By Sunny Hu & Sukie Liu
Link to our presentation recording (Zoom recording): Recording
(Passcode: E5cb=J?3)
Link to our project handout: https://docs.google.com/document/d/1Rd_fGwEP3Jn0GVSfeA7LgSSL3TTCBNcn93IenWjvsDA/edit?usp=sharing
Link to our project slides: https://docs.google.com/presentation/d/19PKjZlaSiwAh-yv8yBW6UJCsy2vzECx12GoUEzOqDRY/edit?usp=sharing
This article introduces a workshop in which teachers use ratios and integer sequences (e.g., the Fibonacci sequence) to make layered drinks. I love cooking, and I also love to make mocktails myself at home. As the article suggests, layered drinks can be made by using different amounts of sugar to achieve different drink densities. I tried a similar activity with my science class before to teach them about density, but I’ve never thought about analyzing the ratio of sugar mathematically by engaging with the concept of the Fibonacci sequence.
The workshop has two parts. Part One introduces the sugar ratios and density. The audience can compare the sweetness of two drinks by calculating the sugar ratio in each layer. The second part of the workshop explores how the monotonic sequence can be represented through the layers of the drink.
STOP 1: “Unlike most examples of food-based mathematical art, where the math is purely visual with no effect on the actual flavor and experience consuming the food [2, 4, 5], the math in our beverages is conveyed entirely by the ingredients and flavor.”
The activity contains two senses --- visual and taste. I think this is a very innovative idea since we don’t usually engage “taste” in learning, especially in math. By adding different amounts of sugar, students can tell that one beverage is sweeter than the other, so the ratio must be higher. This brings the abstract idea of ratio into real-life practice. I can make the connection to develop a bubble tea ordering question. In most stores, customers can request the sugar level of bubble tea. However, since the volumes of medium-sized and large-sized are different, the sugar content in each size must be different. If we want to maintain the same amount or the same ratio of sugar (i.e. 30%), how much sugar should we add? Many of my students love bubble tea, so I think they will be really interested in this question. Perhaps next time they order bubble tea, they will pause and think about it as well.
STOP 2: “Layered beverages are a particularly nice food to play with for this exercise. The layering caused by density is somewhat counterintuitive and can be a nice science lesson in addition to the math lesson described”.
I do agree that this activity works well for the science classes. Since I’ve tried a similar activity with my science class before, I attached some of the drinks I made earlier as the demo. Instead of focusing on the sugar level, I asked students to focus on the “state of matter” and “their density”. For example, in sparkling lemonade, the lemonade at the bottom is denser, while the sparkling water is lighter (because there are bubbles/gas), so it’s on top. Besides that, ice, as a solid, flows on top because it is less dense than liquid water. To develop on this idea, I’m thinking about whether we can also use a sequence with a different ratio of food colouring in water. This may be more accessible at the school lab and produce a clear observation.
Questions:
1. What other food-math connections idea do you have?
2. Instead of focusing on number sequences, can we develop some geometric food-math activities? i.e. different geometric shapes of cookies? Best way to cut and share a pizza?
Activity:
I’m really bad at paper folding. I remember when I was a kid, I was always the last one to finish the paper-folding activity in class (and sometimes I couldn’t even build the final product). I tried the instruction video listed in the doc. To be honest, it took me a long time to figure out how each part gets up and down ---- and I didn’t get it in the end. Instead of being stacked, I decided to try a different method and looked for other videos online. I found this one and it makes more sense to me!
https://youtu.be/9nBWJ3b8i5c?si=H3YFFmHWE6vrMSg5
This really made me wonder sometimes if students can’t do something, it is because they simply can’t do it, or because the instruction is not suitable for them. It also makes me doubt my personal skills. I’m really interested in and good at LEGO building, constructing IKEA furniture, and I majored in Physics so I always assume I’m good at “making things”. However, I always experience some challenges in Origami. I’m wondering if the difference is that Origami focuses more on flat 2D skills, while LEGO is more 3D with the visuals in more space. My final project is about transferring 2D nets to 3D models, so it may be a good idea to research deeper into why someone is more comfortable with 3D models than 2D.
I also asked my parents to try it. They said they tried but failed. One of the reasons is that the paper they used was too soft to fold. After a few rows, it was hard to recognize the pattern. This makes me wonder if we can use a pencil or pen to trace the pattern so it is more visual in the folding process.
Contributors: Sunny Hu & Sukie Liu
Hi, everyone!
This is our draft project! We are working on a project of creating 3D paper models for the secondary math class geometry unit. We attached the project worksheet that students will use and our project slides.
