Introduction

Despite recent efforts to support learners from traditionally minoritized backgrounds in mathematics, inequities in math achievement and participation still exist, particularly for women and people of color (e.g., Boaler and Sengupta-Irving, 2012; Corbett and Hill, 2015). Mathematics has been seen as a gatekeeper to later success in school, high school graduation, and career (Moses and Cobb, 2001), although achievement differences among genders on traditional mathematics metrics and assessments have largely shrunk in recent years (e.g., Boaler and Sengupta-Irving, 2012; Hyde et al. 2008). It has been suggested that problems such as the undervaluing of learners’ diverse skills and abilities (e.g., Bang et al. 2007; Wood et al. 2006; Moses and Cobb, 2001) and a lack of space for diverse materials and activities in math contexts (e.g., Hiebert and Stigler, 2000) have an impact on who feels invited and who is allowed to be successful in STEM spaces. Additionally, much of how math is taught in schools aligns with a particular epistemology that comes from western mathematicians and philosophers (e.g., Joseph, 2017). This may make it difficult for learners to explore different ways of engaging with or understanding math.

Epistemological pluralism (Turkle and Papert, 1990) sits at the intersection of constructionism and feminism. This perspective calls for educators to recognize and value multiple ways of thinking, doing, and engaging in the world, particularly in STEM contexts. An ethic of epistemological pluralism means supporting STEM, and specifically math, participation in more and broader ways, suggesting that doing math looks different across and between people. In maker education, this lens also allows us to explore the relationship between a learner and the technology they use in different ways, wherein the technology can become a negotiation partner in the process of making through exploring new ideas and new ways of knowing and doing (Turkle and Papert, 1990). Designing math educational spaces around this ethic and through the lens of making could open more math possibilities for more learners. For example, we know that weaving - a making activity - involves mathematics in ways that are rigorous, but that weavers may not articulate that math in traditional academic forms (e.g., Thompson, 2022; Saxe and Gearhart, 1990; Dominikus et al. 2023). There is potential in understanding looms as pieces of analog technology, and in viewing weaving as a productive activity for supporting broader ideas about what math can be and look like in educational spaces.

Tangible manipulatives, or physical artifacts that can be moved and rearranged in various ways, have long played a role in math education (e.g., Brosterman, 1997). This use of tangible artifacts in learning environments is also a hallmark of maker education and educational activities through both digital and analog technologies. However, many of these manipulatives have been hard to the touch and block-based (Note: Kafai et al. 2010), and/or geared toward younger learners (e.g., Fennema, 1972; Byrne et al. 2023). While a strong link between weaving and mathematics has been established, fewer studies explore the possibilities of this link for supporting youth’s pluralistic engagement with mathematics in educational settings. This prompts the research question for the present study: In a making-focused intervention designed to expose youth to the mathematical practices inherent in weaving, how do student-created artifacts showcase learning?

Background and theoretical frameworks

Before describing the context and methodological perspective of this study, this section describes some of the literature and theoretical frameworks that inform this work. First, I outline some of the ways weaving has been connected to mathematics to demonstrate that there is promise in exploring middle-school youth’s mathematical engagement as they learn to weave. Next, I explain the constructionist foundations of this work as tied to maker education, tangible manipulatives and epistemological pluralism, allowing a perspective in this study that validates multiple forms of knowing and doing, and views the process of designing and making as a process of learning. Finally, I describe why this work is needed, using past inequities in math education to demonstrate that a new way forward is needed. The current study is not focused on math formalisms or achievement in academic math classrooms, but is a first step toward a broader way to recognize and validate engagement with mathematical ideas through making.

Connecting weaving and mathematics

Weaving and mathematics are deeply connected. While weaving is not the only craft or activity that has connections to mathematics, the depth and complexity of the connection makes it particularly compelling for further exploration. Anthropological research from the 1970s to today has explored this link in cultures and communities around the world. A foundational study showed that children with weaving experience in certain areas of Mexico were able to generalize pattern representations, particularly when those patterns were related to woven cloth that was unique and special to the community (Greenfield and Childs, 1977). Other work has explored the relationships between weaving experience, gender, educational experience, and mathematical representation of children in Brazil (Saxe and Gearhart, 1990).

This work has been especially prominent within the field of ethnomathematics. Authors Barta and Eglash (2009) have explored the mathematical ways we can understand weaving and other art-making in indigenous spaces. They look at students using these contexts to understand mathematics more artfully and personally. For example, in their 2009 chapter, they work with students to use virtual tools to highlight the mathematics embedded in Indigenous craft practices such as bead and loom work, wampum, and rug weaving (Barta and Eglash, 2009). More recently, researchers have explored the mathematics practices of Adonara weavers in Indonesia (Dominikus et al. 2017), and of weavers from the West Amarasi society in Timor Island, Indonesia (Dominikus et al. 2023), finding that weavers in both communities used foundational concepts such as counting and measuring, as well as more complex ideas such as reflection, rotation, and implication logic.

