“Some Angles Are Gonna Be Weird”: Tinkering with Math and Weaving

Abstract

It has been argued that much of how math is taught in schools aligns with a particular epistemology that comes from western mathematicians and philosophers, potentially leading to an undervaluing of diverse skills and abilities. Tinkering, a common STEAM practice, is one way of participating that does not necessarily involve a straightforward path from problem to solution; rather, tinkering may be non-linear, and involve movement back and forth between known and new solutions. This process is not always supported or encouraged in traditional mathematics spaces but may be more available through activities such as crafting. This study examines a weaving workshop with middle-school students, asking the question: When and how do learners tinker in mathematical ways as they learn to weave? Video data were analyzed using qualitative techniques and perspectives informed by interaction analysis and other multimodal analytic techniques. Findings show that youth could be seen tinkering in the workshops in the forms of “negotiating and renegotiating with materials” and “trying unexpected solutions.” Examples from two focal cases break these tinkering forms down in deeper detail and showcase the mathematical engagement made possible by the space to tinker. This work sparks possibilities for designing math learning spaces that honor youths’ personally meaningful ways of doing and being both through the materials used and the practices encouraged.

1. Introduction

Mathematics has long been seen as a gatekeeper to success in graduation and career [1] and has been identified as a primary filter for girls’ and women’s STEM career pursuits [2]. While achievement and ability differences among genders in mathematics do not seem to be a significant factor (e.g., [3,4,5]), many, including people of color, struggle for acceptance and inclusion in math spaces (e.g., [6,7]). It has been argued that much of how math is taught in schools aligns with a particular epistemology that comes from western mathematicians and philosophers (e.g., [8]), potentially leading to an undervaluing of diverse skills and abilities (e.g., [1,9,10]) and a lack of validation for diverse materials and activities in math contexts (e.g., [11]). This may have an impact on who feels invited and who is allowed to be successful in STEM and STEAM spaces.

Constructionism [12] a foundational theoretical perspective within STEAM education efforts, purports that powerful learning can happen through providing opportunities for learners to create and share personally meaningful artifacts, centering process and participation as crucial to learning. This theory of learning also supports the idea that there can be multiple legitimate ways to participate in a given learning context such as math or computing [13]. Tinkering is one way of participating that does not necessarily involve a straightforward path from problem to solution; rather, tinkering may be non-linear, and involve movement back and forth between known and new solutions. Such processes can still lead to productive and even unexpected learning outcomes. Tinkering is not always supported or encouraged in mathematics spaces [3], but may be more available through activities such as crafting.

Weaving is understood as a particularly mathematical craft (e.g., [14,15,16,17]) and it has been seen that experienced weavers invent and create new structures and solutions in their work [18]. However, it is not yet known how tinkering with weaving produces math engagement, particularly with young learners. This study seeks to answer the question: When and how do learners tinker in mathematical ways as they learn to weave?

This exploration may have implications toward economic sustainability in the form of increasing diversity in the math and STEM workforce. However, the more exciting sustainability impact of this work is perhaps in the opportunity to uplift practices typically associated with women and other marginalized and minoritized groups, offering social and societal validation for the types of practices and activities that bring together communities, communicate values, and pass familial knowledge down from generation to generation.

2. Background and Theoretical Frameworks

2.1. Crafting Possibilities

Several research efforts have suggested a history of innovation and possibility at the intersection of weaving and mathematics (e.g., [14,15,16,17]) Additionally, prior work demonstrates that weaving involves elements of invention and experimentation with patterns, sequences, and structure, and that these ideas relate to the essence of seeking, understanding, and building patterns in mathematics. This relationship involves manipulating and inventing mathematical sequences [18], suggesting that weaving is in some ways unavoidably mathematical.

Engaging in mathematics through weaving also has roots in arguments regarding the use of tangible manipulatives in math contexts. Tangible manipulatives, or physical artifacts that can be moved and rearranged in various ways, have played a role in math education for nearly two centuries, from Fröbel’s gifts in the 1800s [19] to Cuisenaire rods and beyond. In particular, a productive suggestion regarding tangible manipulatives at all stages is to utilize them in ways that prompt exploration of new ideas, rather than reifying situations where math is only a set of procedures to memorize and carry out (e.g., [20]. Additionally, craft materials and practices have been used as particularly productive avenues for mathematics engagement, such as in the modeling of hyperbolic space achieved through crochet [21]. This concept had been deemed nearly impossible to model physically until the crochet model was created.

