Indus Valley Civilization

The Indus Valley Civilization was an ancient civilization located in what is Pakistan and northwest India today, on the fertile flood plain of the Indus River and its vicinity. Evidence of religious practices in this area date back approximately to 5500 BCE. Farming settlements began around 4000 BCE and around 3000 BCE there appeared the first signs of urbanization. By 2600 BCE, dozens of towns and cities had been established, and between 2500 and 2000 BCE the indus valley Civilization was at its peak.

Harappa and Mohenjo Daro 2500 - 2000 B.C.E.

•      Well planned towns

•      Main and smaller streets –  cut at right angle

•      Main streets – parallel

•      Straight and wide (about 30 feet) roads

•      Curved corners – easy passage for carts

•      Paved with baked bricks

The most unique objects were square seals made of stone and engraved with symbols and animal motifs.

Paintings and Writings

•      Painted designs on pottery, different types  of clay figurines, toys, seals and ornaments.

•      Trade and exchange of raw materials, finished goods, technological knowledge and food items like precious stones, copper, sea shells, stone tools, grain, wool and livestock.

•      A highly standardized system of weights was used to control trade and also probably for collecting taxes .

The Indus Script: Seals and Writing

•      Abstract or pictographic symbols that were engraved above the animals represent the Indus form writing. The presence of writing on seals, as well as on pottery and other objects, indicates that the Indus people had developed a system for recording the names of deities, or people or materials.

•      On the average, the inscriptions are very short, consisting of about seven symbols. Scholars have not yet been able to decipher these short inscriptions and will not be able to do so until some longer texts or bi-lingual inscriptions have been discovered.

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The Egyptian Civilization || Development of Mathematics in Egyptian Civilization

The Egyptian civilization grew along the Nile River in Northeast Africa. Its Ancient civilization is being considered as one of the oldest and longest lasting human civilizations. Records as old as some 10 to 8 millennium  years old  shows that Egypt passed through all stages of development of human civilization.

                                                          Stages of Egyptian civilization                                                           

Period	Dates
Old Kingdom (The pyramid age)	C. 2649 - 2150 B.C.E.
First Intermediate Period	C. 2150 - 2030 B.C.E.
Middle Kingdom	C. 2130 - 1640 B.C.E.
Second Intermediate Period (Northern Delta Region Ruled by Asiatics)	C. 1640 - 1540 B.C.E.
New Kingdom	C. 1550 - 1070 B.C.E.
Third Intermediate Period	C. 1070 - 713 B.C.E.
Late Period (A series from foreign dynasties, including Nubian, Libyan and Persian rulers)	C. 712 - 332 B.C.E.
Ptolemaic Period (ruled by Greco-Romans)	C. 332 - 30 B.C.E.

•      Government: Egypt developed a highly centralized government. It was ruled by a monarch. The pharaoh, or king, was the most powerful person in empire and inherited his title from his father.

•      Economy: Though Egypt’s economy relied on agriculture, there were large numbers of craftsmen and warriors. Trade was prominent throughout the kingdom, and Egyptian merchants traded with other civilizations.

•      Religion: Egyptians were polytheistic and believed in an afterlife. This belief led them to preserve bodies after death in order to participate in this afterlife. Pharaohs were embalmed (mummified) and buried in tombs with valuable possessions, such as gold, jewels, and items it was thought they would need as they returned to the gods.

•      Egyptian History and People: The history of ancient Egypt is generally divided into three main periods. The Old Kingdom, The Middle Kingdom, and The New Kingdom.

•      People: Warriors, Scribes, Merchants& Craftsmen, Pharaoh, Priests & Nobles, Peasants & Slaves

Science and Technology

Surveying : Determining the position of points and the distances between them.

Geometric shapes : The Egyptian pyramids with a polygonal base and a point, called the apex and triangular faces. Cement was first invented by the Egyptians.

Some Selected Milestones in Ancient Egyptian civilization

•      3300 B.C.E. - Bronze artifacts from this period

•      3200 B.C.E. - Egyptian hieroglyphs fully developed during the First Dynasty)

•      3100 B.C.E. - Decimal system, world's earliest (confirmed) use

•      3100 B.C.E. - 3050 B.C.E. - Shipbuilding in Abydos.

