International Journal of Mathematical Education in Science and Technology ISSN: 0020-739X (Print) 1464-5211 (Online) Journal homepage: http://www.tandfonline.com/loi/tmes20 Various solution methods, accompanied by dynamic investigation, for the same problem as a means for enriching the mathematical toolbox Victor Oxman & Moshe Stupel To cite this article: Victor Oxman & Moshe Stupel (2017): Various solution methods, accompanied by dynamic investigation, for the same problem as a means for enriching the mathematical toolbox, International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2017.1392050 To link to this article: http://dx.doi.org/10.1080/0020739X.2017.1392050 Published online: 27 Oct 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tmes20 Download by: [Chalmers University of Technology] Date: 29 October 2017, At: 00:31 INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY, https://doi.org/./X.. Various solution methods, accompanied by dynamic investigation, for the same problem as a means for enriching the mathematical toolbox Victor Oxmana,b and Moshe Stupelb,c Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 a Western Galilee College, Acre, Israel; b Shaanan College, Haifa, Israel; c Gordon College, Haifa, Israel ABSTRACT ARTICLE HISTORY A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various branches of mathematics and to highlight the importance of providing the students with an extensive ‘mathematical toolbox’. Investigation of the property that appears in the task was carried out using a computerized tool. Received January KEYWORDS Conservation property; multiple solutions/proofs; geometric investigation using dynamic software 1. Introduction The methods of teaching and instruction in mathematics are derived from the programme of studies and its goals. One of the principal goals of the studies of mathematics is to provide the students with methods of reasoning that may assist them in other fields of study and knowledge, and to develop such methods. The meaning of ‘learning to think’ is: the teacher of mathematics should develop the ability of students to apply information, perform analysis and synthesis at an adequate level, of basic properties, rules and theorems that they studied at an earlier stage of the teaching process. Solving different problems is one of the important means aimed at developing reasoning. During their solution of problems in the class, the students are asked guidance questions, such as: ‘can the conclusion of the previous exercise be implemented in solving the present problem?’, ‘Is the problem a particular case of the previous problem?’, ‘Did you encounter a problem of this type before, where?’, ‘Is the wording of the question similar?’. The purpose of such questions is to stimulate and expand the field of reasoning and to prepare the student for independent efforts, where he himself will have to ask these questions and answer them. Impetus for developing reasoning is obtained from solving problems using different methods. By using methods of solution and knowledge from other fields of mathematics, one can obtain simpler and faster solutions. Finding another method of solution by using the same mathematical field, and especially from another field, contributes to the development of reasoning and raises it to a higher level. CONTACT Victor Oxman email@example.com, VictorO@wgalil.ac.il © Informa UK Limited, trading as Taylor & Francis Group Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 2 V. OXMAN AND M. STUPEL Implementation of previous knowledge in a new situation that results in a shorter and simpler solution or a more beautiful solution increases enjoyment and satisfaction from studies of the subject. Integration of fields in problem solution gives the students a wider outlook to mathematics as a comprehensive discipline, while creating connections between its different branches. Successful development of reasoning of students and solution of diverse problems in mathematics depend very much on the teacher’s ability to create and elicit appropriate stimuli in the teaching and instruction process, while accentuating the beauty of mathematics. Solving a problem by a regular method usually leaves the students indifferent and without any particular reaction. However, different solutions of the same problem may cause emotional enthusiasm. The special and the beautiful solution is unexpected, but tangible, and in most cases – simpler and shorter. The ordinary methods are usually technical solutions and they are obtained almost automatically. On the other hand, unorthodox methods permit new initiatives and develop additional skills. In many ways, this is similar to the difference between an algebraic method and solution by trial and error of textual problems. The use of alternative methods highlights mathematics as a magical field of surprising solutions. Indeed, in recent times, one can see from several papers in this field that there is an increasing tendency to make use of alternative methods to develop excellence and creativity and to present mathematics as a specialty composed of a variety of intertwined fields. Mathematics educators agree that linking mathematical ideas by using more than one approach to solving the same problem (e.g. proving the same statement) is an essential element in the development of mathematical reasoning [1–3]. Problem-solving in different ways requires and develops mathematical knowledge , and encourages flexibility and creativity in the individual’s mathematical thinking [4–7]. In