Week 3 Activity Reflections

For this week’s activity I used the outdoor space at the barn where my daughters have riding lessons. Some items took a few attempts for me to sketch, like the paver stone pattern, so I ended up cutting out the best representations and photographing them together at the end.  

     

Sketching forced me to break apart the visual whole and figure out how the pattern worked, where it started and how the shapes fit together to progress it. This aligns with Sam Milner’s thoughts about dancing proofs in our video this week, “the proof takes time, dancing goes step-by-step rather than seeing it all at once. (Gerofsky, Milner, & Duque, 2019) I think this gave me greater understanding and greater appreciation. In the video Duque also mentions that by learning with patterns outside, on and with the land, "knowledge really comes with you; it doesn’t just stay in the classroom.” Noticing recently that transference of skills between subject areas and contexts (like using math in science) is an aspect that could be strengthened, I wonder if this (learning with the land and experiencing parts of the whole) could be key.  

I found myself wanting a ruler when trying to draw the human-made objects - like I needed to measure, to get an exactness to the pattern. It is stark how much the grid that I read about this week in Gerofsky & Ostertag (2018) is very visually present in the human-made patterns. I think that my want of a ruler to get the visual grids “exact” links to the authors’ mention that “the grid is also intimately connected with a sensory bias toward the visual.” (p178) It looks “good” when it is exact and in-line. Knowing that grids are a colonial structure, I wonder if this is culturally embedded? 

The natural patterns that I found and sketched were much more rotational rather than linear. The moss made a star pattern on top, that looked to rotate in layers underneath. The pine branch ended in 2 tips, and had bunches of paired needles that went around the branch. Both patterns still held linear aspects (the arms of the stars and the length on the needles) but rather than gridding they spinned. The oyster shell was interesting, because you could see ridges of growth outwards, but not in a radial symmetry, rather it was elongated on one axis. It also had the least linear lines - moving out in waves or ripples. 

Another thing that occurred to me while making and observing, was that it would be fun to explore these patterns through print-making. Except for the shell, there is a basic unit to each pattern. In the paver walkway it is one rectangular shape that it used. In the chair wave it is a rectangle and a square. In the dog ball, a pentagon and a hexagon. In the moss a star, and in the pine branch a pairing of needles. If one were to design a stamp in these shapes, they could make patterns by repeating and reorienting the stamped shapes. 

References: 

Gerofsky, S., Milner, S., & Duque, C. A. (2019). Dancing Euclidean proofs [Video]. Vimeo. https://vimeo.com/330107264

Gerofsky, S., & Ostertag, J. (2018c). Dancing teachers into being with a garden, or how to swing or parkour the strict grid of schooling. Australian Journal of Environmental Education, 34(2), 172–188. https://doi.org/10.1017/aee.2018.34

Week 3 reading: Dancing teachers into being with a garden, or how to swing or parkour the strict grid of schooling (Gerofsky & Ostertag, 2018)

Summary: Gerofsky & Ostertag point out the dominance of grids in modern society and schooling. Grids enforce a linear, square-box model and in schools can be exemplified by rectangular classrooms, rowed seating, block-time bell schedules, marking spreadsheets, and isolated subject time-tables. While grids can give the feeling of organization, completeness and control, they are problematic because they promote linear, visually dominant, binary thinking. They are a colonial structure that limits ways of knowing and being. In their paper, the authors share experiences that they have had engaging teacher candidates in garden-based environmental education as a way to teach beyond these grids and incorporate more sensory mathematics learning (beyond the visual.) Examples include a drama performance illustrating freedom and control in a grid-grown flax garden, a Nap-In to relinquish control and experience felt time, and a weaving of braids at small, hand-held scales vs. large scale, coordinated multiperson dances to look at mathematical scale. Key findings reveal teachers have a deep entanglement with the grid even when attempting to critique it. While it seems teachers cannot be without the influence of the grid, the paper illustrates how performative and embodied practices can enable a beside-ness to the grid. Working within the grid, but allowing for playful negotiation within it rather than a binary rejection or acceptance. The concept of parkouring off the grid shows use of the rigid grid structure as a platform to pivot and swing off of in a way that turns barriers into launch points for navigating and experiencing a landscape with newfound joy. It affirms that even within rigid secondary school structures and single-disciplinary blocks, there is potential to carve out spaces for multi-sensory maths learning that embraces many ways of knowing and doing.  

