Week 8

 This week I read the article “ Highly unlikely triangles and other impossible figures in bead weaving” by Gwen Fisher (2015).

The article captures the author’s journey to create a 3D sculpture of an impossible triangle.  For those that don’t know, the impossible triangle was first drawn in 1934.  It has since been incorporated into several different artists’ work, and inspired others to create similar optical illusions with other shapes such as squares, hexagons and tetrahedrons.  

Fisher began by using seed beads and a quarter twist on each beam to construct a 3D version of the impossible triangle.  However, the twist in the beads was found to curve the edges of the beams which destroyed the optical illusion.  Therefore, the author decided to use the term “highly unlikely” as opposed to “impossible”.  Here are three of the author’s highly unlikely triangles that were created.

Different coloured beads were used to distinguish the different edges of each beam.  The following diagram shows how the author converted the 2D drawing into its beaded counterpart by cutting the beams into rows of little cubes.  The author then used a beaded cubic right angle weave technique to create each cube.

After mastering the highly unlikely triangle, the author extended her skills and knowledge to create other highly unlikely shapes such as squares, hexagons, and tetrahedrons as seen below.  

The tetrahedron ended up with a pleasant surprise in the form of three distinct paths that twist in a fashion similar to a Mobius strip. As you follow the path of one colour around the shape, you end up on the opposite face.  Therefore you have to make two trips around the shape to return to the original position.

This is the final shape that was included in the article, and represents the most complex shape the author has tackled so far.  The attention to detail that is required to complete such an amazing work of art is nothing short of amazing. Fisher continues to create impossible figures through beading and looks forward to being the first to try beading an impossible staircase.

STOP - As someone that has only ever tried to bead simple earrings or bracelets, I am in awe of the author/artist’s skill and creativity.  I can’t even imagine how hard it would be to create these shapes from a provided pattern, let alone be the one to create the pattern.  I think bringing some form of textile art into the classroom would be a really interesting way to showcase unique skills that students may have but rarely get to show in the classroom.  Whether it’s beading, weaving, sewing, fashion design, screen printing, quilting, knitting or some other form of textile art, there is so much to explore mathematically.  In one of our previous courses, we talked about students doing a project that showcases the math they use in their culture/home/job/community.  I can see these textile arts being a great option for some students to use in that type of project.

Activity

I chose to try Coastal Salish weaving for my project this week.  I assumed (wrongly) that it would be relatively easy.  If high school students can do it, then obviously I should have no problem.  Well was I ever wrong.  The final product, seen below, took hours of work to complete and isn’t even finished because I ran out of purple yarn.  I then created a line graph in Desmos that was similar in slope to the pattern in my project.  The program ended up being the easiest part of this activity, and I can definitely see myself using it in my classroom when we explore linear functions.

Week 7

 This week I read the article “Can zombies write mathematical poetry? Mathematical poetry as a model for humanistic mathematics” by Gizem Karaali (2014).  

What are the three most important things that make us human?  Hint: Both math and poetry embody these attributes

The answer: cognition, consciousness, and creativity.  

STOP - What other attributes would be considered unique to the human species?  Is the author correct in stating that the combination of these three things are what makes our species special?

The author argues that while there are certainly many examples of these attributes found in nature, humans are the only species that exhibit all three.  Math and poetry are connected in that they both require thinking with the mind and the heart.  A relentless pursuit of the artist’s craft, a natural intuition and curiosity, and a creative spirit are also necessary.  This connection between the two art forms, and the author’s own journey into the world of mathematical poetry are the focus of this article.

When thinking about mathematics, most people fail to notice the creative aspect.  They tend to think of the traditional school mathematics, in which students rotely memorize and replicate others ideas.  It’s cold, unemotional, and lacking in human qualities.  However, mathematics is filled with novel ideas, creative problem solving and beauty.  Karaali argues that poetry is a natural vehicle for expressing and showcasing this humanistic side of mathematics. 

STOP - While I appreciate the author’s stance on both mathematics and poetry.  I personally find it difficult to imagine myself embracing poetry as a means of expressing my mathematical understanding.  I find poetry daunting and confusing.  I always find myself wishing the poet would just say what they mean and skip the figurative language.  I just can’t imagine also adding math into that mix. 

