About the Tomb of Menna

The most surprising aspect of the ancient mural at Menna’s tomb is the storytelling capacity of the illustrations. I truly spent 15 minutes re-creating the scene in my mind and trying to discover all the symbolism represented.

Humans have developed technologies according to the development of the economy and society. This wall represents the vibrant commerce activity of the time. Reading Hans Barnard's document about “Surveying in Egypt” helped me to identify measurement representations like the “knotted rope” on the upper level and the angle suggestions on the lower level.

I could almost hear the noise of the place where the image shows how the men on the left offer the measurement of the wheat field plus fruits, birds represented with the three trees and fish, which is carried by one of the characters with a lower hierarchy (if smaller size represents lower category). Encountering these men, there are the people on the right who are offering some grains and animals.

The two questions I have are:

-          Do they represent actual quantities in this mural? You can estimate the number of knots on the rope and the grains inside the containers. Are these containers representing quantities, too? Like weight or volume?

-          The other question is also regarding the symbols and the measurement. What does it mean that in the bottom image, 3 men on the left of the grains, holding 3 white devices. 4 men on the bottom holding 4 devices, and 5 men holding 5 devices, as the men on the left. Does this relate to the 4-3-5 proportions?

-          The last question is about the old man in the bottom right side. What is he doing? He looks older because he is depicted with grey hair and a little belly.

In conclusion, this image represents a complex system of measurements that includes: distances, angles and weights. Nevertheless, it shows only the exchange of commodities; it doesn’t show any symbol that represents money as an element of exchange.

Babylonian Algebra

 Babylonian Algebra.

The reason that Babylonians could formulate mathematical principles without the development of algebra or algebraic notation is because of the practical uses of math allowed them to create hypothetical situations, problems and solutions based on their reality.

Babylonians and other ancient civilizations developed mathematical concepts in response to the need to measure land, register wealth, and track transactions. This practical understanding is derived from the ability to envision scenarios where math can be applied. But this practical usage limited their imagination. The fact that almost all the calculations were made based on real quantities prevents them from thinking about concepts like zero or negative numbers.

When the Babylonians were solving problems, the method didn’t use a second-order polynomial but several versions of simplified quadratic equations, replacing higher orders with combinations of powers of two.

Answering the question of whether mathematics is all about generalization and abstraction, I think mathematics started as a tool for solving problems, and once the problems are solved, the look back of the method inspired the generalization and led to the invitation to play with abstraction

Homework 2 numbers that multiply 45 with base 60

 

The Art of Mathematics

This chapter from "A man left Albuquerque Heading East" written by our teacher confirmed for me the reason why I am always trying to promote Math as an art expression.

My argument for this relationship between art and Math was based on the unique human capacity of creating and appreciating artifacts for the sake of admire them and feel joy during these two stages: either creation or admiration of a piece of art.

The reading illustrates the diverse origins of word problems and their permanence along the civilization evolution. Even with the contrast between the method oriented of Babylonians and the problem-solving of Greeks, the problems accompanied men in their journey of exploring the world and their minds.

The word problems are like pieces of art, like sculpturea made with abstract concepts that connect quantities with reality, this is amazing!! 

I felt also a little intimidating because I've never solved any without help, but after understanding the beauty and joy on the process of trying to solve them, I feel the satisfaction of belonging to this tribe of humans who appreciate the connection with nature through the formulation of problems just because our mind can play around with them.

I want to share this joy with my students as one of the reasons to learn math and try to eliminate the stigmatized vision of the need to learn only practical knowledge.

If we are able to appreciate math we expand our capacity of feeling happiness.

The Arbitrary Way of Measuring Time.

 

 The Arbitrary Way of Measuring Time.

After reading the articles, I am surprised at how recent the arbitrary way of measuring time is. How can we be subjugated to such a mysterious concept? Even with all the theories surrounding the origin of the sexagesimal system, it is based on the observation of nature that we came up with the system. And it is even more surprising that we based our system on our limited appreciation of reality throughout our senses, like the case of measuring according to the number of joints or fingers we have.

After trying to answer the origins of the sexagesimal system and the correlation between the way we measure time, it occurs to me that we could have also used the growth of giant bamboo from Colombia to measure time. (This plant grows 21 cm per day and has horizontal marks that could help to track the growth.)

The beauty of the time measurement system is the number of patterns related to the 60 number. Today we know that the human brain is a machine always looking for patterns*, and this nature can explain why the civilization selects the 60 based on the frequency of coincidences around 60 and its divisors.

 

* https://www.edge.org/response-detail/11498

 

Alternative Perspectives for the History of Mathematics

 

The Crest of the Peacock

Alternative Perspectives for the History of Mathematics

History is written for those who won the war. The English version of Math history is widely known and very popular.

