Dovie started her presentation by talking about the history of mathematics in Africa. A point was how the ability to make fast mathematical calculations in your head is learned. Dovie then had us all play the game Wari, a mathematical counting game. Afterwards, she continued with the Mayan and the Inca civilizations.
I enjoyed the game known as Wari or Mancala. I also find the name omweso in Katz. I did very well. I have been trying to play counting games as often as possible. I enjoy playing twenty-one, or black-jack. I also do not use a calculator to balance my checkbook. It is unfortunate that so little information is available about non-European civilizations.
After Dr. Barsky's commentary on Cray, the computer legend who co-founded Control Data Corporation and unfortunately had been injured in an accident, Dovie Jones's presentation began by pointing out that up unti now, most of the mathematics we've studied was domain of government officials only, and how only a selected few learned the methods. The topics covered consisted of Africa and the Americas, Timbuktu University, Mancala (also called Wari, Ayo, Kpo, and Bao Kiswahili), Mayan mathematics, the Incas, and the Quipu.
Among the topics covered, I found Mancala to be the most interesting. The game of Mancala, which also has other names based on the region, existed as a universal game among the tribes of Africa. Mancala means "to transfer" in Arabic and had the purpose of teaching children how to count. This game has been said to be the most mathematical game, in fact, a group at MIT programmed their computers to play this game to see how they performed mathematical calculations. The game consists of two rows of six cups and a large area at the end for captured pieces, which consist of seeds or ivory balls, and when playing with a total of two players, four pieces are placed in each cup to start the game. To begin, the first player can choose any bin on that player's side to empty and then place, one piece per cup, moving counterclockwise around the board excluding the opponent's side compartment and including the first player's side compartment. If your last piece lands in an empty cup on your side of the board, you can capture all the pieces in your opponent's corresponding cup. If the first player runs out of pieces, the opponent gets to use the remaining pieces. The one with the most pieces at the end wins. This seems to be a simple game, however it has a lot of mathematical considerations. As in chess, one could by assigning values to the moves, calculate the value corresponding to the best choice. It sounds silly but something like it has been done in chess. I believe that Botvitnik, the chess player and mathematician, had used mathematics to calculate some of his decisions. I find it fascinating that someone could calculate the value of the moves of a game of strategy by using mathematics. What a powerful tool!
Student lecturer Dovie Jones started her presentation with a discussion of mathematics in Africa starting around the fourteenth century, which was when the University of Timbuktu was founded. Ms. Jones said that the Africans valued texts written by Islamic mathematicians. Not much is known about African history. The African populations were mostly illiterate, so the written record is rather sparse.
Ms. Jones then introduced a game called Mancala, which can be played with a tray containing ten pockets and a handful of beads, marbles, or nearly any small objects. Ms. Jones used beans. The game is simple to learn and requires no great skill to play, making it easy even for children to play. It does, however, require strategy and logic, making it a good learning tool. Ms. Jones commented that what we have seen up until now has been that mathematics has generally been practiced and studied by the ruling class, or the elite. This is an example of mathematical skills such as logic and counting being used in a purely recreational endeavor. There is also no expensive equipment, so no one is excluded.
Ms. Jones then shifted her talk to the Americas. Here again, the written record is of little help in studying the people of the western hemisphere prior to the arrival of Columbus. Most of the records that were kept by the Mayans were destroyed by the Spaniards. But we do know that the Mayans had an extensive number system, and used it in architecture, calendars, and probably in an economic system based on bartering. Ms. Jones then talked about the Incas and a system of accounting that they used which involved tying knots in strands to represent numbers. The apparatus was called a "quipu" and was similar in appearance to a tree graph.
Dovie covered mathematics from Africa and the Americas. It would seem little is known about the ancient mathematics of these two continents. Since most of the people that lived there didn't have a written language, but relied on oral tradition, we don't have many direct written records. In Middle America, the Mayans did have a written language so there is a little more information, but it is a very difficult language to decipher. Their most complicated mathematics dealt with their calendars. The had three different calendars and had to be able to go from one to the other. The used a base 20 system in their calendars. They might also have used math for their architecture. Another American civilization was that of the Incas. They had a very interesting way of doing their accounting. The device that they used was called a quipu and it consisted of cords and knots. It was used to keep track of things like taxes owed, number of workers needed for a project, and grain records. They used different colors to represent different things. The reason they had this system was because they had to be able to carry it from one place to another easily.
We played a very interesting game in class. It is an African game with many names, the most common being Mancala or Wari. It comes in many variations also. It teaches counting, simple sums, the concept of negative numbers and problem solving. I found the game to be very challenging. It was hard to try and empty out some of the holes on your side and at the same time keep beans on your side. I kept forgetting about the second part, and letting my side run out. I think it was a lot of fun and very challenging. It looks kind of simple, but looks can be deceiving.
This presentation focused on certain mathematical aspects of pre-colonial civilizations in Africa and America. Not much is known about the mathematical knowledge of these civilizations because, with the exception of the Mayans, they did not have written languages.
The discussion began with what little is known about sub-Saharan African mathematics. Since little archeology has been done in Africa below the Sahara, there is not much physical evidence that we can study to gain insight into level of mathematical knowledge these various civilizations possessed. It is known that the Islamic culture to the north had dealings with the people of west Africa and that there was an Islamic university in Timbuktu from about fourteenth century, so it seems likely that many mathematical ideas from the north may have been absorbed. For a refreshing chance of pace, we then had a hands on demonstration. The African board game mankala was distributed to the class and we were invited to play a couple of rounds. The game was a simple but effective way to demonstrate that mathematical principles are present in every culture, whether we have written records or not. The game requires strong counting skills and critical thinking to be played effectively, and is used as a teaching tool for children in Africa.
Switching to the Americas, the Mayans and Incas of Central and South America were both sophisticated civilizations that obviously had some sophisticated mathematics. While the Mayans did have a written language, the Spanish destroyed most of what they found and very little remains of their written mathematics. It is known that they used a base twenty place-value system with numbers less than twenty being represented by groups of dots and lines. The most important use of mathematics for the Mayans was the calendar. They had a sophisticated calendrical system, but again, little is known of the computational techniques used by the Mayan priests as only the results of their computations still exist. The Inca civilization of Peru had no written language, but they used an ingenious numbering system made up of knotted cords called quipus. They were able to keep detailed records of important information by encoding numerical data onto groups of strings by system of knots, spaces, colors, and order. These were then transported back and forth between the central government and all areas of the empire. I thought she did a great job demonstrating this system, and from doing the homework it's easy to see how difficult it would be to encode all but the simplest of information.
In today's lecture, Dovie began by talking about Africa and the Americas. In the African societies, we learned that there was a university in Mali. The empire of Mali had been exposed to mathematics of Islam and valued books that came from the Islamic World. There are actually some writings that have not yet been translated to European languages. When they are eventually translated, we may know more about contributions of mathematics from Africa.
Next, we learned of a game called Mancala. This was one of many of the names and variations. This game helps children learn to count, to add, the concept of negative numbers, and problem solving and strategy. We were allowed to play the game. It was fun.
Dovie then moved her lecture on to the Americas. The Mayans did have a written language and therefore there was more known about this society. Although many of their records were destroyed by the Spanish conquerors, we do know that they had a mixed number system and that they used a 20 base system was used for calendars.
The Incas, on the other hand, did not have a written language. They counted using cords and knots of what are called "quipus" (the quipu makers were well trained). The knots are close to a base ten place value system. Dovie then showed us how to use this counting system and gave us a homework assignment using it.