Course reflection

I understood the importance of contextualizing mathematical concepts within cultural, historical, and artistic frameworks. This can help me make mathematics teaching more meaningful, memorable, and effective by connecting abstract concepts to students' experiences and interests. Understanding history of math is really important. Especially for exploring Non-European mathematicians' achievements. I would like to use this idea in my future teaching and make my classroom more inclusive. 

Artwork presentation reflection

After doing the presentation, I learned how M. C. Escher bridged the gap between art and science. Also his famous artworks and how mathematical concepts are used in them. And I like his "Relativity" (1953) the most. He played with perspective and gravity, presenting multiple gravitational centres in one drawing, which I found really interesting and fascinating. His tessellations demonstrate how shapes can fit together perfectly without gaps or overlaps. 

Through this presentation, I understood that art can visualize complex mathematical ideas. And Escher's ability to represent infinity is brilliant. In my own teaching, I would use Escher's works to demonstrate how mathematics can be both beautiful and creatively expressive. His works can stimulate discussions about geometry and symmetry, making them relevant for students across various age groups.

Art work

Link for our art project slide: 

https://www.canva.com/design/DAF2PmYwJtA/jY4hiddgM57T7JxabeRNRw/edit?utm_content=DAF2PmYwJtA&utm_campaign=designshare&utm_medium=link2&utm_source=sharebutton

Picture of our art project: 

Medieval Islam Mathematics

Innovative Algebraic Concepts and Techniques: Al-Khwrizm's introduction of algebraic concepts, particularly the al-jabr method, is an important moment in mathematics history. It's fascinating to think about how this ancient innovation opened the way for modern algebra. It's remarkable that a 9th-century mathematical concept still serves as the foundation for a subject critical to today's education and technological advancements.

Interplay of Mathematics and Culture: The deep integration of mathematics into Islamic art and architecture demonstrates to mathematics' cultural significance in the Islamic world. The use of geometrical patterns in art, not only for aesthetic purposes but also to reflect spiritual and philosophical ideas, provides a unique perspective on how mathematics can transcend purely scientific realms and become a part of cultural expression.

Knowledge Preservation and Enhancement During Europe's Dark Ages: The Islamic world's role in preserving and expanding on Greek and Hindu mathematical knowledge during Europe's Dark Ages is particularly striking. This emphasizes the significance of intercultural knowledge transfer and preservation. It's a powerful reminder of how civilizations can contribute to the global pool of knowledge at unexpected times and in unexpected ways.

assignment #3: Topic/Draft

The Mathematical Art of M.C. Escher

https://platonicrealms.com/minitexts/Mathematical-Art-Of-M-C-Escher

Drawing

Maya numerals

Introducing these ideas into a math class can be beneficial. It adds a historical and cultural dimension to the subject, making it more engaging and relatable. Students may find the subject more interesting when they see numbers as part of a larger narrative or cultural context.

We could introduce these concepts through storytelling, discussing the history of mathematics, and exploring different cultural perspectives on numbers. This approach can help demystify mathematics and make it more accessible and interesting to students.

In Chinese culture, numbers often have significant meanings, often based on how they sound in the language. For example, the number "8" is considered extremely lucky because it sounds similar to the word for wealth or fortune. I like to include '8' in phone numbers, addresses, and important dates.

Added: 

For storytelling I mean that I can start with stories or myths from various cultures that incorporate mathematical concepts. For example, an ancient Chinese folktale explained the significance of the number '8' and its association with wealth and fortune. These stories not only introduce mathematical concepts but also embed them in a cultural and historical context.

trivium & quadrivium

One of the most surprising revelations of the article is the extent to which medieval arithmetic was influenced by ancient Greek and Roman philosophies. The curriculum of medieval universities was not an isolated development, but a continuation and adaptation of the educational systems of ancient civilizations. The fact that these universities integrated the Greek division of logics (practical calculation) and arithmetic (philosophical study of numbers) into their studies shows a deep-rooted connection to ancient intellectual traditions. It's fascinating to see how the educational frameworks and philosophical perspectives of bygone eras continued to shape new generations of scholars in the Middle Ages.

