This instructional activity was developed in spring of 2024 for PED 3119 "Integrating Technology in the Classroom", taught by Dr. Michelle Hagerman. It is designed to engage students in the Ontario grade 9 mathematics curriculum (MTH1W) by way of technology, using the TPACK framework.
Introduction and TPACK Framework
More information about the TPACK model can be found at https://matt-koehler.com/tpack2/
Activity Overview
In a grade 9 mathematics classroom, weaving can be used to engage students in a variety of ways. Students will be approaching the mathematical concepts outlined in the Ontario curriculum from a range of backgrounds, ability levels, personal identities, and enthusiasm levels. Introducing textile arts, particularly weaving, can provide an entry point for students to engage with mathematical concepts in a fun and exciting way, creating a tangible piece of textile art, and see a variety of perspectives, cultures, and abilities valued as part of the mathematical world.
Contextual Knowledge
Curriculum Connections
This instructional activity is designed for the Ontario grade 9 destreamed mathematics curriculum (MTH1W). It aims to teach the concepts outlined by overall expectation E1 in the Geometry and Measurement strand of the curriculum.
- E1. Geometric and Measurement Relationships:
- demonstrate an understanding of the development and use of geometric and measurement relationships, and apply these relationships to solve problems, including problems involving real-life situations.
Throughout the Ontario math curriculum, and in the MTH1W course, students have engaged or will engage with a wide range of mathematical skills and concepts. In regards to this instructional activity in particular, students have been or will be exploring both 2 and 3 dimensional geometric problems, particularly those involving relationships such as intersecting or parallel lines; triangle properties; circle properties; evaluating or predicting the changes in perimeter, circumference, area, surface area, and volume when changing one or more dimensions; and solving problems involving composite shapes. However, most or all of the students’ mathematical skills will be involved in some way, as despite what the separate strands in the curriculum imply, mathematical processes are not siloed. Working with problems involving geometry and measurement rely on skills and abilities in patterning, building algebraic expressions, numeracy, gathering and representing data, and more. It will also involve the strand A of the curriculum, which outlines the social-emotional learning skills that students should be developing alongside everything else. All of this will be touched on in the instructional activity, and, as has been studied, these mathematical skills are greatly explored and developed through textile arts and weaving activities.
Classroom Context
The Ontario grade 9 mathematics course is a de-streamed course for students of all mathematical skill levels. Students will be experiencing secondary level courses for the first time, and will have, for the most part, a foundational understanding of mathematics concepts such as units of measurements, finding measurements of 2D shapes, and modelling or representing math problems in a variety of contexts. Students will also be coming from a vast variety of cultural backgrounds and contexts, with a range of personal experiences, abilities, and knowledge. There may be students who are English language learners, students who are neurodiverse, students with physical disabilities, or other personal identity factors that can impact how they learn and what they know.
The Hegemony of Math
Many students can feel that they are ‘not math people’ and that mathematics as a concept is simply not a subject area that they can or will ever thrive in. Often, these are female students, students of colour, English language learners, LGBTQIAS+ students, neurodiverse students -- students who are otherwise somehow outside of the white Western hegemony. Classrooms that use activities from a variety of viewpoints and domains to engage with mathematics have positive impacts on students' views of themselves as mathematicians (Thompson, 2023).
Textile and fibre arts are intrinsically connected to mathematics. These skills have long histories of use and association with communities of women and people of colour. Unfortunately, fibre crafts are often seen as "less mathematical" than the activities that occur in a traditional Western math classroom. By bringing textile arts into the classroom, and granting students the opportunity to engage with these fields, the scope of what mathematics is widens, and non-dominant perspectives are brought into the conversation of mathematics.
Pedagogical Knowledge
I draw here on the research and publications primarily of Naomi Thompson, who is prolific and committed to studying weaving as an instructional activity for engaging students with mathematics. As this instructional activity is modelled after the instructional practice of project-based-learning (PBL), it allows for several of the same elements of designing, developing, and constructing hands-on solutions to the problem of creating a woven artifact. There are 3 key components that allow weaving as an instructional activity to be implemented in this way: it creates a ‘Mathland’ environment, emphasizes problem solving through tinkering, and opens pathways for “wonderful and powerful ideas” (Thompson, 2023).
