{"id":13286,"date":"2022-09-29T18:37:13","date_gmt":"2022-09-29T22:37:13","guid":{"rendered":"https:\/\/notes.math.ca\/article\/weaving-together-research-and-teaching\/"},"modified":"2022-09-29T20:04:40","modified_gmt":"2022-09-30T00:04:40","slug":"weaving-together-research-and-teaching","status":"publish","type":"article","link":"https:\/\/notes.math.ca\/en\/article\/weaving-together-research-and-teaching\/","title":{"rendered":"Weaving Together Research and Teaching"},"content":{"rendered":"
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I believe that research and teaching are interconnected.\u00a0 I\u2019ve been fortunate to be part of a research collaboration that has greatly enriched my teaching in diverse and surprising ways. I wanted to share this to encourage others to be open to collaborations outside of their main research interests.<\/p>\n

In 2012, I was asked to join the research group MathWeave,<\/em> by Dr. Eva Knoll (formerly of MSVU, now at UQAM). Eva\u2019s work ranges over many areas including math education and connections between math and art.\u00a0 She was working with a master weaver, Wendy Landry.\u00a0 They asked me to join in the role of \u201cmathematician\u201d.\u00a0 I initially thought that I would be able to provide a mathematical lens to analyze their work.\u00a0 I had no idea how much I would learn from our research collaborations.\u00a0 Our group grew to include other members (an elementary school math and art teacher and a librarian with a math background).\u00a0 According to the group website (http:\/\/mathweave.teknollogy.com\/<\/a>), our work is focused on the question \u201cIs there a \u2018Making Way\u2019 to show or learn mathematical concepts, skills and procedures by engaging with the struggle that is making art?\u201d This includes the creation of a range of artefacts that involve mathematical thinking, research papers discussing the math and art connections we work with, development of curriculum materials for many different levels, workshops and presentations for different audiences (math and\/or art conferences, math and\/or art teachers), and outreach activities to various community groups like Girl Guides.<\/p>\n

For example, consider the series of artefacts in Figure 1.\u00a0 Eva created these artefacts inspired by plaited mats produced in Southeast Asia. Mats can take advantage of two kinds of decorative effects. The first one involves patterns of in-woven holes in the surface. The second one involves patterns incorporating colours. \u00a0Eva showed me this series and I was intrigued by the mathematics underlying the colour choices.\u00a0 There are 14 distinct strips in each artefact. Colour is one way to convey equivalence of strips, and we can use different definitions of equivalence.\u00a0 At one extreme, the strips all have the same colour. At the other extreme, the strips are all distinct.\u00a0 Figure 1(b) displays an artefact where the equivalence is based on which region of the artefact the strip travels.\u00a0 In Figure 1(c), the equivalence is based on which paths are isometric.\u00a0 I think the artefacts are beautiful by themselves but even more so as a series.\u00a0 Different equivalences can help to see different mathematical aspects.<\/p>\n

\"\"\"\"\"\"\"\"<\/figure>\n

Figure 1: Series of Open Squares<\/p>\n

The use of colour is an important consideration for the creation of art.\u00a0 I have found it helpful to incorporate colour in different ways to help students understand mathematical concepts.\u00a0 We want our students to see relationships and patterns.\u00a0 I often teach abstract algebra, and the symmetries of the square is one of the main examples that we see throughout the course.\u00a0 There are 8 symmetries of the square (the identity, 3 rotations, 4 reflections). The corresponding binary table is in Figure 2.\u00a0 The colour helps the students make observations about the structure of the group.<\/p>\n

\"\"\"\"<\/figure>\n

Figure 2: Binary table for the Symmetries of the Square<\/p>\n

The symmetries of the square is a subgroup of the group of permutations on 4 elements.\u00a0 We presented a workshop on another subgroup of the permutations on 4 elements at a math and art conference.\u00a0 This workshop included different ways to experience the subgroup- through music, poetry, visual art and culinary art.\u00a0 The corresponding paper is available at https:\/\/archive.bridgesmathart.org\/2018\/bridges2018-659.html<\/a>.<\/p>\n

The actual act of making art has deepened my own mathematical understanding of different concepts, so I try to bring more hands-on activities into my classes.\u00a0 I introduce topology in a first year math concepts class.\u00a0 In the first few years I taught the course, I would tell the students the joke about how a topologist can\u2019t tell the difference between a donut and a coffee cup.\u00a0 Now I have an assignment that has a component that requires the students to make a donut out of play-doh and transform it to the coffee cup.\u00a0 The students take photos of the transformation and include the photos in their assignment. They generally love this assignment.\u00a0 I have heard students say that they don\u2019t usually get to do anything like that or that they haven\u2019t played with play-doh since they were kids.\u00a0 The photos are amusing too, which helps make the marking more fun.<\/p>\n