addition to the specific roles of proof in mathematics, we suggest that attempts to also prove a certain result (or solve a problem) using methods from several other different areas of mathematics (geometry, trigonometry, analytic geometry, vectors, complex numbers, etc.) are very important in developing deeper mathematical understanding, creativity and appreciating the value of argumentation and proof in learning different topics of mathematics. Our approach, that of presenting multiple proofs to the same problem, as a device for constructing mathematical connections is supported by [1,2,8–11]. Very similar to our notion of ‘One problem, multiple solutions/proofs’ is the idea of multiple solution tasks (MSTs) presented by [7,11,12]. MSTs contain an explicit requirement for proving a statement in multiple ways. Liekin  indicates that the differences between the proofs are based on using (1) different representations of a mathematical concept; (2) different properties (definitions or theorems) of mathematical concepts from a particular mathematical topic; (3) different mathematics tools and theorems from different branches of mathematics; or (4) different tools and theorems from different subjects (not necessarily mathematics). In our case, we apply the third-type of differences between the proofs; we shall present various solutions to a problem using the tools and theorems of Euclidean geometry, analytic geometry, trigonometry and vectors. Adding the concept of multiple solutions/proofs for one problem into the curriculum of mathematics studies, as well as MSTs, allows the development of connected mathematical knowledge not only for students, but for their teachers as well. Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 3 Based on many years of experience in teaching mathematics in high school, in academic courses and from the process of constructing and training pre-service teachers, we have established the major importance of possessing comprehensive command of different mathematical tools, which permit one to deal with complex problems and with challenges that require the use of various solution strategies. Of all branches of mathematics, geometry of the plane is one of the most spectacular fields, due to the large number of methods of solution attainable in it. As a field based on axioms and postulates from which theorems develop in a modular fashion, and in which use is made of auxiliary constructions and alternative constructions, one can find in it many methods of solution for the same task, some of which are standard and without any peculiarities – on the one hand, and where other, unorthodox, very short and impressive methods exist – on the other hand. During the checking of final exams in mathematics of hundreds of high-school graduates, different methods of solution were found to the same questions, in particular questions in geometry of the plane. These, both, point to the fact that different teachers teach using various methods and ways, as well as different emphases with several mathematical tools, and to the fact that the students could implement them in different forms. In the framework of a course in a combination of subjects taught to pre-service teachers in mathematics, at the end of the school year, we presented a geometric task to a group of 20 students, which they were asked to solve using a large number of methods. It was suggested for them to solve it using: geometry of the plane, trigonometry, analytical geometry, vectors and complex numbers. Each of them was required to solve it independently, and to try to find several methods. Indeed, 10 proofs were found, 6 of which were completely different and used different mathematical tools. After presenting the problem, we shall give some of the solutions found by the students. 2. Technology in the classroom: the dynamic geometry environment As is the custom in the modern world which is full of technological tools, today it is impossible to ignore the rapid development of technology and the way it affects almost every facet of life. The education system is no exception, and one certainly cannot disregard the value of technology in teaching mathematics. Empirical/inductive arguments are the traditional methods that are usually used as the first steps for deriving proofs of geometrical problems. However, the advent of dynamic geometry environment (DGE) software serves as an intermediary tool that bridges the gap between a mathematical problem or concept and its symbolic proof by providing a clear, visual representation of the equation involved. It is well established today that DGE has opened new frontiers by linking informal argumentation with formal proof [13,14]. Several researchers conducted extensive exploration into students’ behaviour when using DGE software and its efficacy in connecting the processes of producing conjectures and proving theorems or statements . Introducing DGE software (such as GeoGebra) into classrooms creates a challenge to the praxis of theorem acquisition and deductive proof in the study and teaching of Euclidean geometry. Students/learners can experiment using different dragging modalities on geometrical objects that they construct, and consequently they can infer properties, 4 V. OXMAN AND M. STUPEL Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 Figure . Presentation of the problem: x + y = z. generalities and conjectures about the geometrical artifacts. The dragging operation on a geometrical object enables students to apprehend a whole class of objects in which the conjectured attribute is invariant, and hence, the students can become convinced that their conjecture will always be true . Nevertheless, because of the inductive nature of the DGE, we entitle this process ‘semi proof’. Hence, following the employment of DGE, the experimental–theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical and epistemological concern [17,18]. Students must be aware that they still need to prove rather than rely on the virtual experiment. A link to GeoGebra applet is given which allows the reader to have a dynamic experience of investigating the property. 