2.2. Tinkering in Mathland

The current work views learning from a constructionist perspective. Shifting our understanding of learning out of the head alone and alongside representations out in the world, Papert [12] asserts that learning happens in particularly powerful ways through the construction of artifacts. These artifacts are generally physical, but can also be theories or ideas. It is incredibly important that these created artifacts can be publicly shared and reflected upon [12]. Another crucial element of learning from this perspective is that the artifacts being created have some personal meaning or relevance for the makers. In the current study, these elements of designing, creating, and sharing personally meaningful artifacts are prevalent in the weaving space.

Constructionism also showcases the power of tinkering, or the process of trying multiple solutions while making that may or may not work. This process is said to be crucial to learning as it provides opportunity for learners to see themselves in the learning [22] Tinkering can be seen as a process where shifts in engagement, intentionality, innovation, and solidarity point to learning (e.g., [23]), and youth negotiating between their goals, ideas, and materials leads to discoveries about phenomena [24,25]. Turkle and Papert [13] also refer to this act of “arranging and rearranging, negotiating and renegotiating” (p. 5) as bricolage. They describe people who engage in a bricolage style of making as seeking their goals through collaboration with the materials [13]. Contexts that value bricolage or tinkering as a valid style of interaction are consistent with perspectives that value multiple ways of knowing and doing (e.g., epistemological pluralism, [13]).

Papert also writes about bricoleurs in his book Mindstorms [12], often using the terms tinkering and bricolage interchangeably. He describes learning through tinkering as working resourcefully with whatever is available to build up tools and materials that can be handled and manipulated. He also characterizes this bricolage process as an authentic way that scientific theories and discoveries have been made. Thus, he argues, we as learners owe tinkering processes more respect. Through this respect, Papert [12] describes possibilities of engaging in what he calls “Mathland.” Constructionism envisions contexts where “mathematics would become a natural vocabulary” ([12], p. 39), such that engaging in math is necessary for solving personally relevant problems and is supported through the design of the environment. In a Mathland, mathematics would happen authentically and various pathways of engaging and solving problems would be accepted and valued. Prior work has identified fiber crafts as a space where Mathland can become possible [26]. Papert also argues that ideas about math education and learning tend to assume there is one pathway to success in mathematics. However, designing environments that allow for tinkering may make it possible for learners to follow more alternate routes, such that fewer learners are pushed out of mathematics spaces.

Some classroom mathematics environments have made attempts at incorporating tinkering or other open-ended activity structures into the curriculum and culture. In particular, project-based learning, an approach that involves learners working toward an end goal that requires particular learning along the way (e.g., [27]), has become more common in mathematics and other STEM classrooms over time. One example demonstrates a lesson plan for “tinkering with buoyancy” [28] wherein students must build a boat using a constrained set of materials with a goal to build the boat that can hold the most freight weight. Through the project, students engage in inquiry-based making and document their research as they work [28]. Another example called the “zine machine” [29] brings together zine-making and mechanics in a tinkering project-based activity geared at helping students play and reason with mathematical concepts.

These are just two specific examples of activities designed to encourage tinkering in STEM and mathematics spaces, and many more exist. However, a meta-analysis of literature around project-based learning in math classrooms has shown that these types of activities tend to work best when the overall nature and culture of a math environment shifts toward a new orientation toward math [30]. This highlights the need for not just tinkering-based activities, but also a lens of epistemological pluralism [13] such that what is accepted as mathematical engagement is irrevocably shifted. Weaving and fiber crafts, as activities not typically associated with mathematics, may be useful toward such a shift.

In a similar vein, some educators call for opportunities for productive failure (e.g., [31]) in learning contexts in general and in mathematics specifically [32]. In this paper, a comparison is made between a traditional math class and a math class designed for productive failure. The study found that the students in the productive failure course outperformed the students in the traditional class. Interestingly, the students in the productive failure class also reported they felt less confident in their answers. The classroom in this study provided very complex problems for learners to solve and withheld structure and particular supports for a set amount of time. Alternatively, in open-ended spaces such as craft workshops, learners tend to solve problems, often complex, that they choose themselves. Additionally, such spaces are generally flexible enough to allow for re-defining and renegotiating of the problem through the process, separating these spaces from traditional productive failure contexts. Nonetheless, as mistakes, or failure, are inevitable in open-ended design contexts, some researchers have worked to frame these processes as productive failure [33]. While helpful, productive failure as a framework in open-ended design environments is still different from traditional productive failure environments, as there is no single correct solution to model or achieve. These multiple possible pathways signal possibilities of epistemological pluralism [13] and Mathlands [12], indicating that tinkering remains a useful constructionist perspective in the current work.

