week 7: Math and Poetry

Reading Reflection

Saex de Adana, F. (2018). Surfing the Möbius Band: An Example of the Union of Art and Mathematics. Bridges 2018 Conference Proceedings, 423-426. 

The reading begins with an introduction to the Mobius band, explaining different examples for understanding that the band is a continuous loop without a distinct inside and outside loop. It also explains that the Mobius band is a non-orientable surface, meaning that when the beginning of the band is reached it will be in an inverted rotation. The paper explores the story of Silver Surfer, where the Mobius band is used in the plot of the story as well as for the layout of the comic story. The story takes place on a Mobius band and the reader must follow the band to read the comic in order. Silver Surfer is immersed in a loop that causes him to repeat the same events over and over again. The story shows the reader what a single-sided geometric figure like the Mobius band is. This paper brings forth the excitement that has come from this mathematical figure and has been used multiple times. 

The paper briefly outlines different storylines that use the Mobius loop as a part of the plot, these include Star Trek episodes, films and stories. After reading this I got looking online at different fictional uses of the Mobius band and couldn't find a ton easily. I thought it would be cool to teach students about the Mobius band, have them create one with paper and then look at the story, show or video that has a fictional aspect to it to allow students to understand what the band is a little better. It may also be a fun activity to have students create their own Mobius band comics where the characters are caught in a loop or sorts. As much as I love the idea of these activities and can see how they could easily be used in an elementary or middle school math class, I continue to struggle with how we can implement them into more secondary classes. 

I liked how the comic used the Mobius band as both as an idea for the plot as well as for the layout of the story. I got thinking about other mathematical shapes that could be used in this way to help students understand them better and as a fun way to connect art with math. I was thinking about different 2D and 3D shapes, or perhaps something similar to what was explored in the week 2 Kepler reading with rhombic figures of pomegranate seeds. I'm still pondering about what other objects or shapes could be used in a similar way aside from standard geometric shapes. 

Questions

• What are some other mathematical 'objects' that could be used to shape the plot or layout of a story? 
• How can we bring these sorts of stories, poems, and mathematical art activities into high school classes when there are time constraints for learning curricular concepts? 

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Activity 

For the activity this week I chose to write a Fib poem. I always struggle to come up with a topic to write about when it comes to these sorts of activities. I decided to write one about my dog because she was sitting with me as I did this activity and one about dancing. I had spent my day in a PE class learning hip hop dance moves, this brought a lot of joy to me especially since all of the high school students were engaged and participating without any complaints, it was a great feeling so I wrote about how learning this new activity felt. 

PAISLEY

My 

Dog 

Big ears  

Lots of love

Full of energy 

Always excited to see me 

 DANCING  

 Joy 

 as 

I move

my body, 

Dancing to the beat 

Learning new things is a fun time 

I then thought that first poem writing was fun so tried one of the second method as well, this was inspired by my bluebird day skiing on the weekend. This method of writing a poem was quite a bit harder for me, and I found that as I wrote the same words over and over again they seemed to look weird and incorrectly spelled to me. I don't think my whole poem makes sense with the word combinations but here it is! 

Bluebird Day 

Blue skies over snow 

Skies blue, snow over 

Skies snow, blue over 

Snow skies over blue 

Snow over, skies blue

Over snow, blue skies 

Over blue, skies snow 

Blue skies over snow 

 

Week 6: Mathematics and Dance

In the introduction for the week, Gerofsky explains that we cannot make abstract shapes with our bodies but we do have imaginations, and imagination is where these concepts come from. This made me think about how much math really connects to our imaginations and how using our imaginations at all ages (not just early elementary) can add another level of engagement and excitement to math classes. 

In her Ted Talk, Malke Rosenfled asks herself the question "I wonder if there's math in what I do?" pondering if math exists in her world outside of the math classroom as well. This is definitely a question that I have seen come up over the course of this class and am still asking myself that question as I live my daily life. I really found myself thinking of this question as I read this week. 

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 Reading Reflection

Belcastro, S.-M., & Schaffer, K. (2012). Dancing mathematics and the mathematics of dance. The Best Writing on Mathematics 2012, 79–92. https://doi.org/10.1515/9781400844678-010

This reading explores the connections between artistic dance and mathematics. Sarah-Marie Belcastro and Karl Schaffer are both mathematicians and dancers, they suggest that math is intrinsic to dance as each dance form uses math concepts in their own way. Examples of these connections are explained with aspects for the readers to try on their own and gain a deeper understanding. One focus is on how symmetry can be seen in dance including how symmetries interact with one another and three dimensional symmetries. Some other mathematical concepts explored are geometry, spatial paths of dancers and conceptual problems like game theory. Rhythm is explained as how the math sounds coming from music that repeats its rhythmic pattern. 