In addition to these diverse contexts, some mathematicians have used craft to model mathematical concepts and properties and describe new types of pattern sequences (e.g., Irvine and Ruskey, 2014; Irvine et al. 2020). Weaving technology also helped inspire the invention of automated computation of rapid calculations by way of the Jacquard Loom and the design for an Analytical Engine (Essinger, 2004). These studies suggest there is power in the link between math and weaving. While these studies have primarily focused on pointing out and naming mathematical concepts and properties present in adult craft and weaving work, fewer studies have explored the possibilities of this link for youth learning. For the current study, I draw on a recent exploration of math concepts that emerged through experienced weavers’ practices (Thompson, 2022) to explore the practices and process of youth learning to weave to understand how youth mathematical engagement emerges, and whether it bears any relationships to experienced weavers’ engagement.

This prior work with experienced weavers across the United States and Canada identified three instantiations of mathematics that emerged through their making processes: arithmetic and calculations, image and shape transformations and multiple embedded patternings (Thompson, 2022). These three instantiations showcased variation in the distance between school math and weaving math, as well as in the level of invention and creation that they involved, demonstrating that weavers use complex, overlapping mathematical concepts simultaneously in their work (ibid.). Other recent work focused on middle-school aged youth learning to weave, utilizing video analysis to examine how their tinkering processes became learning processes (Thompson, 2023). The current study seeks to connect these two lines of work by exploring the mathematical concepts and ideas that emerge as youth learn to weave.

Making based in constructionist commitments

In this work, I approach learning from a constructionist lens. Building on Piaget’s constructivism, Papert shifts understanding of learning out of the head alone and places emphasis on representations out in the world. Papert 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. Most important is that these created artifacts can be publicly shared and reflected upon (Papert, 1980; Holbert et al. 2020). Therefore, this work uses constructionism as a way to understand learning through the process of designing, making, and sharing. In the context of weaving, the loom is a piece of technology that allows mathematical engagement to become something beautiful and worth showing to others. Following constructionism even deeper, its concept of epistemological pluralism (Turkle and Papert, 1990) is particularly important here. Epistemological pluralism is a commitment to valuing multiple ways of knowing. As learning from this perspective happens in the externalization of shareable models, epistemological pluralism allows us to see that there are multiple forms that can demonstrate learning and pushes us to think more broadly to recognize those forms as valid. Taking mathematics as an example, much of how math is taught in schools aligns with a particular epistemology that comes from western mathematicians and philosophers (e.g., Joseph, 2017). For example, “truth, order, and stability” (p. 8, Joseph, 2017) stemming from the ideas of Plato have informed the underlying foundations of academic thought, including pure mathematics, its purposes, and how it should be taught. However, these very ideas from Plato have been shown to be rooted in white supremacy and misogyny. Thus, some of our rigid, traditional ideas about what math looks like or should be also have (potentially inadvertent) ties to these inequitable underlying ideas as well (Joseph, 2017). Therefore, it is increasingly important to support students in exploring different ways of engaging with or understanding math.

Constructionism pushes educators to allow space for multiple epistemologies. Turkle and Papert (1990) explored how some women were inclined to engage with computational ideas in more relational and concrete (rather than abstract) ways than commonly considered appropriate and rigorous. They called for the recalibration of STEM fields to accept broader and multiple forms of engagement. An epistemologically pluralistic lens guides this research to be intentional and critical about the practices and processes labeled as math.

Constructionism and its push for epistemological pluralism also opens up possibilities for tangible manipulatives in classrooms, particularly beyond the early grades. 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 (Brosterman, 1997) to Cuisenaire rods and beyond. However, many of these manipulatives have been hard to the touch and block-based (Note: Kafai et al. 2010). Additionally, while the use of tangible manipulatives in schools has mostly been seen as important for younger learners (e.g., Fennema, 1972; Byrne et al. 2023), a focus on multiple ways of knowing and doing makes tangible manipulatives compelling at all stages. Epistemological pluralism also compels work around tangible manipulatives to rethink the historical focus on hard materials. The argument from Turkle and Papert (1990) to revalue and revalidate “soft” ways of reasoning and thinking should also extend to include “soft” objects-to-think-with (Papert, 1980). Finally, tangible manipulatives and soft technologies may also potentially open pathways toward both wonderful (Duckworth, 1972) and powerful ideas (Papert, 1980); these are ideas that may be new just to the learner, or may be entirely new innovations, but that open productive pathways and possibilities through their discovery and engagement. This is again where a commitment to epistemological pluralism is important, as wonderful and powerful ideas may lead to new or unexpected outcomes that should be validated and supported. Learning through the design and creation of tangible artifacts, such as in weaving, may be a particularly important way for youth to discover wonderful and powerful ideas, and to connect closely and meaningfully with these ideas.