2.3. Equitable Math Environments

Further, tinkering in mathematics spaces may not always be available or valued. In a context centered on equitable access to mathematics learning, researchers found that a key facet of mathematical agency was mathematical tinkering [3]. Many of the learners in this context felt left behind or left out of their typical math classrooms and pointed to the opportunity to tinker with mathematics as helping them appreciate and view math as both possible and interesting [3]. These youth engaged in mathematical tinkering through actions and practices such as having more time to work on tasks, choosing more difficult and more complex problems to solve, solving problems using invented rather than prescribed methods, and discovering new problems to solve with tangible manipulatives [3]. In the current study, we can understand tinkering with weaving to be a form of mathematical tinkering due to the presence of similar practices such as choosing complex problems to solve, using invented rather than prescribed methods, and making discoveries with tangible manipulatives. This adds to the link between weaving and math and the possibilities for learning through tinkering. In the current study, I search for examples of math/weaving tinkering practices in the tradition of education research that values practices and participation rather than concepts alone (e.g., [34]). In addition to computational thinking practices outlined by Brennan and Resnick [35], echoes of valued practices are also seen in more standardized contexts such as the Common Core Mathematics Practice Standards. Rather than attempting to search for standardizable practices, however, this work looks for unique and smaller-scale practices.

Considering inequities that still exist in math participation and achievement (e.g., [3,36]), and efforts in the field to address equity and justice more generally (e.g., [37,38,39]), I view this work as one small step toward more equitable futures in STEAM, and in mathematics in particular, for traditionally minoritized learners. I work to consider all learners and their right to learn, take up space, and make a mark in the learning space (see rightful presence, [40]) and work to recognize learners’ contributions as valued and valid (see intellectual dignity, [38]).

3. Methods

This study focuses on a workshop that took place in a middle-school setting, where youth self-selected to spend time weaving as a design activity. I ask the question: When and how do learners tinker in mathematical ways as they learn to weave? I use qualitative video analyses here in an effort to describe and understand the actions and practices that make up tinkering in this space. All names are pseudonyms to protect the confidentiality and privacy of the participants.

3.1. Participants

This study took place in a mid-sized Midwestern city within a charter school focused on creative and project-based learning. In this school, a mixed-age classroom containing youth in grades 6 through 8 (12–14 year olds) holds a weekly session called Design Studio. During these sessions, youth can take on design projects that are of personal interest and meaning to them. Youth had the option to select weaving as their design project; 13 youth (6 boys and 7 girls) opted in to do weaving with me. At the school overall, 77% of students are reported as white, 12% as two or more races, 8% as Hispanic, 2% as Asian, less than 1% as Black, and less than 1% as Native American or Native Alaskan. Additionally, about 29% of students are eligible for free or reduced lunches. Descriptions of the whole group are presented at two particular time points, and two focal youth were chosen for a deeper-level analysis.

3.2. Focal Youth

The two youth were chosen because they were typical for the classroom in that their work was not necessarily exceptional but was consistent and interesting. However, they did verbalize their thoughts more than some others, making more of their thinking visible at times. Additionally, many of their making processes were visible on camera and they could be seen consistently working and making throughout the majority of the sessions. These two youth also had brief check-ins with me during the workshop that were video-recorded and that showcased their thinking. Both of these youth were also seen trying multiple projects and strategies.

Tina is a middle-school girl with no prior weaving experience. She worked with a partner often during the workshop and they talked with one another. Tina was often seen pulling out threads she had already woven or using tools to make sure her woven rows were neat and even.

Sandy is a middle-school girl with some prior weaving experience. She was seen and heard often verbalizing her thinking unprompted. She began the workshop with ambitious plans for a larger project, and ultimately tinkered and improvised with several smaller projects instead.

3.3. Workshop Design

The current study focuses on different aspects—primarily action and process—of the same weaving workshop described in forthcoming work [41]. The workshop took place over six one-hour long sessions during the school day. Two participants were absent from school during one of the sessions; otherwise, the participants attended all six days and stayed for the full session. In these sessions, youth used frame looms that were laser-cut from an open-source pattern to create their woven artifacts and grid paper to plan their designs (see Figure 1 for an example image of the loom and grid paper).