Also included in the article, is the author’s work in the field of humanizing mathematics.  From starting a journal.  The Journal of Humanistic Mathematics released its first issue in January 2011.  In addition to the journal, the author has collaborated with other mathematical poets to organize poetry readings, and has taught several college level courses that bring math and poetry together.

Final Thoughts - Do you enjoy poetry?  If so, can you see yourself embracing mathematical poetry and incorporating it into your classroom?

Activity

I chose to try writing a PH4 poem.  The subject of my poem is my family cat, Smoky.  I tried to pick 4 words that represented both his behaviour and his personality, while also featuring alliteration with all 4 words beginning with the letter “S”.  I found that coming up with the 4 words was by far the hardest part of this activity.  The actual braiding of the words was relatively easy given the predictability of the pattern.  I can see this activity working well in a cross-curricular approach with an English course.  There are so many ways to differentiate the complexity of the activity depending on whether the writer uses words or phrases in the first line.

A Predator in the House

Silent	stalker	stretching	sweetly
Stalker	silent,	sweetly	stretching
Stalker	sweetly,	silent	stretching
Sweetly	stalker	stretching	silent
Sweetly	stretching,	stalker	silent
stretching	sweetly,	silent	stalker
Stretching	silent	sweetly	stalker
Silent	stretching	stalker	sweetly
Silent	stalker	stretching	sweetly

Week 6

 This week I read the article “Reenacting mathematical concepts found in large scale dance performance can provide both material and method for ensemble learning” by Vogelstein, Brady and Hall (2019).

This study used foraging and dissection methodologies to explore how choreographed performances can be used to explore and learn math concepts through ensemble learning and embodied mathematics.

STOP - There were lots of vocabulary terms that were new to me and required some re-reading in order to understand and remember.  I’ve listed them below!

Foraging - the researchers examined rich cultural performances for segments that would allow for people-plus-props mathematical learning opportunities.

Dissection - participants viewed the performances and then reenacted different aspects, along with exploring the use of props within the performance.

Ensemble learning - participants must work together to create the reenactment and problem solve the use of the props.

The performances used within this study were from the opening ceremony of the 2016 Rio Olympics.  I’ve linked a video of the performance below.  To see one of the performances that was used within the study, go to the 12 min mark.

https://www.youtube.com/watch?v=N_qXm9HY9Ro

The results of the study found that dissection and reenactment of group performances can provoke fundamental questions about knowing and interpreting math figures.  The participants often found it challenging to reenact moves that appeared to be “simple” at first.  Group coordination and communication was necessary in order to understand how to best use the prop in order to replicate the movements.  Other findings were that the use of props that were similar to the original performance was important, and that reenactment methods allow for math discoveries within the performance.

STOP - The authors found that quartets were the optimal size group for this particular study, and that the use of similar props was important to the success of the groups.  This made me wonder whether 4 is the optimum number regardless of the performance?  I also wonder about the practicality of replicating performance materials.  I think educators would need to be careful about which performances they chose to replicate, not only in terms of the potential math content, but in the availability of the materials needed.  In my relatively small town, it can be difficult to source specific materials on short notice!

FINAL THOUGHTS - I feel like dance and music are my biggest struggle when I think about incorporating the arts into my classroom.  It’s really hard for me to find connections to the math curriculum, and the thought of dancing in front of students makes me break out into a cold sweat.  Can you see yourself bringing dance into the classroom?  How would you incorporate it?  What math concepts would be the easiest for you to connect with dance?

Activity

I chose to check out the Rope Polygons lesson plan, and was really excited to give it a try.  I convinced my youngest child to try it with me, and we were able to successfully form a square, rectangle, and triangle.  Since it was just the two of us, we struggled to create shapes that had more than 4 corners.  I let her be the “boss” of our efforts, and she was able to direct me on where to stand and how to hold my hands so that we formed the correct shape.  We had a ton of fun, and she’s still talking about it a few days later.

I could see lots of ways to extend this lesson to my older (Grade 8/9) students.  For example, they could investigate the relationship between perimeter and area, or determine what shape would have the biggest area.  They could enlarge shapes from a worksheet and calculate the scale.  They could form complex or irregular shapes and try to determine the perimeter and the area as well.

