Week 7 Readind and Video

Hart, G. W. (2012). What can we say about “math/art”? Journal of Mathematics and the Arts, 6(2–3), 87–91. https://doi.org/10.1080/17513472.2012.679887

 

 

STOP 1: “To reconcile these issues, perhaps what we call an ‘Art Exhibition’ should be rebranded as something like ‘Exhibition of Mathematical Art, Craft, Design, Models, and Visualization.’ This conveniently covers the entire collection without having to be definitionally specific about individual items.”

 

Hart discusses how the definition of art and artistic works may block creators and audiences from engaging with the projects.  He argues the differences among art, craft, design, and model.  For example, crafts/models can be reproduced by competent workers following step-by-step instructions and used for educational purposes, which are not proper art.  I never thought from this perspective before.  In our entire course, we have been talking about math and art, where the word “art” is interchangeable with “craft and design”, in my opinion.  I guess what Hart tries to argue here is that sometimes the fine art is defined as a profession and a high standard of work by the authorities and institutions.  Some works are created as aesthetic objects, but others are intended as mathematical demonstrations.  Thus, mathematicians and educators don’t have to fit their work into the artistic standards, and trying to prove that math is art can limit the creations.  

 

STOP 2: “Mathematics appeals because of the delights to be found in rigorous reasoning and understanding with clarity. I believe mathematical art succeeds in our community because it alludes in various ways to these same pleasures…The art that is mathematical art must bring to mind a landscape of mathematical pleasure.” 

Indeed, some artwork may be hard for the general public to understand.  I think what Hart tries to approach here is to suggest an accessible and explicit way to deliver mathematical messages from artworks.  If the audience is confused and has a hard time understanding the work, it is probably not a good educational math-art work.  Linking back to his idea about how he wants to distinguish professional and educational artwork, I appreciate the suggestion of designing artwork for different settings.  In our classrooms, we always need to think about whether the art project is accessible to students.  For example, what works in elementary classes may not work the best for high schools.  Thus, adaptations are always needed for different circumstances. 

 

 

Questions: 

1.     What is art in your own definition? What can be included as part of art? What is not considered art? 

2.     Do you agree with Hart’s idea to separate art into different categories (i.e. design, model, craft, etc.)?  What is the benefit of doing this? What is the downside of this idea? 

 

 

 

 

VIDEO 

 

Stop 1: I first stopped at around 9:00 – 10: 00 when Nick Sayers said, “Being an artist is more than drawing a picture”.  I really resonate with this idea since each time we talk about art, most people immediately think about drawing and have a conclusion that I CAN or I CANNOT draw.  However, this is not true since we learn that art is so much more than drawing a picture, and there are various aspects of art.  Only viewing art as drawing can restrict many ideas and creativity.  To change the assumptions about art, I guess we need to explore more fields in art to show our students how diverse art can be. 

 

Stop 2: 28:00 I was fascinated by the Christmas tree design with 2000 plastic water bottles. When I first saw the picture in the video, I thought that was a super cool Christmas tree, but I could never imagine that it was made of plastic bottles. I think this is a good example of how recyclable materials can be used in art.  This is a wonderful sustainability project to try with the class.  I can also see the pattern of the Christmas tree, where the hexagon shape is repeated to construct the tree.  We can probably also engage the math part, for example, ask students to calculate and estimate how many bottles they need to construct the tree.  

 

Stop 3: 33:00 I loved to play with spirographs when I was a kid, but I’ve never thought about the math and art behind it.  Nick Sayers mentioned that he liked cycling, so he combined the idea of spirograph drawing with cycling by tracking the wheels.  He created a drawing machine with bicycle parts.  This is something I could never think of, so I really like this creative and original design.  I can tell he tried to engage in his hobbies in art and got the ideas from everyday life.  Probably, this should be something we can engage students to do.  Our students all have different hobbies and come from different backgrounds, so they may also have some creative ideas to combine their interests with math or art.  

 

What does this artist's work offer you in terms of understanding math-art connections, and what does it offer you as a math or science teacher?

 

The thing that inspired me most in the interview was definitely the materials Nick Sayers used in his projects.  Many of them are accessible, unexpected, and fun! As I have mentioned above, sometimes I restrict myself too much to “art is drawing,” so I may not have the chance to explore more creative ideas.  Nick Sayers started with the little things around him and developed the idea by engaging various aspects in different fields.  As a math/science teacher, I guess I can try to observe small things around me in my daily life and take time to think more deeply about the design behind them to see if they can be connected to something else.   