Mathland Environment
Textile arts and mathematics are intrinsically connected, and have been for centuries (Thompson, 2022). Weaving as an activity demands the use of mathematical processes such as patterning, numeracy, and geometry and measurement, meaning that these ideas and concepts are tied directly to the experience of weaving. This is the basis of a ‘Mathland’ environment – a concept developed by Seymour Papert as a place where “mathematics are learned alongside ways of doing mathematics in self-selected contexts” so that a “natural language of mathematics [...] pervades all experiences”, and learning mathematics “occurs as and organically as language learning” (Peppler, 2022). A Mathland environment uses authentic contexts and meaningful activities that allow students to engage with math concepts in real-world applications. Weaving and other textile arts are real-world activities that apply math concepts in a tangible way, where learners have to engage with those concepts in order to effectively engage with the activity. Weaving as an instructional activity de-silos the ideas of academic and crafting math, and applies them in equal measure as relative to each other. By creating this Mathland environment where mathematical concepts are woven into the activity at hand, weaving allows learners “to explore multiple modes of learning” and creates interdisciplinary environments where students can engage with mathematical ideas and concepts in a tangible manner (Peppler, 2022).
Tinkering
Due to the tangible nature of weaving, it becomes an activity through which students are able to engage in the process of tinkering. This is the method of participation in mathematical or STEM concepts that is non-linear, where students move “back and forth between known and new solutions” (Thompson, 2022). Students can try multiple solutions that may or may not work in a flexible and forgiving context similar to play. Often, mathematics can be seen or taught as a rigid discipline, and students who think differently or make mistakes can understand these experiences as failing to enact mathematical processes. However, tinkering through weaving allows students to “play with and push the boundaries of the materials and structures”, make mistakes and try a new solution, which then suggests that “the inherent math in weaving can be flexible” (Thompson, 2022).
Wonderful and Powerful Ideas
Weaving as mathematical tinkering opens a host of possibilities for students, and allows them to engage in math concepts in a meaningful way. This process of tinkering in a ‘Mathland’ environment opens pathways to what Thompson calls ‘wonderful and powerful ideas’; “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” (Thompson, 2023). Weaving as a mathematical activity allows students to engage in a variety of mathematical concepts, allowing them to innovate and discover in exciting ways that map directly onto the mathematical concepts in the Ontario math curriculum. This reveals a deep connection between weaving and mathematics that “makes it particularly compelling for further exploration”, and shows math to be not just a set of content, but, like weaving, “a living, ongoing process” (Thomspon, 2023).
Content Knowledge
As the educator and the lead for this instructional activity, there is a high degree of content knowledge that is needed in order to support student learning in this activity. The foremost piece is knowledge of the process of weaving and the mappings to mathematical concepts, and the ability to articulate how the process that students are moving through directly relates to the concepts that are expected in the math curriculum. Being able to draw students’ attention to the elements of patterning, shape translation, and multiple dimensions of measurement and proportion is a critical skill. One has to be able to see the math that is inherent in weaving in order to be able to support and draw students’ attention to how they are engaging in mathematical learning. If using digital tools to design the woven pieces, knowledge of weaving software like Fiberworks, and the terminology of the software itself, is critical as well.
Technological Knowledge
For those who might believe the act of weaving does not qualify as a technology, I would like to point to a quotation from Ursula K. Le Guin's "A Rant About Technology":
"We have been so desensitized by a hundred and fifty years of ceaselessly expanding technical prowess that we think nothing less complex and showy than a computer or a jet bomber deserves to be called “technology” at all. As if linen were the same thing as flax — as if paper, ink, wheels, knives, clocks, chairs, aspirin pills, were natural objects, born with us like our teeth and fingers — as if steel saucepans with copper bottoms and fleece vests spun from recycled glass grew on trees, and we just picked them when they were ripe...
One way to illustrate that most technologies are, in fact, pretty “hi,” is to ask yourself of any manmade object, Do I know how to make one? [ ... ] That’s the neat thing about technologies. They’re what we can learn to do."
Weaving is a low-floor, high-ceiling activity that students can participate in when bringing these fiber arts activities into the classroom. The basic concept is simple: weaving is the process of making a piece of fabric through the pattern of overlapping threads, using a loom. Looms can (and do) vary greatly, from small handheld models to massive multi-person machines. In this instructional activity, students will use frame looms, which are lightweight frames of plastic, wood, or cardboard that are notched along the inside edges so that threads can be hooked across the frame vertically. These vertical threads are called the warp threads. The weft threads are the threads that students then weave horizontally through the warp threads, using a needle, crochet hook, or their hands, in an over-one, under-one sequence. Once a weft thread has been woven through an entire row of warp threads, the next row starts in the opposite pattern. This repeats until the loom is filled, and the fabric piece can be cut away and tied off.