Inequities in mathematics

While this paper is not about formal math classes, it represents a step toward consideration of more equitable approaches to math learning because inequities in math achievement and participation still exist. Generally, women and people of color tend to be underrepresented in higher level STEM classes and in STEM careers (e.g., Boaler and Sengupta-Irving, 2012; Corbett and Hill, 2015). Specifically, in mathematics, women tend to become less represented as math classes and professions increase in levels. In 2012, it was reported that 45% of math undergraduate degrees went to women while 24% of PhDs in math went to women. Also, only 17% of tenured math faculty were women (Boaler and Sengupta-Irving, 2012). This is not to say that more equitable approaches to math in general, or weaving specifically, are only of benefit to women and people of color, but that current mathematics structures do not work for all learners, and thus educators should consider new paradigms.

It has been suggested that the undervaluing of learners’ diverse skills and abilities (e.g., Bang et al. 2007; Wood et al. 2006; Moses and Cobb, 2001) and a lack of space for diverse materials and activities in math contexts (e.g., Hiebert and Stigler, 2000) impact participation and representation in STEM. Other suggestions include shifting math teaching to be more open and conceptual, increased institutional support and a cultural push against stereotypes and bias (Boaler and Sengupta-Irving, 2012). The current work seeks to address these calls by building on definitions of mathematics that are open and flexible. Schoenfeld (1992) draws on a 1989 National Research Council report to describe math as more about pattern than it is about numbers (National Research Council, 1989, p. 84). Both describe exploring patterns as the better and deeper alternative to memorizing formulas. Schoenfeld continues by claiming that exploring and understanding patterns is thought to be the overarching goal of mathematics. Learners are exploring patterns as they engage in challenges such as inventing sequences of numbers or items or making inferences about data. Thus, a focus on patterns moves mathematics from a set of content to a living, ongoing process (Schoenfeld, 1992). This definition of mathematics maps directly onto actions that take place during weaving, and allows for more flexibility and diversity in what counts as math and who can do math.

Scholars in STEM equity push for educators to design for rightful presence (Calabrese Barton and Tan, 2019). From this framework, all learners have a right to learn, take up space, and make a mark in STEM spaces, as “How youth are legitimately welcomed and positioned as powerful producers of new social futures in their STEM classrooms shapes their opportunities to learn.” (ibid., p. 2). This current work seeks to do this by giving youth space to participate in their own ways and working to recognize and validate their intellectual work. I see this as part of larger effort for STEM equity, resonant with historical efforts to recognize competencies of learners in diverse contexts (e.g., Brown et al. 1989), and contemporary work around equity and justice (e.g., Bang and Medin, 2010; Espinoza and Vossoughi, 2014; Lee, 2017).

Context

Before discussing the data collection and analytic processes of this work, I describe here my researcher positionality and describe some details of the workshop activities that took place, including the tools used for weaving.

Author positionality

This work centers on a workshop I led that took place in a middle-school setting, where youth self-selected to spend time weaving as a design activity. In this work, I position myself as knowledgeable but not expert about weaving, and as learning from youth participants as well as teaching them a bit about weaving. I have been involved in research around crafting since 2015 and have been involved in this particular school setting in multiple ways since 2014.

Workshop design

The weaving 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 an analog technology - frame looms (see Fig. 1 for image) - that were laser-cut from an open-source pattern to create their woven artifacts, and grid paper to plan their designs. 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. The session objectives were as follows (Table 1).

Table 1 Workshop design
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In addition to the looms and 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.

Fig. 1
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Open-source, laser-cut loom (By: TheInterlaceProject: https://www.instructables.com/id/Laser-Cut-Mini-Frame-Loom-Weaving/)

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The frame loom

Figure 2 is a rough and simplified illustration of the type of frame loom used in the workshop. The large black square represents the frame loom itself. The black shapes along the top and bottom are notches in the wood for the warp (vertical) threads to hook into. The red and blue lines represent the warp threads. The curvy purple line represents a weft (horizontal) thread. The weaving builds from the bottom of the frame toward the top. This purple thread takes an under one, over one sequence. Here it can be seen how the thread must follow the opposite pattern on the second row. A weaver must consciously alternate this sequence with each row to ensure the weft thread remains interconnected with the warp threads, and to achieve the desired woven visual effect.

Just as with more complex looms, many sequences are possible with a frame loom. The green line represents a new weft thread following a different sequence. This one attempts to follow an over two, under two sequence. However, there is an odd number of warp threads available. Therefore, the string must modify the pattern at the end of each row. Here, it goes over two, under two, over two, under two, over one. On the following row it shifts again and begins with under two. This irregular sequence will become more and more visible as it occurs over several rows.

This simplified drawing is meant to showcase the basics of weaving on a frame loom such as those used by the youth weavers, that even frame looms produce three-dimensional products, and suggest how the decisions a weaver makes can increase in complexity.

