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)
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| 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)
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| 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)
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| 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.
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.
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
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 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.
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.
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