Stop 1: “The sorting and ordering of things, people, beings, ideas, entities helps humans achieve a kind of power and control over a world.” (p. 174) Oh - I love a good table. I use them in my notes making, I use them to organize and make sense when learning and sharing - as you have seen in my discussion posts for our courses often, I use them in planning my day, week, month. These grids do give me a sense of control, and a visual calmness to settle chaos. I have never paused to think about the limitations that they also bring (like enforcing binaries - not everything needs to be in a box) and the colonial implications that they carry. Should I be abandoning this tool? Exploring and embracing others? It feels a bit tainted now.   

Stop 2: I appreciated the examples of embodied activities for noticing mathematical scale. The authors have teacher candidates experience hand-weaving (small scale) and outdoor weaving as a dance pattern (image below). I would like to try this with students.    

Stop 3: “We think of the garden (or other outdoor place where everything is alive — the forest, beach, park, mountainside, prairie) as having agency as a living co-teacher, rather than as an inert ‘background setting’. (p. 182) I think that this is a beautiful sentiment as we plan for outdoor experiences. While I think that this can start from the narrative that I share with students, I feel like I need to see more examples of how to frame activities and experiences this way.  

Questions: 

1. I have not done much math learning outside. (If I’m outside it's often doing science learning.) Have you had successful math learning outdoors? 
2. What can “parkouring the grid” look like in an 80 minute secondary class with curricular outcomes to meet? With limited access to natural outdoor spaces within that timeframe? 

References: 

Gerofsky, S., & Ostertag, J. (2018c). Dancing teachers into being with a garden, or how to swing or parkour the strict grid of schooling. Australian Journal of Environmental Education, 34(2), 172–188. https://doi.org/10.1017/aee.2018.34

Week 2 Activity Reflections

I recruited my older daughter (grade 9) to help me explore mathematical possibilities with mini candy canes that I had leftover from before winter break. Our analysis ideas included: broken:unboken out of the box (6/36), and even:odd red/green swirl pattern (21:9). When we tried to use them to build geometric shapes, we were disappointed to see that the shapes were not uniform. We envisioned being able to hook and hang them but only some could hook together, some were formed to narrow and could not. Some were alsolonger in length, some shorter. This made shape exploration tricky. But, these differences sparked the idea that they could be weighed to find the average mass of each individually wrapped candy, the range of sizes, etc.      

For the virtual fair in 550 I hosted a workshop with Möbius strips because I had not heard of them before that class. As you can imagine, I was excited to try making one out of a bagel. However, I only had bagels that were presliced, and those did not work. With creative willpower I used cucumber chunks instead. I cut segments and hollowed the center. In thinking about making the paper Möbius strips, I knew that you take a flat 2D strip and then looped it into a ring and made a half twist (180o) before taping it into a loop. This was different because I couldn’t twist the cucumber, that motion needed to be made with the knife instead. So I tried several attempts at cutting around the segment while rotating the knife 180o. Many attempts in, I was worried that I wouldn’t be able to do it with the amount of cucumber I had (which was a whole cucumber) and frustration kicked in. I felt like I needed a knife with two sharp edges - how could I twist one knife without needing to cut backwards? Taking a breath, I took a step back and just practiced moving the knife in my hand 180o. Start with palm up, finish with palm down. In my second to last chunk I got it! Then broke it apart trying to get the photo…I easily reproduced the motion a second time, with my last segment, but it was the tapered end so still not the clearest shot. I was so pleased with myself for persisting. Practicing the movement pattern I wanted first was the ticket. 

In her Supersmarties video, Vi Hart said, “sometimes it’s just more fun to let my hands solve their own problems.” After this week’s experiences, I think this is true. My daughter and I came up with more ideas than I thought we would by dumping the candy out and exploring with it. We also had some ideas (like the hook together building) that didn’t work but we couldn’t tell that until we tried it. My brain conceptually knew how to get a cucumber Möbius strip, but I couldn’t get my hands to replicate that concept without deliberate practice of the movement. (I was doing a 360 instead of 180.) In these cases my hands found more problems, and solved more problems, than my eyes/brain could alone. I see these experiences broadening perspectives and deepening understanding for all learners.   