2.1. Presentation of the problem Given is a square ABCD whose side length is a. A straight line is drawn through the vertex A, which intersects the side DC at the point E. The bisector of the angle ∠EAB is drawn through the vertex A, which intersects the side BC at the point F. See Figure 1. Prove that BF + DE = AE i.e. x + y = z . At the first stage, the students were given an applet that allowed the point E to be dragged along the side BC, including the possibility of dragging the point outside the square to the right of the vertex C. During each stage of dragging, the lengths of the segments and the sum of the relevant segment lengths appear on the screen. The experience with the applet shows that at each stage the properties are conserved. It is possible to reach the applet using the following link: Link: https://www.geogebra.org/m/VnpDPzg4 We denote : ∠EAB = 2α, BF = x, DE = y, AE = z. INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY D y 5 C E G a x z F M x α α A B N Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 Figure . Proof using calculating areas. 2.2. Method A – proof using calculating areas From point F, we drop a perpendicular FG to the segment AE (see Figure 2). From the property of a point on an angle bisector, we have BF = FG = x. We calculate the area of the square ABCD from the areas of the four triangles from which it is composed: SABCD = S ABF + S FCE + S EFA + S ADE , a · x (a − x)(a − y) z · x a · y + + + . a2 = 2 2 2 2 After simplification, we obtain: a2 = x (y + z). By using the Pythagorean Theorem in the triangle ADE, we have: a2 = z2 − y2 . By substituting this relation, we obtain: z2 − y2 = x (y + z). After cancelling (y + z) from both sides, we obtain: x + y = z. 2.3. Method B – proof using similarity of triangles From the point E, we drop a perpendicular EN to the side AB. We denote by M the point of intersection of this perpendicular with the angle bisector AF. By using the angle bisector theorem in the triangle AEN, we obtain: EM z = (∗). y MN The triangles AMN and AFB are similar; therefore, we can write the proportion: MN y = a x ⇒ EM = a − MN = MN = x·y (∗∗), a a2 − xy (∗ ∗ ∗). a We substitute the relations (∗∗) and (∗∗∗) in (∗) to obtain: a2 = x (y + z). Using the Pythagorean Theorem in the triangle ADE, we obtain: x + y = z. 6 V. OXMAN AND M. STUPEL D z −a E a−y C G x a α α A F x a B Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 Figure . Proof using the properties of the angle bisector. Figure . Proof using rotation of triangle. 2.4. Method C – proof using the properties of the angle bisector From the properties of the angle bisector (see Figure 3): AB = AG = a, FB = FG = x. From the Pythagorean Theorem in the triangle ECF: EF2 = (a − y)2 + (a − x)2 . From the Pythagorean Theorem in the triangle EGF: EF2 = (z − a)2 + x2 . From equating the values of EF2 , we obtain: (z − a)2 + x2 = (a − y)2 + (a − x)2 , z2 − 2az + a2 + x2 = 2a2 − 2ay − 2ax + y2 + x2 . From the Pythagorean Theorem in the triangle EDA: a2 = z2 − y2 and by cancelling terms, we obtain: z = x + y. 2.5. Method D – proof using rotation of triangle We copy ABF in such a manner that the side AB coincides with the side AD, i.e. ABF is rotated at 90° in anticlockwise direction around point A (see Figure 4). The triangle F EA is an isosceles triangle since it has two equal angles: ∠EF A = ∠EAF = 90◦ − α. Conclusion : EF = EA ⇒ x + y = z. INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 7 2.6. Method E1 – proof by trigonometry From triangle AFB : (1) From triangle AED : (2) x tgα = . a a2 = z 2 − y 2 . y cos 2α = ⇒ z (3) cos2 α = Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 By substituting (1) and (3) in the trigonometric identity 1 + tg2 α = (4) 1+ 1 , cos2 α y+z . 2z we obtain: x2 2z , = 2 a z+y and by substituting a2 from (2) in (4), we have: (z − y)2 = x2 (5) ⇒ z = x + y. 2.7. Method E2 – proof by trigonometry From triangle AFB : (1) x = atgα. From triangle ADE : (2) y = actg2α. a . z= sin 2α (3) It remains to prove that : a ? = actg2α + atgα. sin 2α We multiply both sides by sin 2α and obtain: ? 1 = cos 2α + sin 2αtgα = 1 − 2sin2 α + 2 sin α · cos α · sin α = 1. cos α 2.8. Method F – proof using analytic geometry We use the rules of analytic geometry. We locate the square in a system of coordinates in such a manner that the vertex A coincides with the origin and the sides AB and AD lie on the axes x and y, respectively (see Figure 5). The coordinates of the vertices are: A(0, 0), B(a, 0), C(a, a), D(0, a). We select a point E on the side DC, whose coordinates are E(b, a), such that b < a. The equation of the side AB is y = 0. The equation of the line AE is y = ab x. a x + b y + c = 0 1 1 , When two straight lines are given by their general equations: 1 a 2 x + b 2 y + c2 = 0 1 x+b1 y+c1 2 x+b2 y+c2 The equation of the angle bisectors are: a√ = ± a√ . 