Week 2: Activity Reflection

 What difference (if any) did it make to actually cut up and observe fruits and vegetables, to observe a honeycomb or a snowflake, and to have an octahedron and an icosahedron, etc. that you made as you were reading the text?

In my opinion, engaging in hands-on activities like cutting and observing fruits, provides a profound and immersive educational experience, which goes beyond standard theoretical learning.

What difference might it make to kids to learn from real 3D living things and/or objects with shape, texture, smell, taste, etc., as opposed to 2D printed images?

I tried the activity of making a 3D Hexahedron/Cube with my kids. When questioned about the properties of the 2D cubic shape, they expressed uncertainty. However, when it came to creating the 3D shape, they found it engaging and interesting. They were able to easily identify the properties of a cube in the three-dimensional context.

What difference might it make for students with sensory impairment (low vision, auditory impairment, etc.)?

I think for students with sensory impairments like low vision or auditory issues, the use of real 3D objects offers a multisensory learning experience. The visual and tactile elements compensate for limited reliance on auditory input. Real 3D objects promote concrete understanding, enhance spatial awareness, and facilitate independent learning. For those with low vision, it provides a hands-on understanding of shapes and textures.

Reading Week 2: Excerpts from Johannes Kepler (1611/ 2010) On the Six-pointed Snowflake: A New Year's Gift.

23-01-20

Weekly Reflection 2

Summary:

In this fascinating essay, the author asks a captivating question about why snowflakes always have six corners and radii tufted like feathers. To unravel this mystery in the 16th century, Kepler delves into various aspects of nature, exploring the geometric patterns in beehives, the shapes of pomegranate seeds, and the spiral structure of snail shells. The author also touches on the formative power of nature itself. Kepler's inquiry extends to the purpose behind hexagonal bee cells, comparing them to regular solids and exploring the tightest possible arrangements of spheres. The author also examines the shape of pomegranate seeds, attributing their rhombic form to material necessity and the growth principle. The discussion extends to hexagonal bee cells' purpose, contemplating whether they align with natural philosophers' views on efficient spatial filling. Throughout the essay, there's a blend of scientific curiosity and philosophical reflections on the intricate patterns observed in the natural world.

Stops:

1. If you should ask the geometers on what plan the cells of bees are built, they will reply, on a hexagonal plan. The answer is clear from a simple look at the openings or entrances, and the sides that form the cells. Each cell is surrounded by six others and Is divided from the next by a shared side. But if you observe the bottom of each cell, you will notice that it slopes into an obtuse angle formed by three planes. This bottom (which you might call the "keel") is joined to the six sides of the cell by six other angles, three higher ones that are trilateral, just like the bottom angle of the keel, and three lower ones, in between, that are quadrilateral.  

I vividly recall a fascinating personal experience that aligns with Kepler's exploration of the geometric intricacies within beehives. During a biology class in school, our teacher brought in a beehive model for us to examine closely. As we observed the hexagonal cells, the instructor emphasized the significance of their shape in terms of efficiency and resource optimization. Looking at the openings and sides, it became apparent how each cell neatly fit with six others, forming a harmonious pattern. Reading the text and looking at the amazing internal structure of the bottom of each cell was particularly intriguing for me, with its obtuse angle and the intricate arrangement of trilateral and quadrilateral angles connecting it to the sides.

My question is how do other living organisms, besides bees, utilize mathematical principles in their behaviors and structures? Whether it's the symmetry of flower petals, the repetition of the angles in the spider web, the spiral patterns in shells, or the way animals navigate, what other examples exist in nature where math seems to play a role? how might these mathematical patterns contribute to the survival and harmony of ecosystems?

2. Having made this observation, one may go on to inquire about purposes: not the one that the bee pursues as it goes about its business, but rather the one that God himself, as the creator of the bee, had in mind when He prescribed upon it the laws of its architecture.

This thoughtful quote made me think about the deep connections between nature and divine design. It led to questions about the higher intentions shaping the precision and order found in the natural world.

Another question is how can the inherent mathematical patterns in nature inspire new approaches or solutions in human endeavors, whether it be in technology, design, or problem-solving?