After reading the document, I was excited about how we can utilize the different paths that humanity has taken to arrive at similar mathematical concepts and tools, thereby igniting curiosity and imagination among our students. Especially nowadays, when we will face classrooms full of diverse backgrounds.

The evidence of the development of concepts of Zero in different civilizations, such as the Mayans and Greeks, which were completely isolated from each other, is proof of how math is an intrinsic need that inhabits human nature. The need to solve problems and the curiosity to build tools are current aspects of human development that can help us convey the concerns of our ancestors to our students in the classroom. As teachers, we can illustrate how there are several methods to approach a problem and different perspectives to understand and solve it.

Another aspect of interest is the collaboration between men who lived in different generations. The evolution of the word “Algorithm” shows how we build on top of previous discoveries, once again, identifying history as a tool for research and expanding our analysis. This part reminds me of the book “The Innovators” by Walter Isaacson, which describes the Digital Revolution as a collection of collaborative efforts from diverse cultures and times until the first computer.

Sometimes kids are afraid of math because they think it has been invented by geniuses for geniuses. This reading provides a wide range of possibilities during a long journey where we can find snippets of history to connect with specific lessons in the curriculum. For example, the need for irrational numbers or the algorithm to solve linear equations.

Why was the sexagesimal system used and why does it still persist today?

 

Without going into further research, I remember learning about how ancient cultures based their calculations on their observations of nature, specifically the sky and the cycles around the sun and the moon. I understand that the 60 comes from fractions of a cycle of 360 days around the sun (that was adjusted in some moment to our 365 days), which also matches the 360 degrees of the circumference path.

EDIT-EXPANDING :

The Sexagesimal System: Origin and Persistence

Three main reasons :

1. The Mathematical meaning of 60

    The primary reason the number 60 was chosen is its unique status as a highly composite number, which means it has more divisors than any smaller positive integer.

Factors of 60:	1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

    The fact that 60 is divisible by the first six counting numbers (1, 2, 3, 4, 5, 6) and by 10, 12, 15, 20, and 30 made calculating fractions simple and accurate.

• In a base-10 system, dividing 1 by 3 results in a repeating decimal (0.3333).

• In the sexagesimal system, dividing 1 by 3 is exactly 20.

This ease of division was crucial for the Babylonians' trade and land division.

2. The Cultural and Historical Trascendance

• Geometry and Astronomy: The Greek astronomer Hipparchus (2nd century BC) and Ptolemy (2nd century AD) adopted the sexagesimal system for their astronomical tables and geometric works. They used it to divide the circle into 360 degrees and further subdivide the degrees into minutes and seconds. This practice was inherited from Babylonian positional astronomy, which connected the approximated 360-day year to the circle's rotation.

• Timekeeping: Through the works of Greek, Indian, and particularly Islamic scholars, the system was integrated into the measurement of time. The division of the hour into 60 minutes and the minute into 60 seconds is a legacy of this transmission, establishing the sexagesimal system as the standard for timekeeping worldwide.

3. Why It Still Persists Today? 

The sexagesimal system has survived because it was integrated into standardized measurements that are difficult to change:

• Time: The structure of time (60 seconds, 60 minutes) is universally recognized and built into every clock and calendar system.

• Angles and Navigation: All international standards for geometry, cartography, and navigation rely on the 360-degree circle and its sexagesimal subdivisions (degrees, minutes, and seconds of arc, or D-M-S).

In conclusion, the sexagesimal system was chosen by the Babylonians for its divisibility, and it persists today not only because of a direct connection to nature, but because Greek and later civilizations adopted it for astronomy and geometry, solidifying its use in the standards of time and angle measurement.

 

Albert Durer. Melancolia

Integrating History of Mathematics in the Classroom

Before reading the chapter, I had already thought about introducing history in Math class to give context to the concept that needs to be learned. I am a member of 3blue1brown.com, a website that illustrates math and shares snippets of history about essential math concepts and problems. I realized that knowing about the original question that motivated past generations to think, research and develop math concepts and notations, increased my curiosity about exploring the different approaches and perspectives of those concepts, and increased my patience to try to understand them.

            Here are the things that made me stop, highlight and think:

·        Learning about math history shows us that failing is part of the math learning process and helps us to understand why some abstract concepts took time to be discovered, and consequently, could be difficult for us and for our students to understand in the first place.

·        Math history is full of collaborative efforts and passion, which brings balance to the curriculum of practical knowledge, and introduces the explanation of the need for proof, rigour and evidence.

·        I was delighted when I found an example about the origins of the “quaternion concept” because I’ve been using this frequently in my career in visual effects for film and television. I used it to create organic rotations and procedural modelling with different computer graphics software, and I remember my need to revisit the concept explanation, looking for videos that helped me to understand such a beautiful tool, but I never looked for the origins of the concept.

Now I am more than convinced and motivated to integrate the math history as much as possible, regardless of the time it takes to achieve it.