2 2 2 2 a1 +b1 a2 +b2 8 V. OXMAN AND M. STUPEL D(0, a) E(b,a) C(a,a) F α α A (0, 0) B(a, 0) Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 Figure . Proof using analytic geometry. D y C E G u z α α A F x v B Figure . Proof using vectors. ax − by = 0 , y=0 = ± y. In accordance with the general equations of the lines AF and AB: we obtain that the equations of their angle bisectors are: √ax−by 2 2 a +b The explicit equation of the angle bisector ∠EAB (the one with the positive slope that corresponds to the straight line AF) is: y = √a·x2 2 . Therefore, the coordinates of the b+ point F are: F(a, 2 √a b+ a2 +b2 a +b ). From the coordinates of the vertices A, B, C, D, E, F, we obtain the lengths of the segments: BF = √ a2 , AE = a2 + b2 , DE = b. √ b + a2 + b2 Using simple algebraic technique (including multiplying the numerator and the denom2 inator by the conjugate of the denominator), one can easily show that b + √a 2 2 = b+ a +b √ a2 + b2 , or in other words, BF + DE = AE. 2.9. Method G – proof using vectors − → −→ We denote: AD = u, AB = v, | u | = | v | = a (see Figure 6). z = u + y, = v + x, z · = u · v + y · v + u · x + y · x. u · v = y · x = 0 (since the vectors are perpendicular). y · v = | y | · | v | = y · a (since the vectors are parallel). INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 9 I y D(0,a i ) E(y ,a i ) C(a,a i ) z2 F(a, xi ) z1 x α α B(a, 0) Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 A (0, 0) R Figure . Proof using complex numbers. u · x = | u | · | x | = x · a (since the vectors are parallel). Hence : z · = z · · cos α = z · · cos α = z · a; Andhence : z · a = x · a + y · a ⇒ z = x + y. 2.10. Method H – proof using complex numbers We place the square in a system of coordinates, where the horizontal axis is real and the vertical axis is imaginary, and the vertex A is at the origin (see Figure 7). z1 = a + xi, z2 = y + ai, z1 2 = a2 − x2 + 2axi. Since z2 and z1 2 both have an angle of 2α, one can write down: z1 2 = k · z2 , where k is real. We equate the expressions: a2 − x2 + 2axi = k (y + ai) and obtain two equations: a2 − x2 = ky, 2ax = ka ⇒ k = 2x. We substitute k = 2x in the first equation and obtain: a2 = x2 + 2xy. Then, z = |z2 | = y2 + a2 = y2 + x2 + 2xy = x + y. 3. The art of asking questions Since the 80s of the last century, methodological aspects of solving mathematical problems moved towards adding a style of questions, such as: overloading data – what if more (WIM)?, replacing data – what if instead (WII)?, removing data – what if not (WIN)? [1,19– 22], general case – is generalization possible? Therefore, as usual in investigation activities, ‘what if ’ questions were asked. 10 V. OXMAN AND M. STUPEL Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 Figure . The case when E is on the continuation of the side DC. Figure . Replacement the square with a rectangle ABCD. 3.1. Question 1 What happens to the relation between the side lengths of the segments when the point E is on the continuation of the side DC outside the square, as shown in Figure 8? From Proof method D, we obtain that the triangle EF A is an isosceles triangle. Therefore, in this case as well, we obtain: BF + DE = AE. 3.2. Question 2 What will the relation between the segments DE, BF and AE be when we replace the square with a rectangle ABCD? (See Figure 9). We denote : AB = a, AD = b. We set a point G on the side AB so that AG = b, and drop a perpendicular H. We rotate the triangle AHG by 90° about vertex A, as shown in Figure 9. The triangle H EA is an isosceles triangle, therefore: AE = H D + DE = HG + DE. From similarity of the triangles AHG and AFB , we have: HG = ab · FB. Therefore, for a rectangle, the relation between the segments is: AE = DE + ab · FB. 4. Methodical discussion with the students Towards the end of the activity, the students presented the different solutions to the entire group. A methodical discussion was held on each solution from different aspects, such as: the degree of difficulty of the solution method, the elegance and simplicity of the solution, the creativeness of the solution method, integration between different fields in mathematics Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 11 and their tools, the extent to which the solutions match the knowledge of the students and the programme of studies. Subsequently, we presented to the students the concept of ‘conservation’, based on the conserved property of the sum of the lengths of two segments along a third segment by shifting the point E on the side DC. From this point on, a discussion was held concerning the importance of finding several methods of solution of the same task, the fact that in each mathematical theorem, and especially in Euclidean geometry, there is a single property for several properties that are conserved. A comprehensive discussion was held on the geometry conservation properties. By applying GeoGebra applets, other examples of conservation properties were presented and their contribution to solving problems in geometry has been emphasized . In addition, a discussion was held on the importance of using the computerized technological tool for testing the correctness of different hypotheses. 4.1. Feedback of the students • The different proofs found for the task reinforce the perception of mathematics as a field built from different intertwined branches. • At first glance, this seems to be a task in geometry, and without the requirement of solving it using traditional methods, we would not have tried to solve it using analytical geometry, vectors and complex numbers. • The proof that the students liked the most, of those shown at the end of the activity, is method D (by rotating a triangle), followed by method A (by calculating areas). • There is much similarity between the proofs using vectors and the proof using complex numbers. 