Week 7: Reading Reflection

A basketmaker’s approach to structural morphology

Alison G. Martin

 
Alison Martin explores how traditional basketmaking techniques provide structural algorithms for modern computational design. By hand-building models from bamboo and paper, she demonstrates how 2D tessellations can be deformed into spherical or hyperbolic surfaces by varying connectivity.  These woven structures mirror nanostructures and minimal surfaces found in nature, such as sponges. Ultimately, these physical models serve as powerful pedagogical tools for visualizing complex geometry and mathematical principles in architecture and engineering.

STOP #1: "The geometry principle of introducing singularities in a mesh - each pentagon in a hexagonal mesh introduces positive convex curvature to a fullerene shell or a basket shape while a heptagon introduces concave curvature - applies equally in molecular structure and woven baskets. The shape of a basket can be defined in mathematical terms; a 3-D representation of hyperbolic functions (saddles) results from adding extra weavers, while subtracting from the numbers of weavers produces closed shapes." (p. 2)

Although I had to read this several times to begin to understand what was being describes, I was hooked after the first read. I really loved the idea of the transition from 2D to 3D and what had to be done to achieve that shape and formation in the basket. You can tell in this excerpt just how mathematical basket weaving really is. This was the first time I truly saw a deep connection between the two. The use of the pictures that go along with this text, really helped solidify my understanding of the concept.

STOP #2: "Porifera (sponges) are simple aquatic invertebrate whose shape seems to depend on nature’s ability to find the simplest algorithm which will generate the most extensive surface with minimal materials. As cells multiply their surface arrangement adds extra area to the organism. This is a matter of life and death for filter feeders like sponges and corals. Evolution is a great optimiser." (p. 6)

As soon as I am introduced something where math is being done in nature, without the intervention of humans, I am so HOOKED! This concept here is beyond cool. In the last 2 years of being in my MEd, I have been expose to a variety of ways that math is being done in nature, every day, by nature itself. Ever since I was introduced to Fractals, I see them everywhere. And now sponges... WHAT!? This lead me to run to my washroom, where I have a natural loofah, and observe its complex net. Although I failed to pin point it's pattern, I am convinced that it is in there somewhere. Of course a loofah is a plant, and not a Porifera, but I thought perhaps there was a connection. 


Martin, A. G. (2015). A basketmaker’s approach to structural morphology. In Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam: Future Visions.

Week 7 - Nick Sayers Interview

STOP #1: I really enjoyed how Nick Sayers spoke about how creating spheres out of recycled materials can be a representation of the end of waste (around the 20 minute mark). Since spheres have an endless surface, than the end of waste can be had. Nick Sayerscreated a type of shelter with these spheres when created at a large scale (like with the signs), but I wonder what other purpose (besides a really neat piece of art with a story) a smaller sphere could have.

Stop #2: Nick Sayers mentioned (around the 29 minute mark) how numbers have meanings and related many numbers to OCD and superstitions. I had to laugh when I heard him say this. I don’t let volumes (radio, TV) be on odd numbers. My partner and parents laugh at me for this, but there is something about odd numbers that feels off, whereas even numbers are just superior in my head. Perhaps because they can be paired? Unsure.

I was curious if this was just a “me” thing, so I did a quick google search. AI’s response was:

People tend to prefer even numbers due to cognitive ease, perceiving them as more stable, balanced, and orderly than odd numbers. This preference is rooted in the ease of dividing, pairing, and processing even numbers—particularly multiples of 2, 5, or 10—which reduces mental effort and feels more "complete" or symmetric.

And that makes sense to me. Even numbers and multiples of 5 really do reduce mental effort when any kind of math is needed. There is also a beautiful degree of symmetry, perhaps relating to my idea of pairing.

Stop #3: The entire idea of pinhole cameras, which was a great section at the end of the video. I loved how there was an extensive discussion around how the image ends up upside down when it comes through the pin hole. I immediately thought about how our eyes work. The science teacher in me found this fascinating and how this would be a great visual, hands-on activity to describe the biology and functionality of our eyes. So cool.