Due to the accessible nature of the looms, youth were able to keep their looms and continue work on their projects after the end of the workshop. The six sessions were designed to provide time for the youth to steadily work on their projects while also prompting them to engage with mathematical ideas through explaining their work and showing their processes on paper. Briefly, the workshop objectives were as follows:

• Day 1: introduce examples of weaving designs and grid paper planning; warp the looms
• Day 2: learn to weave (without pre-planning)
• Day 3: plan weaving designs on grid paper; complete initial project
• Day 4: begin weaving second project
• Day 5: continue weaving second project
• Day 6: complete second project; reflect on process; return to examples of weaving designs and grid plans

In addition to the looms and the tools that came with them, youth also had access to yarn in multiple colors and thicknesses, string meant for setting up (or warping) the looms, scissors, pencils, tapestry needles, and grid paper. Youth were encouraged to make at least two projects during the workshop and to try new techniques and styles in their second design. This plan is based on my prior experience hosting multiple crafting workshops, including two pilot weaving workshops I led with middle school youth.

3.4. Data Sources

The weaving workshops were filmed using four video cameras that ran continuously throughout 6 workshop hours. Two 360° cameras were placed on the table to give a view of hands and faces—these cameras provided the best view, and the bulk of the analysis was from these cameras. This view showed youth’s faces, hands, and the positioning of their bodies in a way that shows the construction of the artifacts as well as whether and how the projects are positioned and seen by others. This supported analyses from a constructionist perspective that values action and artifacts. Additionally, two wide-angle cameras were placed at the corners of the table for backup filming and provided another angle on the participants. Last, individual check-ins with participants centered on their progress and how they were working through problems were filmed on a handheld camera focused primarily on the youth’s hands and projects. This again centers action and artifacts.

3.5. Analytical Techniques

The video data were analyzed using qualitative techniques and perspectives informed by interaction analysis (IA; [42]) with additional inspiration from multimodal interaction analysis [43] and mediated discourse analysis (MDA; e.g., [44]). These methods are rooted in commitments to the social nature of knowledge and action and place importance on the ways practices, actions, and interactions signal to larger discourses such as math, weaving, or tinkering. Focusing on interactions among youth and materials aligns with a constructionist perspective on learning that centers actions and artifacts.

In the analysis process, I first watched through all the data in search of repeated actions, focal students, or focal moments. This allowed me to start identifying high-level actions among the students, including preparing yarn, designing on paper, counting, warping the loom, weaving, and un-weaving. I also identified focal students as those who participated consistently but seemed to represent typical patterns of interaction. I needed to explore more deeply what was happening and what made it possible, and thus took another analytical pass.

Next, I used repeated views of the data to identify what actions the focal students were doing and when. I tagged segments of data for their primary activity (e.g., weaving, preparing the loom, talking with friends) and their flow of participation (i.e., order of actions). From this, I was able to note patterns of action and participation for focal students. Through repeated viewings of these actions and interactions, it emerged that youth were often seen doing, undoing, and redoing their work, which could be understood as negotiating and renegotiating with the materials—an element of bricolage or tinkering. It also emerged that youth were trying solutions that were surprising or unexpected as I observed, another aspect of tinkering. Focusing in on two days (Day 2 and Day 4) where the youth were largely seen working consistently, I looked more carefully for unexpected solutions and renegotiations with materials to gain a sense of the prevalence of these actions across all the youth.

Last, I did a separate pass through the individual check-in videos to pull out further verbalizations of the focal youth’s plans, processes, and progressions. Focusing on the aspects of tinkering outlined above, these individual check-ins provided information from the focal youth’s talk that helped locate and specify their tinkering practices. Each focal youth engaged in negotiating and renegotiating and trying unexpected solutions in varying and specific ways. These check-in videos led to further naming and narrowing of tinkering practices around materials, pattern, precision, tools, sequence, and calculation.

4. Results

Viewings of the data for overall tinkering practices pointed toward Days 2 and 4 as times the youth were working consistently on different stages of the work. On Day 2, the first projects were well underway, and youth were getting the hang of weaving rhythms. On Day 4, the second projects were underway, and youth were beginning to experiment more as well as moving on from their practice projects (see

Table 1. Overall tinkering.

	Timepoint Seen Prominently during the Workshop	Number of Youth Seen at the Prominent Timepoint	Seen Specifically in Focal Youth Through
Negotiating and Renegotiating with materials	Day 2	6/13 (46%)	tinkering with materials, sequences, calculations
Trying unexpected solutions	Day 4	5/13 (38%)	tinkering with pattern, precision, tools

).

