Week 5 Activities + Connections

This week we get to choose either a dance or art that embody various concepts of mathematics. I have chosen to explore Sarah Chase's dancing mathematics. 

In the video, Chase (2018) shows various ways of embodying multiplication, lowest common multiple and factors using concepts of relative prime numbers. She assigns various body parts a different number (which translates to a movement sequence of the same number), and then simultaneously runs through the sequence with each assigned body part. 

Chase demonstrates this three ways:

1. multiplying 3 (left arm) x 2 (right arm), producing 6 unique full body movements before returning to the beginning of the sequence. Chase extends this version by assigning a narrative quality to the values 2 (summer/winter) and 3 (mother/father/myself) to more easily engage with the sequencing of movements.
   
   Here's my version:
   
   
   
   
2. multiplying 7 (legs) x 11 (left arm) x 13 (right arm), producing 1001 unique movement combinations.
   
   
3. multiplying 4 (right arm) x 12 (left arm), this time as a measurement device for discovering which element is associated with each Chinese Zodiac animal by year.

Four ideas of extending this activity:

1. a sequence of notes could be performed by individuals on a monochord. Then, as they play their notes, we could listen for how when the cycle starts repeating again. It would be very interesting to have students silently choose a number (between 1-6 to start) without telling anyone and develop their number into a note sequence in the same key signature (let's imagine "C+"), then we all play together. A instrument-less student would keep a tally of how many notes were played before the sequence repeats itself. The music that would come about would be very interesting - especially if we started putting restrictions or extra criteria upon it (ie. Lydian mode only! or Blues scale notes only!).
   
   
2. Using hand-clapping games from childhood. It would be interesting to develop a two person sequence where one individual has a set of 15 movements (ie. Kit-Kat Bar hand clapping), and the partner has a 8 movement sequence (ie. Miss Mary Mack hand clapping) and see when these patterns would mesh. 
   
   
3. Instead of thinking about the numbers in which the number sets share common multiples, we could imagine each number set as a cycle: a toothed gear. Then we could talk about ratios in how many revolutions through a movement set until the other set is complete. When one small gear turns through its set, how far into the next number set have we gone (or what portion of the cyclical gear has turned)?
   
   
   Source: https://www.notesandsketches.co.uk/images/Wheel_and_pinion.jpg
   
   
4. As another cross-curricular extension, we could connect this idea with chemical interactions in developing how many of each atom are required for an ionic chemical formula.  Part 1: For a multi-party activity, each student is assigned a different charge based on a cation (ie. Ca 2+, Fe 3+, Pb 4+, Va 5+), and anion (ie. (Br 1-, O 2-, N 3-, Si 4-), then students would approach their ionic opposite, and they would perform their movement together until they're both done counting how many times they needed to complete the sequence while the other person was still moving. This number is how many of them are needed in the chemical formula. Part 2: this idea can further be extended by using only anions, and determining how many of each are needed in a covalent compound. In this situation, anions would interact with anions and complete their movement sequence.

Source: https://chemistry.mtsu.edu/wp-content/uploads/sites/57/2024/07/Chapter-3Ions-Ionic-Compounds-and-Nomenclature.pdf

The potential sequence of events in a classroom lesson for my Math 9 students on LCM: 

• Lesson hook: Chase's (2018) 3 x 2 arm set. Students would try. I would ask them:
  What do you notice? 
  What do you wonder about this sequence?
• I would extend the complexity to then involve a third number (4), by having them turn their bodies in the four cardinal directions (N, E, S, W) and ask them when do they end up back in their same spots facing "forward" or "North"?
  
  
• Exploration: I would ask my students to choose numbers between 1-5 to develop into movements for each body part or body direction/orientation. 
  
  
• Extension: Map a phrase to each position for the individual movements. Then, express verbally their movement while dancing. (This must be the equivalent to rub your head & pat your belly and will be very tricky for coordination). 
  
  
• Think-Pair-Share: In your partnerships, explain your attempt and what you noticed/experienced. 
  
  
• Class discussion: Share out to the rest of the class what you've rehearsed through explaining to your partner.
  
  
• Lesson: Lowest Common Multiple - a pen and paper lesson with various methods of determining the lowest multiple among a set of numbers. 
  
