Week 3: Sustainable mathematics in and with the living world outdoors

 

Reading Portion:

In this week's reading, Doolittle's exploration of the failures of the grid and its implications on spatial and temporal organization, three stops along this intellectual journey stand out and spark curiosity.

First Stop: Seductive Simplicity of the Grid

The journey begins with the seductively simple nature of the grid. This system effortlessly extends into domains like neighborhoods and towns and three-dimensional spaces like buildings and malls. Doolittle's analogy of gardening leading to geometry and the connection between linen and line resonates deeply. With its straight lines and right angles, the grid provides a sense of control, predictability, and mastery. However, the simplicity that makes the grid appealing also raises critical questions about its adaptability to the complexities of real-world applications. The tension in the fiber, which creates linearity, becomes a metaphor for the tension between imposed order and the organic growth and diversity of specific places.

Second Stop: Failures and Control

As the grid exploration progresses, Doolittle astutely observes that failures in the grid system have been apparent since its inception. However, responding to these failures raises essential questions about control and ownership. The grid, deeply ingrained in our thinking, leads to rules governing behavior, shaping our sense of control. The failure to let go of the grid results in the subordination of a place's unique qualities to the grid geometry's imposed uniformity. The grid is not merely a geometric construct but a manifestation of cultural and human tendencies toward order and control.

Third Stop: Giving Thanks and Liberation from the Grid

A fascinating turn in the journey comes with the Indigenous perspective, introducing the idea of giving thanks and offering an alternative solution to the Königsberg Bridges problem. Acknowledging what has been missing from the discussion, recognizing the river as a foundation, and giving thanks through offerings present a profound departure from the rigid confines of the grid. The narrative weaves through philosophy, bringing in Immanuel Kant and his habitual walks, drawing parallels to the mental abstraction imposed by accepting Euler's negative result. The call to widen our perspective, move off the grid, and see the world as it truly is bringing a refreshing perspective to mathematical problem-solving.

Engaging Questions for Reflection:

1. How do the failures of the grid, as highlighted by Doolittle, resonate with your understanding of spatial and temporal organization in various contexts?
2. How can alternative geometries, described as "geometries of liberation," find a place in modern mathematics education, and what challenges might arise in incorporating them into the classroom?

Reference:

Doolittle, E. (2018). Off the grid. In Gerofsky, S. (Ed.), Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7

 

Activity Portion:

My son's drawing                                                               My spouse's drawing.

The activity described in Doolittle's "Off the Grid" gave me a unique perspective on observing and sketching the world around me, even if I could not go outdoors. I was sitting indoors with my spouse and two-and-a-half-year-old son. We looked through the window; we focused on living beings and human-made things, using watercolors and brushes to document my observations. I did the observation, and my spouse and son made the drawings. As I found a spot to sit and observe, I noticed various lines and angles in the natural and human-made things. I observed natural flowing lines like plants, trees, rocks, and the sky. The natural world seemed to express itself through gentle curves and irregular shapes. On the other hand, human-made things, like buildings, roads, fences, and street lights, displayed more structured and geometric lines and angles. Therefore, the contrast between the organic and the structured was apparent.

Reflecting on these observations, it became evident that specific patterns emerged in both living and human-made elements. While living things displayed a sense of irregularity, human-made objects adhered to a more ordered and intentional design. However, there were exceptions, as nature sometimes exhibited symmetrical patterns, and human-made structures occasionally embraced organic forms. The patterns I observed raised questions about why such distinctions exist. I pondered whether these patterns are inherent like things or influenced by human perception and design choices. The contrast in lines and angles reflects a fundamental difference in the creation processes, one guided by nature and the other by human intention.

Considering how this activity could be translated into teaching, close observation, and sketching could be powerful tools for helping students understand lines and angles. By encouraging students to explore the world around them through drawing, educators can foster a deeper connection between their eyes and hands, promoting a more intimate understanding of geometry in their surroundings.