Fig. 2
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Simplified illustration of a frame loom

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Methods

In this study, I ask the research question: In an intervention designed to expose youth to the mathematical practices inherent in weaving, how do student-created artifacts showcase learning? The analyses of artifacts created by youth here seek to showcase how math can show up in weaving through a lens that values multiple ways of knowing and doing. Other published work (Thompson, 2023) utilized video analysis to more deeply explore processes and actions. All names are pseudonyms to protect the privacy of the participants.

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 participate in the weaving workshop due to personal interest in exploring weaving as a design activity. Most youth had little to no weaving experience prior to this workshop. Two had learned to weave previously. Race, ethnicity and income demographics were not collected for the 13 workshop participants. This was done in part to help protect their confidentiality and reduce chances of their identities becoming known as they were a small subset of students within a small school. However, some demographics for the school as a whole are included here for additional context. At the school overall, 77% of students are reported as white, 12% as two or more races, 8% as Hispanic, 2% Asian, less than 1% Black, and less than 1% Native American or Native Alaskan. Additionally, about 29% of students are eligible for free or reduced lunches.

Case selection

Of the 13 workshop participants, I present here four cases selected to show a range of engagement with image and shape transformations and multiple embedded patternings; these ways of engaging in math relate to ways experienced weavers see math in their craft and signal reasoning around shape, measurement, pattern, and the relationships between them (Thompson, 2022). I show two cases for each mathematical instantiation, one that is at an earlier stage of sophistication and one that is at a later stage of sophistication. These four cases were chosen for several reasons. First, compared to their peers, the mathematical connections in their work were clearly accessible and visible. Second, these cases demonstrated a range of sophistication that is representative of the group of participants as a whole. Finally, each of these four cases showed a slightly different personal approach to designing and making, allowing the epistemologically pluralistic analysis to showcase that mathematical engagement is possible and visible across a range of ways of thinking and doing.

Below, I briefly describe each of the four focal youth based on researcher observations and the ways their actions and interpersonal interactions guided their approaches to making and crafting. These descriptions are to provide an image of the classroom and interaction and individual types within it.

Image and shape transformations cases

  • Marg: Marg is a middle school girl with some prior crafting experience. She was seen to be highly interested in activism and politics, which influenced her crafting designs.

  • Amy: Amy is a middle school girl with prior crafting experience. She was seen to be highly creative and artistic in her crafting.

Multiple embedded patternings cases

  • Heidi: Heidi is a middle school girl with some prior craft experience. She was seen as collaborative in her crafting, often consulting with a partner or friend.

  • Ezra: Ezra is middle school boy with some prior crafting experience. He was seen as inventive in his craft techniques and strategies.

Data sources

The data center on images of the artifacts the youth made. Other work (Thompson, 2023) focuses on video and dives more deeply into making practices. The artifacts here include both the items that they wove and the plans they made on grid paper. Multiple images were taken of these artifacts during and after each workshop session, providing records of design documents, in-progress work, and completed or nearly completed work for each project. Images of each youth’s first and last projects at their most completed stages serve as a parallel to “pre” and “post” timepoints. Viewing the work at these timepoints makes visible learning that occurs through changes in youths’ techniques, designs and strategies.

It is important to note that youth sometimes created small projects, abandoned projects (in the case of first projects), or wanted to continue working beyond the scope of the workshop (in the case of second projects). Thus, projects described as in their final state indicates the most complete state the project reached during the scope of the workshop.

Analytical techniques

Data were analyzed using perspectives informed by multimodal analysis (Jewitt, 2011), multiliteracies (New London Group, 1996), and with some perspectives from artifactual literacies (Pahl & Rowsell, 2010). While there is not a deep focus on a historical analysis of craft and materials in this work, the artifactual literacies perspective has informed an approach here that views artifacts as texts that tell stories and showcase lived experiences and embodied knowing. Additionally, multimodal analysis and multiliteracies inform the current work by putting forth a perspective that there are multiple modes through which communication takes place and that modes such as gesture, gaze, artifact creation and artifacts themselves are valuable for understanding a wide range of cultures, communities, and epistemologies.

From these perspectives, I viewed each of the available artifacts and first sought to understand how each piece was constructed. This included writing memos about each artifact regarding how the loom was warped, how many weft colors were used and how many rows of each color were woven, as well as the patterns and sequences used. Multimodal perspectives indicate that these features of an artifact’s construction can be understood as forms of communication that can tell a story about both the creator and the artifact. Thus, I came to understand these details about the artifact construction as design moves and choices. Rather than simply given facts about the artifacts, these features indicated active choices made by the youth that can convey messages about what they knew, what they wanted, or how they experimented with the materials. The design moves viewed and noted in memos included choices such as warping the loom using every notch or every other notch; weaving with the weft thread using an over one, under one, or over two, under two sequence; switching from one color to another; or embedding shapes into the woven artifact.

