Activity Week 7

This activity is based on Ali and Colin's activity of representing binary numbers as different coloured concentric circles. I have extended this activity and I have done for base 2 and 5 addition and subtraction. It is really an engaging activity for students to understand binary arithmetic operations.

The concentric circles activity combines visual elements with hands-on engagement, providing a multi-sensory learning experience for students. Through this approach, students not only grasp the abstract concepts of binary arithmetic but also retain the knowledge by associating it with a tangible and visually appealing representation. 

To extend this activity, I think about doing the binary operations like the braiding we have done in our previous class.

Curriculum Sketch

Issues/Guiding Questions

How can we implement binary operations more artistically through embodied activities like braiding?

Story

 The story of integrating the art of braiding and embodied learning for students to understand the concept of binary operations. 

Integrate embodied learning & other learning

Introduce the basic binary operations addition and Subtraction. Practice some problems on the paper.

Connect it artistically to the art of braiding  for students who face challenges in binary arithmetic operations.

Emphasize the steps of braiding and its connection with the binary operations.

Make them understand the patterns that have been seen in the binary braiding process.

Make them to solve the earlier problems differently in braiding style.

Students can engage in different binary problems in different groups by braiding with differnt colour patterns.

Extensions

Extend the braiding activity to include artistic elements such as color choices, patterns, creative designs, music and dance.

 

 Week 7 

Reflection on Reconfiguring mathematical settings and activity through multi-party, whole-body collaboration

Molly L. Kelton1  Jasmine Y. Ma

This research looks at how doing activities that involve moving the whole body in a group setting can make learning math more fun and understandable. The study compares two cases: one where middle school kids participate in a special program in their gym called the walking scale number line, and another where elementary school students do a math activity called "whole and half" during their regular math class. Both activities use the space and the body to make learning math more interesting. In the first case, seventh and eighth-grade students act as points on a giant number line on the gym floor. They engage in activities that explore displacements and mathematical concepts like doubling and opposites. The goal is for students to move efficiently and safely to their opposites on the number line, combining movement, spatial awareness, and math understanding in a dynamic learning experience. The school gym is transformed with taped number lines and activities, creating a dynamic math space. Students use their bodies to represent quantities and movements on large-scale walkable number lines made with colorful tape. This turns the familiar concept of number lines on paper into a tangible, shared experience, challenging traditional ideas like left and right and making students' movements meaningful interactions within the setting.The second case focuses on ratio and proportion through interactive and immersive experiences. It is a task called whole and half (W + H) where students used their hands and bodies to understand mathematical concepts. In this activity, one person created a space with their hands, and the other had to respond by placing their hand halfway in the interval. This interactive task aimed to engage students in embodied learning, emphasizing the use of the body as a tool and object of measurement. While similar to some existing math training tools, W + H didn't require any technological apparatus, relying solely on manual actions to explore mathematical concepts within a broader context of bodily and spatial understanding.

Stop 1

"How we think, learn, and communicate about mathematics depends a great deal on our opportunities for physical movement, interaction, and expression ."(Hall & Nemirovsky,2012).

As a math teacher, this sentence struck me because we used to depend on traditional methods of teaching which follow the curriculum. But, there are other options to teach them math more engagingly. Students think about mathematics when they get real-life experiences like physical movement, interaction and expression. When they touch and feel math in the outdoor setting with embodied activities, they feel it. Otherwise, math appears to be an abstract thing for them.

Stop 2

Maggie and Thad solved the problem from their respective physical and mathematical perspectives in the material arrangements of the space - Maggie from nine units to the right of Thad, needing to get to nine units to the left, and Thad needed to stay put but everyone on either side of him swap to the other.

(p 186).

This situation in case 1 really struck me because it shows a fascinating exploration of logical thinking and problem-solving. When the students got a situation from their life experience, they tried to solve it. We can see that the problems in textbooks are repeating and become boring for students. But, when they are really involved in a physical real-life problem, they connect it with math and get the solution for it.

Question

Reflect on a time when you integrated hands-on activities or visual representations to teach a mathematical concept. What was the impact on students' engagement and understanding?

Reference

Kelton, M. L., & Ma, J. Y. (2018). Reconfiguring mathematical settings and activity through multi-party, whole-body collaboration. Educational Studies in Mathematics, 98(2), 177-196. https://www.jstor.org/stable/45184633

 Activity

2013 Bridges Conference

This artwork is done by Jane Alder, a fibre artist, math tutor and programmer based on the angles of a pentagon.