4.2. Concluding remarks and implication for teaching Geometry is a goldmine for MST. Proofs may be derived by applying different methods within the specific topic of geometry or within other mathematical areas such as analytic geometry, trigonometry and vector spaces. The authors claim that multiple proofs foster both better comprehension and increased creativity in mathematics for the student/learner. The multiple solutions that were presented herein for one geometry problem demonstrate the connectivity between different areas of mathematics, and show how geometry can serve as a base for topics such as trigonometry, analytic geometry and vector spaces. For example, solving the problem by analytic geometry methods (besides the clear employment of Euclidean geometry) reveals a connection to trigonometry by the slopes of the sides of the triangle and the resulting need to solve trigonometric equations. When doing the proof by vectors, the connection to trigonometry is shown by the requirement to apply the scalar product formula and again, the need to solve trigonometric equations in order to arrive at the conclusion. This occupation with the connection between different mathematical domains builds among students a vision of mathematics as a linked science and not as a collection of discrete, isolated topics . In most school textbooks worldwide, mathematics problems are organized by specific topics that are presented in the curriculum: students tend to understand that certain problems are connected to specific topics and hence assume that Downloaded by [Chalmers University of Technology] at 00:31 29 October 2017 12 V. OXMAN AND M. STUPEL for each problem, there is one, and only one, method for its solution . Also, while the National Council of Teachers of Mathematics (NCTM) standards document emphasizes that teachers should find tasks that exhibit connectivity between different mathematical domains, it also indicates, nevertheless, that locating such problems is time-consuming and calls for special initiative from the teachers. Certainly, our experience also shows that while this mission is not an easy one, there is certainly a need to identify additional problems that can be solved by various methods and demand the application of proofs from different realms of mathematics. The authors see it their task to continue searching for such appropriate problems and encourage their colleagues to do so, too. They believe that mathematics teachers should present to their students problems that must be solved in more than one way, requiring, as much as possible, the application of knowledge from different domains in mathematics. In his study, Bingolbali  indicated that applying multiple ways of solving problems has the potential to develop students’ ‘relational understanding’ (a term attributed to Skemp ) and contributes to the development of their autonomy. Furthermore, when testing students, teachers should occasionally allow their students to solve problems by employing proofs from any mathematical area, and not insist on a proof from the specific subject under study. In this day and age, it is impossible to ignore the rapid development of technology and the way it affects almost every facet of life. The education system is no exception, and one can certainly not disregard the value of technology in teaching mathematics. The authors firmly believe in the advantages of incorporating DGS in mathematical problem-solving and actually demonstrated our problem using GeoGebra software. In a special issue of The International Group for the Psychology of Mathematics Education (PME), Jones et al. , writing a guest editorial entitled ‘Proof in dynamic geometry environments’, stated that ‘there is a range of evidence that working with dynamic geometry software affords students possibilities of access to theoretical mathematics, something that can be particularly elusive with other pedagogical tools’ [23,p.3]. Inductive exploration with DGS can lead students to develop their own conjectures about the solution of the problem and then to deal with the deductive proof. This is in addition to the contribution of visualizing different graphical representations of concepts and other related situations to the problem. We further recommend, therefore, that mathematics teachers initially allow their students to cope with problem-solving by working in such an environment until they reach a solid conjecture for deductive proof. In conclusion, our efforts in coping with the presented problem afforded us a real feeling of working as mathematicians who look for multiple solutions to a problem, especially those that are short, elegant and mathematically aesthetic. By encouraging student/learners to do the same, they, too, will learn to appreciate the connections between the various branches of mathematics and discover how one problem can be tackled from different points of view. In addition, the incorporation of DGE software will add a complementary technological tool to assist the students in their investigations, while also providing teachers a means with which to base pedagogical action and in-class discussion. 4.3. Summary The authors believe that this activity suggests that students need to absorb the importance of solving tasks using several different methods. 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