Project Outline

 Letting go of the grid in an urban school garden!

By: Courtney Lepetich (just me, myself and I)

Grade 9 Math and Science.

Students aged 14 /15 years old.

SD #28 - Quesnel Junior School

Outline:

*A cross-curricular project between grade 9 math and science

*Math concepts include: measurement, scale, proportional reasoning, and a review of surface area, volume and 3D shapes.

*Math skills include: estimation, communication, and modeling through diagrams

*Science concepts include: ecosystems, sustainability, interconnectedness, and Indigenous peoples ecological knowledge

*Pedagogies featured within my project include: Outdoor education, garden based education, multisensory learning, and embodied measurement

*Students will visit local forests with an Indigenous knowledge keeper from our community to learn about local plants and their uses.

*Students will visit local community gardens and meet with the volunteers to learn about the needs of plants, our local climate, seasonal growing and companion planting, the purpose of community gardens, and their plans for the future.

*Students will assess the current school garden and create a proportional diagram based on these measurements.  

*Students will work in groups to design a plan on how to improve the current garden.  Things they will need to consider include: the available area, plant spacing, movement of the sun, companion planting, climate, native species vs. non-native species, perennials vs. annuals, design elements such as patterns, themes, and colours.

*A possible extension for students is to reimagine the empty field space around the garden and transform it into a food forest that is self-sustaining and low maintenance.

Look at that grid! So much empty and unused space!

Annotated bibliography:

Almers, E., Askerlund, P., & Kjellstrom, S. (2018).  Why forest gardening for children?  Swedish forest garden educators’ ideas, purposes, and experiences.  The Journal of Environmental Education, 49(3), 242-259.

This article explores the differences between education within a forest garden and other outdoor education settings.  The authors found that forest gardens provided children with the opportunity to feel a sense of belonging, experience self regulation and systemic dependence, co-create with non-human organisms, and imagine possible transformation of local places.

Khan, M., McGeown, S., & Bell, S. (2020). Can an outdoor learning environment improve children’s academic attainment? A quasi-experimental mixed methods study in bangladesh.  Environment and Behavior, 52(10), 1079-1104.

This study compared the academic performance of students who experienced a barren school ground, and those who experienced a school ground that featured a garden and outdoor learning modifications.  It was found that students not only performed better academically, but that they also had higher levels of collaboration, experimentation, and exploration.

McCarty, J. (2013). REAL school gardens program: Learning gardens and teacher training to improve student engagement and academic performance in low-performing elementary schools.  The Journal of Applied Research on Children, 4(2).

This article outlines the REAL garden program, which brings parents, schools and the local community together to build school gardens in low income neighborhoods.  The program provides help with designing and building the garden, as well as extensive teacher training around outdoor learning.  Results show improved academic performance and overall health in the students of these schools.

Merritt, E., Peterson, A., Evans, S., & Marston, S. A. (2021).  Learning about culture and sustainable harvesting of native plants: Garden-based teaching can foster appreciation of Indigenous knowledge.  Science and Children, 58(4), 69.

This article provides an example of how educators in Arizona collaborated to develop lessons based on local Indigenous knowledge of native plants, sustainability, and multi-sensory learning experiences.  The lessons aim to inspire educators to create similar learning experiences using their own local knowledge and ecosystems.

Mitchell, J., Niras, J., & Niefeu, L. (2020). Cultivating well-being: Young people and food gardens on Tanna, Vanuatu.  Engaged Scholar Journal, 6(1), 1-17.

This article analyzes a collaborative project between youth and their communities on the island of Tanna, Vanuatu in the Southwest Pacific.  The project aims to address the declining interest in traditional knowledge among youth by having young people conduct research on customary food gardens and to document Indigenous customary knowledge, practices,  and stories about their sustainable food practices.

Peach, L., Richmond, C. A. M., & Brunette-Debassige, C. (2020). “You can’t just take a piece of land from the university and build a garden on it”: Exploring indigenizing space and place in a settler canadian university context. Geoforum, 114, 117-127.

This study examines the tensions that exist between universities, environmental repossession, and local Indigenous peoples since the release of the report by the TRC.  It found that Indigenization projects, such as gardens, rely heavily on a strong relationship between Indigenous peoples and university administration, as well as the ability of the Indigenous people to be self-determining within these spaces.