On Day 2, one of the ways tinkering manifested was through youth negotiating and renegotiating with materials like pulling out woven threads in order to re-weave them. This view of tinkering comes from the way Turkle and Papert [13] describe the act of bricolage. It also necessitates a safe space for making and repairing mistakes, which may not always exist in typical math classrooms. Renegotiating with materials allows discoveries about the materials and the mathematical underpinnings of the structures. On this day, six of the thirteen youth were seen doing this at some point. This does not mean that other tinkering or non-tinkering practices were not done by these youth, or that renegotiating with materials did not take place on other days. This finding does demonstrate that pulling out threads was a practice done by about half the participants on this day, suggesting it is a practice to look at more deeply on other days and in the focal cases to better understand its importance and connections to mathematical thinking.

Later in the workshop, on Day 4, one of the ways tinkering manifested was through youth trying unexpected solutions, such as using pencils as needles and incorporating unintended occurrences into their designs. This view of tinkering is related to the way tinkering has been described as working with materials in ways that lead to new discoveries about phenomena [24,25]. Tinkering in this way also requires that multiple ways of thinking, doing, and solving problems are allowed and seen as valuable, which may not always be the case in typical math classrooms. Trying unexpected solutions allows youth to play with and push the boundaries of the materials and structures, as well as suggesting that the inherent math in weaving can be flexible. Five of the thirteen youth were seen doing this on Day 4.

This does not mean that other tinkering or non-tinkering practices were not done by these youth, or that unexpected solutions did not appear on other days. This finding does demonstrate that trying unexpected solutions was a practice done by just below half the participants on this day and that it is a practice to explore more deeply on other days and in the focal cases for its importance and connections to mathematical thinking.

In the remainder of this section, I explore in more detail the participation of two focal youth to show more closely what tinkering with math through weaving can look like (see

Table 2. Focal youth tinkering.

	Tinkering Example	Tinkering Explanation	Math Explanation
Tina	With materials	Tina decided to start over so that her yarn was long enough to work with but did not get tangled in the process. She cut and pulled out all the threads she had already woven. “This time I’m doing it shorter!”	Implications for considerations of two- and three-dimensional space and the impacts of different materials on measurements such as length and width.
	With pattern	Tina thought about embedding a shape into the center of her project: “we would start and then we would like do like one color then stop midway and do that one color and then like keep on moving it.”	Implications for understanding how shapes are built and how measurements such as length, width, and area are both related and separate.
	With precision	Tina was trying to switch from using blue yarn to white yarn and wanted the transition to be seamless. “Oh! Look at this transition from color to color, you can barely even realize that that’s the blue and white!”	Implications for understanding the importance of precision in elements such as shape, size, pattern, and sequence.
Sandy	With tools	Sandy wanted to use the loom tools to simulate the moving parts of a different type of loom. “This won’t work!” She explained to me, “I realized that the last loom I had worked on where this had worked, it was 3D and the pieces could pass through each other.”	Implications for Sandy’s understanding of the relationships between mathematics and engineering, as well as the intersection between physical materials and mathematical principles.
	With sequence	Sandy found a way to hack the over one, under one sequence “On this one it’s easy ‘cause I just go underneath this one, underneath the first one and then over the rest of them.”	Implications for engagement with the underlying mathematical principles of weaving by seeking, understanding, and building patterns and relating those patterns to structures and materials.
	With calculations	Sandy realized her design had not taken into account the actual size of the loom. She planned to flip the orientation of her design “I’m gonna make this [horizontal] 29 instead of having this [vertical] be 29, so I don’t have to completely redesign it. But that does mean that some angles are gonna be weird.”	Implications for her ability to relate multiple dimensions of measurement such as length, width, and area to one another.

).

4.1. Tina

Tina had no prior weaving experience and was often seen during the workshop seeking ways to achieve a certain level of aesthetic precision. Earlier in the workshop, she was seen pulling out woven threads to re-weave them to her liking. As she appeared to become more comfortable with the techniques and materials, she used the loom tools to flatten and smooth each row as she wove. Tinkering with each row to make it just right, she also often worked and talked with a friend and partner, comparing designs and plans throughout the sessions.

Tina warped, or prepared, two projects during the unit. When the grid paper was introduced, she initially planned a joint project with her friend so that they would each weave a segment and attach them together later. At the end of the workshop, it was unclear whether the resulting pieces were part of this joint project or had shifted into individual projects. As mentioned above, Tina shifted from pulling threads out to using loom tools to achieve her desired aesthetic. The following short vignettes showcase Tina tinkering with materials, pattern, and precision.