Another striking aspect is the central role that arithmetic played in religious and everyday life during the Middle Ages. In particular, the calculation of the date of Easter, a major religious event, as a primary practical application of arithmetic, underscores how deeply mathematics was integrated with religious practice. It's interesting to think of mathematicians as key players in determining religious observances.

The shift from Roman to Hindu-Arabic numerals in Europe marked a significant change in the history of mathematics. This transition wasn't just a change in the symbols used for numbers; it represented a fundamental shift in the approach to computation and the conceptual understanding of numbers. This change is a testament to how the exchange of knowledge between different cultures can have a profound impact on intellectual development.

Dancing Euclidian Proofs: a short film and an article

Two concepts:

Embodied Learning: The idea that mathematical concepts, often viewed as abstract and intangible, can be better understood through movement is fascinating. Having university students dance Euclidean proofs on a beach brings ancient geometric concepts to life and challenges the traditional notion of mathematics. This approach resonates with the broader trend in education towards experiential and embodied learning, where students learn by doing and feeling, rather than just by listening or reading.

Humanizing Mathematics: The article introduces the concept of "humanizing mathematics". By dancing the proofs, the authors argue that they make the mathematical concepts more accessible and relatable. This idea is profound. For many students, math can seem cold and impersonal. By adding a human element: the movement of bodies in space, the authors break down barriers and make the subject more approachable.

Helpful:

Deeper understanding: Physical embodiment can lead to a deeper, more intuitive understanding of mathematical concepts. When students move their bodies to represent geometric shapes or relationships, they are likely to internalize these concepts more effectively.

Engagement: Such activities can make math classes more engaging and fun. Students who may be intimidated by abstract mathematical symbols may find it easier to relate to and enjoy the subject when it's presented as a dance.

Constraints: 

While some students may love the idea of dancing in math class, others may not want to dance.

Such activities can be time-consuming, and teachers may find it difficult to balance them with the need to cover the curriculum.

Euclid and beauty

• Why is Euclid and Euclidean geometry still studied to this day? Why do you think this book has been so important (and incredibly popular) over centuries?

Euclid's Elements provides a fundamental understanding of geometry and mathematics. It introduces basic concepts, theorems, and principles that form the basis for more advanced mathematical studies.

Euclid's clear presentation and rigorous proofs set a standard for mathematical writing. His method of presenting definitions, postulates, propositions, and proofs became a model for later mathematical works.

• Is there beauty in the Euclidean postulates, common notions and principles for proofs? How can we define beauty if these are considered beautiful?

The way Euclid presents his ideas, from basic postulates to complex theorems, shows an elegance of thought and structure. Also, the fact that these principles are universal and have stood the test of time. They represent unchanging truths in a changing world.

Revised: 

Euclid starts with very basic assumptions (postulates) and builds up to more complex theorems. This progression from simple to complex is not just a methodical approach to teaching geometry but also reflects a deeper philosophical beauty – the idea that complex truths can be derived from simple, self-evident principles. The logical structure of "The Elements" – starting with definitions, followed by postulates (assumptions), and then theorems (conclusions) – is a testament to a clear and methodical way of thinking. This clarity and structure bring a form of aesthetic pleasure, akin to the beauty one might find in a well-composed piece of music or art. The elegance of Euclid's proofs lies in their minimalism and efficiency. Each proof in "The Elements" is crafted to demonstrate a geometric truth in the most straightforward and unambiguous manner possible. This elegance is not just in the final result but in the process of reasoning itself. Euclid's work is deeply rooted in the philosophy of his time, particularly Platonism. The idea that geometry, a discipline concerned with ideal forms, can provide insights into the nature of reality itself, adds a layer of philosophical beauty to Euclidean geometry. The lasting impact of "The Elements" on numerous fields – mathematics, science, philosophy, and even art – over centuries underscores its inherent beauty. Something that remains relevant and admired over such a long period carries a timeless beauty, reflecting universal truths that resonate across ages and cultures.