  
• Brain Break: 5 minutes at different physical stations around the classroom (2000 pushup challenge, plank hold, desk yoga, partner squat hold, hand-clapping games, competitive games of arm wrestling/thumb war/rock-paper-scissors-lizard-spock)
  
  
• Skill development: Practice worksheet - try questions that look challenging to you. Check your work. Ask questions of your partner and teacher to gain more understanding if confused. 20 minutes.
  
  
• Reflective Journal: Prompt...
• What was your big take-away from today? 
• Where have you experienced this concept of Lowest Common Multiple in everyday life? 

Videos referenced:

Sarah Chase: Dancing combinatorics, phases and tides (long version 13:35);

New teachers riff on Kandinsky in Binary (5:00)

Week 5 Readings: Developing full-bodied modalities

Reference: Kelton, M. L., & Ma, J. Y. (2018). Reconfiguring mathematical setting and activity through multi-party, whole-body collaboration. Educational Studies in Mathematics, 98(2), 177–196. https://doi.org/10.1007/s10649-018-9805-8

Summary: Kelton and Ma's study helps explain how designs for mathematical activity enhance or lessen mathematical possibilities and relations by demonstrating the effect that the social learning environment has on the organization, opportunity and productivity of embodied mathematics in learners. The researchers focused on a few key distinctions outside of "traditional" (or "container") mathematics: 

• Full-body movements in mathematics (as opposed to gesturing with hands-only).
• Mathematical environment = arena, but the arena is experienced differently by each individual so an individuals mathematical environment = setting. 
• Mathematics as a non-hierarchical and non-dualist conceptualization of mind-body relations. So, math is not a transcendent ideal that is separate from the body, but instead it can be performed with the participation of bodies.
• Multiple bodies as a whole interacting in a space together, rather than emphasis on an individuals single body for mathematics.  

They present two cases of mathematical multi-party, whole-body activities for analysis and comparison: The walking scale number line (WSNL) and Whole + Half (W+H). They conclude that repurposing space could create resonances or dissonances depending on the history built in that space and this consideration should be taken into account while designing learning experiences involving whole body multi-party mathematical activities.

Stop 1: "Students' moving bodies also became meaningful aspects of the setting themselves and each other, beyond simply performing individual quantities moving and operation along the number line. Quantitative relationships were understood and talked about as spatial relations between students' home positions and bodies" (Kelton & Ma, 2018, p. 184). 

Upon hearing about this walking scale number line (WSNL) activity, I didn't think it was particularly interesting. However, reading about the interesting operations and movements students were able to do in relation to each is generating some intrigue. I imagine every child standing on their own "home-base" (ie. a point on a numberline without numbers) and then being asked to do operations. Addition and subtraction would be viewed as everyone marching together up or down the numberline. Multiplication would see an expansion of coordinated bodies from centre. I am now curious as to how this activity would look on a large scale.

Student buy-in is a concern for me, but seeing that grade 8 students participated in this number line aspect and seemed to enjoy this full-body activity, there could further be value added when taking this to include imaginary numbers too with a rotation about the x-axis. That would be an interesting extension onto this activity having two intersecting axis and using a complex number.  

Another extension idea would be to replace the blue tape with a flexible knotted rope. Arrange the rope in a spiral, or create a connected ring and see what kinds of mathematical relationships arise when thought of as a cycle. 

Stop 2: "Thad suggested that if he held onto Morgan (two to his right) and Kian (two to his left) with either hand, he could just turn around and rotate them to their opposites. He then revised this to include the whole group... Maggie and Thad solve the problem from their respective physical and mathematical perspectives in the material arrangements of the space" (Kelton & Ma, 2018, p. 186).

This was very interesting. Thad was centre on the numberline, and swinging his partners around him allowed for a "multiply by -1" scenario. He suggested everyone link arms and swing instead of each student individually count. This physical (re)arrangement without the need for counting placements could've only been accomplished with this space-oriented, full-bodied activity. Very interesting connection.