4.1.1. Tinkering with Materials

Tina initially cut a very long piece of yarn for weaving. Additionally, the tool she chose for weaving meant that she had a long tail of yarn hanging that needed to be pulled all the way through with every pass. This piece of yarn proved too long and became tangled many times within itself and with her neighbors’ yarn. After attempting to work with the yarn, she realized there was an ideal length at which to prepare her yarn so that it was long enough to work with but did not get tangled in the process. Tina decided to start over, cutting and pulling out all the threads she had already woven. As she did this, I checked in with her, asking if she would do something differently next time (see Figure 2, left). She replied, “This time I’m doing it shorter!” as she finished pulling her threads out. Changing her strategy in the future was not enough; she needed to start over right away, negotiating with the materials to reach her desired outcome. This process allowed Tina to engage with and consider both two- and three-dimensional space alongside the ways different materials relate to measurements such as length and width.

4.1.2. Tinkering with Pattern

After weaving for a while, Tina wanted to add something additional to her design. She thought about embedding a shape into the center of her project and asked me how she should do it. Rather than telling her, I asked her to think through it aloud to consider how it could be done. She told me, “Instead of doing the whole row one color, we would start and then we would like do like one color then stop midway and do that one color and then like keep on moving it, like where it is” (see Figure 2, center). I told her that this sounded great and encouraged her to also think about how the over, under sequences of weaving would impact that plan. Another way to do this would have been to weave all of one color first, then go back and fill in with the second color to avoid switching back and forth between yarns. This solution signaled an understanding of how shapes can be created in weaving at the intersection of over, under sequences and measurements such as length and width. This may impact the way Tina understands how shapes are built and how measurements such as length, width, and area are both related and separate.

4.1.3. Tinkering with Precision

Tina often worked with and talked to her friend who sat next to her during all the sessions. While I was checking in with the friend about a separate topic, Tina was solving her own problem. She was trying to switch from using blue yarn to white yarn, but, as she valued a certain level of aesthetic precision, she wanted the transition to be seamless. After working for a while, she discovered a way to tie the two pieces of thread together that did not show a knot or seam on the front of the project (see Figure 2, right). She interrupted my discussion with her friend to proclaim, “Oh! Look at this transition from color to color, you can barely even realize that that’s the blue and white!” This method was a unique way to change colors that signaled some understanding of the over and under sequences and their relationship to the movement of the yarn. This kind of tinkering helped Tina experience the importance of precision in elements such as shape, size, pattern, and sequence.

4.1.4. Tina Summary

Throughout the weaving workshop, Tina’s specific tinkering practices were related to some of the more general ways tinkering was seen across all the participants; her tinkering with materials signals negotiating and renegotiating, while her tinkering with pattern and precision relate to producing unexpected solutions. Tina’s tinkering also demonstrated avenues for math engagement, allowing her to tinker with mathematical ideas through the weaving activities. Her tinkering with the materials had implications for considerations of two- and three-dimensional space and the impacts of different materials on measurements such as length and width. Her tinkering with patterns had implications for her understanding of how shapes are built and how measurements such as length, width, and area are both related and separate. Tina’s tinkering with precision had implications for her understanding of the importance of precision in elements such as shape, size, pattern, and sequence. Together, these tinkerings suggest Tina was moving back and forth between known solutions and strategies and her own ideas and experiments. Boaler and Sengupta-Irving [3] (invoking [45]) discuss the importance of opportunities for learners to exert their agency in adapting and extending problems, suggesting that these opportunities, such as the ones Tina had here, can propel learning. Not only does Tina’s mathematical tinkering seem to create additional opportunities for learning and reasoning in mathematics beyond typical classroom activities, this tinkering does so in a way that allows for mistakes, echoes the three-dimensional spaces and problems present outside of classrooms, and is flexible rather than hostile as some math learning experiences have historically been.

4.2. Sandy

Sandy had some prior weaving experience with a different type of loom and was often seen and heard verbalizing her thought processes as she worked. Sandy had a pattern of starting new projects rather than continuing existing ones, and, as such, started over several times and tried different techniques. She appeared to be content to engage in the process regardless of the final products.

Sandy warped, or prepared, four separate projects during the workshop. Only two of these were taken off the loom as complete at some point. When grid paper for planning was introduced, Sandy planned an ambitious project with complex patterns and shapes. She did not complete this project as planned; some of her reasoning around this will be discussed below. Her later projects were created largely without designing on grid paper. Sandy generally worked alone but compared designs with peers twice during the workshop and chatted with friends throughout. The following short vignettes showcase Sandy tinkering with tools, sequences, and calculations.

