Math History Final Project: Women in Computer Science

 Link to presentation slides

The Marshall Islands and Embodied Mathematics

The most interesting part of this article was the idea of maps as mathematical abstractions and analogical spaces. I find this interesting because of mapping’s connection to comics and graphic novels; artists and writers Scott McCloud and Dylan Horrocks have described comics as maps of time and space. Prior to starting the B.Ed program at UBC, I was finishing up my last English requirement where I wrote my final paper on Dylan Horrock’s graphic novel Hicksville. Written in the 1990s, Hicksville is the semibiographical work of the Pakeha New Zealander navigating time, pre and post-colonial New Zealand, and space, Aotearoa, or New Zealand, in relation to the world. Despite being a fan of Randall Munroe of XKCD and Ben Orlin of Math with Bad Drawings, I had failed to make a deeper connection between mathematics and comic form. 

I think the significance of embodied mathematics in relation to the history of mathematics is shown in the origins of what we call mathematics today. Mathematics came into existence from the world as a way for us to communicate with each other and represent ideas, concepts, and knowledge. Math teachers often feel the need to cover the prescribed curricular content that students’ understanding of mathematics is mostly procedural and they lack a conceptual understanding of mathematics. From experience, the fast-paced coverage of material often leads to a low level of knowledge retention, so the teacher’s efforts are in vain. Luckily for us as teachers in BC, our Ministry of Education has recently shifted the emphasis of the curriculum, making content a vehicle to drive competencies. In our current situation, it makes it easier for us as math educators to have students focus on the process of contextualization, discovery and embodying of mathematics. Hopefully, this facilitates a higher level of retention and a deeper understanding of mathematics. 

In teaching secondary mathematics, embodied mathematics can take the form of a relational learning approach to conceptualizing pi of the primary trigonometric ratios of sine, cosine, and tangent through measurement of varying-sized circular and right-triangular objects, teaching operations with fractions through coins, making clinometers for indirect measurements of tall objects, or seeing quadratic behaviour through collecting and modelling height and distance data of a thrown ball or launched pneumatic rocket. I don’t think it’s a lack of creativity or ingenuity of the teacher that results in a lack of embodied mathematics in high school classes. From my experience, teachers often reserve experiences of embodied mathematics for their honours classes since they can’t get through content faster and are afforded these richer learning opportunities. Additionally, institutional norms and pressure from students, parents, other teachers, and administration to prepare students for written tests is another major factor that we do not see more of this in high school classrooms. 

Math Art Project (Check-in)

 Topic: Women in computer science or a woman in computer science (Lovelace, Hopper, or Hamilton)  

Artistic Form: Interactive 3D display  

Reference List: 

• https://dl.acm.org/doi/abs/10.1145/543812.543853

• https://www.proquest.com/openview/698bc6cc298c3e4918e0f9de3ad2f461/1?pq-origsite=gscholar&cbl=37905

• https://ieeexplore.ieee.org/abstract/document/1702333?casa_token=Uj1sBm7PiQAAAAAA:TXoTNoCakL_UvDySV8qL9d4Jz2eE_GID4leOkT945mlkBH_I4nNBE629Q30rrQX25E5_UfjS

• https://dl.acm.org/doi/abs/10.1145/609784.609818

• https://dl.acm.org/doi/pdf/10.1145/129852.214846

• https://ieeexplore.ieee.org/abstract/document/4051189?casa_token=UQ-orSzi9DIAAAAA:IRE1Rgln3I4mxby5tUTaRqEizA3ws8-TswYnl7NuCOeJAlXUKss29E7CRrzr9rXsYVk5bFYY

Education in Medieval Europe

    The first quote that made me stop and think was “The very word ‘liberal’ implies that these arts belonged to the education of free men, not to the technological training of slaves” (Schrader, p.264). I was reminded of my privilege in receiving an education. I had assumed that liberal referred to free thought and not to people who are free. This is an interesting take to me because that kind of access to education is prevalent, to a lesser degree, in most countries around the world. Liberal arts education is available to you if you can afford it and are not overburdened by other responsibilities. I think it speaks to the inequity in our societies are continue to make higher education inaccessible. 

    The second quote that made me stop and think was “Those who did go on into law or medicine did so for profit” (Schrader, p.271). I find this quote rather amusing since growing up in a Chinese community, this was definitely a pervasive belief, to the point where parents only viewed school and education as a means for future profit. I have a lot of respect for people in law and medicine, but when I hear about dentists performing procedures that their patients don’t require, and billing the insurance companies, I can’t help but think that they’re grifters. Existing in a capitalist society makes it difficult for any to pursue education purely for the sake of education and knowledge and I wish that were not the case.  

    The third quote that made me stop and think was “There were no examinations in the modern sense of the term. The student had simply to swear that he had read the books prescribed and attended the lectures. To qualify for a degree, he was required to participate in public disputations, either defending a proposition or opposing one defended by another student” (Schrader, p.272). This brings up important questions regarding assessment. At some point, possibly when schooling became compulsory, the purpose of exams became weaponized as an accountability tool in addition to determining students’ understanding. What’s interesting is that the alternative to the exam seems like what could be considered a thesis defence. One of the benefits of a system like this includes allowing adequate time for students to think about ideas and synthesize something meaningful, showing a higher level of understanding according to Bloom’s Taxonomy. What I also find interesting is how this also adheres to the modern idea of authentic assessment, in particular the need to defend your stance. 