I then viewed and re-viewed these images of artifacts from across the workshop’s duration, clarifying, naming and re-naming the moves and choices in analytic memos. Returning to the multimodal perspective, two related stories emerged from the data (e.g., Glaser & Strauss, 1967): (1) that each youth made different choices over time from their first project to their last project and (2) that these changes represented changes in mathematical principles, insights, and instantiations implemented. Further, this second story was recognizable and valid regardless of whether the youth verbalized these changes, because the epistemologically pluralistic, artifactual, and multimodal lenses that inform this work hold that actions are a form of knowing and that artifacts are a form of communication. Using a spreadsheet, I then tracked how these choices were different between the first and last projects the youth made. From here, I explored how these changes aligned to mathematical principles and instantiations, as outlined in the next section. As a solo researcher, I engaged in frequent checks of the analysis for quality and trustworthiness with colleagues and members of my larger research team.

All the data were analyzed, and changes from first to last artifact were identified and marked in a spreadsheet for each of the thirteen youth weavers. These results are reported briefly at the beginning of the Findings section. Then, in order to provide rich, illustrative descriptions of the various design choices and mathematical instantiations taking place, focal cases were pulled out and are described in the Findings section in detail.

Operationalizing math in weaving

My prior work interviewing experienced weavers (Thompson, 2022) found that:

…weavers described math in three simultaneous, overlapping mathematical instantiations… These instantiations are (1) arithmetic and calculations, (2) image and shape transformations, and (3) multiple embedded patternings… Further, these instantiations are additionally interrelated and occur simultaneously, suggesting that mathematics can have multiple definitions at once.

-- (Thompson 2022, p. 9)

In the current work, I describe two of these instantiations, image and shape calculations and multiple embedded patternings, that emerged in the prior work and that became apparent during the analysis in the current work. Please see prior work for more detail on how these mathematical instantiations were initially identified and articulated (Thompson, 2022). Here, I describe the two instantiations that inform the current work:

Image and shape transformations deal with designing with regard to space, shape, proportion, symmetry. This instantiation involves a certain level of invention, as solutions to problems are not necessarily known in advance. This requires a weaver to reason with multiple concepts at once, such as both unidimensional (e.g., length and width) and two-dimensional (e.g., area) measurements. In the type of weaving used in the workshop here, image and shape transformations may involve moving from one form - such as a design on paper, to another - such as a woven artifact. It may also involve transforming the gridded structure of weaving into the appearance of curved lines, or other shifts between materials and images. Although engaging in math in this way does not require academic formalizations, we can also see that this instantiation relates to Common Core Math Standards such as “Create equations that describe numbers or relationships” (High school: http://www.corestandards.org/Math/Content/HSA/CED/) and “Apply geometric concepts in modeling situations” (High school: http://www.corestandards.org/Math/Content/HSG/MG/). It also moves beyond content to encompass Common Core Math Practice Standards such as “Make sense of problems and persevere in solving them” and “Look for and make use of structure,” (http://www.corestandards.org/Math/Practice/).

Multiple embedded patternings encompasses invention and experimentation with patterns, sequences, and structure in weaving. This instantiation points to the essence of mathematics, seeking, understanding and building patterns. These ideas signal that weaving is in many ways unavoidably mathematical. As weavers engage with multiple embedded patternings, they participate in manipulation and invention of mathematical sequences rather than solving math problems. Creating a woven artifact necessitates making decisions around multiple possible patterns, such as order, sequence and time. Each vertical thread can be over or under the thread moving horizontally, and each of these binary choices exists in relation to all other possible choices, meaning that even a visually simple outcome has an incredibly complex underlying mechanism. This type of mathematical engagement may look less like typical classroom math, but does indicate core level engagement and exploration with mathematical thinking. Although this is more difficult to align with formalized concepts learned in school, there are some relationships to Common Core Math Practice Standards such as “Look for and express regularity in repeated reasoning” (http://www.corestandards.org/Math/Practice/). Multiple embedded patternings also moves beyond these standards to include both invention and experimentation. Engaging with multiple embedded patterns means immersion in the underlying mechanics of patterns and sequences, playing with their relationships, and experimenting with new ones. This type of engagement is not easily recognized as formalized academic math, yet is deeply mathematical from a stance of epistemological pluralism and when analyzed from a multimodal perspective.

These two mathematical instantiations - image and shape transformation and multiple embedded patternings - show up in experienced weavers’ work (Thompson, 2022). In this study, I used a constructionist theoretical stance as well as a multimodal and multiliteracy analytical stance to look for evidence of these instantiations in the youth learners’ work. For learners, engagement with image and shape transformation often looks like embedding shapes within other shapes as it is necessary to consider multiple dimensions of measurement, symmetry, and proportion to do this. Multiple embedded patternings often looks like shifting away from over one, under one weaving sequences and experimenting with the outcomes of these shifts.