Stop 3: "...Jeff, was presenting a whole interval to Ms. Collins that involved crossing his hands. Jeff explained that , in this case, half would need to 'go to the other side of the world' in order to complete the task. This solution imaginatively expanded beyond the walls of the classroom, wrapping and bending the whole interval in a great circle around the world while entailing an impossible journey for half." (Kelton & Ma, 2018, p. 191)

While reading about this partnered activity of half (H) keeping their hand in between whole's (W) interval, I had this exact wonder... where would H place their hand to keep in between? I wondered about he limits of the boundaries, assumptions, etc. Claire's alternative solution of putting the hand behind W's back was an interesting and more practical solution incorporating realistic boundaries. How creative! 

Stop 4: "Bringing this comparison into broader dialog with the field, we suggest that researchers and practitioners attend more closely to the ways in which different patterns of mobilities in mathematical activity might affect the negotiation and development of reconfigured mathematical practices in any instructional design" (Kelton & Ma, 2018, p. 193). 

These whole bodied activities produced some interesting and unanticipated results that developed from the use and limitations of the students' bodies (not necessarily the mathematics part) leading to interesting conversation that were brought about organically. There is much value in this kind of discovery through experimentation. I would attribute this quality to higher level students if they could do this kind of hypothetical thinking and pose a problem with pen and paper... but with whole-body movements, each child is capable of this higher level thinking by associating their movement with the concept and asking "what does this mean?" and "what now?". My big take-away from this is whole-bodied mathematics is an accessible way to engage mathematics concepts at exploratory/experimental (higher) levels for a wider range of students when done right. 

Wonders: In this article, they have used really very dense vocabulary to describe some of these ideas, but aren't explicit in the concerns that they have for space and how it may influence movement-based mathematicians. Are there spaces, based on past experiences/history, in and around the school where movement mathematics would be severely hindered by their space? (I'm imagining the examples from the paper where a gym was used which, based on experiences, encouraged the students to move, walk and explore, whereas the classroom experience for W+H activity a student had to be coaxed out of her desk to start the exploration in a crowded room with its limitations actually leading to more theoretical imaginings.) I couldn't imagine even an area, even a home economics classroom with kitchens to impede whole-body mathematics. 

Perhaps they are mentioning something more on a personal level? Each student has had an experience in a space, and that will influence how they interact in it. Probably why the researchers used new terms like "arena" and then "setting" that required "editing" (or certain ways that students interacted in the environment). 

Assignment 1: Draft Outline and Annotated Bibliography

Activity: 

The Geometry of the Blade: Mathematical Principles of Historical Swordplay
by Kristie Truell with Oliver Podwysocki
(Click title for link to document featuring summary of key findings, research question, search strategy, and unabridged annotations)

Photo by Lance Reis on Unsplash

Grade level/course:

Foundation of Math 9-12, Workplace Math 10-11, Precalculus 11

Age of students for this lesson:

13-16 years old

Schools: 

Port Moody Secondary School (Olly) and Richmond Christian School (Kristie)

Photo by Tima Miroshnichenko

Topic Outline: 

Measure (spacing to opponent)

• Activity: Footwork (advancing step, retreating step, passing step, lunge, triangle step)

• Inquiry: How many of each type of step does it take to touch your partner? (Ratios, distance, measurement (body units))

•  Inquiry: With no steps, how far can you reach? (Graphing, Circle formula, number line (domain)

• Inquiry: With one step, how far can you reach? (number line, integer operations, displacement)

• Activity: With sword in hand, using footwork, how far can your sword reach in each direction. Draw it. Then, overlay a coordinate system.

The Disengage

•  Activity: Keep your sword overtop of the other

• Inquiry: How do you force your opponent to draw a bigger circle than you? (geometry, angles, measurement (distance), focus on angle of partner sword & distance of disengaging opponent = both have impact)

• Inquiry: How to minimize your time making the disengage?