At first, I tried to make a replica of this with craft paper with glue. But, it did not go well. And then made it with craft paper and sketch. This picture has a lot of pentagons connected with each other and was quite difficult to sketch . It contains angles 90,120,60,150, and 120 degrees. In the centre, three pentagons are connected at 120-degree angles. Unfortunately, I could not draw it properly like this picture.

                                  

If we use and help the children to draw these kinds of artworks then they can relate it with mathematics and geometry. In grade 8 of the mathematics syllabus in India, there is a chapter called construction of quadrilaterals. At that time, I used to give children these kind of activities. For them, this chapter seemed more interesting than other chapters. 

 Grade 8, Kerala State Syllabus, India

  

Art and math are always related to each other. We have to break the rules of traditional teaching methods which involve only a grid type of teaching. Art is beneficial for children's development. It fosters creativity, expression, and emotional intelligence. Math helps students to recognize the pattern, and angles in a painting or artwork. integrating art with math allows cross-disciplinary learning, logical thinking, and problem-solving skills.

  

 Reference

https://gallery.bridgesmathart.org/exhibitions/2013-bridges-conference

 

 Week 6

Reflection on "Bridges: A World Community for Mathematical Art"

By

KRISTO´F FENYVESI

The mathematical community column has explored the intersection of mathematics and art in the Bridge organization's 2005 conference. This conference was held in the Canadian Rocky Mountains at Banff. It was titled "Renaissance Banff"  because it brought together all mathematicians, artists and enthusiasts in a celebration of the interconnectedness between math and art. In addition to conferences, lectures and the theatre performance, the program included an international mathematical art exhibit, a mathematical music night, and a math art workshop series developed for teachers by teachers.

  After a professional musical performance, participants, whether they were experts or not, joined in playing instruments. This collaborative effort created a vibrant mathematical art community, where everyone, regardless of expertise, worked together, sharing the joy of a collective creative experience. The conference highlighted the idea that anyone could contribute, emphasizing equality among participants.

    The Bridges conferences at Southwestern College, initiated by Reza Sarhangi, stemmed from his multidisciplinary background as a mathematician, graphic artist, and theatre enthusiast. Sarhangi's interest in medieval Persian mathematics and arts, coupled with experiences at Art and Mathematics conferences, led to the formation of Bridges. These conferences, influenced by ISAMA(International Society of the Arts, mathematics, and Architecture), aimed to foster collaboration between mathematicians and artists, exploring the aesthetic and mathematical dimensions of their work. This integration of disciplines, pioneered by Sarhangi, contributed to the development of an innovative and interdisciplinary approach to mathematical art.

Bridges

Organization’s founders Reza Sarhangi (right), Sarhangi’s wife

Mehri Arfaei (middle), and Carlo H. Se´quin (left). (Photo: Reza

Sarhangi.)

 Stop 1

 ‘‘Mathematics creates art’’; ‘‘Mathematics is art’’; ‘‘Mathematics renders artistic images’’; ‘‘Hidden mathematics can be discovered in art’’; ‘‘Mathematics analyzes art’’; ‘‘Mathematical ideas can be taught through art.’’ After eighteen consecutive years of Bridges gatherings, we can say that the inverses are also true: Art creates mathematics; Art is mathematics; Artistic images render mathematics; Hidden art can be discovered in mathematics; Art analyzes mathematics; Artistic ideas can be taught through mathematics. (p 36)

   Reading this paragraph made me pause and reflect. As a math teacher, I hadn't thought much about the connection between math and art before. The passage also suggests that even after 18 years of gatherings, there was a recent realization about the mutual relationship between math and art. At first, they thought that math was an art and we had to look at everything through a mathematical lens. But, now they realise that art creates math so we have to see things artistically too. That is , there is a deep and interconnected relationship between math and art. Mathematics can inspire and create art and vice versa. As teachers, we have to implement this idea of integrating math and art in our classes in effective ways.

Hungarian sculptor, Istva´n Bo¨szo¨rme´nyi and

some of his mathematical artwork is based on his collaboration

with the mathematician, Lajos Szilassi

  Stop 2

Reza Sarhangi draws on his past in theatre to evoke the personable atmosphere and wealth of experiences at these first gatherings: ‘‘Theatre involves making connections with the audience that go beyond just the script […] So at Bridges, I—and the other three board members—want the conference attendees to get more than just the content of the papers, but to have an enjoyable experience that integrates art, dance, and other performances’’

Mathematics is always considered as a tough subject among students and teachers because it always aligns with a "grid style". In the past, teachers tried to go with the prescribed syllabus and tests.

