Week 4 (Mathematics and Art)

 Reading Reflection:

 Berezovski, Cheng & Damiano (2016). Spinning Arms in Motion: Exploring Mathematics within the Art of Figure Skating. Bridges Math and Art proceedings. 

This paper accompanied the workshop presented in Bridges 2016 conference where participants (pre-service teachers) explored the mathematics related to a figure skater's arm movements while performing an upright spin in relation to trajectories in which hands and arms travelled. The presenters used a geometry software to depict the arm movements during the spin. The mathematical concepts explored in this activity included proportional relationships, regression, circular geometry, and trigonometry.

STOP -This is a very practical example which demonstrated the real life application of mathematics. It is a very short paper and I recommend looking through this if you are interested in learning more about how the art of figure skating has been analysed from a mathematical perspective. It got me thinking about how I can use something similar in my classroom. I am thinking of exploring the mathematics of yoga and various poses. It will be something that brings various subject areas - science ( as in the movement of joints and muscles), PE and  Mathematics. 

        Wondering:  I am very interested in learning more about activities that involve movement in a mathematical classroom. Do any of you have any examples to share particularly for secondary level?

STOP - It was worth noting how the same activity has been adapted as per mathematical understanding of participants.

The activities in the workshop were divided into two parts. Elementary teachers analysed the animation with focus on proportional relationships to calculate scale factors b/w skaters actual measurement in relation to the animation drawing. Secondary teachers analysed the same animation with focus on trigonometry, and geometry in terms of measuring the angles and finding the length of arm movement and areas of the movement.

Activity (Bridges 2017)

I am not into art, I feel like I am a perfect fit for the category of people who say “ I am a Math person, not an art person”.  I really have to extend my thinking to understand a piece of art. When it comes to drawing, sketching or making a replica or even coloring - thats not my comfort zone. I appreciate the aesthetics, beauty and the effort that goes into the beautiful pieces of art that were created by different artists in Bridges 2017 page. But understanding and meaning behind each artwork is not something I am able to make direct connections with or comprehend right away.

I chose the art work by Regina Bittencourt as I was able to make direct Mathematical connections simply based on first look and in addition it was something that I thought I would be able to replicate as the design involved only straight lines, and I could use a ruler to draw those. 

As I was measuring and drawing my squares, it gave me so many ideas about how we can use this activity in our math classrooms in terms of building understanding of shapes, proportions, symmetry and extending to understanding areas. What a great lesson it would be in terms of understanding the different sizes of squares and how no squares sharing a boundary have the same color.

Week 3 Post (Mathematics in Outdoor Settings)

  This week I chose the following reading

Williams, D. (2008). Sustainability education's gift: Learning patterns and relationships. Journal of education for sustainable development, 2(1), 41-49.

In this paper, the author has presented a case study from  United States about establishing the Learning Garden model in schools in Portland. Students in grade levels all the way from kindergarten to grade 8 are learning to grow, harvest, and cook food. The program is based on a multicultural, interdisciplinary and multisensory curricular and instructional framework. Students' learning has been assessed through their written reflections of gardening experience. Their writing reflections show that learning through gardening projects has led to students’ understanding of patterns and relationships in nature.

Stop # 1

“The whole is more than the sum of its parts. The essential properties of a living system are properties of the whole which one of the parts have; these properties arise from the interactions and relationships among the parts” ( Williams, 2008. p.42)

William (2008) described the above understanding of learning as an essential part of sustainability education.  This just got me thinking about our current education system especially at high school level which is so much divided and compartmentalized into different subject areas. The demands of curriculum are so intense that as teachers, even if we try to establish interdisciplinary connections by collaborating with teachers from other subject areas, it can only be limited to a few certain aspects of the curriculum areas.  For a true understanding of the understanding of patterns and relationships, we need to move beyond the walls of our subject areas. A true shift is only possible if a major change happens the way our curriculum has been written or designed.

I wonder if it is possible to revamp our education system so that the curriculum is designed with a more interdisciplinary approach without any compartmentalization between different subject areas?

Stop # 2

This is a great example provided in the table as a sample of curricular goals and integration of subjects and shows how the different lessons and activities in the learning gardens address the curricular benchmarks at different grade levels and different subject areas. Most of the examples I have seen in the past are limited to elementary levels, inclusion of grade 8 examples is giving me ideas to ponder and it is something very practical that I can present to my colleagues at school for interdisciplinary collaboration projects. 

I am interested in learning more about collaborative teaching approaches (so as to build relationships between different subject areas) at high school level. If anybody has any experience to share or any resource that you can guide me towards, that will be much appreciated.

Activity Experience and Reflection

Disclaimer - I can not draw. I should have probably had my 4 year old draw for me - which would have been lot more presentable. But here it is - my best attempt at drawing what I Observed in the backfield at my school.

I noticed that man-made things (Flood light poles, grass field, markings on the grass field) are clearly very symmetrical and  consist of straight lines, right angles. Even semi circles drawn on the field are very symmetrical. The rough path created around the field was also very clearly defined and well carved. 

In natural living things e.g. humans, squirrels, dogs, tree trunks- I noticed everything is upright and standing/sitting perpendicular in relation to the ground level. Humans, and trees have a common feature in terms that they are vertically symmetrical.

Despite my clear dislike for drawing, I enjoyed the observation and reflection part of this activity, and I am thinking that there is so much opportunity for students to learn about different shapes and angles in an outdoor setting. Rather than having them draw what at 45 degree or 75 degree looks like on paper, we can have them go outdoors and do a walking activity and have them figure out what is the angle or direction in relation to a particular object. And on sunny days, just learning about shadows and angles, position of sun and directions of sun rays can be so much more engaging and help them in learning patterns and relationships in the real world.