On the topic of the pin holes photos, I really enjoyed this visual (around 1 hour 10). The explanation of the description of seasons shown throughout the photo, with this morse code look representing the sunny and cloudy days, and the description of the differences of light coming through the tree depending on the season, I was so in awe the entire time Nick Sayers was talking about this photo.

Nick Sayers really highlighted how there are so many different forms of art, and that there are millions of entry points into math. I believe that watching this interview reminded me that math does not always need to be formulas and complex tasks, it can include beauty and adapt to different forms.

Video Link: https://vimeo.com/1166172275/3a7a243bce?share=copy&fl=sv&fe=ci

Week 6 - Activity Reflection

 I chose to focus on Math Hello's!

I started by following along in Karl Schaffer's video. This brought back memories from when I was a kid, I remember doing this "flipping your hand without turning your elbow" activity. I decided to explore this a little further. Do we have to use 90 degreed? What happens when we cut that angle in half, and move only in 45 degree angles, still in the x-y-z planes?  

https://www.youtube.com/watch?v=c1AqopgzlSU

In 90: Takes 6 x 90 degree movements to flip the hand

In ~45: Takes 12 (if i can count) to flip the hand. you end up on a diagonal a few times. My video will not load. But I challenge you to try this and also count. PS: It takes way more brain power than I thought, and actually, doesn't fall directly on the x-y-s axis but instead ends up in spaces in between. I kept sill 3 movements per round. 

Round 1: palm starts by facing me, parallel to my face (movement 0), ends on a diagonal, I can still see my palm (movement 3)

Round 2: Starts at the diagonal (movement 3), ends with hand perpendicular to my face (movement 6)

Round 3: Starts perpendicular to my face (movement 6), ends on a diagonal to me, palm facing away (movement 9)

Round you, Starts on diagonal, palm away (movement 9), ends with palm facing away from me, parallel to my face (movement 12)

I really enjoyed moving in the x-y-z space, with simple-ish movement. This was a great way to explore a 3-D space, mathematically. As well as using mathematical language like parallel, perpendicular, and diagonal to help explain my movements. 

I think this activity could be used to explore many other angles, as long as they are easy to visualize, such  as 30, 120, 180. 

As mentioned, this could be a fun activity to help introduce and understand a 3-axis graph, as well as how to translate lines/functions in this spaces. Your palm can act as a plane, to show how planes can rotate in this space. 

Do you have any other possible extensions? 

Week 6 Reading Reflection

Learning to love math through the exploration of maypole patterns

By Julianna Campbell & Christine von Renesse

The authors explore how maypole dancing inspires liberal arts students to engage with mathematics through inquiry-based learning. This pedagogy shifts the focus from delivering information to student-driven problem-solving, where learners collaborate to develop their own conjectures and proofs. Co-author Julianna Campbell recounts her personal transformation from math anxiety to research confidence, highlighting how curiosity and perseverance enabled her to find joy in the subject.
Mathematically, the study investigates the total number of non-equivalent ribbon patterns generated by different combinations of dancers and colors. The class developed two primary models for analysis: the tree representation, which visualizes diagonal crossings on graph paper, and the screen representation, used to automate and view patterns horizontally and vertically in Excel. Patterns are classified as equivalent if they are related through rotations, translations, reflections, or color changes. The authors prove four specific theorems—Letter Rotation, Ribbon Swap, Leader Rotation, and Pair Swap—to identify these symmetries. Their findings for a six-ribbon dance identify five unique patterns using two colors, seven using three colors, and five using four colors. The research concludes by posing open problems regarding how these structures generalize to larger groups of dancers.

STOP #1: 

" - Students will strengthen their reasoning skills and become better problem solvers.
   - Students will strengthen their skills in reading, writing, argumentation and speaking.

[...]
   - Students will improve their mathematical confidence. 

   - Students will develop awareness of the negative impact of broadly held societal views. " (p.132)

These goals are beautiful to say the least. These goals definitely work towards becoming more fluent in mathematics, but debatably more importantly, these goals work on create better humans and better and more confident members of our society. These goals work towards exposing students to really important topics while enhancing their math skills. 