As a math teacher, Sarhangi's perspective on the Bridges conference aligns with the idea that math goes beyond just numbers and formulas. It emphasizes the importance of making math engaging and relatable by incorporating creative elements like art and dance. This approach encourages students to see math in real-world contexts and fosters a more enjoyable learning experience.

Questions

• Why is it important for schools to promote interdisciplinary activities that combine both arts and math in the curriculum?
• Can you provide examples of how technology and art can be integrated to foster creativity in math education?

 Week 3 Activity

Drawing Living and Non-living Things Around You

I usually go to Central Park, Burnaby near to my apartment. I just take a walk, sit there with my friends and listen to music. Right in the middle of the park, there's this awesome lake. It's super pretty and reflects all the greenery around it. You can take a slow walk around it and even see ducks chilling in the water. It's like a nature show happening right in front of you! My friend Aiswarya and I drew the living and non-living things we saw there.

The park has these really nice gardens with lots of flowers and cool art. As you walk around, you might find hidden statues and cute bridges. It's like a treasure hunt but with nature and art! If you love to write or just want a quiet place, Central Park is perfect. Find a comfy spot, maybe under a big tree, and let your thoughts flow. The mix of city and nature around you will make your ideas pop!

 As I was sitting there I was thinking about the mathematics in nature. Math is around us in the nature. We have to explore it and as math teachers, take children outside to learn more things through hands-on activities. In both living and human-made environments, various lines and angles can be observed. Living things often exhibit curves, branching patterns, and spirals, while human-made structures commonly feature straight lines, angles, symmetry, and repetition.

 As an example, we can see that each human-made thing like in the picture you can see the bench, light, cap and dress the man-made after exact measurements by scale, tape and so on. And the natural things like trees, leaves, dogs, birds, water everything has its own essence and beauty. Those have their own angles, measurements and patterns. The thing is While general patterns exist, exceptions abound due to architectural innovations and advancements. The reasons for these patterns are rooted in natural growth processes for living organisms and considerations of functionality and aesthetics for human-made structures. Teaching about lines and angles can be effectively done through close observation, drawing, and outdoor activities. Sketching allows students to visualize geometric concepts, while outdoor experiences provide opportunities to explore and experience lines and angles in nature through whole-body movement. The combination of these approaches creates a comprehensive learning experience connecting geometry to the real world.

 Week 3

Reading Reflection on "Sustainability Education's Gift: Learning Patterns and Relationships" 

by Dilafruz Williams

This article deals with the pressing challenges of sustainability. It has been seen that traditional schooling methods, their mechanistic and technocratic systems are away from a sustainable approach. We need a shift from these methods and systems by integrating principles of systems thinking and holistic learning. David Orr and Fritjof Capra are the most important people who have postulated systems thinking and holistic learning as a means to shift our modern culture to new models and metaphors for a more sustainable world. Sustainability education would need to include the following three understandings embedded in such thinking and learning:

1. Wholeness Principle: Living systems exhibit unique properties as a whole, emerging from interactions among their parts. Understanding individual components requires grasping the entire system, and emphasizing the importance of contextual thinking in systems.

2. Network Dynamics: Life operates as interconnected networks at all levels. Nature lacks hierarchies, relying instead on networks nested within networks, forming a complex web of life.

3. Nonlinear Ecology: Relationships within ecological communities involve nonlinear dynamics and multiple interdependent feedback loops. To understand life's essence, focus on processes and relationships among living organism components.

 For this sustainability, the author presents a case study from Poland, Oregon, where the Learning Gardens model is sowing the seeds of "change". This study is based on students from kindergarten to eighth grade, where they are not just learning from textbooks but actively engaging in the process of growing, harvesting and cooking food. Additionally, the Learning Gardens program utilized two pieces of land where students, including those from diverse and low-income backgrounds, learned about growing, harvesting, and cooking food while integrating various subjects. Both locations foster students' appreciation for the environment, enhance their knowledge of the cultivation and applications of edible and medicinal plants, and provide education on nutrition and the advantages of adopting healthy eating habits. It also helps them to understand multicultural values, learn through interdisciplinary approaches, promote connections between different generations and embrace multisensory learning. Furthermore, students gain a sense of connection and empathy by participating in service-learning projects, such as working at a shelter for the homeless, encouraging them to think critically about the importance of local food production amid global challenges in energy and transportation.

Stop 1

"As I was walking down by the creek at JEAN’s farm, I noticed my good spider friend looking worried. I asked what was the matter and Spidey (the spider) told me that everywhere he tried to make his web someone would accidentally or purposely knock it down. This was bad because spiders’ webs are not only their homes, but also help them catch their food, which are nasty bugs like mosquitoes, aphids and other bugs that are bad for our gardens."(page 47).

