Assignment 3 Presentation - Jacob and Jasmine

 This is the link to our presentation on Chinese/ Japanese abaci and its accompanying artsy puzzle.

Reflection on Assignment 2: History of Normal Distribution

My presentation on the history of the Normal Distribution was a fulfilling experience, allowing me to explore and share a topic I’ve previously avoided. My challenges with Probability and Statistics in college, exacerbated by the limitations of online instruction during the COVID-19 pandemic, had made me hesitant to revisit the subject. However, I am determined to offer my future students an engaging and meaningful experience in this area. To prepare myself, I’ve taken a different approach—exploring its history, conducting research, forming my own connections, and collaborating with others. I believe my presentation achieved two key goals: informing my classmates about the historical development of the Normal Distribution and deepening my understanding of how to approach Probability and Statistics creatively and confidently.  

Learning about the history of the Normal Distribution was particularly valuable. While the development of the concept reflects a Eurocentric perspective—dominated by British and French mathematicians—it also highlights the collaborative and iterative nature of mathematics as a discipline. The formulas and principles we use today emerged over decades, shaped by contributions from multiple thinkers. It was fascinating to discover that its origins were rooted in error analysis in astronomy and later expanded into social sciences, underscoring the interconnectedness of different fields.  

I learned a lot from the research process itself. Analyzing diverse sources, synthesizing information, and organizing it into a compelling narrative were deliberate and enriching steps. If I were to recreate this assignment for my classroom, I would scaffold the skills necessary for each stage, helping students engage with the history of mathematics while aligning the activity with mandated core competencies.  

Here’s a fun insight I gained during my research: there are contrasting views about the field of Probability and Statistics. Jordan Ellenberg, in "How Not to Be Wrong: The Power of Mathematical Thinking", describes probability and statistics as humanity’s attempt to control the randomness of the world—framing it as a response to uncertainty rather than a necessity. Conversely, astrophysicist Neil deGrasse Tyson argues that human perception is inherently biased, making probability and statistics essential for providing an objective lens. Interestingly, the Normal Distribution and Probability and Statistics didn’t emerge until the 18th century, suggesting that this method of thinking is not intuitive and was not considered essential for survival in ancient societies. A classroom discussion on this topic would invite reflection on the nature of mathematical knowledge and its role in our understanding of the world.

Response to Milner et al., 2019: Dancing Euclidean Proofs

Reading Milner et al. (2019) after watching the dance video was both surprising and surprisingly amusing—the written rationale was so much more comprehensive and thoughtful. The authors really took into account various pedagogical tools they had been studying, including the use of movement for engagement, humanizing mathematics, and promoting land-based learning and decolonization. These themes were deeply embedded in the entire process, from conceptualizing to executing the dance. They all fit together like puzzle pieces, forming a larger picture that connects mathematical concepts to our bodies, the land, and our surroundings.

In addition to encouraging movement to help students visualize and internalize mathematical concepts, the article’s emphasis on recognizing patterns in nature was particularly inspiring. Just as the golden ratio and Fibonacci sequence appear throughout nature, other patterns are also present and can be explored in similar ways. Helping students recognize these patterns can engage them while integrating various pedagogical tools. I also see a connection to environmental and social justice here, as teaching students to appreciate nature can be the first step in inspiring them to protect it.

In my practicum, I heard about a teacher candidate in Biology who incorporated movement and visualization in a lesson on the circulatory system. The school’s central area has a balcony overlooking the ground floor, so he had half the class holding balloons and running around below to mimic blood circulation, while the other half observed from the balcony. Then the groups switched, so everyone experienced both perspectives. This approach helped students engage with and retain the content while deepening their understanding.

Some constraints to consider with this approach are classroom management and ensuring all students participate comfortably. Some may feel self-conscious about dancing or other creative activities. Clear instructions and fostering a supportive, safe classroom community are essential to encourage full participation. Additionally, practical constraints like limited time, space, and resources might hinder the use of dance activities in some schools. Grounding the activity with a follow-up discussion, sketching, or writing down observations could help solidify connections between the movement and the abstract concepts, ensuring students don’t lose sight of the mathematical ideas behind the physical enactment.

Response to Gustafson (2012): Was Pythagoras Chinese?

Recognizing and teaching mathematical knowledge from diverse cultures has a profound impact in schools. Connecting to Dave Stuart Jr.'s five key beliefs about student motivation—teacher credibility, value, efficiency, effort, and belonging—it can foster in students a sense of pride and connection to the field of mathematics. By showcasing mathematics as a collaborative human endeavor that spans civilizations, students see its value as intellectually stimulating, relevant, and deeply woven into daily life.

The naming of mathematical theorems and concepts, often after Greek mathematicians, is likely a legacy of imperial colonization, where European powers imposed their worldview and intellectual contributions while overlooking rich knowledge from other cultures. For instance, as we discussed in class, Pascal’s Triangle, attributed to Blaise Pascal in the 17th century, was originally devised by Chinese mathematician Jia Xian in the 11th century. Similarly, Pythagorean triples have clear precedents in Babylonian civilization, and the Pythagorean theorem itself was rigorously proved in China by mathematician Liu Hui, known as the Gougu Theorem and documented in the "Zhoubi Suanjing". 