In summary, the beauty in Euclid's work can be seen in its simplicity, logical structure, elegance of proofs, philosophical underpinnings, and its enduring impact on human thought. This beauty is not just in the content of the work but in the way it reflects a deeper order and harmony in the universe.

presentation reflection

For me, this presentation was not just about teaching how to prove a mathematical formula, but about instilling a sense of curiosity and exploration. It reinforced my belief that education is about understanding and applying, not memorizing. It made the learning process more interactive by presenting proofs and encouraging students to consider different cases. 

Was Pythagoras Chinese?

• Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

Yes, it does. While acknowledging non-European sources of mathematics can foster a more inclusive and holistic understanding of the subject. We will also be able to challenge the Eurocentric narrative of understanding mathematics education. We also want to give students from diverse backgrounds a sense of belonging and engagement during their studies.

• What are your thoughts about the naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle...check out its history.)

Most mathematical theorems are named after their inventors. However, the Pythagorean theorem was known in ancient India, China, and Babylon long before Pythagoras. Pascal's triangle was also known as the "Yang Hui triangle" before Blaise Pascal. It is important to know the true origin of the theorems other than the Eurocentric view.

2 visual/animated proofs of the pythagorean Thrm

 1. 

https://www.wikihow.com/Prove-the-Pythagorean-Theorem

2. 

https://www.youtube.com/watch?v=cXSfHE1ZE1k

(Presentation)Proof of the volume formula for pyramid

https://www.canva.com/design/DAFv22PPjig/r7Wj7I0dm6nyMoQUBbkSwA/view?utm_content=DAFv22PPjig&utm_campaign=designshare&utm_medium=link&utm_source=publishsharelink

Surveying in Ancient Egyptian

Questions: 

How did the ancient Egyptians ensure the accuracy of their surveying tools, given the technological limitations of their time?

Were there any surveyors in ancient Egypt who played a pivotal role in advancing the field of surveying?

Despite the lack of advanced technology, the ancient Egyptians used ingenious methods and tools to conduct surveys. This surprised me. 

Ancient Babylonian Numeration System VS. Ancient Egyptian Numeration System

The differences between the ancient Babylonian numeration system and the ancient Egyptian numeration system are:

1. Base: The Babylonian system used a base 60 number system, but the Egyptian system used a base 10 system. 

2. Symbols: The Babylonian system used a Y shape to represent the number 1 and an arrow shape to represent the number 10. Egyptian system used more different shapes to represent 1, 10, 100, and so on. The symbols are different for different powers of 10. 

Egyptian System: 

Affordances: 

1. Base 10 system can count with 10 fingers. 

2. Changing only the symbols for powers of 10 made it easy to read and understand. 

Constraints:

 Large numbers may require more time to write down the symbols. 

Babylonian System:

Affordances:

Base 60 system has many factors and easy to calculate fractions and divisions.

Constraints:

May be difficult to understand.

Ancient Egyptian multiplication example

 

Ancient Egyptian and Russian Peasant

Russian peasant and Ancient Egyptian multiplication are both based on the principles of doubling and halving. They all work with powers of two, which makes the process simpler and more efficient. 

Revised: 

Russian Peasant Method:

Doubling and Halving: This method involves writing two numbers to be multiplied at the top of two columns. One column is repeatedly halved, ignoring any fractions, and the other is doubled.

Eliminating Even Numbers: Any row where the halved number is even is eliminated.

Summation: The remaining numbers in the doubled column are summed up to get the final product.

Example:

Multiplying 9 by 13:

Halve 9 and double 13: 4 (ignore the fraction) and 26.

Halve 4 and double 26: 2 and 52.

Halve 2 and double 52: 1 and 104.

Eliminate the rows with even numbers in the halved column (4 and 2).

Add the remaining numbers in the doubled column: 13 + 104 = 117.

Ancient Egyptian Method:

Doubling and Representation: Similar to the Russian method, it involves doubling one of the numbers. However, the other number is represented in terms of powers of 2.