4.2.1. Tinkering with Tools

Sandy had some experience with a different type of loom that has moving parts that allow the weft thread, or the working thread, more space to pass over and under the vertical threads. She started this workshop trying to take a similar approach and wanted to use the loom tools to simulate the moving parts of the other loom (see Figure 3, left). Sandy quickly realized that she would not be able to move parts around as easily as she had hoped. After exclaiming, “This won’t work!” she explained to me, “I tried to do this where it would be the opposite, where the ones that were going up were going down so that I could flip, but I realized that the last loom I had worked on where this had worked, it was 3D and the pieces could pass through each other.” Sandy’s negotiation with her loom here led to a deeper understanding of the relationships between woven structures and the tools that help create them. It also primed engagement with relationships between mathematics and engineering, as well as the intersection between physical materials and mathematical principles.

4.2.2. Tinkering with Sequence

Sandy was working on her first project and had created a motif that was surprising to me. Her rows did not appear to alternate as typical; instead, they all seemed to stack directly on top of one another. Usually, such a sequence would result in the thread unraveling as the horizontal threads would not be hooked onto the vertical threads. Sandy, however, found a way to hack the over one, under one sequence (see Figure 3, center). When I asked how she got the threads to not pull through, she explained, “On this one it’s easy ‘cause I just go underneath this one, underneath the first one and then over the rest of them.” This means she looped the thread around the end warp thread, and then continued on with her desired sequence. This was a unique solution that produced a striking appearance and suggested Sandy had some understanding of the ways weaving structure works and can be pushed. Tinkering with this solution also prompts engagement with the underlying mathematical principles of weaving by seeking, understanding, and building patterns with structures and materials.

4.2.3. Tinkering with Calculations

Initially, Sandy used grid paper to plan an ambitious weaving project with multiple shapes and angles. When she attempted to move from the paper to the loom, she realized that she had not taken into account the actual size of the loom. Her design assumed she would have space for 42 vertical wrap threads, but there was only space on the loom for 29 threads. Sandy explained to me her plan to flip the orientation of her design (see Figure 3, right), “I’m gonna make this [horizontal] 29 instead of having this [vertical] be 29, so I don’t have to completely redesign it. But that does mean that some angles are gonna be weird, like it’s not gonna be a specific slope of one or two or whatever, it’s gonna be really weird.” This reasoning showcases understanding about the impacts of shape measurements on angles and images. This also showcases Sandy’s ability to relate multiple dimensions of measurement such as length, width, and area to one another.

4.2.4. Sandy Summary

Throughout the weaving workshop, Sandy’s specific tinkering practices were related to some of the more general ways tinkering was seen across all the participants; her tinkering with sequences and calculations related to negotiating and renegotiating, while her tinkering with tools related to trying unexpected solutions. Sandy’s tinkering also demonstrated avenues for math engagement, allowing her to tinker with mathematical ideas through weaving activities. Tinkering with tools had implications for Sandy’s understanding of the relationships between mathematics and engineering, as well as the intersection between physical materials and mathematical principles. Her tinkering with sequences had implications for engagement with the underlying mathematical principles of weaving by seeking, understanding, and building patterns and relating those patterns to structures and materials. Sandy’s tinkering with calculations had implications for her ability to relate multiple dimensions of measurement such as length, width, and area to one another. By tinkering, Sandy was moving back and forth between known solutions and strategies and her own ideas and experiments. Turning back to Boaler and Sengupta-Irving [3] again, Sandy had opportunities to exert her agency and independence in these activities, allowing her to adapt and extend existing problems in ways that potentially lead to the production of new knowledge. Sandy’s mathematical tinkering created opportunities for mathematical learning and reasoning beyond what may be typical in classroom activities and allowed for active practice in three dimensions, making mistakes in a safe and accepting environment, and solving problems that resemble out-of-school contexts.

5. Discussion and Implications

Weaving gave youth opportunities to tinker, such as by negotiating and renegotiating with materials and making use of unexpected solutions throughout the workshop. These young crafters also tinkered with mathematical ideas by tinkering with weaving. By tinkering with materials, patterns, precision, tools, sequences, and calculations, the focal youth here engaged with mathematical ideas, such as by considering the relationships between two- and three-dimensional space and physical materials; considering precision in relation to elements such as shape, size, pattern, and sequence; seeking, understanding, and building patterns; and relating multiple dimensions of measurement to one another. Such mathematical engagement through tinkering has implications for supporting learners’ mathematical agency in the classroom and for validating learners’ intellectual work in math contexts [3]. Many of these math concepts and practices also mirror ways that experienced weavers engage with math in their craft, such as shape and image transformations and multiple embedded patternings [18] These mathematical instantiations not only relate to United States Common Core Math Standards in Concepts and Practices (http://www.corestandards.org/Math/; accessed 13 February 2020), but move beyond them due to the opportunities tinkering provides for invention and production of new knowledge.