Week 6 Reading + Activity

Belcastro, S.-M., & Schaffer, K. (2011). Dancing mathematics and the mathematics of dance. Bridges: Mathematical Connections in Art, Music, and Science, 99–106.

 

In this article, Belcastro and Schaffer explore the connections between mathematics and dance by showing how dance movements can express mathematical ideas. Multiple aspects of dancing are analyzed mathematically, including symmetry, transformations, patterns, permutations, and spatial reasoning to explore the connections between dance and math.  

STOP 1: “Karl often uses simple props as giant math manipulatives. He recently created Fragments, a dance involving. oversized tangram pieces that are a serious exploration of the fragmentation and destruction of war. He has explored polyhedral in dance using various props: PVC pipes, loops of string, and even fingers”. 

 

One question I wondered about last week was how to engage students who are not good at dancing in “dancing activities”.  I used my personal example: I’m not good at dancing or making body movements, so many mathematical dance activities are extremely hard for me to try.  I asked in my last week’s post how we can adjust the activities to be accessible and inclusive to everyone.  In this week’s article, I found this amazing idea that answered my question very well!  Karl developed activities that can be done with various accessible materials, and even just fingers!  Those are good starts for students to collaborate and be engaged in the activities without being left behind.  I’d like to try one or two examples introduced here and see how it goes! 

 

 

STOP 2: “This pattern can be demonstrated by having one person clap a five-beat. rhythm and another clap a seven-beat rhythm with the same tempo, and at the same time. Both people clap loud sounds on beat 1of each phrase. This is a way to create interesting syncopations in which the accented beats don't fall in the expected place.”

 

Connecting to the previous week’s reading on how the use of different senses can be applied to math learning, I remember we discussed that sound may be something hard to apply to math.  However, here in the article offers a great example of math rhythm.  The picture I attached is a record of one person clapping a third beat against another person clap eight-beat.  At the 24th time, both of them are clapping together since 24 is the lowest common multiple of both numbers.  Many students are confused about how to find the lowest common multiple of two numbers and why we need the lowest one.  This would be a good activity for students to try so that they can clearly see the multiples between two numbers, and the first time both of them clap together is the lowest common multiple.  

 

Questions: 

1.     Can dance also be connected with functions (i.e. linear, quadratic, etc.)?  If so, do you have any creative ideas on that? 

2.     What challenges do you anticipate during the dancing activities?

Activity: 

 

I chose to try the activity "Making Stars" by Scott Kim this week.  In the video, three people altogether used both their hands and fingers to create a five-pointed star.  I want to connect this idea with “symmetric dancing in front of a mirror” introduced by Belcastro & Schaffer (2011).  

Thus, I tried to do a “solo finger geometry” by using the mirrors to create the symmetric parts from reflections.  

 

TWO fingers in front of ONE mirror --- a two-pointed star.

TWO fingers in front of TWO mirrors --- a three-pointed star. 

 

We can develop from this and let students observe what happened.  By getting different numbers of mirrors, the reflections will be different, so there must be a pattern! 

 

Ask students to take a guess: What will happen if I put two fingers in front of three mirrors? If I want a five-pointed star, how many mirrors do I need? Are you able to conclude any patterns? 

 

 

Week 5 Reading + Activity

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

 

Dietiker draws on Einser’s (2002) idea of applying an artful lens to some educational challenges to develop art-based learning in math education.  Dietiker focuses on connecting the mathematics learning with storytelling by creating a sequence of tasks in story form from a Grade 7 math textbook.  This approach transfers the abstract math concepts into “verbal art” so readers/students can be inspired through the interpretation of the stories.  “Math stories” start from:

1)    The beginning (introduction of the story and discussion of what happened)

2)    Problem-solving (try the games and activities with classmates or teachers)

3)    Make a decision (analyze the information and use critical thinking)

4)    A resolution (predict or get the results)

 

STOP 1: “Stories integrate both logic (e.g., Does the story make sense?) and aesthetic (e.g., Does the story move me to continue reading?). While each of the other art forms offer elements of value that draw attention to particular aspects of mathematics, a narrative perspective also combines the temporality (i.e., how a story unfolds) and the message (i.c., the moral of the story) of curriculum. Stories conjure fictional worlds for which truth is self-contained, much like mathematics.”