•      3000 B.C.E. - Copper plumbing

•      3000 B.C.E. - Egyptian medicine

•      3000 B.C.E. - papyrus, world's earliest known paper

•       2700 B.C.E. - Surgery, world's earliest known

•      2700 B.C.E. - precision Surveying

•      2600 B.C.E. - Great Sphinx of giza, still today the world's largest single-stone statue.

•       2600s-2500 B.C.E. - Shipping expeditions:

•      2600 B.C.E. - Pyramid of Djoser, world's earliest known large-scale stone building

•      B.C.E.- great Pyramid Giza, the World's tallest structure until 1300 C.E.

•      2400 B.C.E. - Egyptian Astronomical Calendar, used even in the Middle ages for its mathematical regularity

•      B.C.E. - Alphabet, world's oldest known

•      1800 B.C.E. - Berlin Mathematical Papyrus,2nd order algebraic equations

•      1800 B.C.E. - Moscow Mathematical Papyrus, generalized formula for volume of frustum

•      1650 B.C.E. - Rhind Mathematical Papyrus: geometry, cotangent analogue, algebraic equations, arithmetic series, geometric series

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Babylonian Civilization || History of Mathematics || Development of Mathematics

One of the first river valley civilizations is the civilization developed in Mesopotamia, the fertile land between the Tigris and Euphrates Rivers around 5,000 to 6,000 years ago. The earliest people of Mesopotamia were the Sumerians. The Babylonians also excelled in the field of science. The priests watched the sun, moon planet and star very carefully and forecast the future. Like the Sumerians, the Babylonians also adopted lunar calendar. They divided one year into 12 months and each month was divided into 30 days. Early systems of writing was developed and cuneiform was used for writing stories, myths, and hymns that told of their religious beliefs. The priests also predicted future. They had acquired knowledge in geography, life science and astrology. All these things prove the love of the Babylonians for science.

The Babylonians, one of the first civilizations, existed about 4000 to 2500 years ago. They were very skilled in the arts, science and mathematics. A very intelligent people and extremely advanced for their time, they valued both mathematics and literature. It has been estimated that Babylon was the largest city in the world from 1770 to 1670 BC, and again between 612 and 320 BC. It was perhaps the first city to reach a population above 200,000.

They used sun dial and water clock to know time and prepared the first map of the Babylonian world. They also knew the use of numbers from 1 to 9.

•      The main contribution of the Sumerians and Babylonians was the development of writing with their cuneiform script, an advance that allowed record keeping and knowledge to be preserved and passed down through the generations.

•      Many of these records, preserved on clay tablets, have been discovered by archaeologists and translated, revealing information about the daily life of these ancient people

•      The Babylonian king, Hammurabi, created the first written code of laws. The Code of Hammurabi contained laws pertaining to most things that happened in daily life, including marriage, religion, consumer products, and criminal law. However, Hammurabi’s Code treated nobles and commoners differently, and some of the laws punished criminals very harshly.

•      Economy: Agriculture was the basis of the Mesopotamian economy, since farming and herding supported most other economic activities. Sumerian artisans and craftsmen made items to support Mesopotamian cities, and  the peoples of Mesopotamia traded with other civilizations as far away as India.

•      Religion. The Mesopotamians believed in as many as 2,000 different gods, making them polytheistic. Many historians believe that Mesopotamian religious beliefs were the basis of the world’s oldest faiths. Since Mesopotamian rulers were often priests, a city-state was a theocracy (a society governed by religious leaders). Amazing temples (ziggurats) were built to honor and worship the gods.

•      Society: Men in Mesopotamia were responsible for farming, herding, building, and trade. Most boys worked with their fathers. A few wealthy boys attended school where they were trained to be priests or scribes.

•      Most girls stayed at home with their mothers and learned to do household chores. Women were responsible for helping to process grain and taking care of the children and homes. Wealthy women enjoyed much greater freedom than commoners. They could go to the market to buy goods, own property, take care of legal matters in the absence of their husbands, and even seek a divorce. A few noble women enjoyed even greater privileges.