Findings

Experimenting with shape and sequences are two practices that experienced weavers do in their own work (Thompson, 2022). Engaging with shape in weaving is related to mathematical image and shape transformations, and engaging with sequence in weaving is related to mathematical multiple embedded patternings (Thompson, 2022). Here, I tracked youth first and last projects looking for increased sophistication around these mathematical instantiations. Findings show that as youth progressed from their first to their last project, 76% of participants shifted their practice to experiment with embedding shapes into their designs. This demonstrates engagement with image and shape transformations and suggests increased reasoning around considering multiple dimensions of measurement, shape, and proportion in concert as well as increased invention of unknown measurements such as slope and area. Additionally, 30% of participants shifted toward either planning or implementing new “over, under” sequences. This demonstrates engagement with multiple embedded patternings and suggests increased reasoning around seeking, understanding, and building underlying mathematical patterns as well as increased core level engagement with mathematical thinking.

To illustrate how these instantiations are enacted in youth projects, I show four cases here that highlight a range of sophistication (Table 2). For each case I present an image of the youth’s first project in its most final state to demonstrate and describe initial math engagement, an image of the youth’s final project in its most final state to demonstrate and describe ending math engagement, and two images of the youth’s planning or in-progress work to demonstrate and describe math thinking in development as well as to discuss the steps youth took to reach the final state of math engagement.

Table 2 Four cases and explanation of math in final projects
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Marg

In Marg’s first project (Fig. 3, left), she used an over one, under one weaving sequence. There are six passes, or rows, of one color, followed by two passes of a new color. In her last project (Fig. 3, right), she continues using an over one, under one weaving sequence. However, she progresses to embedding a yellow shape within a blue background. In order to create this artifact, Marg needed to progress in her attention to symmetry as well as in her imagining and transforming with regards to shape.

Fig. 3
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Marg’s first and last project

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Additional evidence of this is visible in Marg’s in-progress work as well. Her design plan (Fig. 4, right) indicates her plan to use an over one, under one sequence. She clearly planned out evenly spaced shapes that were centered on the background. During construction (Fig. 4, left), Marg did not have immediate access to the yellow yarn; she decided to continue with blue yarn behind and around where the yellow would eventually go. In order to accomplish this, she needed to imagine how the yellow would fit into the space and into the established over/under sequence. She also needed to understand how the shapes would change in the actual size of the project compared to the planned size.

Fig. 4
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Marg’s work in-progress

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Table 2, row 1 further outlines the mathematical thinking around image and shape transformations seen in Marg’s work. Her shift to explore shape in her last project suggests increased reasoning around considering multiple dimensions of measurement, shape, and proportion in concert; this also signals increased invention around measurement, slope, and area. This task of visualizing multiple dimensions of measurement (length and width in addition to over/under sequences) at once is difficult to approach in schools. This type of engagement relates to but goes beyond some Common Core concepts such as: Apply geometric concepts in modeling situations (High school: http://www.corestandards.org/Math/Content/HSG/MG/). This goes beyond what may be typical in classrooms as it deals with invention of shapes and concepts, and involves reasoning in real-world contexts. Additionally, building rectangles on paper in a classroom is essentially 2-dimensional. However, the rectangle in this context may seem 2-dimensional, but in fact exists in 3-dimensional space as the strings move over and under one another. Inventing shapes in this way is not only more complex, but is more applicable to challenges that exist in real-life and real space. Although the rectangles used here appear somewhat simple, this exploration bears similarity to more complex transformations with image and shape that experienced weavers instantiate in their work (Thompson, 2022).

Amy

In Amy’s first project (Fig. 5, left), she uses an over one, under one weaving sequence. She creates this piece with stripes of various heights; there are nine passes of one color, nine passes of another, ten passes of the first color, and finally, two passes the second color. Her last project (Fig. 5, right), however, takes an entirely different approach. Amy begins with an over two, under two sequence for the first five passes. She then moves to partial passes, or rows, changing the density of the weft yarn and the number of passes to create curved shapes and motifs in her design. This last project appears more adventurous and seems to follow conventional rules of weaving less closely. Amy uses multiple new methods to create curves in an inherently gridded structure.

Fig. 5
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Amy’s first and last projects

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Evidence of this can be seen in Amy’s in-progress work as well. Her design plan (Fig. 6, left) served as a close guide, although it was not replicated exactly. The design indicated her plans for sequence, rows, colors, shapes, and the overall impression of the piece. Figure 6, right, provides an up-close look at the ways Amy used length and location of rows of yarn to create curves. This demonstrates an innovative approach to creating freeform shapes, using shorter and longer passes such as with the blue yarn, with single passes in between such as the single white row seen here. To accomplish this, Amy needed to understand how shapes are constructed and visualize the relationship between those shapes and the weaving grid structure.