Mathematical Concepts and Processes in Weaving
Weaving with frame looms allows students to engage with patterning, numeracy, and geometry concepts in a small, constructionist context. They can select colours, patterns, and sizes that appeal to them creatively, and in doing so, can self-select the level of complexity that they would like to attempt. Weaving on a frame loom, particularly the small, handheld models that would be used for this instructional activity, allows students to create individual projects while still working with or alongside their peers, sharing ideas, problem solving issues or mistakes, and making discoveries. Weaving is a deliberate process that allows for correction or redirection at any given moment, meaning that there is imbedded flexibility for students to play or tinker with their designs and their work, so they can practice those mathematical skills of problem solving, creative thinking, and modelling in a forgiving and focused environment.
Activity Constraints
There are a set of constraints to be considered with this activity as well. Weaving requires a degree of mobility, fine motor skills, and focus in order to actually engage with the process in a manner that will result in a final product. Because of the individual nature of a small frame loom, students who struggle with these skills can work with a partner, but may find it difficult or isolating to engage with the process in this way, particularly because of the highly tangible nature of the activity. Weaving also simply requires a degree of materials to be on hand: a classroom set of frame looms (even if using a set of homemade cardboard looms), enough thread/yarn to afford students a variety of choice and the ability to make mistakes and start over, and other tools like needles or scissors. It can also be a time-intensive process – meaning that it will likely require multiple instructional periods for students to be able to set up the loom, design their pattern, and actually create a finished product.
Rationale
I plan to teach weaving in connection with the concepts of geometry and measurement in the grade 9 MTH1W curriculum, as stated. In exploring shapes and patterns, and determining relationships between shapes, measurements, and dimensions interact, students will construct a piece of woven fabric. Students will be encouraged to adapt patterns from global communities and cultures, and to create something that represents themselves in their design process, while using shapes and geometrical concepts we have explored in class. If desired, digital tools like Fiberworks can be used to digitally imagine these designs and navigate these concepts in a new software. They will then use their knowledge of and experience in patterning and sequencing to actually weave the design, changing the concept, process, and method as needed to be able to create a finished piece. They will be asked to track and highlight the changes they make both in their design and technique, as well as the ways that they solve problems with their woven artifacts, so that a conversational assessment can take place where they describe and reflect on their work and relate what they have done to the math concepts they have learned in this class or previously in their education journey.
Works Referenced
Peppler, K., Keune, A., Thompson, N., & Saxena, P. (2022, October 26). Craftland is mathland: Mathematical insight and the generative role of fiber crafts in Maker Education. Frontiers. https://www.frontiersin.org/articles/10.3389/feduc.2022.1029175/full
Crafting is a domain that is often seen as distinct from higher-level or formalized mathematics, despite the mathematical concepts that are woven throughout the crafting process. In particular, fiber arts, and their gendered associations, can be dismissed as less mathematical than other fields. However, crafting as a Mathland (an environment where the learning of mathematics occurs as immersively and organically as language learning and in authentic contexts) allows learners to engage with mathematical concepts in a variety of ways. Students can construct their own interests, self-select their own complexity level, and investigate cultural contexts that are often absent from math classrooms. All of this works to increase participation in STEM fields, particularly for women and girls.
Thompson, N. (2023a, April 28). “Some Angles are Gonna be Weird”: Tinkering with math and weaving. MDPI. https://www.mdpi.com/2071-1050/15/9/7363
Tinkering, the process of non-linear movement between known and new solutions when engaging with a problem or activity, has the potential to open many and varied pathways for mathematical thinking. Weaving as a way to allow for tinkering in the mathematics classroom allows students to engage with mathematical concepts like patterning, numeracy, measurements, sequences, and calculations. Particularly, weaving allows for students to engage in the making process, and brings those math concepts to life as they learn a hands-on skill and experience the flexibility of math as they construct a woven object. In the math classroom, project based learning is less effective when the general culture around mathematics is the same – by tinkering through weaving, students engage with a field that is typically recognized as less mathematical. This shifts the core of what counts as mathematical, meaning that learners who are hesitant about mathematics have another entry point, and the element of tinkering, of trying solutions and problem solving, allows students to learn in a low-stakes, flexible environment. Weaving presents an opportunity for students to tangibly engage with mathematical concepts in a flexible manner, where they can develop a range of mathematical skills and take part in a wide manner of mathematical concepts.
Thompson, N. (2023b, November 6). Weaving in: Shifts in youth mathematical engagement through weaving. SpringerLink. https://link.springer.com/article/10.1007/s11423-023-10316-y
Weaving is a constructionist activity that allows for high engagement with mathematics in a highly pluralistic sense. Wonderful and powerful ideas are those that may be new to the learner, or new innovations, but that open pathways and possibilities through discovery and engagement. From observing middle-school students in a multi-day weaving workshop, it can be seen that when engaging with weaving, students are inherently engaging in mathematical concepts, and with wonderful and powerful ideas. The ability to make artifacts when weaving means that students are discovering and innovating mathematical ideas in a way that is meaningful to them.