While I grew up figure skating, I found that much of my reading this week was connecting mathematics to my experience skating. While I skated in a variety of different disciplines including free skate, another aspect is ice dance. In this we learned different foxtrots, waltzes, tangos, etc. These all had music that accompanied them and involved a lot of counting to stay on the correct timing as well as the beautiful patterns that the dance leaves on the ice. I don't do this kind of skating anymore but I will always remember the counting and discipline involved in ice dance to ensure that everything was smooth. 

Towards the end of the reading, Belcastro and Schaffer state that "even when a dance has a strong mathematical element, we let the dance take on a life of its own" (p. 20). This stood out to me as dance is not fully viewed through a mathematical lens however the math can be seen in the art. There isn't a need to always look at dance in a mathematical way as it is its own art and beauty. I think this applies to many different art forms or other mathematical connections - while we can see them in a mathematical way it is equally as empowering to acknowledge the beauty of the art on its own and allow the art to take its own shape regardless of the mathematical intent. 

In the PE class I am teaching right now, we have a Jamaican dance company coming to dance with the students next week. I am excited to see how they use math in their dance, if it is presented intentionally or more subtly.  

Questions 

• Have you used dance in your teaching? (either with math or without) 
• What math have you noticed appears in the activities you do in your daily life? 

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Activity 

The lesson plan that I chose to look at was Math in your Feet which is an exploration of math where students work together to figure things out themselves. 

I think this would be a good introduction activity for all ages to incorporating dance and math together as it gets students moving and dancing in an activity that doesn't require a lot of math work and allows students to participate in small groups.

I don't have my own classroom or group of students and my nephew who I often have tried activities with is quite sick this week so I tried the activity on my own. While this lesson requires groups of people to complete this activity I thought about how it could be done with only one person. I found that while it is possible to do the activity of moving in the square on your own it definitely is not as fun, I tried to put a few of the cards together in longer sequences and see if I could memorize and finish the sequence without messing up - this was quite fun and I could see it being fun with a group of people too. 

Some more extension ideas could be to create different shapes on the floor and have students try to create patterns that could exist within those shapes. For a more high school math class students could explore the different combinations and patterns of using a few movements in the squares and four movements. 

In their Ted Talk, Stern and Schaffer explain that physical activity in the classroom can be an opportunity for all ages and all disciplines. This is something that I think is often forgotten when students enter higher and more academic grades. Using an activity such as this one can be incorporated into math lessons but also can be used to encourage students to get moving in the classroom even if for a short amount of time. These allow students to use their imaginations as well as see math beyond the sheet of paper or textbook work that they may be used to. 

Week 5: Developing Mathematics Pedagogies that integrate embodied, multisensory, outdoors and arts-based modalities

This week I really connected with the reading but found myself puzzling over the activity. I chose to do an activity that put me outside of my comfort zone with the 'dancing' aspect and found that it took me a lot longer to complete this time. I've mentioned this in previous weeks but I wouldn't consider myself a really artsy person and find that any sort of body moment where others are watching and I am without explicit instruction is a stretch for me. I grew up figure skating and dancing but was not overly dramatic or comfortable with the 'interpretive' freestyle when there were others around. I chose the movement activity this week in hopes that I would be pushed outside this comfort zone even if my housemate thinks I'm crazy as I do random body movements in the kitchen.  

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Reading Reflection

Dietiker, L. (2015). What mathematics education can learn from art: The assumptions, values, and vision of Mathematics Education. Journal of Education, 195(1), 1–10. https://doi.org/10.1177/002205741519500102

While I was reading this week I found that I had a lot of stops where I was putting stars on my notes either because what I was reading stood out to me or connected with me. 