STOP #2: "I realize that K-12 math education failed me. I was one of millions of kids my age who had grown up to believe that math was just a formal, numeric, and logic-based subject." (p.134)

As a senior math teacher, this sentence hurts. I struggle when I hear this, as I was always a lover of math, and math always came easy to me. I got to where I am today because I purely loved the subject and always found joy in it. I see most complex math problems as puzzles, and find joy when finally finding solutions. BUT, I am 100% aware that this is not the case for most of my students. I work towards integrating more "fun" into my class, but find myself often restricted by time and the giant pile of curricular topics I need to cover before the final Ministry exam that is worth 50-100% of their year! Because the Ministry is still using a HUGE standardized test at the end of the year, we are limited as senior math teachers when going rogue and off the "traditional trail". As much as we don't want to admit it, we are kind of "teaching to the test" and need to be cautious of potential "wasted time" because we are already so limited. This is not me being close-minded. I am still working on incorporating different things into my classes, and have had some success since starting this program, I am working towards continuing this process even after the program is done! I am just highlighting the unfortunate obstacles I feel, and my colleagues feel when there is such a high pressure from the Government for that final exam. As Barney Stinson would say " Challenge Accepted!" Because it is not fair for students to feel that the system failed them, it is not fair for students to sit through the majority of their life in a subject that doesn't "click" for them. AND it is not fair that these students do not get to enjoy all of the joys and extremely amazing things that math has to offer! 

Campbell, J., & von Renesse, C. (2019). Learning to love math through the exploration of maypole patterns. Journal of Mathematics and the Arts, 13(1-2), 131–151. https://doi.org/10.1080/17513472.2018.1513231

Week 5 Reading Reflection

Movement-based Mathematics: Enjoyment and Engagement without Compromising Learning through the EASY Minds Program

The EASY Minds program (Encouraging Activity to Stimulate Young Minds) addresses the global decline in mathematical achievement and low levels of physical activity among primary school children. This intervention integrates movement into mathematics lessons, utilizing movement-based learning to enhance student engagement. A six-week trial involving Grade 5/6 students found that this approach significantly increased enjoyment and engagement without compromising the quality of learning.

This study features activities that combine procedural fluency like practicing multiplication while skipping, with real-world applications, for example measuring shapes on a netball court. Results demonstrated significant positive effects on on-task behavior and moderate-to-vigorous physical activity. Students reported that "doing math with their bodies" made difficult concepts like averages (mean, median, mode) more accessible and helped them focus better. Teachers also expressed high satisfaction, noting that the program prompted pedagogical reflection, reduced discipline issues, and encouraged more innovative teaching styles. Overall, the sources suggest that embedding movement-based lessons is a feasible strategy to improve both student health and academic attitudes

STOP #1: Although this was not news to me, it is important to highlight, and thus takes the stage for stop #1 in this blog. The significant decrease in mathematics motivation with a parallel or increase in obesity and health issues in children is ALARMING! And this study makes it sound like there is an easy-enough fix to this problem. I know that here in Quebec, we often talk about how student don't get enough Physical Education periods in their schedules, often only getting 3 for every 9 school days. But if we start incorporating these types of learnings into other subject areas, we are working towards a unified and justified solution. 

STOP #2: The prep work that comes with this. Teachers expressed concern for prepping, but found that the pros outweighed the con here. Students being WAY more engaged and involved meant that there was much less behavior issues and thus more learning and fun. I think that like any other, the initial year(s) will take a lot of work, but once a repertoire of movement lessons have been added to your bank, then this becomes actually easy and much more enjoyable for students and teachers. 

QUESTION: I plan on adding more movement into my teaching when we can be outside more. I hope to be able to bring nature and movement into lessons, together. How will you make your students move this year!?

