About You Can Count On It! Why have a counting activity at a math night? One of the main goals of primary math education is to develop in students a sound understanding of the base ten place value numeration system. Looking at how other cultures represent base ten numerals can provide insight into the Hindu-Arabic numeral system. The counting activities in this module offer historical base ten representations from a variety of cultures, none of which involve paper and pencil. Quipu The ancient Incas used knots in ropes to record numbers. This is not unique to their civilization; knots were used in Germany and in China as well. The shape of the knot indicated the numerical value in Germany. What is interesting about the Aztec system is that it is a base ten place value system, with 0 in a place indicated by absence of a knot. The activity The materials needed for this activity are several skeins of chunky yarn in different colors, scissors to cut the yarn, and twine for the strand to which the knotted pieces of yarn are attached. A sample quipu should be made up ahead of time, and children can be encouraged to read the numbers on the strands. An explanation of how numbers are represented on the quipu is on the display panel for the module. The volunteer in charge of this activity can encourage children to think of numbers which they would like to record, and help them to place the knots on the string. The authentic knot-tying technique for making quipu looks somewhat difficult; we had children make simple overhand knots. Chinese number rods Number rods came into use in China sometime before the fourth century BCE. This system is a base ten place value system with an empty square representing zero in a place. The sticks were thin and short. Descriptions vary: some sources give a length of 9 cm and others somewhat more or less. Instructions for their use are found in several ancient Chinese texts. Chinese mathematicians were able to do sophisticated arithmetical and algebraic calculations with them. The activity The equipment needed for the Chinese number rod activity is as follows: number rods: either lollipop sticks, coffee stirring sticks, small pretzel sticks, or shoestring licorice cut into 5 or 6 cm lengths. number board from printable materials deck of number cards from printable materials Children will draw a card from the deck and make the number on the number board, using the rules on the display. We used licorice for number rods and the activity was very popular. Answers to display questions top row: 913 second row: 6033 third row: 5000 bottom row: -488 Medieval finger counting Most cultures have or have had ways of expressing numbers by body language. The finger counting described here was used throughout medieval Europe. In this system the fingers are used for numbers up to 9999, and higher numbers are represented with whole body movements. A few medieval and renaissance European math texts, including Luca Pacioli's Summa di arithmetica (15th century) and the Venerable Bede's On calculating and speaking with the fingers (8th century), give instructions on finger counting. The version presented in this display is based on Bede's instructions. The activity The panel on finger counting is mainly intended as interesting information. Children may enjoy drawing a card from the deck used for the Chinese number rods and showing the number on their hands, using the finger counting chart. Answers to display questions left: 3308 right: 275 Books to have available Many other cultures have finger counting systems. Some children's books on the history of counting would be a nice addition to the display. Here are a couple of suggestions: From One to Zero: A Universal History of Numbers by Georges Ifrah The History of Counting by Denise Schmandt-Besserat Web Resources Chisenbop tutorial by Andy Harris describes a Korean tradition of finger counting and computing. Quipu for You by Alice Storey describes how to make a quipu. References Ascher, Marcia and Ascher, Robert. (1981) Code of the Quipu: a Study in Media, Mathematics, and Culture. University of Michigan Press: Ann Arbor. Flegg, Graham. (1983) Numbers: Their History and Meaning. Andre Deutsch: London. Karpinski, Louis Charles. (1925) The History of Arithmetic Rand McNally: Chicago. Menninger, Karl. (1969) Number Words and Number Symbols: a Cultural History of Numbers. tr. Paul Broneer. MIT Press: Cambridge. O'Connor, J. J. and Robertson, E. F. "Chinese numerals." in MacTutor History of Mathematics Archive http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Chinese_numerals.html. Smith, David Eugene and Ginsburg, Jekuthiel. (1937) Numbers and Numerals: a Story Book for Young and Old. Teachers College, Columbia University: New York. Zaslavsky, Claudia. (2001) "Developing Number Sense: What Can Other Cultures Tell Us?" Teaching Children Mathematics vol. 7 no. 6 p. 312. Written 1/21/06 Janet A. Dixon