When I went through these lines, it was really striking because as teachers when we teach something to our students, they connect it with their own life experiences. Children are curious about nature and they see everything like insects and plants as their friends. When a spider got into a problem, he identified it and connected it with his previous knowledge and grasped patterns and relationships within the ecosystem. It shows a holistic understanding of nature and the interconnectedness of components.

Stop 2

"They are clearly in need of a good meal, and if they could afford it, they wouldn’t be there…. When we serve, we show people that we care about them. When people know that someone cares about them, they are generally happier and it gives them hope."(page 47)

These lines are filled with emotions and empathy of the students towards everyone around them. I think when students see and realise the issues around their surroundings they become more empathetic and ready to help. This line might evoke a pause because it captures a significant moment of realization for the students. It signifies a transformative understanding that goes beyond academic knowledge, touching on the emotional and social aspects of learning. The awareness of societal issues seems to have triggered a sense of responsibility and a willingness to make a positive impact on them.

These pictures are from my school in India where students grow their own food for their noon meal. They use the school garden for cultivating vegetables and different herbal plants.

Question

How can educators create effective strategies to instil empathy, foster social interconnectedness, and nurture a sense of responsibility among children?

Reference

Williams, D. (2008). Sustainability Education’s Gift: Learning Patterns and Relationships. Descriptive Reports.

WEEK 2 Reading reflection on  

"TACTILE CONSTRUCTION OF MATHEMATICAL MEANING: 

BENEFITS FOR VISUALLY IMPAIRED AND SIGHTED PUPILS" by 

Angeliki Stylianidou, Elena Nardi 

Summary

This reading discusses a study focused on integrating tactile mathematics into a grade 5 classroom and promotes inclusivity for visually impaired and sighted pupils. The study is based on drawing theoretical frameworks including Vygotskian sociocultural theory and the theory of embodied cognition. The mathematical task highlighted in this paper involves the teacher asking the class to close their eyes and describe two shapes, one of which is referred to as "Shape X." These shapes were constructed using Wikki Stix, a flexible teaching tool made of wax and yarn suitable for VI pupils' learning. The focus is on Shape X, and during the task, the teacher also provides circles of various colors and sizes, prompting the class to identify differences between Shape X and the circles. This study records the contribution of two students in 2 episodes, Zak and Luke, one is visually impaired, and the other one is a sighted pupil. 

Episode 1: Zak's Tactile Exploration: 

Zak, a sighted pupil, explores Shape X through touch and vision. He constructs different meanings of Shape X through touch and vision. While he confidently states the existence of a straight-line segment when he feels the shape with his hands, he does not see a straight-line segment when he sees the shape with his eyes. His tactile experience reveals a straight-line segment, contrasting with uncertainty when visually inspecting the shape. The study interprets this through Vygotskii's theory of mediation, emphasizing the impact of different sensory tools on mathematical constructions. Zak's positive response to tactile exploration highlights potential benefits for all students. In the evaluation form of the lesson, Zak wrote that he liked “the hidden facts on the shapes” 

 Episode 2: Luke's Practical Insight: 

Luke, a visually impaired pupil, differentiates between Shape X and a circle through touch. He makes different meanings of the circle and of Shape X. He feels that the circle is going to roll more – while Shape X is not; it is instead going to “bob up and down. The study attributes Luke's varied constructions to different material tools, emphasizing the interplay of sensory and material factors in mathematical understanding. By inviting the entire class to experience mathematics through touch, the study challenges ableism, fostering a more inclusive and diverse mathematical environment. It sets the stage for a future where tactile learning is not just an accommodation but a fundamental approach to teaching mathematics. 

Stop 1 

"The hidden facts on the shapes"(Stylianidou, A., & Nardi, E. 2019, p.348) .

In this study, Zak mentioned this in his evaluation form. As a math teacher, this statement is a thought-provoking one. In Geometry, there are different shapes which involve different hidden properties which we cannot see directly. As a math teacher, this might resonate as an encouragement for students to go beyond surface-level observations and engage in deeper analysis of shapes. It prompts students to think beyond what they see initially. 

Stop 2 

"Luke makes different meanings of the circle and of Shape X through touch. He feels that the circle is going to roll more – while Shape X is not; it is instead going to “bob up and down."(Stylianidou, A., & Nardi, E. 2019, p.348). 