Week 5 Activity

Week 5: Developing mathematics pedagogies that integrate embodied, multisensory, outdoors and arts-based modalities

Sarah Chase: I really enjoyed watching Sarah Chase, what she is doing is extremely impressive. I have tried the 3 - 2 pairing before, and it has taken be FOREVER to master. I am extremely blown away at how she does a 13-11-7 sequence and can remember it. I feel that when I was learning this, I memorized the 6 different movements that come in a 3 - 2 pairing. 6 movements is not a lot to memorize, but 1001 is a TON. Although I am not focusing on Sarah Chase's work for this blog post, I felt the need to highlight how fascinating her work and skills are.
https://vimeo.com/251883173

I have found throughout this course that I am really enjoying the relationship between visual arts and math, which is why I was pulled towards focusing on Ali and Colin's activity this week. 
https://vimeo.com/217231056

I asked so many questions as I was watching this. I am still not 100% certain that I understand the logistics or quite how to create a piece like this yet, but I was able to think of a few extension that could be interesting. 

    - What if we worked with subtractions that resulted in negative numbers, we could use inverted colors to demonstrate negatives. This could results in the addition of more colors, that are related and continue to represent different things. 

    - What if we attempted to do this with multiplication, would we mix colors, or would we add rings in the "product" square? Perhaps we use this to represent exponents, say we use a base of 2, and then represent 2 to the power or 0, then 1, then 2, then 3, and so on. 

    - Could fractions be integrated into a visual arts project like this? How....? (looking for answers!)

I really like the idea of showing negatives. Perhaps a grid containing 33 squares (3x11) representing -16 to 16 could be really interesting to see, with 0 being in the very center. The inversion of colors shows a connection between the numbers. This would have to be taught to students beforehand. The question that lies now is how many rings are needed...? It took some trial and error and after getting through half, I realized I was missing a ring, so I started again. Here is what I created. 

I really love the color scheme and how the inverted colors look. This was extremely confusing to me at first and I had to go find the art on the Bridges Website and read the authors blurb to better understand the art. (https://gallery.bridgesmathart.org/exhibitions/2016-bridges-conference/chamberland) Once I understood and began the creation process, it was extremely easy to follow the pattern of numbers and colors throughout the piece. After creating this, I realize that the inverted colors work well, and I feel with the "zero" tile in the middle, it is much clearer now what is being represented. 

As seen in Ali and Colin's video, I was interested by the adding that he showed. I think this could be really fun to do with my art, including positives and negatives. Perhaps with a variation of adding and subtracting, as well as multiplying (as long as the answer is under 16 - or perhaps would you have 2 squares in your answer after?) 

-1 x -1 = +1

-15 x 3 = -45

I don't know if that is how we would want to show it, but I feel that this method emphasizes the two negatives make a positive and that as soon as there is only 1 negative, that the answer will also be negative. 

Although I did not answer everything in an organized brainstorm sketch, I do believe that I have replied to all aspects to this weeks activity in my messy thoughts written here, and through my art. 

Thanks for reading, and please! Extend on these ideas with me, as perhaps I could try this with my students! 

Annotated Bibliography

Teaching Fractions through Multi-Modal Approaches

Amanda Wheeler and Taylor Faille

Life And Work Skills (LAWS) Program (aged 15-19), Chateauguay Valley Regional High School (Quebec, Canada)
Grade 11 (aged 16-17), Chateauguay Valley Regional High School (Quebec, Canada)
Grade 7 (aged 12-13), Comox Valley School District (British Columbia, Canada)

Rationale

There are many mathematical strands that can be infused with the arts, nature, and physical embodiment to improve student learning, engagement, and retention of skills. In deciding the direction to begin on our project, we wanted to pick a mathematics topic that can be conventionally difficult for some students. Additionally, as our teaching assignments differ greatly from each other (Faille teaches high school math and Wheeler provides K-7 numeracy support to her district), we also wanted to choose a mathematics topic that is applicable to a wide range of grade levels. The decision to settle into the topic of fractions came around mutually.