Ray, R., Fisher, D. R., & Fisher-Maltese, C. (2016). School gardens in the city: Does environmental equity help close the achievement gap? Du Bois Review, 13(2), 379-395.

This study aimed to compare the differences between traditional and garden based learning programs amongst low income and minority students.  The authors concluded that the presence of a school garden was associated with higher test scores and that school gardens can be used to create more environmental equity in urban areas.

Tsinajinie, G., Kirboyun, S., & Hong, S. (2021). An outdoor project-based learning program: Strategic support and the roles of students with visual impairments interested in STEM.  Journal of Science Education and Technology, 30(1), 74-86.

This qualitative study was conducted to understand how students with visual impairments engaged in STEM education.  The students took part in a week-long outdoor, inquiry based STEM education program with multisensory experiences and assistive technology.  It was found that the inclusion of purposeful accessibility allowed students to act as apprentices, collaborators and independent researchers within the program.

Williams, D. (2008). Sustainability education’s gift: Learning patterns and relationships. Journal of Education for Sustainable Development, 2(1), 41-49.

A case study of the Learning Gardens model in Portland, Oregon where students participate in a multi-disciplinary approach to learning within the garden context.  The article focuses on how sustainability education provides students with a holistic worldview, while simultaneously promoting the use of local cultural knowledge, encouraging multisensory learning, and bringing together the multiple generations within a community.

Week 5

This week’s article was titled “What Mathematics Education Can Learn from Art: The Assumptions, Values, and Vision of Mathematics Education” by Leslie Dietiker (2015).  

Traditional math classrooms are “monotonous”, “flat-lined” and “dry as dust”, at least that’s what the author of this article is claiming.  Even with the advancements in technology and countless studies investigating how students best learn mathematical concepts, little has changed in the last few decades.  Students still spend the majority of their time listening to the “expert” at the front of the room followed by replicating what they’ve been told through endless practice problems.  Textbooks and curriculum are designed to be unemotional, sequential, and efficient.

STOP - I paused here to consider my current and past classroom practices, the technology that I use, and the issue of textbooks.  I have to admit, that this year I’ve fallen back into old habits of lecturing from the front of the room, and then handing out worksheets with similar problems.  While I use manipulatives extensively, I don’t really use any other technology to support my lessons.  Just a couple of years ago I was more inclined to try new things and I was using Peter Liljedahl’s methodologies quite heavily.  After reflecting on this change, I think I went back to old habits as a coping mechanism to help me deal with how stressful my current students are in terms of their behaviour and academic ability.  Approximately ⅓ of my students are diagnosed with FASD and ASD.  They do better with the predictability of a more structured, traditional classroom, and are better able to regulate their emotions and behaviours.  As for textbooks, my school stopped purchasing textbooks over 10 years ago because the students couldn’t read them or take care of them responsibly and it was just too costly to continue to replace them.  I realized that I haven’t actually opened a textbook in so long that I’m no longer able to comment on how effective/ineffective they are.

So how might educators shake up the conventional math curriculum and invite the imagining of rich new mathematical stories?  

Dietiker proposes that making use of sensory experiences, engaging imaginations, risk-taking, and movement within the math classroom enhance the learning experiences of students.  They are more likely to feel a sense of wonder, pride, and joy when learning.  While there are many ways of bringing these types of activities into the classroom, Dietiker believes that making use of stories provides a natural avenue for incorporating all of the above.

These stories could be in the form of historical origins of math problems, narratives, or mathematical (similar to a rich task, or 3 Act task).  The best stories are the ones that draw students into the setting with interesting characters and unexpected plot twists.  These stories compel students to investigate alternate endings, take risks, and ask questions.

STOP - I’ve used stories a few different times within my math classroom, and while I enjoyed them, I’m not sure my students would say the same.  Perhaps I didn’t choose the right stories, or I failed to help students connect the math concepts with the story.  Either way, it didn’t seem to work for me and I moved on to trying other strategies.  

Have you tried using stories in your math classroom?  In what context have you used, or could see yourself using, stories to engage students in their math learning?