Schrader, D. V. (1967). THE ARITHMETIC OF THE MEDIEVAL UNIVERSITIES. The Mathematics Teacher, 60(3), 264–278. http://www.jstor.org/stable/27957550

Here's Looking at Euclid

    I believe that Edna St. Vincent Millay’s poem describes Euclid as an unparalleled genius who transcended the mediocrity of the general population to create something original; a new way of seeing the math world. The allegorical poem seems to compare Euclid to Jesus in that sense. In regards to Beauty, I think Millay is referring to it as something with great importance since she chooses to capitalize the B, making it a proper noun. She suggests that it’s subjective for everyone except for Euclid who is able to anatomize Beauty and see her bare. Euclid is able to see objective truth and does the world a service by showing Beauty to others. 

    I think Euclidean geometry has been so popular over the centuries partly due to how well 

organized and visual the math is in Euclid’s Elements and partly due to the mystery and intrigue behind Euclid. Euclidean geometry’s foundational building blocks only rely on five postulates, none of which are incredibly complicated. It starts with a point, line, and circle which makes it incredibly accessible for students starting out in math. The way that the book progresses, it becomes increasingly obvious how similar math is to art. It’s the construction of patterns and structures using basic building blocks. Euclid’s Elements is an exemplar of good mathematics that continues to inspire students. 

    My experience of learning Euclidean geometry has unfortunately not been as inspiring as my description above. I learned it mostly through lecture-style delivery. There was no discovery or creation. It focussed mostly on exercises that prepared us for a test. I was luckier than some of my peers because I had better short-term memory, at the time than others so I did well on tests, however, I would say that geometry was and still is one of my weaker areas in mathematics. The current BC curriculum gives teachers a lot more flexibility in how they teach, and I believe building a strong foundation of Euclidean geometry through exploration, discovery, and synthesis would be beneficial to students.

Embodying Mathematics

The first thing that made me stop and think was the concept of embodying mathematical proofs. My worldview is highly influenced by mathematics, but it has been more passive and focused on observations so it’s quite inspiring to see a more active approach to seeing people, math, and their connection to the world. Related to the first thing, the second thing that made me stop and think was the idea of human agents becoming the mathematics we do. I find the idea of instrumentalizing our bodies for the pursuit of math to be quite provocative. One does not simply do math, but they become math and become part of the canon of the development of mathematics. 

The dancing Euclidean proofs activity in class was quite interesting. Since we needed to choreograph the proof, there was additional care in deciding the sequence in which we would perform each step. This was evident when watching the other groups rehearse and perform. Our group decided to perform a rap, which similarly required careful sequencing. However, where other groups felt the math with their bodies, we engaged with the proof using another sense, hearing. 

This kind of activity may be helpful for mathematics learning and understanding mathematics history in a secondary school context such that it contextualizes the mathematics students are learning. Math is no longer decontextualized numbers on a page and becomes something that you experience or something you become. As one is doing and thinking about such an activity, it also provides students with some reprieve from worksheets and gives them an opportunity to bridge the empathy gap between themselves and the many contributors to mathematics. Regarding constraints and challenges, the biggest one that comes to mind is the cultural hegemony and the pervasive idea that mathematics is answering carefully designed questions. There will be pushback from parents and students who claim that we’re not teaching math or that it’s not rigorous enough. Another concern will be from your colleagues who question your practices and worry that you won’t cover enough content to adequately prepare students for further math courses.  

His Name was Liu Hui

Speaking from personal experience as a non-white, whose education in Canada was mostly Eurocentric, I think it makes a difference when teachers acknowledge non-European sources of mathematics because representation matters. It is a political choice to not acknowledge it and perpetuates an image of superiority and cultural hegemony. Research supports the adoption of culturally sustaining pedagogy as helpful for all students, however, we need to do this in a way where we are being respectful and celebrating achievement and not tokenizing or fetishizing the “other”. Even in this week’s paper, Gustafson repeatedly misspells Liu Hui’s name. This does not create a classroom environment where students can feel safe to be their authentic selves. 

The inclusion of non-European sources of math can also help students get a more holistic perspective on how math is developed and also view math in relation to the world outside the classroom. We can draw from a wider range of examples of ingenuity and creative problem-solving in real-world applications across multiple cultures. Acknowledging non-European sources of math also allows for the integration of social studies with math, which are two subjects that are often isolated from each other except by the means of statistics, by using the development of math as the lens to view cultures and civilizations.  

As for the naming of the Pythagorean Theorem and other named theorems and concepts, I view them as relics from antiquity which perpetuate cultural hegemony and institutional racism. There is insufficient evidence of Pythagoras discovering the concept or being the first to provide proof, so having his name associated with something that was used in cultures outside of Greece is an issue given the European dominance in math education. Also, from a practical perspective, calling it the Pythagorean Theorem makes less sense than calling it the RIght Triangle Theorem. The Right Triangle Theorem is more descriptive and is able to help students associate it with right triangles. Lastly, named theorems are a form of hero worship that I think is unjustified. Mathematical ideas and concepts are not things that people develop independently. There is a high degree of collaboration and I believe the naming of theorems should reflect as such. In general, I think we should be celebrating the idea and not a singular person.

Gustafson, R. (2012). Was Pythagoras Chinese- Revisiting an Old Debate. The Mathematics Enthusiast, 9(1-2), 207–220. https://doi.org/10.54870/1551-3440.1241