• Activity: What shape will minimize time spent under the blade? (geometry, graphing (parabola), optimization (shortest path))

The Yield

• Activity: At tai chi speed, partner thrusts, opponent yields (both sides)

• Inquiry: How do you maximize your safety? (optimization, leverage, body measure, arm & shoulder positional geometry (pentagon acute vs obtuse), direction of footwork)

Meyer Square Drill

• Activity: Divide the body into four quadrants

• Inquiry: What are the different permutations that you can do to cut into each quadrant. Choreograph a combat sequence with these movements. (Permutations)

• Activity: Six cuts total (Right-hand on top)

• Inquiry: Can you replicate the cuts on the other side with Left-hand on top? (reflection on y-axis of symmetry)

Fendente angles

• Activity: Partners experiment with cutting angles vs. opponent cutting angles

• Inquiry: Which cutting angle has the most leverage against your opponent? (Forces, Vectors)

Source: wikimedia commons, public domain

Annotated Bibliography

Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds. International Journal of Child-Computer Interaction, 2(1), 1-16. https://doi.org/10.1016/j.ijcci.2014.07.002

This article summarizes and synthesizes findings from studies related to perception-based and action-based lesson design and demonstrates why embodied activities like swordplay are more than a fun distraction and can lead to genuine mathematical understanding. In the appendix there is a framework for embodied design which will be a valuable resource for intentional design of lesson activities rather than mining the movements of swordplay for relevant mathematical concepts.

Alkhateeb, M. A. (2018). The Effect of Using Performance-based assessment Strategies to Tenth-Grade Students’ Achievement and Self-Efficacy in Jordan. Cypriot Journal of Educational Science, 13(4), 489-500.

In this 2018 quasi-experimental study in Al-Zarqa city, Jordan, 72 grade ten students participated in two types of assessment: traditional vs. performance-based with statistically significant results in favour of the performance-based assessment. This connects with our topic by encouraging the use of a performance-based assessment to demonstrate understanding of geometrical concepts through swordplay in order to increase student self-efficacy and performance.

Allen, J. (2025). Di Grassi Drill Book (1st ed., pp. 1–40) [Review of Di Grassi Drill Book]. https://www.dropbox.com/scl/fo/26scwli34cmwqskk2ii97/ABrp9lqJqwznYmV81ql_Sfc/di%20Grassi%20Drill%20Book%20July%20Preview.pdf?rlkey=t4s6s8fhhxlruggv8x950vsxl&e=3&dl=0

In this 2025 instructional drill book for students of historical swordplay, Allen explores concepts of movement and space using figures and diagrams from historical swordplay manuals and treatises. This drill book connects with our topic by being an excellent example of foundational concept instruction for swordplay that we could simultaneously introduce concepts of mathematics.

Anggraini, S., Setyaningrum, W., Retnawati, H., & Marsigit. (2020). How to improve critical thinking skills and spatial reasoning with augmented reality in mathematics learning? Journal of Physics: Conference Series, 1581(1). https://doi.org/10.1088/1742-6596/1581/1/012066

In this literature review, Anggraini et al. explain how to improve critical thinking skills, spatial reasoning, creativity, and collaboration with teammates through augmented reality in mathematics learning. In addition, implementing a constructivist theory in mathematics learning allows students to experiment, apply previous knowledge and experiences as strategies for new problems to test their ideas and develop new understandings. This paper connects with our topic because the advantages of working with augmented reality can be applied to using swordplay to teach mathematical ideas.

Chelak, G. (2005). Italian circle theory: A study of the applied geometry of the Italian Renaissance. In S. Hand (Ed.), SPADA II: An anthology of swordsmanship (pp. 57–76). Chivalry Bookshelf.

This article analyses primary renaissance Italy combat theory sources to make explicit connections between mathematics and Historical European Martial Arts (HEMA), considering the hypothesis that if geometry is intrinsic to swordplay, then geometry can be used to describe swordplay. A central topic is the “Italian Circle Theory” which considers the interplay of distances, angle of attack, and optimal timing, all of which are concepts taught in secondary mathematics. 

Hardianti, D., Priatna, N., & Priatna, B. A. (2017). Analysis of Geometric Thinking Students’ and Process-Guided Inquiry Learning Model. Journal of Physics: Conference Series, 895(1). https://doi.org/10.1088/1742-6596/895/1/012088

The aim of this study was to analyze students’ geometric thinking ability and theoretically examine the process-oriented guided inquiry (POGIL) model. POGIL is a theoretical model with three phases (exploration, concept discovery, and application of concepts) which align with the five phases of Van Hiele’s model potentially providing a new framework for deeper geometrical discovery and understanding. The paper provides a framework to develop our swordplay lessons in geometry.