Selecting Representations: The method identifies which powers of 2 can be combined to represent the second number.

Summation: The corresponding values from the doubled column are added to get the product.

Example:

Multiplying 9 by 13:

Write down 1 and double it to get 2, 4, 8 (stop at or before reaching the multiplier, which is 9).

Double 13 alongside: 26, 52, 104.

Since 9 can be represented as 8 + 1, select the rows with 8 and 1.

Add the corresponding numbers in the doubled column: 104 + 13 = 117.

Similarities and Differences:

Similarity: Both methods are based on the principle of doubling and halving. They utilize the decomposition of numbers into sums of powers of 2.

Difference: The Russian peasant method involves halving one number and eliminating rows based on even numbers, while the Ancient Egyptian method focuses on representing one number as a sum of powers of 2 and selecting the corresponding rows for summation.

Equivalence: They are equivalent because both methods essentially break down the multiplication process into additions of numbers that are powers of 2. This is a fundamental principle in binary arithmetic, where numbers are represented as sums of powers of 2.

In summary, while both methods use the concept of doubling and powers of 2, the Russian peasant method is more about halving and eliminating even rows, whereas the Ancient Egyptian method is about representing a number with powers of 2 and selecting the corresponding rows for summation. 

Babylonian word problems

The Babylonian word problems are practical in nature. They deal with real world problems. These problems were deeply rooted in the daily life and social needs of the Babylonians. While the Babylonians focused on solving specific problems, they began to develop general methods for approaching similar types of problems. They worked on algebraic problems without the symbolic notation or abstract symbols that we use today.

Babylonian Algebra

The Babylonians did not use abstract variables like we do today. Instead, they asked questions about specific quantities, such as length, area, or volume. Modern mathematical symbols were not used, and their notation was more like written language. 

The article emphasizes that the Babylonians were able to solve complex mathematical problems without using the abstractions we use today. Their approaches were practical and based on actual issues, indicating that while abstraction helps modern mathematics, it is not the only strategy for solving mathematical problems.

 

For number theory, we can explore properties of numbers by listing the numbers and finding the patterns and sequences.

Revised:

Babylonian Algebra:

Babylonian mathematics was highly practical, focusing on specific problems rather than general formulas. They dealt with concrete quantities like lengths, areas, and volumes directly.

Problems were often stated as word problems. They used tables of values to aid in calculations, such as tables for multiplication.

Instead of variables like x and y, Babylonians described problems using everyday language, referring to specific quantities and objects.

Many algebraic problems were solved geometrically. For example, solving a quadratic equation was approached by manipulating areas and lengths in a geometric context.

General Mathematical Principles Before Algebra

Before the development of algebra, many mathematical concepts were expressed through geometry. For example, the Pythagoreans used geometric diagrams to represent and explore numerical relationships.

Tables were a common tool for calculation. For instance, multiplication tables helped in performing complex calculations without the need for general formulas.

Problems and their solutions were often described verbally, using everyday language. This method was more narrative and less symbolic.

Mathematical principles were often taught and understood through practical examples and real-world applications, rather than abstract theory.

Stating General or Abstract Relationships Without Algebra: 

Number Theory: Instead of using variables and equations, patterns and properties of numbers could be explored through listing and visually examining sequences of numbers, much like the way prime numbers or perfect numbers were identified.

Geometry: Geometric shapes and figures can represent mathematical concepts. For example, the concept of the Pythagorean theorem can be demonstrated using a diagram of a right triangle with squares on its sides.

Calculus: The fundamental ideas of calculus, like limits and rates of change, could be explored through geometric slopes and areas under curves.

In summary, while modern algebra provides a powerful and efficient language for expressing and solving a wide range of mathematical problems, historical approaches demonstrate that it's possible to explore and convey complex mathematical ideas through more concrete and practical means. The Babylonians, for instance, effectively handled what we now recognize as algebraic problems, but they did so using specific, real-world scenarios and a more narrative style, without the abstract symbolism that characterizes modern algebra.

Babylonian-style base 60 multiplication table for the number 45