Reading Week 1: Mitch Nathan (2021) Foundations of Embodied Learning

23-01-14 

Weekly Reflection 1: 

Summary: 

The provided text discusses how young children effortlessly learn balance and spatial navigation, drawing a comparison with the challenges faced in programming robots to acquire similar skills. It points out the surprising difficulty in teaching computers to understand speech, contrasting it with how infants naturally develop speech skills by their first birthday. The main concern highlighted is the absence of a well-defined learning theory in educational systems, leading to decisions based on personal ideas about learning and resulting in inefficient exercises and ineffective teaching methods.

The author argues for an integrative framework that recognizes diverse learning styles across different time scales. Emphasizing the importance of embodied learning, where individuals use their bodies to make sense of new ideas, the text provides an example of early algebra education. It contends that current education systems often undervalue embodied forms of knowing, relying too much on mentalistic approaches that separate the mind from the body. The text identifies two obstacles to improving educational systems: misguided practices and policies, and resistance from scientific fields. It criticizes the overemphasis on abstract concepts, limiting students' access to valuable cognitive resources and potentially discouraging them from pursuing certain career paths.

In the discussion of Mathematics Education, the authors explore the idea that mathematics, often perceived as abstract and disembodied, is fundamentally connected to embodied experiences and conceptual metaphors. Drawing from the work of Lakoff and Núñez (2000), he argues that mathematical ideas, such as quantity, construction, length, and motion, are grounded in basic metaphors derived from human experiences. These grounding metaphors serve as the foundation for understanding and applying mathematical concepts.

The authors emphasize that while grounding metaphors are powerful for foundational concepts, they become challenging to apply as mathematical phenomena grow in complexity. To bridge this gap, they introduce the concept of linking metaphors, which enables individuals to offload cognitive operations from a grounded domain to a new and more complex domain. The linking metaphor approach demonstrates significant generativity for mathematics, allowing for the extension of concepts like analytic geometry and infinity. The discussion highlights the importance of embodied experiences in mathematical learning and suggests that effective teaching strategies involve connecting mathematical ideas to learners' lived experiences.

Stops:

1. "One program, in particular, has argued that mathematics is not abstract and disembodied at all, that mathematics ideas are fundamentally grounded in conceptual metaphors of embodiment of movement and space, and that this has guided the major advancements in mathematics." (pg. 147)

After reading about the embodied nature of mathematics, I draw connections to my teaching experiences in elementary school, particularly when introducing the concept of symmetry, which proved challenging for some students to visualize. In my approach, I leverage the use of students as tangible examples of symmetrical objects. I encourage them to stand upright and rotate their bodies, fostering a discussion about how their bodies exhibit rotational symmetry and lines of symmetry. Hence considering the diverse learning styles within the classroom, combining visual aids with embodied mathematics creates a multisensory learning experience that can enhance comprehension and retention of the concept.

My question is how to balance the need for traditional, abstract representations (such as formal notation) in the context of Asian teaching style with the embodied learning approach?

Are there any specific strategies you've found effective in helping students transition between these different modes of understanding?

2. "Despite the enormous importance of effective educational systems for social mobility, individual opportunity, and a healthy and secure nation, educational institutions are not guided by a coherent, evidence-based theory of learning." (pg.3)

I can identify instances during my teaching career, where the lack of a clear and evidence-based approach to learning has impacted the overall effectiveness of the educational experience. For example, recalling the Cambridge mathematics checkpoint exams where the teaching methods seemed disconnected from a comprehensive understanding of how students learn best. This disconnection has hindered the ability to fully grasp and apply concepts.

3. "Educators specifically design classrooms to restrict students’ physical and social interactions. This practice only increases with students’ age. The education community creates testing situations that restrict children’s ability to move their bodies in ways that can help them think, interact with objects, and interact with other people. For many students, schools with traditional instruction are a unique setting where they are blocked off from access to some of the most useful and flexible cognitive resources they have, resources that people ordinarily use while thinking and learning in nearly any other setting (Resnick, 1987)." (pg.4)

I can relate to that based on my daughter's experience in kindergarten. When she began learning addition and subtraction, her teacher discouraged her from using her hands as a method, instead emphasizing different and specific approaches to perform these operations. In the context of the South Asian teaching style emphasis is given on the students' testing through different grade level exams and emphasis is given on teaching traditionally.

Are there alternative assessment methods that could better accommodate embodied learning in the Canadian context?