I have learned a great deal about pedagogy from being in the group.\u00a0One thing that was new to me is the idea of embodied cognition (using the whole body in the learning process).\u00a0 One example I have used is in the proof that the exterior angles of a convex polygon always add to 360.\u00a0 I expect the students to be able to prove it algebraically, but to also feel what it really means.\u00a0 They can use painters\u2019 tape to make big polygons on the floor and then walk around them.\u00a0 No matter how many sides there are, their body will do one complete rotation as they move around.<\/p>\n

There are other ways that my teaching has benefitted from my involvement with MathWeave<\/em>.\u00a0I am a complete novice when it comes to weaving.\u00a0 I can get frustrated by how slow I am and how much of a struggle it can be.\u00a0 I had an epiphany about this though- it helps me to imagine what many of my students are going through when they are struggling with new concepts.\u00a0 In my first year math concepts class, they have a final project where they have to research a way to connect math to something they are interested in.\u00a0 Originally it had to be a paper.\u00a0 Then it could be a paper or poster.\u00a0 Over time, I\u2019ve gotten more flexible in how the students present their projects (and the pandemic also forced me to be more creative).\u00a0 Some will create music or visual art.\u00a0 I\u2019ve also noticed that the more artistic activities often show different strengths of students compared to what we typically assess.\u00a0 This helps the students see themselves and each other differently.\u00a0 It has also helped me to see many of them in a new light.\u00a0 Every year I am blown away by what they come up with.\u00a0The MathWeave<\/em> group continues to be a respectful community of learners, and that is one my goals when I am teaching- that the classroom is a community of learners where we can all contribute and we can all learn.<\/p>\n","protected":false},"author":9,"template":"","section":[56],"keyword":[322,85,200],"class_list":["post-13286","article","type-article","status-publish","hentry","section-education-notes","keyword-education-2","keyword-math-education","keyword-mathematics-education"],"toolset-meta":{"author-4-info":{"author-4-surname":{"type":"textfield","raw":""},"author-4-given-names":{"type":"textfield","raw":""},"author-4-honorific":{"type":"textfield","raw":""},"author-4-institution":{"type":"textfield","raw":""},"author-4-email":{"type":"email","raw":""},"author-4-cms-role":{"type":"textfield","raw":""}},"author-3-info":{"author-3-surname":{"type":"textfield","raw":""},"author-3-given-names":{"type":"textfield","raw":""},"author-3-honorific":{"type":"textfield","raw":""},"author-3-institution":{"type":"textfield","raw":""},"author-3-email":{"type":"email","raw":""},"author-3-cms-role":{"type":"textfield","raw":""}},"author-2-info":{"author-2-surname":{"type":"textfield","raw":""},"author-2-given-names":{"type":"textfield","raw":""},"author-2-honorific":{"type":"textfield","raw":""},"author-2-institution":{"type":"textfield","raw":""},"author-2-email":{"type":"email","raw":""},"author-2-cms-role":{"type":"textfield","raw":""}},"author-info":{"author-surname":{"type":"textfield","raw":"Taylor"},"author-given-names":{"type":"textfield","raw":"Tara"},"author-honorific":{"type":"textfield","raw":""},"author-email":{"type":"email","raw":"ttaylor@stfx.ca"},"author-institution":{"type":"textfield","raw":"St. Francis Xavier University"},"author-cms-role":{"type":"textfield","raw":""}},"unknown":{"downloadable-pdf":{"type":"file","raw":"https:\/\/notes.math.ca\/wp-content\/uploads\/2022\/09\/Taylor_Education_Sept2022.pdf","attachment_id":13304},"article-toc-weight":{"type":"numeric","raw":"52"},"author-surname":{"type":"textfield","raw":"Taylor"},"author-given-names":{"type":"textfield","raw":"Tara"}}},"_links":{"self":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/13286","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/users\/9"}],"version-history":[{"count":6,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/13286\/revisions"}],"predecessor-version":[{"id":13303,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/13286\/revisions\/13303"}],"wp:attachment":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/media?parent=13286"}],"wp:term":[{"taxonomy":"section","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/section?post=13286"},{"taxonomy":"keyword","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/keyword?post=13286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}