Pappus in a Modern Dynamic Geometry: An Honest Way for Deductive Proof Hee-chan Lew Korea National University of Education hclew@knue.ac.kr Abstract. This study shows that dynamic geometry using the "analysis" method systemized by the Greek mathematician Pappus in the 3rd century AD can provide a good learning environment to teach deductive proof for secondary students. Traditionally, in teaching deductive proof the axiomatic or synthesis method to deduce a new result from assumptions has been far more emphasized at the expense of the mathematical discovery process. The method systemized in Euclid s Elements is not an honest way for teaching deductive proof in that it shows only final results by mathematicians and does not help students to appreciate why and how to prove. To improve deductive proof abilities through the analysis method, a dynamic environment in which geometric figures can be easily manipulated are required for an "active justification" to find the heuristics for proof. This paper suggests four phases to solve construction problems in dynamic geometry: First is the understanding phase to recognize problem conditions and goals. Second is the analysis phase to assume what to be solved is done and to find the proof ideas by the analysis method. Third is the synthesis phase to construct a deductive proof as a reversed process of the analysis and finally, the reflection phase to reflect on the problem solving process as a whole. Deductive proof is a process used to deduce a new result from assumptions existed in the problem, axioms, what was previously proven, etc. Since the 6th century BC, it has been the flower of mathematics and marks a distinction between mathematics and other science. Euclid s Elements written in the 3rd century BC has been used as a textbook to develop students deductive proof over 2000 years. However, the axiomatic method used in Euclid s Elements has been criticized in the sense of showing none of mathematical activities as imagination, intuition, experiment, thoughtful guessing, trial and error, making mistakes, etc. (Clairaut, 1741; Lakatos, 1976; Vincent, 2005). Similarly the proof process that appears in secondary mathematics textbooks shows only the final result produced by some mathematicians and does not help students to see why and how to prove. Students have few meaning in the proof explained by teachers based on textbooks and lose confidence in constructing proof. eventually. This paper argues that the axiomatic method of Euclid s Elements and current mathematics textbooks are not honest ways for teaching deductive proof, and in order to improve students proving abilities, an active
justification to find the heuristics for proving by students themselves should be required rather than a passive justification that occurs through teacher s explanation or persuasion about the process. This paper introduces one strategy for active justification to improve students deductive proof: Analysis with dynamic geometry software. Dynamic geometry provides a good environment for students to develop deductive proof abilities when it is combined with the analysis method systemized by Pappus, who criticized Euclid s synthesis method which shares the same order as the proof process that appears in secondary geometry textbooks. Analysis and SynthesisMathematical heuristics related with proof goes back to the Greek era. In the 3rd century, the Greek mathematician Pappus systemized in his book Collection the analysis which was also emphasized by Euclid but did not appear in his book Elements. The analysis method, which is the oldest mathematics heuristics in the history of mathematics, assumes what is sought as if it were already done and inquire what it is from which this results and again what it is the antecedent cause of the latter and so on, until by so retracing the steps coming up something already know or belonging to the class of the first (Hearth, 1981, p.400). The synthesis as the reverse of the analysis take as already done that which was last arrived at in the analysis and arrives finally at the construction of what was sought by arranging in their natural order as consequences what before were antecedents and successively connecting them one with another. Greek mathematicians thought the dialectic integration of analysis and synthesis to be a substance of mathematical thought. However, Euclid s Elements considered only synthesis to reduce theorems from the foundation as a way to guarantee the truth of mathematics. As same as Euclid s Elements, current secondary geometry textbooks introduce only the synthesis. The analysis also should be introduced in order to develop students proof abilities. Design of an instructional scheme to improve deductive proof ability Dynamic geometry One problem is that the analysis method is very difficult to be applied in the paper and pencil environment because various dynamic operations such as manipulating geometric figures are required for the method. Particularly, the paper and pencil environment is worse for normal level students than for high achievement studentsit might be because of the lack of proper dynamic tools that the analysis known well by such Greek mathematicians as Plato and Euclid had not emphasized in schools since the Greek era. Dynamic geometry, which has been developed since the late 1980s, can provide a circumstance for the revived use of the analysis in that it allows students to drag and transform geometric figures.