Moreover, I contemplated integrating whole-body movement into the learning experience. For instance, the video of two UBC undergraduate students, Sam Milner and Carolina Azul Duque, demonstrates the  Dancing Euclidean Proofs. Indeed, it suggests one-way students could experience lines and angles through activities that involve exploring the outdoors. I wonder what other ways we can engage students with the living world to offer a holistic understanding of geometry, connecting abstract concepts to tangible, real-world experiences. This approach could make learning more immersive and memorable for students.

 

 

 

Week 2: Multisensory mathematics

As I delve into this insightful exploration of embodied mathematical cognition, several stops along the way have caught my attention. Here are two key moments that resonated with me.

1. Antagonistic Nature of Education: The introductory passage about living in an antagonistic society and the parallel drawn to education sets a powerful stage for the paper. The contrast of the real and the imaginary, the concrete and the abstract, creates a lens through which we view the challenges in education. The authors highlight the tension between what teachers are trained to perceive as "normal" and the diversity within their classrooms. This makes me wonder: how can educators learn to recognize and appreciate the diverse ways in which students perceive and engage with mathematical concepts? Indeed, this stop raises awareness about the need for a shift in perspective within the educational system, urging teachers to adapt our methods to embrace diverse learning styles.

2. Multimodal Resources and Embodied Learning: The paper's core argument revolves around the embodied perspective on mathematical cognition. The experiences of two blind students navigating symmetrical figures and geometrical transformations become a lens through which we understand the interplay between the body and cognition. One of the pivotal moment is when the discussion transitions from symmetrical figures to reflection. Edson, with visual memories, leverages his past experiences with mirrors to conceptualize and articulate mathematical properties. In contrast, Lucas, without visual memories, relies on tactile exploration and hand movements. This makes me wonder: how do the embodied experiences of individuals shape their approaches to mathematical understanding?  Certainly, this stop encourages a deeper exploration of how personal histories and sensory modalities influence cognitive processes.

Questions for Discussion:

1. How can educators cultivate an inclusive learning environment that recognizes and accommodates diverse embodied experiences in the teaching of mathematics?
2. In what ways might the reliance on sighted learners' trajectories hinder the development of effective learning scenarios for blind students, and how can educators overcome this challenge?

Lulu Healy & Solange Fernandes (2013), Multimodality and mathematical meaning-making: Blind students' interactions with symmetry. 

Week 2 activities

Engaging in hands-on mathematical activities, inspired by the videos and readings, has been a transformative experience for me. The combination of visual, tactile, and sensory elements brought the mathematical concepts to life in a way that simply reading or watching videos could not achieve.

Firstly, experimenting with data analysis using candies allowed me to apply abstract concepts in a concrete and delicious way. Creating a Bar Graph and Pie Chart with Rockets and Smarties not only reinforced my understanding of data representation but also made the learning process enjoyable. It was a reminder that mathematical activities can be both educational and fun, especially when they involve something as universally appealing as candy.

Bar Graphs:

Pie Chart

The exploration of hexaflexagons, following Vi Hart's instructions, provided a hands-on experience with geometry that went beyond traditional methods. The tactile sensation of folding and flipping the hexaflexagon enhanced my spatial awareness and geometric intuition. Decorating the surfaces added a creative dimension to the activity, making it a personalized learning experience.

Cutting a bagel into a mathematically correct breakfast, inspired by George Hart's video, shifted my perspective on geometry. The tangible experience of creating interlocking rings with a real-life object like a bagel added a practical dimension to the abstract concept. This hands-on activity demonstrated the relevance of geometry in everyday life, making the learning experience more relatable and memorable.

Considering the reflection questions, the hands-on experimentation with these mathematical activities made a significant difference in my understanding. It not only solidified abstract concepts but also made the learning process enjoyable and memorable. For students, learning from real 3D objects with various sensory elements could enhance their understanding and retention of mathematical concepts. The engagement of multiple senses, such as touch, smell, and taste, can make the learning experience more immersive and appealing. For students with sensory impairments, hands-on activities become even more crucial. Utilizing real-life objects with distinct shapes, textures, and sensory qualities can provide a richer learning experience. For example, exploring geometric concepts with objects that can be touched, smelled, or even tasted can compensate for the lack of visual or auditory stimuli. This approach not only promotes inclusivity but also recognizes the diversity of learning styles and preferences among students.