 

Many people assume math is the opposite of the arts/literature.  I guess one of the reasons is that in our education system, we tend to separate STEM from the arts (like how university degrees are designed).  However, Dietiker argues that stories contain both logic and aesthetic parts, and the logic part has the same nature as doing math.  I totally agree with this since all the fictional stories designed by the author must have a reasonable plot in order to make sense to readers.  This is similar to doing math, where each step must flow reasonably.  

 

STOP 2: “The meaning and effect of sequential temporal experiences have been theorized and rigorously studied in terms of novels and short stories alike but have so far been ignored in regard to mathematics instruction. Although it may be unorthodox to consider mathematical objects and activity in these novel ways, conceptualizing the unfolding of mathematical content in a textbook as a mathematical story allows new questions to emerge.”

 

When I read the idea of connecting math to stories, I immediately recall the novel “Mr. Tompkins in Wonderland”.  If you are familiar with physics education, this is a great book that teaches physics concepts through stories --- a physics version of “Alice’s Adventures in Wonderland”.  Mr. Tompkins, the protagonist, entered a strange world where all the physics laws are visible and exaggerated.  I have to say I had a lot of fun reading this book when I was a child, and this is definitely something that encouraged me to go into the physics field (plus I’m a big fan of sci-fi novels and movies in general).  From my own experiences, I can tell the power of storytelling.  Stories in math are something people always ignored, as Dietiker said, but they may have great potential to benefit students’ learning. 

 

 

Questions: 

1.     Is reading skill essential in doing math? If so, what can we do as math teachers to help students improve their reading skills (especially for English-learning students)? 

2.     Why do students always find word problems hard to understand?  Will “math stories” help students better understand the concepts? 

 

 

Activity: 

For this week’s activity, I want to try something similar to option b) Ali and Colin’s activity.  

One of the examples I have is to try the base of 4 and do a growing base in tiles activity.  I will create 3 tiles: large, medium and small, as shown in the picture.

 

 

Then, students can try a drawing task.  For example, represent 27 in base 4.

 

Exponent calculation and factoring are big parts of Math 10, and I have found that many students have difficulty understanding those two topics.  This is just an initial idea of connecting the concepts with drawings.  I’m thinking about whether I can develop from the tile drawing to create a factoring activity for students to do.  

Project Outline: 3D Paper Model

This is the project idea we created (Sunny & Sukie).  We want to focus on the math topic of surface areas, volume and nets of prisms to engage students in designing a template to form a 3D paper model.  We researched our idea from academic papers, websites, and videos (annotated bibliography is attached as a Google Doc).  We wish to try the idea by ourselves first and introduce it to the class --- and of course, if you are willing to join this 3D paper model design challenge, feel free to let us know!  We are happy to hear from you! 

**Complete project outline with annotated bibliography is attached as a Google Doc: Link to complete outline

Contributors: Sunny Hu & Sukie Liu 

Name of the math project: 3D Paper Model: From Nets to Spatial Structures

Grade level: 8 – 10 mathematics, Richmond School District 

Project Idea:

In this project, students will design and draw a 2D template composed of multiple geometric shapes that can be assembled into a 3D paper model of their choice, such as an animal, fruit, or vehicle. Through the process of planning, measuring, and constructing their designs, students will apply their understanding of nets of prisms, surface area, and volume to combine different geometric forms into a coherent structure. The transformation from a flat template to a completed 3D model emphasizes spatial reasoning and embodied learning, as students physically fold, assemble, and refine their designs. By integrating mathematical accuracy with creative choice, this activity positions mathematics as both a problem-solving tool and a medium for artistic expression. 