Four phases of problem solving In this paper four phases of problem solving is suggested as an instructional scheme to improve deductive proof ability: First is the understanding phase to clearly recognize problem conditions and goals. Second is the analysis phase to assume what solving is to be done and to find the construction ideas by using the analysis. Third is the synthesis phase to construct a deductive proof as a reversed process of the analysis and finally, the reflection phase to reflect on the whole problem solving process. Problem situation There are two kinds of geometry problems: Proof problems or construction problems. There are two kinds of analysis. The first kind of analysis is to find proof process by getting a series of previous sufficient conditions of the conclusion to be proven under the assumption that what is required to be proven is already proven. The second kind of analysis is a problem solving strategy for construction problems. This is a strategy to find the construction process by extracting a series of necessary conditions from the assumption that what is required to be constructed has already constructed. In this paper, geometric problems are limited to problems of construction although the analysis method can be applied to both construction and proof Analysis & synthesis phase for a proof problem There is a triangle ABC. Make three equilateral triangles ABD, BCE, AFC by using each side of the given triangle ABC. Then prove that quadrilateral BEFD is a parallelogram The following axiomatic or synthesis proof process usually appears in secondary school textbooks: 1. ABD, BCE, AFC are equilateral triangles 2. FCA and ECB are 60 and FCB is common (DAB and FAC are 60 and FAB is common) 3. ACB = FCE and AC = FC, BC = EC (BAC =DAF and AC = AF, AB = AD) 4. ABC FEC (ABC ADF) 5. AB = FE (DF=BC) 6. BD = FE (DF=BE) 7. Quadrilateral BEFD is a parallelogram Here, thought flow 1 2 3 4 5 6 7 is a series of sufficient conditions in that 1 is a sufficient condition of 2 and 2 is a sufficient condition of 3 and and 6 is a sufficient condition of 7. However, this synthesis process does not explain to students why they have to consider FCA and ECB are 60
and FCB is common from ABD, BCE, AFC are equilateral triangles. Actually it is for showing ACB = FCE. Therefore students should consider ACB = FCE first, then they have to find the reason why ACB = FCE. The reason is ECB is 60 and FCB is common. Similarly, students should consider ABC FEC before considering ACB = FCE, AC = FC, BC = EC and AB = FE before considering ABC FEC, BD = FE before AB=FE and finally quadrilateral BEFD is a parallelogram before considering BD=FE. Thus, the most natural thought in order to prove this theorem is to assume first that quadrilateral BEFD is a parallelogram. Then to find a series of sufficient conditions 6, 5, 4, 3, 2, 1. That is, thought flow 7 6 5 4 3 2 1 is more natural than 1 2 3 4 5 6 7. Here, Pappus called this natural thought flow analysis and the reversed thought flow synthesis. Analysis & synthesis phase of a construction problem There is a circle O and there are two lines m and n which are perpendicular to each other. Construct a circle whose center is located on the line n and to which the circle O and the line m are tangent According to the textbook, the synthesis proof process is as follows (See Fig 1): 1. Draw a circle H with radius OG.. Then let F be the intersection point 2. Draw a perpendicular bisector of OF. Let P be an intersection of the perpendicular bisector and line n. Then OPF is an isosceles triangle. 3. Then, TP = PH 4. Then a circle can be drawn with the center P and the radius PH. Draw a circle P with radius PH. That is the circle to find.
Fig. 1 Here, thought flow 1 2 3 4 is a series of necessary conditions in that 1 is a necessary condition of 2 and 2 is a necessary condition of 3 and 3 is a necessary condition of 4. However, in this synthesis process students cannot understand why they have to draw the circle H with radius OG in order to obtain the circle P tangent to the given circle O and the given line m. What is important is how to find point F such that OG = HF. Students have to find the way to get point F by themselves. Similarly, they cannot appreciate why they have to draw a perpendicular bisector of OF. Students also cannot appreciate the relation between point F and Point P. Through only the analysis method, they can understand the relation between the circle H and the tangent circle P. The analysis process of the problem is as follows (See Fig. 2 and Fig. 3): 1. Assume that the circle P is constructed satisfying the given conditions. Let T be a tangent point to the circle O and H be a tangent point to line m. 2. Then TP=PH 3. Draw a circle P with radius OP and Let F be an intersection point of the line n and the circle. Then, OPF is an isosceles.4. Then OT=HF. Draw a line perpendicular to OF passing through P. Then E is a mid point of OF Of course, this analysis process is not an easy route. Students have to find a series of necessary conditions starting from line 1 by using operational activities in dynamic geometry. In this phase, students need special help from their peers and teacher through discussions or dialogues with them. And, after finishing the analysis phase, students have to go to the synthesis phase as the reversed process of the analysis to improve their deductive proof abilities.
