ࡱ> :<9x @ ;bjbj00 4`RbRby1,8T<<,5 %+%+%+g4i4i4i4i4i4i4$6RT8r4--&<%+--4E 4G4G4G4-0g4G4-g4G4G4^G4x `k=".HG4g4405G48 0*8G4,,8G4 %++^G4,LS,B%+%+%+44,,74,,Applicable Problems in History of Mathematics; Practical Examples for the Classroom Behnaz Savizi bsavizi(at)yahoo.com Azad University, Science & Research branch, Tehran, Iran Problem and problem solving have a central role in current mathematics curricula. Problem solving has been also at the core of historical development of mathematics.[1],[3],[6] Teachers have looked for really good problems,"that is, problems whose solutions require application of certain mathematical concepts and techniques, whose contents demand a certain amount of interpretation, and presentation can capture and hold the interest of a student." [5] Among mathematics problems, those which have some applications in other branches of science and technology or the ones which have been essentially aroused from real problems in life, might be more attractive for students, since they bring life to the abstract concepts of mathematics which they learn and make them more tangible. Teaching applications in mathematics education and how to apply mathematics in real world are both essential and important. Today it seems that the process of applying mathematics in real problems is more important than teaching applied mathematics, i.e. modeling is more important than models in mathematics education [4]. According to Hans Freudethal (1973), we could say that it is not so important that students learn applied mathematics, but that should learn how to apply it. Teachers should spend more time in finding suitable answers to the question: How to teach mathematics so as to be more useful? If teachers like to teach "useful" mathematics, they have to show some instant problems in real world that mathematics has been used to solve them, and it is not sufficient to introduce such problems, the process of applying mathematics in modeling and solving problems must be analyzed in classroom by teachers. Actually it is not always possible to find practical examples for students' mathematical knowledge. Teachers may bring some examples in which statistics concepts and formulas are being applied to show that how students do the similar way in their own experiences. They also may encourage students to do some practical examples to experience the properties of analogous triangles. As a matter of fact, most of the practical problems that in which students' mathematical knowledge could be applied are so complicated that the process of applying mathematics could not be analyzed and explained for them; think about transistor simulation in electronics that needs a grate amount of knowledge in linear algebra, differential equations, numerical methods etc. Focusing on how to apply mathematics, we would better to say that teachers should put their attempts on practicing the process of modeling and solving those problems of real world that fits the classroom. If we put Freudethal's idea about teachers' task in teaching how to apply mathematics, together with Swetz' view of the properties of problems that teachers look for, we can say that teachers look for those practical examples and problems of real world: 1- that in modeling and changing them to the mathematical language a certain amount of interpretation and presentation is needed, 2- whose solutions require application of certain mathematical concepts according to students' knowledge 3-whose solution could be related to the main real problem by students and 4- presentation can capture and hold the interest of a student. In modeling the problems of real world, the content of problem may be interpreted in different ways, the interpretation may lead to a proper solution or not. The cycle of interpretation, modeling, solving and relating the solution to the problem, may happen several times, and there might be always some mistakes or misunderstandings in each of the mentioned stages. There is no limitation for choosing the mathematical concepts and procedures. But in classroom the situation is different; students should be able to interpret the content of the problem in a right way after an acceptable amount of attempts. Their interpretation should lead to (usually) a right kind of modeling which it will then lead to a right solution by the help of certain amount of techniques and concepts. (Figure 1)          Relating solution to problem Relating solution to problem by student  Figure 1 There are two main sources of such examples with aforementioned specifications which could be used in classroom, first is examples of problems of modern world and the second is historical examples, relating to past real world. It seems that in not very advanced levels of education, finding such problems in history of mathematics is easier than finding them in modern world, since in earlier stages of education, students' resources and ideas of mathematical concepts and procedures are not very rich and transferring problems of our modern world to mathematics language and then solving them, most of the time is more complicated than doing the same thing with old problems in history. The nature of problems relating to real world in past, most of the time is more tangible and understandable than modern problems for students. Considering the process of modeling and solving an old problem in history shows students the old ways and techniques of solving the problems. They may get some ideas from historical problems which could be useful for solving their own problems. They can also compare the old and new methods of solving the problems. Historical problems may be interesting for students, but another important reason for dealing with such historical problems is that through solving and considering these problems, they get a sense that for applying mathematics in our life or for solving the real problems, simple tools and techniques together with human's thought and initiative, may work better than mechanical application of several complicated methods and high amount of information. This will increase students' self confidence and cause them to believe in their own abilities as human beings. An example from history: Al-Biruni's measurement of the earth For showing how history may help us to find a suitable example relating to the real world, here we bring the method that Al-Biruni measured the earth circumstance; actually it is an elegant application of elementary trigonometry in real world.