Fig. 6
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Amy’s work-in- progress

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Table 2, row 2 further outlines the mathematical thinking around image and shape transformations seen in Amy’s work. Her shift toward playing with curves and shapes in her last project requires understanding the impacts of number and density of threads on the shape and symmetry of the project. This equates to holding multiple dimensions of measurement in mind at once, a task that is difficult to approach in schools. This has ties to some Common Core Math Standards [e.g., Apply geometric concepts in modeling situations (High school: http://www.corestandards.org/Math/Content/HSG/MG/)] but ultimately moves beyond these standards due to the level of invention and experimentation involved. Again, this invention and experimentation is more related to the real-world contexts in which geometric concepts might be necessary. Working in real space and in three dimensions may be especially useful in preparing youth to solve authentic problems in the future. Additionally, building rectangles on paper in a classroom is essentially 2-dimensional.

Heidi

For her first project (Fig. 7, left), Heidi employed an over one, under one weaving sequence for twelve passes. For her last project (Fig. 7, right), she shifted to using an over two, under two for 22 passes. She began trying to embed a yellow shape into the background, also using an over two, under two sequence. In this project, she also used a different type of yarn for the warp or vertical, threads, altering the texture and look of the artifact compared to her earlier work. The new texture that was produced sits at the intersection of experimentation with sequence and material. Heidi also started playing with shape in this project but had not yet seen it through.

Fig. 7
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Heidi’s first and last projects

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Evidence of these experiments can be seen in Heidi’s in-progress work as well. She created a design plan (Fig. 8, left) that included a general idea of the embedded shape she wanted, but her project did not follow all of the elements exactly. She was also flexible with the colors she used, opting to use a blue warp, white background, and yellow shape rather than the colors she wrote on her grid paper. Additionally, her over two, under two sequence is not indicated on her plan, although she kept the plan nearby as she started her work (Fig. 8, right). This suggests that her use of this sequence may have been a spur-of-the moment experiment, producing results that compelled her to keep moving forward.

Fig. 8
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Heidi’s work-in-progress

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Table 2, row 3 further outlines the mathematical thinking around multiple embedded patternings seen in Heidi’s work. To be able to create and follow-through with an over two, under two weaving sequence, Heidi needed to manipulate and invent mathematical sequence. This type of mathematical practice may look less like learning formal classroom math, but does relate to the underlying core of mathematics that seeks to recognize and create patterns in the world (Schoenfeld, 1992). This type of engagement also leads to mathematical discoveries; Heidi began to explore creating curved shapes from a perpendicular space and discovered that the introduction of new sequences complicated this effort. Such discoveries may be wonderful, and even powerful ideas (e.g., Duckworth, 1972; Papert, 1980), that take the relationships between number, pattern, shape, and angle out of conceptual spaces and into real spaces of practice, likely moving beyond much classroom instruction.

Ezra

For his first project (Fig. 9, left), Ezra used an over two, under two weaving sequence, continued throughout his piece. He did about seven passes, or rows, of one color with a shape embedded, followed by five passes of another color, five passes of a third color, and two passes of a fourth color. This first project was already quite varied and experimental compared to many of the first projects by the other participants. In Ezra’s last project (Fig. 9, right), he used an over two, under two sequence for four passes, followed by an over three, under two for around 14 passes with a few inconsistencies throughout the work. This last project required Ezra to shift toward an exploration of sequence and its effects. After discovering this sequence, he continued it much longer than any of his previous sections of work. This allowed the visual impact to play out more fully, showcasing the effects that sequence has in weaving.

Fig. 9
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Ezra’s first and last projects

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This exploration of sequence can be seen in Ezra’s in-progress work as well. Ezra did not refer to a design plan as he worked, suggesting that his unique sequence was invented in-the-moment. This experimentation process has traces throughout his work, both in the way he unconventionally ties his weft thread to the warp (Fig. 10, left) and used his needle to foreshadow the coming invented sequence. This needle placement seemed to demonstrate either the inspiration for or practice around the over three, under two sequence to follow (Fig. 10, right).

Fig. 10
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Ezra’s work-in-progress

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Table 2, row 4 further outlines the mathematical thinking around multiple embedded patternings seen in Ezra’s work. Ezra invented a new motif that was unlike others in the room, and continued this longer than any of his other design elements. This sequence had to remain constant across different rows even when the image being created was not yet clear. His ability to do this suggests significant reasoning around seeking, understanding, and building the underlying patterns that comprise mathematics. Engaging with multiple embedded patternings in this way looks like playing with mathematical sequences and their outcomes. It may look less like learning math in school but does indicate mathematical thinking. It also leads to wonderful and powerful (e.g., Duckworth, 1972; Papert, 1980) mathematical discoveries like Ezra’s - such as the multiple outcomes possible within a series of binary options. Discovering in real-time between number, pattern, shape, and angle in physical, 3-dimensional space may be beyond the possibilities of many classrooms, but may be a crucial step toward future stronger mathematical thinking and reasoning.