Fig 2 Fig 3 Reflection Phase in a construction problem In the reflection phase, students look for another proof. The analysis process is as follows (See Fig 4): 1. Assume that the circle P is constructed satisfying the given conditions. Let L be a tangent point of two circles O and P. 2. Draw OP which passes through L. And, draw OQ which is perpendicular to m. And, make two isosceles triangles OQL and PHL 3. Then segments QL and LH makes one segment QH i.e. Angle QLH is 180 The synthesis proof is as follows (See also Fig 4): 1. Draw a perpendicular line to m passing through O and let Q be an intersection point of the given circle O and the line. Then draw QH and let L be an intersection point of the line and the circle O. 2. Draw a line OL and let P is the intersection point of the line OL and line n. Then LPH is an isosceles triangle. That is, LP=PH. 3. Draw a circle P with the radius PH In the reflection phase, students can check whether the construction process by the synthesis is right or wrong. If the relation among components of the figure constructed by synthesis is preserved while dragging it, the construction process can be considered as a right procedure.
Fig 4 Fig 5 The functions of dynamic geometry for the analysis method Three functions of dynamic geometry for the analysis method are as follows: First, dynamic geometry can help students draw a precise figure. To find more easily a series of necessary conditions to a final conclusion, students have to show relations among components in the problem situation. In a figure roughly drawn on the paper it is very difficult for students to find the relation. Second, dynamic geometry has a measure function. It makes students determine a good starting point for analysis by continuously measuring length or angle while dragging the point continuously. In a paper circumstance, it is very difficult for students to make a precise enough measurement to find a good starting point for analysis. Third, dynamic geometry is dynamic. It can make students perform various experiments to find necessary conditions by drawing, erasing and manipulating figures easily as well as dynamically. In a paper and pencil circumstance, it is almost impossible to perform analysis because the figure drawn on the paper cannot be manipulated. Finally, dynamic geometry is a reflective tool. If the relation among components of the picture constructed by synthesis is preserved while dragging it, the construction process can be considered as a right procedure. In a paper and pencil circumstance, there is no way to check whether the construction process is right or not. Conclusion In the late 1980s, Cabri and GSP were designed as dynamic tools for students to investigate the properties and relations within and between figures through operating figures on the computer screen directly. It is a very powerful environment in which much more activities than in traditional construction using normal compasses and rulers made possible: Construct, erase, drag and transform figures, measure segments and angles etc. A dynamic method for Euclidean geometry was proposed by Clairaut, a French mathematician of the 17th century (Laborde, 1999). However, then neither did he have the proper tool for the dynamic method. In dynamic geometry, students can make a conjecture to geometric properties and
confirm them informally and feel the need to prove the conjectured and informally confirmed geometric facts. In dynamic geometry, students can improve their proof abilities by using the analysis method. Dynamic geometry is a very excellent tool for the analysis method which is a good mathematical strategy proposed by Greek mathematicians but that has been forgotten for a long time, perhaps due to the lack of a proper tool. In mathematics textbooks, the conclusion is a conclusion. In mathematics education, the conclusion should be a starting point rather than a conclusion. Dynamic geometry can provide a better and safe route from the starting point to the development of deductive proof by normal students. References Clairaut, A.C. (1765). Elements de geometri. (translated in Korean by H.W. Chang). Lakatos, I. ( 1976). Proofs and refutation - The logic of mathematical discovery (edited by J.Worrall and E. Zahar). Cambridge University Press. (translated in Korean by J.H.Woo) Hearth, A. (1981). A history of Greek mathematics(v. 2). Dover Publishications, Inc. Laborde, J.M. (1999). Some issues raised by the development of implemented dynamic geometry as with Cabri-Geometre. Plenary Lecture at the 13 th European Conference on Computational Geometry (held in Antibes, France, INRIA, March 15-17, 1999). Polya, G. (2004). How to solve it: A new aspect of mathematical method. Princeton University Press. Vincent, J.(2005). Interactive geometry software and mechanical linkage: Scaffolding students deductive reasoning. In W.J. Masalski & Elliot, P.C.(Eds.), Technology-supported mathematics learning environments, Sixth-seventh yearbook. Reston, VA: the National Council of Teachers of mathematics.