The basis of this initial research had two main targets. The first goal was to gain a deeper understanding of the impact that art integrated fractions teaching can have on students' learning and overall experience of mathematics. To achieve this, we sought academic research and journals which demonstrated impact. The second goal was to begin ideating about the many ways in which the arts can be integrated into fractional concepts. Much of this research was conducted through watching videos and gathering resources such as lesson plans online.

Our project is going to have us designing a series of activities that integrate aspects of the arts, integration with nature, and authentic embodiment of the fractional skills. As we are working with varying grade levels, we are interested in how these activities can be developed in an open-ended enough way to provide extension opportunities whilst also allowing entry for all students.

Annotated Bibliography

        Adler, I. (1998). The role of continued fractions in phyllotaxis. Journal of Algebra, 205, 227–243

Adler uses points on a cylinder to explain why leaf patterns follow number rules known as continued fractions. He proves that the angle between leaves (the divergence) determines which spiral patterns are visible to the eye. He also identifies "points of close return", which are leaves that line up closely to each other which also shows how they naturally match the Fibonacci sequence

        Azaryahu, L., Broza, O., Cohen, S., Hershkovitz, S., & Adi-Japha, E. (2024). Development of creative thinking via fractions and rhythm. Thinking Skills and Creativity, 52, Article 101514. https://doi.org/10.1016/j.tsc.2024.101514

The researchers looked at the connection between learning fractions and learning rhythm in music. By breaking students into groups receiving separate teaching and comparing these groups to a control that is only taught the fractional concepts, student achievement is assessed. Success was measured for both musical and mathematical concepts. There was a notable enhancement in student performance in the groups of students who engaged in a creative way when compared to those in the control group. This serves as further evidence of the efficacy of creativity in mathematics. Music is something that so many humans enjoy and could prove to be an engaging tool to use when teaching fractions.

        Chahine, I. C. (2013). The impact of using multiple modalities on students’ acquisition of fractional knowledge: An international study in embodied mathematics across semiotic cultures. The Journal of Mathematical Behavior, 32(3), 434–449. https://doi.org/10.1016/j.jmathb.2013.04.004

The researcher investigated how an embodied approach to teaching fractions impacted the understanding students had of various outcomes. The findings were that the 5th grade students who were exposed to multiple modalities of learning including hands on manipulatives, gestures, and movement performed stronger in a post assessment than their peers who utilized a more traditional pencil and paper style of learning. For the purpose of our work, we can use this research as further evidence of the efficacy of utilizing embodied math in fractional learning. We could also look at the activities that were performed to gain inspiration for our own project.

        Goral, M. B., & Wiest, L. R. (2007). An Arts-Based Approach to Teaching Fractions. Teaching Children Mathematics, 14(2), 74–80. http://www.jstor.org/stable/41199065

The authors write about an arts-based approach to teaching fraction concepts through poetry, physical movement, and music. The fractional concept being addressed with these grade 4 and 5 students in the USA is part-whole relationships and equivalence. This foundational understanding is often a difficult one to master which is why an integrated approach was taken. This was not a formal data collecting research enterprise, but they did note that they observed the students demonstrating strong engagement and improved reasoning through the lessons. For our project, it would be interesting to look at a multimodal approach to fractional learning as it seems the more branches and connections that can be developed, the more concrete student understanding will become.

        Lovemore, T. S., Robertson, S.-A., & Graven, M. (2021). Enriching the teaching of fractions through integrating mathematics and music. South African Journal of Childhood Education, 11(1), Article a899. https://doi.org/10.4102/sajce.v11i1.899

Lovemore, Robertson, and Graven (2021) describe an action research study that integrates music note values into Grade 5 fraction lessons to address learning challenges in South Africa. By employing multiple sensory representations and the American note-naming system, the authors demonstrate how musical rhythms can clarify concepts like equivalence and the inverse order relationship of unit fractions. The study concludes that this integrated approach fosters a deeper conceptual understanding of fractions while simultaneously increasing student motivation and confidence in mathematics

        Maclean, Lauren. (2020, September 12). Mentoring Nature Connections: Fraction Nature Walk [Video]. YouTube. https://www.youtube.com/watch?v=IJwQ-PQk6Ms