While it may be challenging to change globally established names, sharing with students that other civilizations independently discovered, confirmed, and contributed to mathematical knowledge can cultivate appreciation for diverse intellectual heritage. Even posing the question, “What are your thoughts about the naming of the Pythagorean Theorem and other named mathematical concepts?” can spark valuable discussions, enriching students' perspectives and honoring their cultural identities.

Response to St. Andrew's University article on Euclid

Euclid and his work in "The Elements" laid down the basic definitions, rules, and methods that continue to shape mathematics today. This book has remained significant for centuries because it builds a strong foundation for math, introducing key ideas such as definitions, common assumptions, axioms, and postulates. From these starting points, a wide range of mathematical theorems are logically developed, setting a standard for what we now consider "mathematical reasoning." "The Elements" doesn’t just compile facts; it teaches a systematic way of proving ideas step-by-step. Euclid’s approach has endured because its logical, clear structure is timeless and remains relevant in many areas of study. Acting almost like a dictionary for math, "The Elements" provided the language and approach needed for discussions and applications in mathematics and beyond.

There is undeniable beauty in Euclid’s postulates, common assumptions, and the principles he used in his proofs. The clarity and simplicity of these ideas make complex concepts accessible and easy to apply, while the universality of Euclidean geometry shows how these ideas hold true across different times and cultures. Studying "The Elements" is also mentally stimulating and rewarding, as it enhances our logical thinking and ability to follow structured arguments. Additionally, it’s likely that "The Element"* was created and refined by several mathematicians over time, showing that math is often a collaborative effort. This collective aspect of mathematics, where ideas build on each other across generations, highlights how shared knowledge and cooperation drive progress in the field.

The Dishes Puzzle

• Initially, I approached the problem by considering a small group of guests. For a group of 4, I found that there could be 6 dishes of rice (using "4 choose 2"), 4 dishes of broth (using "4 choose 3"), and 1 dish of meat (using "4 choose 4"). Adding them up gives 11 dishes, meaning each group of 4 would have 11 dishes. Since there are 65 dishes in total, I estimated there could be about 6 groups of 4, which suggests there are 24 people. However, this approach falls short because it doesn't account for guests who eat "across groups," meaning the guests can't be neatly divided into groups of 4. This led me to shift toward an algebraic approach.
• I then assumed a fixed number of guests and divided that number by 2, 3, and 4 to represent the number of rice, broth, and meat dishes, respectively. Adding them together should equal 65. Letting "a" represent the number of guests, the equation becomes a/2 + a/3 + a/4 = 65, which gives a = 60.
• Using examples, puzzles, and mathematical histories from diverse cultures can make a significant difference in engagement. This problem was especially engaging for me because I could visualize the setting—a bustling local restaurant with only a few items on the menu. I thought about how the cook might have gathered the information. With this context, I found myself more invested in the math. I also appreciated seeing a math word problem from a non-Western culture, as these are valuable resources that offer unique insights into different ways of thinking mathematically.
• The story and imagery in a word problem or puzzle are crucial. When the imagination is sparked, I explore different ways the problem can be interpreted and solved. The only insight I have is that, if I give students a puzzle with a specific answer in mind, I would let them know upfront to limit their imaginative scope and help them focus on the puzzle’s main point.

Reflection - Solving ancient puzzles in ancient and modern ways

TsáKtalay’pa and I tackled the proof of the formula for volume of a pyramid: V=1/3*base area*height, in ancient and modern ways. The original problem suggests investigating through two special cases, dividing a cube into six congruent symmetrical pyramids, and dividing it into three congruent oblique pyramids. We argued that the Ancient Egyptians likely derived the formula not from geometric dissections of a cube but through practical building methods. They likely began with a frustum for structural stability and efficiency of materials, and continued building upward to form a complete pyramid, perhaps aiming to connect with the stars. For the modern proof, the intuitions from these cases were helpful. We provided clay models for our classmates to explore the two divisions of the cube, along with paper templates to visualize the second case of three asymmetrical pyramids. We extended the discussion by introducing the Mayan pyramid, El Castillo, which resembles stacked square prisms and provided intuition for the formal proof using integration. This comparative approach enriched our understanding of the formula, blending practical, historical, and mathematical perspectives.

I learned a lot from both the process and content of this presentation. It sparked my curiosity about formulas I had previously taken for granted. Taking the time to explore and question them helped strengthen my relational understanding and appreciation of math, making me feel more prepared as a future math teacher. Understanding these concepts through ancient methods added historical context and narrative to the formulas, enriching the learning experience. I especially enjoyed the "extension" part of the presentation—not only was the research helpful, but it also encouraged me to connect and expand on different areas of my knowledge.

It was also really inspiring to see my classmates' presentations. I learned a great deal from this project, and I plan to adapt it into a class activity in the future.