•      Innovations and Technology: The Sumerians were the first builders of cities. They made buildings from mud bricks enforced with thick reeds, since they lacked stone and lumber for building. Mesopotamian cities were protected by high, thick walls.

•      Their ziggurats (temples) were stepped pyramids that featured arches. They were usually the largest and most notable buildings in the city.

In Babylon, the king built a great palace for his wife. Known as the “Hanging Gardens of Babylon, it was considered one of the seven wonders of the ancient world because of its impressive architecture and engineering.

•      The Sumerians invented the wheel and the sailboat and created weapons and tools made of copper and bronze - a tin and copper alloy.

•      They devised a calendar by dividing the year into twelve months, as well as a number system based on 60.

•      Perhaps their greatest innovation was the development of the world’s first writing system, cuneiform. Cuneiform featured using a stylus to make marks in wet clay tablets. When the tablets dried, the writing was permanently etched into the tablets. Only priests and scribes were able to read and write. 

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The Research on Mathematics Education ||| Philosophical Views |||

    

From some decades now, research in mathematics education has been emerging as a field of study in its own right. In building theoretical frameworks specific to its domain, while still productively drawing on cognitive and instructional psychology, this field is also dedicated to the mission of examining and improving the institutions and practices of school mathematics education. Accordingly, there is increasing interest in other social science disciplines and recognition of the cultural embedding of mathematics (education). After a long struggle for its identity, today one can say that mathematics education has become a full-fledged, interdisciplinary field of research and analysis, aiming at a better understanding of the processes underlying the acquisition and development of mathematical knowledge, skills, beliefs, and attitudes applied to the design of valuable tools and powerful environments for teaching/learning mathematics.

      There are different views applying research in mathematics education, they are:

      a) The Empiricist's view

      b) Intuitionist's view

      c) The Teacher's view/Constructivist's view

      d) Alternative View's on research in mathematics education.

     

      The trends in mathematics education research is variate in different stage of time period. It depends on the world views and development of mathematics. Some trends in mathematics education research are as follows:

      a) Historical trends in math education research.

      b) Modern trends in math education research.

      c) Ethnographic research trends in math education.

      d) trends towards action research

 

      The research area in math education are related to mathematics education as well as students, teachers, society, curriculum, pedagogy etc. Generally, following areas can be taken:

      a) Research on curricula, methods and materials

      b) Research on learning and the learning

      c) Research on teaching and teacher

      d) Research on cultural mathematics

      Finally, in my personal view, research in math education is to explore the mathematical knowledge in different areas which are mentioned above. Also my interest research area is ethno-mathematics.          

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Major Issues in Education of Nepal

                                                       

National Report of Nepal (2004) states the students of the marginal and disadvantaged groups have multi-religious and cultural backgrounds. However, the “available educational opportunities cannot address the diversities”. There are several issues in Nepalese education. Mainly, it can be categories in three major issues in Nepalese education. They are as follows:

1) Different epistemological discourses.

2) Quality in education

3) Gender equality in education.

 1) Different epistemological discourses: Epistemological discourses is the back bone of the education system in every countary. Epistemological discourses is the guided principle of education system. Nepalese school level curriculum and pedagogy guided on the opposite way of Nepalese needs and reality.

• Nepalese educational discourse is influenced from the western epistemology which is not applicable for Nepalese society.
• Influenced curriculum is the main problem for Nepalese minorities and indigenous children.
• Nepalese education is mono-cultural based mainstream but multi-variate culture there.

2) Quality in education: The word quality is not specified itself, what is the criteria of the quality? Quality education focuses on equity and diversity. Many indicators reflect quality of education including investment of the government on education. However, some indicators must be meet for quality education. Following indicators can be discussed:

   a) Curriculum and learning materials:

• Nepalese curriculum is De-contextualized.
• Education planner and curriculum makers are unfamiliar to communities.
• Content and medium of instructions are mostly decided based.

    b) Medium of instruction:

• Main factor of drop out of school level child is language.
• Lack of trained and experienced teachers.
• Lack of strong physical infrastructures in the school.
• The language which use children in their home and communities is not use in the class room.
• Not proper provision for marginalized children.