This video brings students on a nature walk and allows them the freedom to find fractions in leaves. Whether it's color differences or broken leaves. The small project is to set up a fraction clothes line, where a leaf that represents a half, would go half way on the clothes line. There is an element of environmental care in this video as well. The teacher introduces ratios/proportions by teaching students when it is okay to take a piece of nature and when it must be left, if you can find 7 samples, then you may take 1, if not you must leave it. Other options for observing include a technology component, such as taking pictures, or an artistic component such as sketching. I would love to use this as an example with my group as we have a trail through a forest on campus, therefore we could easily do this project to help students understand fractions while appreciating nature, if weather permits (we currently have a lot of snow!), this could be done and incorporated into this project. Something similar may also be done that accommodates the season of winter.

        Marotta, Barbara. (2022, February 3). If You Were a Fraction by Trisha Speed Shaskan [Video]. YouTube. https://www.youtube.com/watch?v=PL7Vc-v8Lus

This is a read aloud of the book “If You Were a Fraction” by Trisha Speed Shaskan. A use of everyday items (such as food) and animal characters to demonstrate fractions through literature and illustrations. Perhaps a good introductory tool to a fractions unit.

        Narsh's art. (2021, January 21). Fun Fraction Art [Video]. YouTube. https://www.youtube.com/watch?v=PS-wYCuM8Gk

This video demonstrates an art options to better understand fractions. Students cut pieces of shaped paper into smaller symmetrical pieces where each piece represents a fraction of a whole. With these smaller pieces of paper, the students glue them onto a square to create a colorful symmetrical design. Once the whole class has made their piece, they are put together as a sort of quilt. This would be an activity we would like to integrate into our classes to show the connection between visual arts and fractions.

        Richmond, H. (2021). Fraction action: An embodied approach to teaching fractions (Curriculum unit). Charlotte Teachers Institute.

This resource is a multi lesson unit that puts into action how embodied learning through movement can occur within the learning of fractional content. The lessons are designed for grade 3-5. The author suggests utilizing strategies like human number lines, walking research, and kinesthetic modelling in the teaching. There is an emphasis on making movement a natural part of this teaching and not just a token for motivation. There is no research data connected to this unit which can aid in deciding whether these choices would be impactful in a classroom, however, it does provide great ideas on how to integrate movement. This resource provides clear, concrete examples.

        Scaptura, C., Suh, J., & Mahaffey, G. (2007). Masterpieces to mathematics: Using art to teach fraction, decimal, and percent equivalents. Mathematics Teaching in the Middle School, 13(1), 24–28.

This is a journal article where middle school students created visual artwork to explore the connections between fractions, decimals and percentage. There is an emphasis placed on representational fluency and also is supportive of students who are English language learners. This is a fantastic source of an idea of how to integrate artwork and fractions.

Next Steps

With the goal of utilizing these lessons within our teaching roles, we have decided upon the following timeline for completion to ensure this is achieved.

We want to develop around 4 activities that effectively integrate fractional thinking with the arts, nature, or kinesthetic movement (or a combination of them). We will each work on and develop two.

February 20th - Each have 1 activity developed and ready to use

February 27th - Each completed 1 activity in classes

March 6th - Reflect and include first activity into presentation, and develop a 2nd lesson each

March 13th - Lesson 2 completed in class(s)

March 20th - Reflect and include the second activity into the presentation. Video recording done

March 24- Must be handed in by

Week 4: Reading Reflection

Spinning Arms in Motion: Exploring Mathematics within the Art of Figure Skating

by: Tetyana Berezovski, Diana Cheng, and Rachel Damiano

SUMMARY: The article "Spinning Arms in Motion" explores mathematical modeling through the art of figure skating, specifically focusing on the upright spin. The study uses a bird’s-eye view to simplify arm movements during the spin’s acceleration phase. The authors present two dynamic models using Geometer’s Sketchpad (GSP) software. In Model 1, Only the lower arm moves. This model teaches proportional reasoning, scale factors, and regression by having students calculate real-life dimensions from diagram measurements and create scatterplots of arm positions over time. In Model 2, both upper and lower arms are in motion. This advanced model incorporates trigonometry and circular geometry, requiring students to solve triangles, calculate arc lengths, and analyze the symmetry of the "pentagonal" position formed when hands cross. Ultimately, the activities demonstrate how middle and secondary-level mathematics, including algebra and geometry, can explain the physical and aesthetic components of skating performance. By simplifying complex real-life movements, students learn to apply mathematical concepts to athletic phenomena. 