Luke's description of the circle as feeling like it will "roll more" indicates an understanding of spatial properties. As a math teacher, this observation aligns with concepts related to shape dynamics and spatial relationships, providing insights into how students intuitively grasp these ideas through touch. His words like "roll" and "bob up and down" suggest a kinesthetic understanding of shapes. This emphasizes the value of kinesthetic learning in mathematics, where students physically interact with shapes to enhance their understanding. Math teachers might reflect on incorporating more hands-on, movement-based activities to reinforce geometric concepts. 

Questions

1. How might including tactile mathematics tasks benefit all students in your math classroom, including those with visual impairments? 

2. How can you encourage your students to explore mathematical concepts through touch, like Luke's, to enhance their understanding? 

Reference 

Stylianidou, A., & Nardi, E. (2019). Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. Journal of Inclusive Education in Mathematics. 

Activity

Hexaflexagon

Making a hexaflexagon was a cool math activity that helped me understand shapes better. Watching videos gave me the basic idea, but actually folding and flipping the hexaflexagon with my hands made it more real. Even though I struggled at first, every attempt taught me something new. Creating the hexaflexagon changed it from something in my head to a real thing I could play with.

This hands-on experience made geometry more fun and easier to understand. It wasn't just about learning angles and folds; it was like playing with a math toy. Now, I'm thinking about how this kind of activity could help other students too. Touching and moving real shapes might be a better way for everyone to learn, especially students who can't see or hear well. It could make math class more fun and fair for everyone. Moreover, we can make the students understand math concepts more clearly. When I was teaching in India, students loved to do paper crafts and 3D shapes like pyramids, cones, rectangular prism and cubes. These hands-on activities make them engage in math class.

 

 

 

 

 

 

 

 

 

 

 

 

 

Reflection on the Research Report “Gesturing Gives Children New Ideas About Math”

by Susan Goldin-Meadow, Susan Wagner Cook, and Zachary A. Mitchell

 

Mathematics is often considered as a difficult subject for most students I have seen. As a math teacher, I have witnessed this situation where students have difficulty in solving problems and analyzing math concepts. This research “Gesturing Gives Children New Ideas About Math” is based on the significance of gestures in math teaching and learning. The study mainly focused on fourth-grade students, and they were provided with a set of problems. Researchers studied students' responses, hand movements, gestures, and ways of solving problems by placing their fingers. Some of the students used different correct gestures for solving the problem and they got the correct answers, some of the students were not showing the correct gestures and even if they got answers, the others did not show any gestures and did not get the answers. Students who showed correct gestures learned more than others in the class. Moreover, these experimenters state that gestures are not only useful for communication but also beneficial for mathematical learning. The study's implications extend beyond the traditional learning methods, and it proposes that by instructing learners on how to move their hands, educators may lay the foundations for new knowledge.

Stop 1

“Can the children’s hand movements really be considered gestures? Gestures tend to be meaningful movements produced along with speech” (Goldin-Meadow, 2003; McNeill, 1992,p.271).

When I read the article, I was struck with this question. Sometimes it is difficult to consider students' gestures as we do not know the nature of gestures. Traditionally, it is strongly connected to the spoken language. In this part, we as math teachers need more clarification about the nature of gestures.

Stop 2

“When comprehending an action word that is semantically related to a body part (e.g., lick,pick, kick), the motor area in the brain that is associated with that part (the face, hand, or leg area, respectively) is routinely activated (Pulvermuller, 2005; see also Pulvermuller, Hauk, Nikulin,& Ilmoniemi, 2005; Pulvermuller, Shtyrov, Ilmoniemi,2005). (Goldin-Meadow et al., 2009, p. 271).

 

When I read about this, it had some connection with my life as a math teacher. One of my students in grade 6 was always active in class and always solving problems by shaking legs and hands. In the first week of the class, I was quietly surprised by his actions. But he found correct answers. As I previously learned in my BEd class, learning always involves different bits of intelligence like spatial intelligence, musical intelligence, kinesthetic intelligence and so on. When I read this article, the learning involves gestures that can connect with kinesthetic intelligence, and it is especially useful for mathematics learning. However, as a math teacher, we face difficulties in implementing and teaching math with useful gestures. Only a small percentage of the class effectively uses gestures while others see math as just an abstract subject which does not involve any movements, gestures and so on.

 My questions are:

·       How might the findings of this study be incorporated into our mathematics teaching to enhance every student's engagement, promote a deeper understanding of math concepts, and contribute to a positive learning environment for every student?

·       Can you recall any instances in your classroom where gestures played a significant role in your understanding of the lesson?

Reference

Goldin‐Meadow, S., Cook, S. W., & Mitchell, Z. A. (2009). Gesturing gives children new ideas about math. Psychological Science, 20(3), 267–272. https://doi.org/10.1111/j.1467-9280.2009.02297.x