3) Gender equality: Gender equality is also a major issue for Nepalese education. Following concepts and domains effect the gender equality:

• Female dominated Nepalese culture.
• Male pre-dominant society.
• Girls are treated from lower class and caste communities.

 

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Nepali Shepherds and Their Lifestyle

According to Oxford Advanced Learner’s Dictionary, A person whose job is to take care of sheep is a Shepherd. A Shepherd is a person who tends to feeds or guards sheep, especially in flocks. Shepherding is one of the oldest professions, beginning some 6,000 years ago in Asia Minor. Sheep were kept for their milk, meat and especially for their wool. Over the next millennia sheep and shepherding spread throughout Eurasia. To maintain a large herd, however, the sheep must be able to move from pasture to pasture, this required the development of a profession separate from that of the farmer. The duty of shepherd was to keep their flock intact and protect it from wolves and other predators. The shepherd was also to supervise the migration of flock and ensured they made it to market areas in time of shearing. In ancient times,  shepherds also commonly milked their sheep, and made cheese from this milk; only some shepherds still do this today.

In many societies shepherds were an important part of the economy. Unlike farmers, shepherds were often wage earners, being paid to watch the sheep of others. Shepherds also lived apart from society, being largely nomadic. It was mainly a job of solitary males without children, and new shepherds thus needed to be recruited externally. Shepherds were most often the younger sons of farming peasants who did not inherit any land. Still in other societies, each family would have a family member to shepherd its flock, often a child, youth or an elder who couldn’t help much with harder work; these shepherds were fully integrated in society.

In modern times shepherding has changed dramatically. The abolition of common lands in Europe in the eighteenth and nineteenth century moved shepherding from independent nomads to employees of massive estates. Some families in Africa and Asia have their wealth in sheep, so a young son is sent out to guard them while the rest of the family tends to other chores. Wages are higher than was the case in the past. Keeping a shepherd in constant attendance can be costly. Also, the eradication of sheep predators in parts of the world have lessened the need for shepherds. In countries like Britian, hardly breeds of sheep are frequently left alone without a shepherd for long periods of time. More productive breeds of sheep can be left in fields and moved periodically to fresh pasture when necessary. 

Typically shepherds were responsible for the sheep during the day, while a hut-keeper watched them during the nights. The sheep were taken out to graze before sunrise by the shepherd and returned them to brush-timber yards at sunset. The hut-keeper usually slept in a movable shepherd’s watch box placed near the yard in order to deter attacks on the sheep. Dogs were also often chained close by to warn of any impending danger to the sheep or shepherd by dingoes or natives. In Nepal, especially, Khas Chhetri is related to shepherding. It is believed that a huge flock of men were come from Iraq, Iran, Afghanistan and Kashmir. They migrated on north-west mountain area of Nepal, especially, Taklakot, Sinja (Jumla) etc. In that time they feed sheep and goats. They had no harited land for cultivation. Their main occupation was shepherding. It is also believed that Khas Chhetri were migrated in Nepal very before of 11^th  century. 

The mountain area of Nepal, there is less suitable soil for cultivation. So must of the people who live in that area are shepherd. From generation to generation they are shepherding that’s why it can be said that shepherding is like as a culture to them. They depend totally on their sheep. They can exchange their sheep with needed things. They can also sell their sheep and get money. Especially, they get meat, money and wool from their goats and sheep. Sheep are also used for transportation. They carry foods and other needed things through strong sheep. Sheep are the soul of the people of this region. Every household has a dozen to hundreds of sheep which are assembled into groups and taken to grazing fields and jungle. This group has sheep from five families that were mixed in the month of Jestha and will be separated in Asoj. When the sheeps are mixed one person from one household is enough to go after them but two are required when they are separated.

 Those people who are shepherd in Jumla, their whole life depends on sheep and they learn that things where they interact with jungle, hills, planet, sheep and their senior shepherd. They use their own mathematical concepts and process. So it become a great topic of research for researchers.

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