Week 2 Reading: Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils (Styianidou & Nardi, 2019)

Summary: In this article, Stylianidou and Nardi investigate how a universally designed math task involving senses of touch and sight might benefit all learners in a classroom, not just those with visual impairments (for whom tactile adaptations are often targeted.) Observations presented in the article were from two students, one with a visual impairment and one without, in a grade 5 class. The task had the class close their eyes and describe shapes made out of Wikki Stix (Figure 1). Students were then asked to compare the bottom shape (a circle with a straight segment embedded into the circumference) to true circle shaped manipulatives of different sizes and colours. It was found that students could feel aspects of the shape that they could not differentiate visually - they saw a circle, but felt a section with a straight segment. This was contributed to tactile perception providing a gradual way of knowing "allowing the exploration of the object from its individual parts to its whole,” while visual perception is more wholistic, and “at once.” (p. 347) Furthermore, the visually impaired student demonstrated embodied imagination offering practical insights unique to their experience (imagining whether a shape would "roll" or "bob") that enriched the entire class's discourse. The key takeaway is that tactile tasks are not merely accommodations for disability but valuable learning tools for all learners to develop and deepen mathematical thinking. By universally using with the whole class, it also fosters a more inclusive and non-ableist environment.

     

Stop 1: The original shapes that students explored with their eyes closed were constructed of Wikki Stix “a flexible teaching tool made of wax and yarn.” (p. 346). I had not heard of these before, but they look fun and adaptable to lots of different activities. I might try ordering a few on Amazon (they also have many non-brand-name options) to try out. Have you used them for learning tasks before?  

Stop 2: “Vison is wholistic and and touch is gradual, allowing for exploration of an object from its individual parts to a whole.” (p. 346) I really liked this articulation of why the non-visually impaired student constructed different meanings of the shape with vision vs touch. To the student the shape looked like a circle, but it did not feel like a circle because it had a flat spot. The student shared that touch exposed “hidden facts” about the shape. (p. 348) I feel like knowing this helps in designing when/how tactile learning opportunities may best fit into a lesson.  

Stop 3: Stylianidou and Nardi cite Gallese and Lakoff (2005) for the term “embodied imagination.” Looking into this more, it is a concept used to describe how imagination is grounded in bodily experience. They say when we imagine we re-activate neural systems originally developed through body action and perception; therefore, imagination is not simply making things up, but simulating experiences shaped by live action.  

Questions: 

1. Knowing that imagination/creativity are precursored by embodied experiences makes me want to include them even more! This article focuses on tactile experiences specifically, which I have not embedded into any of my math teaching before. Have you used any tactile learning in your math classrooms?
2. I am noticing that embodied learning often involves purchased materials like Wikki Stix, various candies or food items, or other physical manipulatives. My school does not have a budget for our math department. Should I start advocating for one? How do you envision the cost of materials playing out in your workspace if this was a regular part of your practice? 

Resources: 

Gallese, V., & Lakoff, G. (2005). The brain's concepts: The role of the sensory-motor system

in conceptual knowledge. Cognitive Neuropsychology, 22(3-4), 455-479.

Stylianidou, A., & Nardi, E. (2019). Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 343–350). University of Pretoria, South Africa.

Week 1 Activity Reflections

In his TED Talk, Antonsen relays that to really understand something we must be able to appreciate it from multiple perspectives. This reminds me of Diamond Paper from Jo Boaler (Mathematical Mindsets, YouCubed) where your concept goes in the center, say 4/3, and then you represent that with words, pictures, symbols, and a real-life situation. However, like Dr. Gerofsky states in our week’s introduction, it still requires students “to ‘just think’ their way into understanding new mathematical ideas, while sitting statically and silently in a chair.” What this activity lacks after watching Antonsen is bodily ways of knowing and imagination. I loved that Antonsen says we can make up the language to represent the mathematical patterns we see. It feels like permission to explore and create in a math world that can feel full of strict rules. I wonder if Diamond Paper could be adapted to include these aspects better - maybe four quadrants is not enough…? Maybe a class breaks into small groups that each create a new perspective to share out? I am excited to open this level of imagination up to students and see what they can come up with. 

 

Both Antonsen and Nathan connect perspective and experience to creating metaphors and analogies to deepen understanding. In a reality where students are looking for an instant answer, I will be looking for opportunities to develop this cognitive skill. 

I grew up on the water and have never known what a fathom is, only that it is about six feet. I found the traditional measurements fascinating. Noticing that there were no volume measurements to calibrate on the list, my daughter Ella (10) and I tried this out while baking some homemade granola bars :)  

   

My palm divot holds ½ tbs                 Ella’s palm divot holds 1 tsp

My hand scoop holds ¼ cup                Ella’s hand scoop holds ⅛ cup

My two hands hold ½ cup                Ella’s two hands hold ⅓ cup

I have always respected those who cook by feel. They just know how much to add here and there, no measuring tools or recipe needed. Muscle memory built by experience over time. There is something very grounding about not needing formal tools to measure. It can just be us and our body - no rules, formality, or external tools required. Next cooking by feel goal: gain the muscle memory needed to estimate the amount of spaghetti noodles needed when cooking for my family. I always seem to be way over or under!   