Project Worksheet:
https://docs.google.com/document/d/1Rd_fGwEP3Jn0GVSfeA7LgSSL3TTCBNcn93IenWjvsDA/edit?usp=sharing
Slides for the project:
https://docs.google.com/presentation/d/19PKjZlaSiwAh-yv8yBW6UJCsy2vzECx12GoUEzOqDRY/edit?usp=sharing
Karaali, G. (2014). Can zombies write mathematical poetry? Mathematical poetry as a model for humanistic mathematics. Journal of Mathematics and the Arts, 8(1–2), 38–45. https://doi.org/10.1080/17513472.2014.926685
This article engaged with the author’s personal experiences to introduce how he started to connect his interests in math with poetry, which he was not very familiar with at first. Karaali got interested in the ideas introduced by the Humanistic Mathematics Network Journal (HMNJ) and stepped out of his comfort zone to explore how poetry in English can be connected with math. He believes that mathematics and poetry share the three most important ingredients of what makes us human: cognition, consciousness and creativity. Thus, he developed a course titled “Can Zombies Do Math” to encourage students attempt to write mathematical poems with creativity.
STOP 1: “In my native language, Turkish, sentence structure is quite different from that in English. Correct grammatical forms exist, though they are not as rigid as one might expect. The speaker may and will often move the main parts of speech around in order to make a statement, emphasize a point, underline a concept or simply to convey the spontaneity of the particular conversation… But with poetry, I could only hear my voice in Turkish. Mathematics was different. My mathematical language was definitely English…”
This is a really interesting point, and I resonated with it a lot as a second language speaker. Same as Karaali, I use English to do math a lot, but I’m more comfortable in writing or reading poems in my first language (which is Chinese). I’m wondering if there is a reason why people feel more attached to their first language in poems. Probably it is because of the culture? Or how familiar you are with the poetry styles when you were kids, since lullabies and songs are all different variations of poetry. Karaali mentions that grammar in Turkish is not as rigid as English, so you can change the order of words to form a sentence. I guess this gives people more flexibility and creativity in writing poems. I found Chinese grammar has a similar nature as well, where we can arrange different characters to make new words each time without affecting the grammar or meaning of the sentences. This really makes me wonder whether multilingual people have more ideas for exploring poems and make more innovative connections in mathematical poems from various perspectives.
STOP 2: “Referring to our students’ course work, we can inquire explicitly: what does it mean to be creative in the mathematics classroom? The answer will come rather easily to most teachers of mathematics: students can think of novel ways of approaching problems. But what is novel is most often simply some idea or technique that they have seen elsewhere, now taken and used out of that original context. A concept that is often considered in tandem with creativity is divergent thinking, which captures thinking processes that lead to multiple possibilities. So it is fundamental to creativity in general, and to creativity in the mathematics classroom in particular, to be flexible and open to many interpretations and approaches.”
When I first thought about being creative in math, I had the same idea as Karaali mentioned here: to have more approaches to solving problems means you are creative. However, Karaali argues a new idea: that students may use different approaches because they have seen or learned the questions elsewhere. This may not reflect the students’ creativity, as the idea may not be original. This aligns with his statement in ‘zombie-learning’ about following the rules without any human creativity, interpretation, or flexibility. Karaali introduces the concept of divergent thinking, which means generating more possibilities. I can tell he tries to link this idea to poem-writing, since writing is very personal, and the idea of a poem can only come from your own mind, which aligns with the use of divergent and creative thinking.
Questions:
1. How do you support English Learning students in your class to try the mathematical poem activity? How can you make the poem-writing activity fun but not too challenging for students?
2. In your opinion, what makes a poem a mathematical poem? Besides structuring the poem by numbers or sequences, what other ways can you think of? Can geometry, function and graph also be used as part of the poetry? If so, what would that look like?
Week 8 Activity
I’m not really good at poem-writing. Poetry was always the unit that caused me the most headaches when I was in high school. However, I’d like to challenge myself this time to try the Fib poems.
Similar to what Karaali experienced in his article, I came to Canada in grade 10 and studied most of the math in English, but my first language is Chinese. I am definitely more comfortable in reading or writing poems in Chinese. In the Chinese language, one word (character) is one syllable, so we never distinguish words and syllables. This makes it extremely hard for me to write English poems when I have to be aware of the differences and count syllables in words.
I really like the flexibility of the Fib poems, which can be counted as syllables per line, or words per line, or lines per stanza, or any other countable thing connected with the poem. This offers more creative ideas. You also don’t have to be perfect at English or poem-writing, since it is easier to start by counting the number of words.
Since I’m working on my project this week on 2D-to-3D models, I had the idea for a geometry-themed poem.
Geometry (count by the number of words: 1,1,2,3,5,8)
Point
Line
An angle
Forms a surface
Enclose edges of a figure
New dimension is formed to create more possibilities
Element (count by the number of words: 1,1,2,3,5,8)
I
Am
An Element
Starts with H
Have one proton, no neutrons
Lightest element in the world—who am I
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.
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?
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.