Summary

Overall, twelve of the thirteen youth did some experimenting or leveling up either in their planned or implemented designs. The cases here showcase a range of sophistication related to instantiations of mathematics relevant to weaving: image and shape transformations and multiple embedded patternings. The data suggest that planning and implementing weaving designs prompts engagement with mathematical structures and principles in interesting ways. These ideas bear some relationships to Common Core Mathematics Standards in both concepts and practices, but also go beyond these toward experimentation and invention. Space to experiment and invent may not always be possible in classrooms, but echoes uses of mathematics in real-world contexts in ways that allow for mistakes as well as enacting math in 3-dimensional space.

Discussion and implications

Reflections on learning

In this work, engaging in making and crafting with loom technology allowed youth to create designs that mirror experienced weaver designs; similar engagement with image and shape transformations and multiple embedded patternings were seen in adult weavers’ work. This shift signals a more central and legitimate form of participation (Brown et al. 1989) in communities of both crafting and mathematics. While these youth are still quite new to weaving and have not reached a comparable level of expertise to experienced adult weavers, the inventions the youth were able to create suggest that maker-centered activities with the tools and technologies of weaving open up possibilities for leveling-up and for creative mathematical engagement.

Recalling the notions of wonderful (Duckworth, 1972) and powerful (Papert, 1980) ideas, the youth weavers here were seen making discoveries about how math is enacted in weaving that were exciting and productive. For example, Amy found that curved lines could be created either by changing the density or the length of woven rows. This discovery has potential to influence Amy’s understanding of curved lines in general and may give her new wonderful ideas about creating curved lines in future art and design contexts such as making clothes, baskets, or pottery. Also, Ezra discovered that sequences in weaving structures can create both small- and large-scale visual effects. This discovery might impact how Ezra understands number and sequence in the future. He may now be primed toward more wonderful ideas about number both in math classes and in out-of-school contexts like art and video games. As the youth in this workshop were enacting mathematics through their weaving, current and future opportunities were created for expansive mathematical ideas.

It is also important to note that youth work also showcases something powerful around playfulness and the freedom to experiment and make mistakes. I explore the concept of tinkering more deeply in other work that focuses on the actions and processes that took place during the workshop (Thompson, 2023). Here, I will simply note that these mathematical discoveries about shape, number, and more may not have been possible in a traditional classroom setting without an approach that allows for experimentation as well as space and time to solve complex challenges in multiple ways. It seems that wonderful and powerful ideas can be facilitated by open-ended, making-focused, (analog) technology-rich environments such as learning to weave.

Reflections on epistemological pluralism and broader definitions of math

It is important to note that in this work, the constructionist theoretical stance as well as the multimodal and multiliteracy analytical stance come together to create a perspective that asks us to rethink what we count as mathematical engagement. Must math look like solving problems on a worksheet or verbalizing processes, or can it look like the act of crafting? If we commit to epistemological pluralism and to validating multiple ways of knowing and doing, we can come to recognize performing mathematical ideas through crafting as a form of doing math. While this form of mathematical doing may not result in increased test scores or traditional academic discussion of math, it is a worthwhile phenomenon to examine and analyze. Epistemological pluralism helps us see that math skills in weaving and math skills in academics are both legitimate, and allows more types of skills to be seen as valid. Just as experienced weavers talk about the math embedded in their craft in ways that are both similar to and different from the ways math is discussed in school (Thompson, 2022), the youth crafters here do not always verbalize their mathematical understanding in traditional academic ways, and often not at all. The focus of this work is to take a view of learning that can glean the necessary math behind and within the action of crafting and validate it as consequential. This may be a step for educators and researchers toward making math more accessible and welcoming to more diverse learners.

Understanding math knowing in broader ways, such as through weaving and as present in designing and making crafts, could help reframe a culture of exclusion in math education (Louie, 2017). This culture permeates, and even teachers dedicated to equity may still frame math knowledge as fixed and students as deficient in math knowledge (ibid.). Making visible the mathematical engagement that youth undertake through weaving may be a step toward frameworks and classroom activities that work to combat these disciplinary issues. Further, when we understand the loom as an analog technology, we can facilitate new (and old) types of engagement with mathematical and technological ideas, creating fertile spaces for discovery and innovation.

The goal of this work is not to suggest that all math classrooms should take up weaving as a method for teaching mathematics, although the findings here do point to some ways that weaving may be a productive classroom activity. Rather, the goal is to provide an example that supports youth and educators to think more broadly about what math is and can be. Expansive notions of math and other STEM contexts could open up more potential future pathways particularly for minoritized learners who may feel they have limited options (e.g., Bang and Medin, 2010). Such notions may also have implications for identity development: What does it and can it mean to be a “math person”?

Future work in this area may consider possibilities for weaving in more traditional mathematics classrooms. In such work, a comparative research design would allow for additional exploration of the pluralistic ways math can emerge through craft and design. Further future research should also continue to explore other practices and activities through which to recognize broad and plural ways of engaging in mathematical doing.