STOP #1: “Contexts of interest to students include art and sports” (Berezovski et al., 2016).

Although this may not be 100% true, it is definitely try for a grand majority of students. Many are either fully into the arts, or sports, or both. This short article has made me start reflecting of the math involved in other sports and how bringing that into my classroom could impact the interest and motivational rates in math.


STOP#2: I have always loved watching figure skaters and I have close friends who compete in this sport. I have never thought about the mathematical component of figure staking and how the symmetry and slight change in angle could change the result of a spin. This was a really interesting article to read and learn about the math behind something I have been watching my whole life.


QUESTION: Have you ever taught math through a sport? Is yes, which sport and how? If no, which would you choose and what component and mathematical concept would you focus on?

Berezovski, T., Cheng, D., & Damiano, R. (2016). Spinning arms in motion: Exploring mathematics within the art of figure skating. Bridges Finland Conference Proceedings, 625–628

WEEK 4: Activity Post

 I was assigned the 2012 Bridges Conference, and I decided to choose John Hiigli's Piece, shown below. 

I really enjoyed this piece, and honestly felt that it was within my abilities to recreate with the tools that I had. I read the excerpt that went with this piece, it mentioned how the artist started "with the surfaces farthest from the observer, working [their] way forward to the front surfaces". I though that this would help me when drawing the Hypercross II. I attempted to start at the back, but then got so confused that i figured I would start at the top instead. 

After several minutes of attempting this, I was getting so beyond lost in all the lines. I was showing colleagues and they were laughing at me as some choice words may or may not have been coming out of my mouth as I struggled my was through the chaos of lines. After a while, I was able to get a good enough sketch of the Hypercross II. It was far from perfect, but it was the best I was able to do. 

Since this original piece was done with paint, I thought I would use markers, as that was the closest thing I had available to me. I was really nervous to start as this had taken me close to an hour to create. So in the nature of "I don't trust myself" I photocopied my piece a few times to be able to test out colors and markers overlapping before committing to finishing my piece. Thank goodness I did. The markers ruined this piece. I quickly decided to switch over to pencils. 

This piece was so frustrating but fun to create. I realized that it was not perfect and that eye-balling parallel lines was not my best choice. After walking into the grade 11 art class and realizing that they are currently working on a 2-point perspective project, I realized that the technique to create those masterpieces would have really helped me for mine. Attempting to have all the same angles in the same places of each cube was really hard considering I was doing it all by eye. The more I looked at it the more I realized that there were small imperfections. It was also interesting that the more lines I got on the page, the easier it became. Was this because I was getting use to it, or was it because there was actually a picture being created? 

Regardless, back to the pencils. When I started coloring, I was now at home at my kitchen table, that is made out of wood, therefore has a texture to it. This would not seem relevant to my art piece, except i quickly realized that it textured my coloring. 

At first I thought I better find a smoother surface to work on, but I quickly began to admire the "rougher" look and decided to carry on. As soon as I started coloring all the tops of the cubes green, and all the right sides red, it was getting easier and easier to see what was being created. 

I must say, that I was proud when this product was complete. I am friends with the art teacher at school who lent my fancy pencils to help make my final product better. When I was done, I sent him my project and he told me I could have gotten a 90%. I know that the point of this activity was not to create a beautiful piece of art, it was more to reflect on the process, but the fact that my art did turn out to be something that gave me pride, I think is a bonus. This feeling also allows me to realize and reflect on why we should link art and math more often, students can learn to appreciate the math behind art, while gaining a deeper understanding od mathematical concepts and creating something to be proud of.