The reading begins with an introduction of how math has been referred to as "dry as dust" "monotonous" or "flat-lined" and how the arts can help to re-inform math experiences. There is an explanation of aesthetic ways of knowing and learning. To challenge current assumptions of math this paper explores how math can be artful and aesthetic. Aesthetic is described as "an individual's response to a mathematical experience" (p. 2) which is said to help mathematicians sense if a result is correct or not. This describes aesthetic to be essential to mathematics whether it is a positive or negative response. Through understanding mathematical aesthetic, the paper discusses using different art to structure aesthetic math. Dietiker uses stories as an example, outlining the various parts of a mathematical story and how it can make math a verbal art. Mathematical stories include characters, setting and plot, much like a regular story. Using stories in math classes can help to 'shake up' the familiarity of traditional math classes. 

The idea of using aesthetic in math was interesting to me, when I first think of aesthetic I think of the way I want things to look and if they look nice then I would consider them aesthetically pleasing. In explaining her experience with math Dietiker (2015) explains a fond interest for math and puzzles growing up which led her to studying math in college and developing a desire to share this with students. This resonated with me as this is largely similar to my experience with math. I used to love doing puzzles with my dad and now as I have spent the last month helping my mom clean out some of his things I found a lot of the puzzles and mind-teasers that we used to do together and am reminded of my love for the puzzle aspect of math. None of my siblings enjoyed these but it was something that my dad and I did together all the time. This was a happy memory that stayed with me as I read and did the activity this week. 

Dietiker (2015) states that math classrooms where students and teachers work is tame and harnessed are something other than mathematical as math is generally untidy; she says that "creating aesthetically-rich mathematical classrooms requires untidying the mathematical experiences of students" (p. 3). 

This stood out to me, mainly because I had not thought of math as tidy or untidy before. When I think of my general math work - in high school or university - I think that my work was generally tidy and "aesthetically pleasing" because I like the neatness of lining up my equal signs and erasing mistakes. When I think of my teaching experience though I love engaging students in math classes where they are able to move around, use different materials and try to come up with a solution to the problem on their own (as we have learned the benefits of this throughout our courses). To me this means embracing mistakes and using these as learning opportunities.

Finally my last connection was to using stories in math. When I was completing my undergrad I was involved in an applied study where I had the opportunity to go into schools and tell students stories. I ended up doing a research project connected to this and explored storytelling and math. I explored different oral stories as well as picture books that tell stories about math concepts. This sparked my desire for including stories in math classes and have used them a few times but find that I often get caught up in needed to get things done and forget about this research and work that I did and how impactful it was. 

Questions

• What do you think it means to untidy the math classroom in your own classroom? 
• Do you use stories in math lessons or is it something you are willing to try? 

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Activity 

I chose Sarah Chase's illustration of moving 3 against 2 as the basis for my mini lesson idea. I found myself trying many different combinations of this and becoming confused as I tried to count and move both arms at the same time. While I was doing the movements I was also spending time trying to draw them out to see if my movements were correct with where the arms should line up. I then thought of how students could do this together, having one person do the first number and the other doing the second. I found myself trying to think of how these types of movements could connect to multiplication or division however am still questioning that one. 

Here are some of the ones I tried to draw out, it got a little confusing when I tried to include larger numbers so didn't draw those ones. 

I found that some of them worked with multiplication

 but not all of the movement combinations. 

After moving my arms over and over again I connected this movement to help understanding angles or the unit circle - not directly with what was shown in the video but with the way the arms were used. 

I really struggled with this activity and wanted to give up many times but here is an idea I came up with. 

Unit Circle inspiration

Ideas 

Students will learn about the unit circle on paper and then have a chance to explore it with their arm movements, this will allow them to add a multisensory aspect to the math concept and potentially help them understand or remember it better. 

Guiding Questions 

• How can we understand the unit circle with our bodies? 
• What connection does the unit circle have to angles? 

The Story 

• I am still brainstorming ideas for this part... I can think of stories to go with elementary math concepts but am struggling to come up with one for this. 

Integrating Embodied learning and other learning 

• Students learn about the unit circle and triangles, how to construct the unit circle on paper 
• Imagining that their arms are equal to the radius 1, they can create a larger scale unit circle 
• This can help them to 'memorize' the unit circle with their body being the y-axis and arms stretched out to the sides being the y-axis 
• Using this new tool for understanding the circle students can work together to try and solve math problems that require the unit circle 

Possible extensions

• Ask students to think of what other math concepts they can understand with their body 

Idea for unit circle with body

Final Project Proposal

Hello! 

Here is the proposal for my final project, completed with Kaitlin Burns. 

 EDCP 554 Final Project Outline