. In Handbook of research on transforming mathematics teacher education in the digital age. IGI Global166–189. 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S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350. Thanheiser, E., & Jansen, A. (2016). Inviting prospective teachers to share rough draft mathematical thinking. Mathematics Teacher Educator, 4(2), 145–163. Thompson and Yoon (2007). Why build a mathematical model? Taxonomy of situations that create the need for a model to be developed. In R. A. Lesh, E. Hamilton, & J. J. Kaput (Eds.). Foundations for the future in mathematics education (pp. 193–200). Mahwah, NJ: Erlbaum. Tramway Engineering, LTD (2017). Advanced cableway equations. Retrieved on October 12, 2017 fromhttp://www.tramway.net/Advanced%20Equations.pdf. Wagner, J. (1993). Ignorance in educational research or, how can you not know that? Educational Researcher, 22(5), 15–23. Zbiek, R. M., & Conner, A. (2006). 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