By understanding weaving as rooted in mathematics, we can understand weaving tinkering as a type of mathematical tinkering. This work suggests that tinkering can be an important part of creating personally meaningful artifacts. The tinkering process may not always end in the desired result, as can be seen in the case of Sandy. Her ambitious design did not come to fruition, but the process led to mathematical reasoning and discoveries. As opportunities to tinker in math classrooms may be few, modeling these classrooms in some ways after weaving engagement to support tinkering and agency could potentially lead to more emergent mathematical discoveries and more opportunities for learners to see themselves as doers of math.

Flexibility to make mistakes with low-stakes consequences is an important element of tinkering spaces. This makes tinkering similar to frameworks of productive or safe failure (e.g., [31]). In both cases, making and resolving mistakes is part of the learning process and works best when consequences like grades are not at stake and when the activity is meaningful or relevant enough for learners to be interested enough to move past frustrations and roadblocks. For example, when Tina realized that she had made errors in her over one, under one sequences, she pulled out row after row and started over. If there had been external pressures such as grades or time limits, the frustration of this might have been too much. Instead, she was able to collect herself and begin again, determined to achieve a high level of neatness and precision.

This may be a defining feature that separates open-ended tinkering environments from some productive failure environments, and even other expansive approaches in mathematics classrooms. One such approach is project-based learning (PBL), where learning happens through active construction of a project [27]. Some work has demonstrated the usefulness of such approaches across multiple stages and levels of math education. However, the positive impacts so far have seemed to be fairly localized and have yet to make lasting impacts across curricula [30]. One reason might be that there are still externally set outcomes and goals as end points for PBL lessons. With crafting, the outcomes may be more internally determined. Again, for Tina, the goal of perfect over one, under one sequences was not external. No one told Tina to do this, and she was not going to be evaluated on her weaving skills. Tina wanted a particular outcome for personal reasons, likely tied to beauty and aesthetics. The desire for a certain look or level of craftmanship is a driver for some learners and crafters and led Tina to engage with mathematical ways of thinking to achieve her goals. This suggests that personally meaningful design activities can lead learners into disciplinary ways of thinking and doing as they work to achieve their goals. This resonates with sociocultural theories in the learning sciences that look at learning as participation in authentic and community-driven contexts (e.g., [46]).

This also circles back around to the power of epistemological pluralism. As noted above, project-based learning initiatives in math classrooms seem to be less effective when the general culture around math remains the same [30]. This suggests that it is not merely changing the math activities, but rather the nature of what we recognize as mathematical, that has an impact. For Tina and Sandy, it is not just that they were tinkering with traditional mathematical concepts. Instead, their tinkering with the elements of crafting was itself mathematical in ways that are less typically recognized as such. By shifting the core of what counts as mathematical, we open up possibilities for more learners to engage in math in authentic and meaningful ways.

Finally, this opening of the gates for more learners to engage authentically in mathematics has the potential to impact sustainable workforce opportunities, potentially making STEM and math more welcoming environments for women and people from other marginalized groups to contribute and work. A more diverse workforce would likely lead to more knowledge production and more creative solutions to the problems of the moment. Additionally, aside from their economic impacts, practices such as weaving have long been undervalued in mainstream educational spaces and society in general. Honoring and validating those practices would uplift marginalized and minoritized members of society, serving as a step toward more just and sustainable social environments for all.

6. Potential Areas for Future Research

Future work in this area could compare the prevalence and impacts of tinkering in math classrooms to tinkering with math through weaving. This could further illuminate ways that classrooms could build on weaving activities to support tinkering, as well as support designers of out-of-school activities in thinking about how to more explicitly support mathematical engagement. Such comparative work could also further explore how epistemological pluralism is or is not supported in classrooms and in weaving, with additional implications for design for learning.

Future work should continue to explore STEAM as a transdisciplinary space [47] where practices such as tinkering become possible and valued. This work could help illuminate what makes a STEAM learning environment uniquely powerful for learning and provide additional recommendations for transforming more traditional learning environments into spaces that take up STEAM practices to improve experiences and outcomes for learners from traditionally minoritized and marginalized communities.