Activity - Sorry there's no pictures! I tried to get my kids to take pictures, but they were laughing so hard at my attempts that none of them turned out.

After watching both videos, I ended up choosing to try Sarah Chase’s dancing combinatorics to try.  This surprised me, as I am not a dancer. I was quite intimidated by the idea of creating my own movements, so I started by copying her example of 2 and 3 with just my arms.  As I predicted, I struggled for a long time to get my arms to move simultaneously within the pattern.  I tried using her idea to connect the movements to my emotions but that actually felt harder because then I was concentrating on moving my arms and speaking at the same time.  While I’m not sure if I could confidently demonstrate this activity to my class, I could still invite students to try this activity.  

I could see this activity working well with a lesson on finding the lowest common multiple.  Even though my students are in grade 8/9, they still struggle with this concept.  Partly because they are not great at their multiplication facts, which makes it difficult for them to find common multiples.  

Issue - Students struggle to find the lowest common multiple through knowledge of their multiplication facts

Guiding Question - How can dancing combinatorics be used to support students understanding of lowest common multiples?

Integrated Embodied learning - students create sequences for pairs of numbers using whole body movements as a means of investigating how many times the sequence needs to run in order to cycle back around to the first movement

Possible Extensions - Students use their knowledge of dancing combinatorics to explore greatest common factor.

Week 4

 This week I chose to read the article “Bridges Stockholm 2018” by Eve Torrence.  

The article is a summary of the events and exhibits of the 2018 Bridges conference in Stockholm Sweden.  It begins with a detailed description of the location the conference was hosted at, including the mathematical architecture such as the pentagonal tessellation on the ramp leading to the main entrance (see photo below).  

The main events, presenters, workshops and the winners of the art exhibit are all related in great detail.  Events included a play about 3 female mathematicians from different centuries, a family day, formal and informal music nights, a fashion show and a film festival.  I’ve included a few pictures of the events and winning art exhibits below.  The Bridges conference is an amazing representation of what can happen when mathematics and the arts are no longer thought of as separate entities, and instead, are considered to be one and the same.

STOP - I am stunned by the creativity of all the participants in this conference!  It’s amazing how they’ve brought together so many different disciplines with the common theme of mathematics.  It really highlights for me how isolated math is at the highschool level.  It also reminds me of how my own mathematical creativity has stagnated.  I’ve been teaching the same 2 grades for over 10 years.  I used to try really hard to bring in artistic and creative elements into my classroom and I’ve lost that over the last few years.  

Has anyone else brought the arts into their classroom?  What did you do?  How did it go?

  Activity

For this activity, I was assigned to the Bridges 2021 art gallery.  I chose to replicate the piece by Conan Chadbourne, as it reminded me of quilting.  It was also one of the only pieces that I felt I actually understood the gist of the mathematics that were used to create it!  Below is a side by side comparison of the original piece and my own.

The mathematics of symmetry (especially in the forms of rotation and reflection) were used to create these 12 grids.  It turns out that when working with 4 x 4 grids, you can partition the grids into nearly 213,000 different ways.  However, there are exactly 12 that have a four fold dihedral symmetry.  If you're like me and aren’t too sure what exactly four fold dihedral symmetry is, I’ve included a definition below!  

Even though I thought this would be a doable challenge for myself, I still found myself stuck on how to recreate the entire piece.  It seemed overwhelmingly complicated for some reason.  I started by trying to fit the entire piece onto a single sheet of graph paper, and quickly gave up.  My daughter, who was watching me struggle, suggested that I look at it as individual pieces rather than one big picture. This was so much easier!  I ended up creating each individual piece and then cutting them out and arranging them on a table in the same orientation as the original artwork.  This process really helped to solidify the concepts that were used by the original artist.  I had thought about creating a black/white/grey version, but ended up going with coloured grids as well.  I felt it enhanced the symmetry features more than the monochromatic version.

Four fold dihedral symmetry - eight subunits are related to each other by one 4-fold axis and two 2-fold axes.

References

Conan Chadbourne. (2021) Bridges. https://gallery.bridgesmathart.org/exhibitions/2021-bridges-conference/conan-chadbourne

Eve Torrence (2019), Bridges 2018, Nexus Journal