 

Examples of final product

(source: https://www.polypapercraft.com/products/fox-low-poly-papercraft-kit)

Mathematical topics:

-       Nets of 3D shapes and transformations from 2D to 3D 

-       Surface area and volume 

-       Geometric solids (prisms and related polyhedral)

-       Spatial reasoning and visualization

-       Mathematical communication through design and construction

Embodied and arts-based pedagogical approaches:

-       Students design, cut, fold, and assemble 3D paper models to physically experience mathematical structures 

-  Using origami-inspired techniques, students fold and form-making to emphasize precision, symmetry, and geometric relationships 

-       Learning through touch and manipulation of materials support conceptual understanding

-       Combining mathematics, visual design, and craftsmanship encourage creativity

Week 4 Reading + Activity

Torrence, E. A. (2019). Bridges Stockholm 2018. Nexus Network Journal, 21(3), 705–713. https://doi.org/10.1007/s00004-019-00455-2

 

Bridges Stockholm 2018 is the 21^st Bridges Conference on Mathematics, Art, Music, Architecture, Education, and Culture, held at the National Museum of Science and Technology, shared with the Ethnographic Museum, the Maritime Museum, the Swedish Sports Museum, and the Police Museum in Stockholm, Sweden.  The main themes and activities can be concluded in the following categories: 

1.     Interdisciplinary Mathematical Connections: how mathematics is involved and connected in artistic and cultural contexts

2.     Papers, Workshops & Presentations: works contributed by the participants are displayed and presented during the session. 

3.     Cultural & Public Engagement: public workshops encouraged visitors to attend and participate in the activities (i.e. Family Night, Formal Music Night, etc.)

 

 

STOP 1: “Marjorie Rice, a woman with no mathematical training beyond high school, wrote to Martin Gardner in 1976 claiming she had found a new pentagonal tiling. Gardner sent Marjorie’s work to Doris for verification, and so began a mathematical friendship that lasted 30 years. Doris explained how Marjorie invented her own notation, and ultimately discovered many new pentagonal tilings”. 

 

In the picture, I saw that the entrance to the Tekniska Museet is paved with a pentagonal tile discovered by amateur mathematician Marjorie Rice.  I have to say I have seen this pentagonal tile path before in many other locations, but I’ve never thought about the mathematics behind it, and I can never imagine that this tile was created by someone with no math training after high school.  I’m glad the author uses this as an introduction to the conference report, which encourages people who are not math professionals or who are afraid of doing math to step into the math world.  This is an excellent example we can share with our students that math is not necessarily something you learned from your academic class --- it can be anywhere, and everyone can do math!  

 

STOP 2: “The Family Day public workshops were well attended by many visitors to the museum (Fig. 9). There were over 20 workshops to choose from, with opportunities to make paper geometric models, build and color kaleidocycles, learn origami, use a bicycle cog spirograph, try mathematical virtual reality, explore hyperbolic puzzles, play with Zometool, jump into an educational “sandbox” created by the band OK GO, build 4DFrame robots and use them to play soccer, and help build a large mathematical sculpture”.  

 

I really like the idea of Family Day that brings the abstract math idea into something tangible for children and families.  Many people may think an academic mathematical conference is far away from their lives and they rarely get the opportunity to participate.  The Family Day engaged the idea from the conference projects and presentations, and offered the general public an accessible way to get involved in math learning.  I can tell from the picture that there is a diversity in visitors from different ages, backgrounds, cultures, etc.  It is wonderful that everyone can find something they are passionate about and explore some new ideas in these sessions. 

 

Questions:

1.     Can teachers do something similar in their classrooms? Like organizing a mini-conference to let students present their final projects? (I will also answer this question myself first ---- absolutely YES!  I remembered I had a project-sharing day with my science 10 class before.  Students presented their Rube Goldberg machines, and students from other classes came to visit and vote for their favourite.  Students in my class who designed and presented needed to think harder about how to introduce and promote their machine.   Students who came to visit learned about new science ideas and engaged their interest in future science study.)

2.     What are some art-math projects inspired by the conference that teachers can do with students in the class with accessible materials? 

Activity: Bridges 2014

 

I tried to replicate the C318 carbon nanotube trefoil knot by Chern Chuang on Bridges 2014. The original work is made of beads and traces a connected graph of the molecule, using a fishline to represent each chemical bond: https://gallery.bridgesmathart.org/exhibitions/2014-bridges-conference/chern-chuang

 

The first idea that I have in my mind is to use colored plastic balls to make the knot.  Unfortunately, I don’t have any materials with me at this moment (I also thought about using candies to make it).  In the end, I used my bracelet to make a knot (obviously, I can't rearrange each bread, so a lot of details got lost).  

After I made the trefoil knot, I think the main difference between my replica and the original one is: individual carbon atoms are well represented in the original one by using beads, and the one I made focused more on the overall shape in general.  This makes me reflect that if the material choice will make a difference in how people think and view the project. Would there be a better material than another for certain mathematical representations?