Week 2 Post (Multisensory Math)

 Reading Response and Reflection

This week I chose the following reading:

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

In this paper, Fernandes & Healy examined how the mathematical practices of exploring symmetrical figures compared in two research objects - first object being the student who had no visual memories and second was the student who had more recently lost their vision.

Stop, Wonder & Reflect- One major point that stood out for me throughout reading this article is how much we rely on our sense of vision in basically all areas of mathematics. The use of other senses including that of smell ,touch, and hearing is very rare in our classroom. I was very much intrigued by the use of touch in the examples provided in this article.

“ We noticed, in particular, that both subjects explored the geoboard and the figures displayed upon it in similar ways, moving their hands successively from the outside to the inside and back again, with an initial tendency to move their hands together, following symmetrical trajectories – something we had by no means anticipated when planning the tasks.” ( Fernandes & Healy, 2013)

This made me stop and wonder about the importance of use of touch and other multifaceted ways of learning for our students. Especially when it comes to symmetry we are so much dependent on visual identification of the axis of symmetry that we hardly pay attention to what symmetry could feel like through touch. It was interesting to read about the hand movement of participants as they explore the symmetry on the geoboard. 

As an extension to this wondering, I asked my 4 yr old to close her eyes and presented her with two different shaped pieces of clothes (pajamas and shirt) and asked her to show me how she will fold them into half. Folding into half is a relatively new concept for me as she has just begun to assist me in folding clothes after laundry. It was very interesting to see how we used her hands to try to feel the texture and identify the design and pattern to identify the lines of symmetry to fold them into half. The pajama that I had given her had a particular design on one leg and not the other. She was very frustrated when exploring the pajama with her hand, she kept on repeating that this leg feels very different so she can't fold it.  I was very impressed by her ability to use her hand movements to touch and explore the clothes without having seen them. 

Activity Reflection

For this week’s activity, I really enjoyed the process of making hexaflexagons with paper.

 I havent had a chance to make Flex Mex hexaflexagon burrito, but I would definitely like to try that next time we are making burritos.

Adding hands-on mathematical activities and having them learn from real 3D living things and/or objects with shape, texture, smell, taste, etc makes the learning more accessible to all students including the one that may have sensory impairment. Such activities provide them opportunities to use their physical senses other than vision or hearing. Senses of touch, taste and smell are very rarely explored in a mathematical classroom. My 4 yr old daughter definitely prefers eating a heart shaped sandwich as opposed to traditional square or rectangular ones. Its about what appeals to her eyes - that makes it more tasty. And the opposite of this is true as well. 

The whole process of making a symmetrical hexaflexagon shaped burrito or wrap is so much more involving and hands on than simply drawing a 2-D image of hexaflexagon. It draws students’ attention and sparks interest in not just simply learning about the shape but also about the size and the angles involved as well. 

** I am getting an error message when I am trying to upload the images. I will try adding them in a different post.

Week 1 Post ( Mathematics and Body)

 Reading Response 

Goldin-Meadow et al (2009). Gesturing gives children new ideas about math.

The focus of the article was on whether gesturing influences learning. For the purpose of this article, the authors did a test and divided children into 3 groups: first group where children were taught a particular math concept using correct hand gestures, second group where they were taught using partially correct gestures, and lastly where they were taught using no-gestures. The data from this study suggests that gesturing can indeed facilitate learning as it helps children extract information more effectively when they are pointing at the numbers they were supposed to regroup.

This following quote in the article made me stop and ponder more about teaching correct gestures to young children: Gesturing has been manipulated in studies of memory––children told to gesture when trying to recall an event do, in fact, remember more about the event than children prevented from gesturing (Stevanoni & Salmon, 2005).

I found this particularly interesting as this is something we experience at home with my own pre-school age daughter. Oftentimes we model and encourage her to do the thinking gesture as when trying to recall any event. In my mind, the gesture simply gives them time to process and recall information. But now after reading more about gesturing, this definitely goes beyond a simple gesturing, but holds more value as it might be triggering thoughts about the gestures related to past event which helps recall. 

I personally have never thought of gesturing as something that can be actively taught. I have also considered gesturing as a secondary act that accompanies our speech- it definitely adds impact to what we are saying. But I have always considered it to be a passive learnt behaviour. But now after reading this article, and reflecting back on how we have been passively teaching /correcting hand gestures for my own two young children, I definitely see the importance of Teaching correct hand gestures. 

Activity Experience and Reflection

The activity aimed at contextualising  Mathematics- so that students can see the real life application of Math without disregarding it as too abstract. The activity of using the body as a tool of measurement is a perfect example of how Math is part of everybody's life and is embedded in our everyday bodily experiences of the world. The TED talk by Roger Antosen also highlights that how a change in perspective helps us realise how we are using and can use mathematics to understand the world. There are so many aspects of our life which use the underlying concepts of mathematics without us even fully realising. 

This activity was relatable to me as I grew up on a farm in a rural community. My grandparents often used body parts as measurement references for everyday items and activities. References to these measurement tools have been an integral part of my everyday life and doing this activity made me reflect about how my young children have also picked up a few references from us in regards to measuring things. A few days back, my 3 yr old made a comment about how she could cover the naan with her both hands. That was a teaching moment for us to tell her how she can use hands to measure things. 

We are currently in the process of rearranging our 3 yr old’s room as she transitions from crib to a toddler bed. I used my footsteps as a tool to measure the various dimensions of her room to figure out if we can fit a queen bed in her room and is there will be a room for a dresser. I involved my daughter in the process too and it was a good learning experience for her too watching me measure the room with my steps and then doing the same with her steps. This led to a very long discussion about why her step count of the room is more than mommy’s? I have recorded this information and hope to come back to this next year and by that time she will have a more evolved and deeper understanding of using steps as a measurement tool.

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