  

References: 

Antonsen, R. (2016, December 13). Math is the hidden secret to understanding the world [Video]. TED. YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o

Boaler, J. (2015). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Jossey-Bass.

YouCubed. (n.d.). YouCubed: Inspire ALL students with open, creative mindset mathematics. Stanford Graduate School of Education. https://www.youcubed.org/ 

Week 1 Reading: Foundations of embodied learning: A paradigm for education (Nathan, 2021)

Summary: Nathan points out that our current educational practices are developed from a poor evidence-base of how learning works. He argues that choices for instructional design should return to the roots of how we actually learn, which is through embodiment. His book presents an evidence-based framework of this; I have included a table of his examples of embodied math learning below. The idea is that learning activities maximize first-order experiences that live out the world (playing store), as opposed to the second-order experiences that dominate traditional teaching today, which describe the world (word problems.) Action schemas (I gave you $5 to pay for a $2 item and got $3 back) intuitively build mathematical reasoning (5 take away 2 leaves 3 leftover.) These types of embodied experiences develop grounding metaphors (object operation of removing to arithmetic operation of subtraction.) To these simple foundations, we can then link to more complex topics (like the taking away of negative numbers.) Grounding metaphors are often learned through culture by playing games at home or doing everyday family tasks, but where they are not, teachers must explicitly teach them in order to create the foundations to link more complex topics.     

Examples:

Make a collection of things - Idea of quantity (what is 5)   [this reminds me of hundreds day collections!]

Using a balance scale - Equated quantities (2+3=5)

Playing games where pieces are moved a number of spaces along a path - Number lines and directionality of operations

Making angles with arms - Magnitude of angles

Stop 1: “Educational institutions are not guided by a coherent, evidence-based theory of learning” (p. 3) and “we regularly make educational choices and implement educational programs with a poor understanding of how people learn.” (p. 4) These statements had me feeling a bit defensive, thinking that all the work we put into designing quality learning experiences and programming isn’t effective and lacks evidence-base. Can this be true? But, then I remembered a discussion in an earlier course about John Hattie’s work in ranking teaching practices effect, and how he started this project due to lack of evidence-base. It is interesting that in scanning the list of the visual representation, I could not find any discrete mention of embodied practices (other than broad “music programs” or “drama/arts programs.”) It would be interesting to see where some of the practices Nathan mentions would fall in.  

Stop 2: Nathan mentions children often coming to school with lived experiences, like games played or measuring to prepare home meals, but “when it is not learned culturally, it must be taught explicitly.” (p. 148) This makes sense, but caused me pause to reconcile with my current teaching experience. It seems that some of the basic, grounding metaphors Nathan mentions are not as foundational for students as they used to be, at the start of my career. (For example, my grade 9s really struggled with ordering rational numbers on number lines and thinking about directionality with integers.) Is this because home life is changing? Families are busier or the activities that they do together are different? Or maybe the foundations are there, but we (teaching at the middle/secondary years) are not linking to the foundations well enough for the new concepts to attach on? I would like to explore how to “teach” the linking aspect better. 

Questions: 

1. Nathan remarks that “education is basically about engineering learning experiences,” (p.3) is embodied learning kinda the same thing as experiential learning?  
2. Nathan mentions our current educational culture tends to steer “away from concrete, hands-on thinking about mathematics” and that restriction of “students’ physical and social interactions…only increase with students’ age.” (p.4) If any, what embodied practices do you already use when teaching math? What challenges do you see in trying to bring in more first-order experiences and embodiment into your classroom?     

References: 

Nathan, M. (2021). Foundations of embodied learning: A paradigm for education. Routledge. 

Waack, S. (2018). Hattie Ranking: 252 influences and effect sizes related to student achievement. Visible-Learning.org. https://visible-learning.org/hattie-ranking-influences-effect-sizes-learning-achievement/

Hello Cohort!

I am excited to share my journey exploring the teaching and learning embodied mathematics with you! I teach high school math and science on Vancouver Island in BC. When I am not teaching I am happy to get outside with my 2 daughters and our pup, Benson.