Iolanda Guevara Secondary Institut Badalona VII Carme Burgués Faculty of Education, University of BCN 2012 NCTM: Annual Meeting & Exposition April 27, 2012 1
History of Mathematics and Education In this presentation, we propose a sequence of activities implemented by pupils of secondary education (9 th grade), based on the problems in chapter 9 of The Nine Chapters on the Mathematical Art. 2
The construction of the third square Construcció 3r quadrat 3
The proof The largest area of the square is the sum of the areas of two other squares Demostració teorema
Some examples: problem 5 Suppose we have a tree of 2 zhang as height of and 3 chi as a perimeter. A climbing plant that grows from its base surrounds the tree seven times before reaching the top. It asks how much is the length of the climbing plant. Roll up a sheet of paper forming a cylinder, simulating the trunk of the tree. Draw the climbing plant around it. Expand the sheet. 5
Construction of key figures It is about building figures, which the ancient Chinese used to infer relationships between measures of the sides of the triangle, and sums and differences between sides. 6
Problem 6 Suppose we have a square pond of 1 zhang side in the center of which there is a rod protruding from water level one chi. When you pull the rod toward the bank, only the tip reaches it. How much is the depth of the lake and the length of the rod? zang, chi, cun are units in decimal base 7
Problem 6 With visual aids, the 2nd figure a=5 c-b=1 b=? c=? In an analytic form, for example a 2 + b 2 = c 2 c b =1 c = b +1 25+ b 2 = (b +1) 2 25+ b 2 = b 2 + 2b +1 25 1= 2b 12 = b c =13 8
Problem 11 Suppose we have a single leaf door where the height exceeds the width of 6 chi 8 cun and where two opposite angles are at a distance of 1 zhang from each other. It asks how wide and high the door is. b a = 6 chi 8 cun = 68 cun c = 1 zhang = 100 cun a =? b =? 9
Problem 11 Resolució problema 11 10
Problem 12 Suppose we have a bamboo 1 zhang high and that its end, after breaking, touches the ground at a distance of 3 chi from the base. It asks how high the fracture site is. c + b =10 a=3 c =?
Problem 12 c+b =10 a=3 With visual aids, the 3rd figure 10 100 9 2c =109/10 2c
A classic book of ancient Chinese mathematics Jiu zhang suan shu (s. I dc) The Nine Chapters on the Mathematical Art or The Nine Chapters on the Mathematical Procedures
Two classical books Euclid's Elements Nine chapters Scholars belive that The Nine Chapters has been the most important mathematical source in China for the past 2000 years, comparable in significance to Euclid's Elements in the West.
The commentators of the classical text The classical text (s. I) ü The statement with specific numerical data. ü The questions. ü The answers. ü Brief description of the calculation algorithm to find solutions. The commentators Liu Hui (263) and Li Chunfeng (656): ü They provide the algorithms needed to solve the problems, and explanations of how the algorithms work. Liu Hui 220-280 15
Final remarks Visual aids Ø The visual aids were introduced by the commentators,as a tool to support the justifications of classical text. Ø The arguments used are based on the procedure of cutting and pasting and the conservation of areas. Ø The use of key figures (the first one, the second one and third) allows a demonstration mode that combines numeric and geometric reasoning. 16
Final remarks Visual aids in other context Ø The application of this resource for teaching activity expands the possibilities of understanding the students because it opens two alternative ways of demonstration and at the same time, complementary (the numerical and geometrical). Ø The resolution of the second degree equation with visual aids as do Al-Khwarizmi. Ø A project research: The use of historical context in the secondary education. The visualization in the relationship between geometry and algebra. 17
Final remarks Using history to teach math Negative numbers and zero Finding π with Archimedes method Pythagorean theorem in Euclid s Elements Systems of Linear Equations in ancient China Solving 2n degree equations, al-khwârizmî Menelaus Spherical. A construction with GeoGebra Regiomontanus and the triangles resolution http://upcbarcelona.academia.edu/iolandaguevara 18
非常感謝 Thank you!!! Moltes gràcies!!! Iolanda Guevara <iolanda.guevara@gmail.com> Carme Burgués <cburgues@ub.edu> 19