Magnani, C., & Defrasne Ait-Said, E. (2021). Geometrical analysis of motion schemes on fencing experts from competition videos. PLOS ONE, 16(12), e0261888. https://doi.org/10.1371/journal.pone.0261888

The purpose of this 2021 study conducted in France was to use footage of fencing matches from the Rio Olympic games to create a mathematical schematic representing the movements of fencers in order to identify universal geometric and kinematic patterns. This article provides data evidence of the geometry concepts of fencing movements and supports the embodied mathematics learning activity because it demonstrates that by performing a lunge (for example), the students are physically executing a geometry concept.

Misnasanti, & Mahmudi, A. (2018). Van Hiele Thinking Level and Geometry Visual Skill towards Field Dependent-Independent Students in Junior High School—ProQuest. Journal of Physics: Conference Series, 1097(1). https://doi.org/DOI:10.1088/1742-6596/1097/1/012133

In this 2018 survey, Misnasanti & Mahmudi concluded that most students have a field dependent (FD) cognitive style which means they prefer learning in a group, require more discussion with classmates and teachers to process information, and use the teacher to guide and motivate them. This connects with our topic in creating a baseline for cognitive style in geometry in that most students are field dependent, and could seriously be lacking in geometry skills at a Van Hiele level of zero indicating that they are capable of recognizing shapes pictorially but cannot state properties or deduce shapes of objects given certain properties.

Riley, N., Mavilidi, M. F., Kennedy, S. G., Morgan, P. J., & Lubans, D. R. (2021). Dissemination of "Thinking while Moving in Maths": Implementation barriers and facilitators. Translational Journal of the ACSM, 6(1), Article e000148. https://doi.org/10.1249/TJX.0000000000000148

This 2021 study is conducted to evaluate Australian  teacher perceptions of implementing a “Thinking While Moving in Maths” program after attending a professional development session. A key takeaway that is relevant to our lesson design is that teachers found classroom management to be a real struggle, with some students just playing instead of meaningfully engaging in the learning part of the activities. This is an important consideration for our own lesson design project and will need to be carefully managed through intentional design of the swordplay lesson.

Setiadi, D. R., Mulyana, E., & Asih, E. C. M. (2019). Learning trajectory of three dimensions’ topic through analytical geometry approach. Journal of Physics: Conference Series, 1157(3). https://doi.org/10.1088/1742-6596/1157/3/032109

In this 2019 study in Indonesia, the researchers completed an observational study examining in-class teaching and learning experiences, creating a didactic framework that they applied to eight lessons: Start with a three dimensional problem, draw it in Cartesian coordinate system, then apply the concept of distance, angle or angular measures, identify the related vector, and finally determine the answer through calculation.  This study connects to our topic by providing a didactical structure and sequence to the geometry produced in three-dimensions via swordplay.

Smith, C. P. (2018). Body-based activities in secondary geometry: An analysis of learning and viewpoint. School Science and Mathematics, 118(3-4), 134–143. https://doi.org/10.1111/ssm.12279

In this study researchers compared the use of body-based geometry lessons to a control group, finding that those in the body-based groups showed learning gains compared to the control group, wrote longer descriptions that included more mathematical language, and used more first and second person perspectives compared to the control group which used more third person perspectives. The authors propose that shifting between perspectives helps students develop deeper understanding and contributes to learning gains seen. These ideas of line of sight and shifting perspectives are relevant to our topic, as students will be considering the geometries from a first person perspective, an opponent's perspective, as well as a birds-eye view when incorporating concepts of circle theory.

Suárez, Y. M. S., & González, M. M. (2025). “Curricular Integration of Mathematics and Dance to Improve Geometric Reasoning in Secondary School Students.” Revista de Gestão Social e Ambiental, 19(1), 1-41. https://doi.org/10.24857/rgsa.v19n1-102

In this 2025 study in Columbia, researchers investigated how integrating dance into mathematics education improved geometric reasoning and interest in STEAM disciplines among Columbian secondary students. Of particular value to our project, the authors highlight the similarities and differences between thinking and reasoning processes in mathematics and dance which can be applied to swordplay. The paper also provides a practical model for developing and implementing a curriculum unit that combines movement with geometry education.