[2] Al- Biruni introduces his method,"Here is another method for the determination of the circumference of the earth. It does not require walking in deserts." Since the method assumes one knows how to determine the height of the mountain, al-Biruni first explains how to do that. The problem is nontrivial since a mountain is not a pole and therefore we cannot easily measure the distance from us to the point within the mountain where the perpendicular from its summit hits ground level. To measure the height of a mountain al-Biruni first requires that we prepare a square board ABGD whose side AB is ruled into equal divisions and which has pegs at the corner B,G. Then ,at D, we must set a ruler, ruled with the same divisions as the edge AB and free to rotate around D. It should be as long as the diagonal of the square. Set the apparatus as in Fig.2 so that the board is perpendicular to the ground and the line of sight from G to B just touches the summit of the mountain. Fix the board there and let H be the foot of the summit of the perpendicular from EMBED Equation.3 . Also, rotate the ruler around  EMBED Equation.3 until, looking along it, the mountain peak is sighted along the ruler's edge EMBED Equation.3 . Now  EMBED Equation.3 is parallel to EMBED Equation.3 ,  EMBED Equation.3 , and therefore the right triangles  EMBED Equation.3 and  EMBED Equation.3  are similar. Thus EMBED Equation.3 , and since of the four quantities in this proportion only  EMBED Equation.3 is unknown we may solve for  EMBED Equation.3 . However, since both  EMBED Equation.3 and  EMBED Equation.3  are equal to right angles it follows that EMBED Equation.3 , and thus the two right triangles  EMBED Equation.3 and  EMBED Equation.3 are similar, so that  EMBED Equation.3 This means that we may solve for the single unknown  EMBED Equation.3 , which is desired height.  E  A   D B  Z G H Figure 2 Then al-Biruni uses this method of measuring the height of a mountain to determine the circumference of the earth as following, illustrated in Fig.3: Let  EMBED Equation.3  be the radius of the earth and  EMBED Equation.3  the height of the mountain. Let  EMBED Equation.3 be a large ring whose edge is graduated in degrees and minutes, and let  EMBED Equation.3 be a rotatable ruler, along which one can sight, which runs through EMBED Equation.3 , the center of the ring. An astrolabe, would be perfectly suitable for this, using the ruler and scale of degree on the back of this instrument. Now move the ruler from a horizontal position  EMBED Equation.3 until you can see the horizon, at T, along it. The angle BEZ is called the dip angle, d. From L on the earth, imagine LO drawn so LO is tangent to the earth at L. By the law of sines applied to  EMBED Equation.3  (ELO)  EMBED Equation.3  Since the two angles, as well as EL, the height of the mountain, are known, we may determine LO; but, TO=LO, since both are tangets to a circle from a point o outside it. Also, since EL and LO are known, it follows from the Pythagorean Theorem that  EMBED Equation.3 is known, and hence ET=EO+OT is known. Again, by the law of Sines, EMBED Equation.3 Since KT, the radius of the earth, is the only unknown quantity in this proportion, we may solve for KT and so find the radius of the earth.  A H   B Z D G  Z  O G   T 4CD - 2 - Figure 3 As al-Biruni said, he tried the method on a mountain near Nandana in India where the height, EL, was 652;3,18 cubits and the dip angle was 34'.(Note the very small angles and the rather optimistically accurate height.)These give for the radius of the earth 12,803,337;2,9 cubits. Al-Biruni takes the value  EMBED Equation.3  for  EMBED Equation.3  and arrives at the value 80,478,118;30,39 cubits for the circumference of the earth, which, upon division by 360 yields the value of 55;53,15 miles/degree on a meridian of the earth.[2] References [1] Barbin,E.(1996).The role of problems in the history and teaching of mathematics. In R.CalingerO(ED.), Vita mathematica: Historical research and integration with teaching, (pp. 17-25). Washington, DC: The Mathematical Association of America. [2] Berggren, J.L. (1986). Episodes in the Mathematics of Medieval Islam, p. 141-143 [3] Ernest,p. (1998).The history of mathematics in the classroom. Mathematics in school,27(4), 26-31 [4] Humenberger H.(1997) "Applicable Mathematics" in Mathematics education, selected Results of a Viennese Research Project [5] Swetz, F. (1986). The history of mathematics as a source of classroom problems. School Science and Mathematics, 86(1), 33-38 [6] Swetz, F. (2000b).Problem solving from the history of mathematics. In V.Katz (Ed.),Using history to teach mathematics: An international perspective, (pp.59-65).Washington, DC: The Mathematical Association of America. 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