Embracing Embodied Mathematics to Connect the Dots between Body, Mind, and Classroom Beauty.

In our exploration of mathematics, it is crucial to question the perceived separation between mathematical concepts and the physical world, challenging assumptions about how and where math learning occurs. Traditional classroom settings, confined within four walls, often promote passive learning and portray mathematics as isolated mental calculations. However, our recent experiences with embodied mathematics have opened up new avenues for understanding, bringing beauty to the classroom.

This week's reading, inspired by Nathan's insights from "Foundations of Embodied Learning," demonstrates that the challenge in mathematics lies not in understanding ideas but in grasping the meaning of formal notation, which resonates deeply (Nathan, 2021, p. 147). Our bodies, with their unique measurements, offer an untapped resource for learning and teaching mathematics.

I engaged in a body measurement activity this week. The exercise is on calibrating my body, recording the results, and comparing them with a tall adult man(www.vendian.org).

The variability in body size, as evident in my measurements, challenges preconceived notions about standardized learning tools. When introducing the activity to my grade 8 students, their initial enthusiasm vanished when they realized it was not a graded task. They questioned the utility of measuring body parts without traditional tools like tapes and rulers. However, my grade 12 service student embraced the activity passionately, showcasing the potential for embodied learning.

Here is my grade 12 service student measuring:

His approach, guided by the conceptual metaphor of "NUMBERS AS PLACES ALONG A PATH," as discussed by Nathan, involved using body parts, including the index and hand span, to measure dimensions.

After calibrating his body measurements, he made measurements of an Elmer's Corrugated Tri-Fold Display Board-White (big rectangle) and French vocabulary sheet (small rectangle) using his body parts:

 

Then, he made calculations to find out how many small rectangles could fit into the big rectangles.

 

This embodied exploration showcased the potential of multisensory learning and integrated artistic expression into mathematical concepts. Therefore, the resulting Elmer's Corrugated Tri-Fold Display Board became a mathematical creation, exploring dimensions and measurements and a practical and aesthetic addition to the classroom.

 

His work is a fantastic opportunity to teach and learn concepts of dimensions and measurements. On one side, the product of his work serves as a board to display students' activities and, on the back side, the French vocabulary they need to learn. Moreover, it is a decoration for my classroom. For instance, my stand desk is a storage for my classroom games, dividers, and dictionaries that are not visible anymore when students sit on their desks.

Roger Antonsen's TED talk further emphasized the hidden beauty of mathematics in understanding the world. He argues that exploring mathematical concepts from various perspectives offers a transformative approach to learning (Antonsen, 2015). By engaging all our senses, incorporating movement, and observing the living world, we aim to unlock new pathways for understanding mathematical patterns and ideas.

The goal is clear – experimenting with alternative teaching methods beyond conventional approaches. This reflection does not advocate discarding traditional mathematical tools but aims to explore ways to augment them with embodied, arts-based learning experiences. In doing so, we challenge the perception that precision and accuracy in mathematics are divorced from the physicality of our bodies and the artistic expressions of the world.

Reflection Questions:

1. How can integrating embodied, multisensory learning experiences in mathematics address the challenges of understanding different math concepts?
2. In what ways do you envision the fusion of artistic forms of expression and exploration beyond the traditional classroom contributing to more comprehensive and enjoyable learning and teaching of mathematics?

Top of Form

 

 Hello

My name is Tony Domina. I live and teach in North Vancouver. This is the territory of the Coast Salish people. I express our gratitude to the Squamish Nation (Skwxwú7mesh) and Tsleil Waututh Nation (Səl̓ílwətaʔ/Selilwitulh), and I value the opportunity to learn, live, and share educational experiences on this traditional territory.

This blog investigates concepts and ideas for teaching and learning mathematics.