[font=Arial][justify]The National Council of Teachers of Mathematics (2003) of USA recommends in “The Principles and Standards for School Mathematics”:[/justify][/font][list][*][justify]all students should have the opportunity and support, necessary to learn significant mathematics with depth and understanding;[/justify][/*][*][justify]the mathematical instructional programs should offer opportunities to students to “use visualization, spatial reasoning, and geometric modelling to solve problems” and, also, to be “encouraged to reflect on their thinking during the problem-solving process so that they can apply and adapt the strategies that they develop to other problems and in other contexts”;[/justify][/*][*][justify]students should make and investigate mathematical conjectures and learn how to generalise and extend problems by posing follow-up questions.[/justify][/*][/list][justify]Problem-posing activities might be organised before, during, or after the problem-solving process. Stoyanova (1997, pp. 18–33) marked the impact of students' problem-posing activities on mathematical instruction, аs:[/justify][list][*][justify]a way of extending students’ understanding of important mathematical concept;[/justify][/*][*][justify]a means for improving students’ skills in problem solving;[/justify][/*][*][justify]an important component of students’ assessment;[/justify][/*][*][justify]a way of changing the nature of the communication in the mathematics classroom;[/justify][/*][*][justify]a window into students’ difficulties in mathematics;[/justify][/*][*][justify]a way of investigating the highest level of students’ mathematical performance;[/justify][/*][*][justify]a way of preparing students to be intelligent users of mathematics in their everyday life;[/justify][/*][*][justify]a way of linking students’ own interests with their mathematical education.[/justify][/*][/list][justify]This article presents the possibilities of GeoGebra usage to create mathematical problem-posing and problem-solving environment, which stimulates the mathematical creativity of students - future teachers of Mathematics and Informatics. [/justify]

[justify]The term “[i]problem posing[/i]” is used in different meanings, such as [i]problem finding, problem generating, problem creating, problem sensing, problem formulating, creative problem discovering, problematizing, problem envisaging, problem modelling, [/i]etc. Problem posing is frequently defined as: [/justify][list][*][justify]engaging a person in a task, the goal of which is to generate a new problem with a given set of conditions (Silver, 1994, p. 19); [/justify][/*][*][justify]the formulation of novel problems with solution unknown at least for its creator (Van den Heuval-Panhuizen et al. 1995);[/justify][/*][*][justify]reformulation of an existing problem (Cohen & Stover, 1981); [/justify][/*][*][justify]the generation of new problems and the reformulation of given problems (Singer, Ellerton, & Cai, 2013, p. 3); [/justify][/*][*][justify]a special case of problem solving (Kontorovich, Koichu, Leikin, & Berman, 2012, p. 151); [/justify][/*][*][justify]a process by which, based on mathematical experience, students construct personal interpretations of concrete situations and from these situations formulate meaningful mathematical problems (Sriraman & Lee, 2011, p. 9; E. Stoyanova & Ellerton, 1996, p. 518).[/justify][/*][/list]

[justify][font=Arial]Some researchers of the school training in mathematics have developed the possibilities of GeoGebra as a dynamic environment for using and generating strategies for solving and posing mathematical problems:[/font][/justify][list][*][justify]Problem solving and posing, using Dynamic Geometry Software (DGS), involves the direct and indirect effects of the software’s interface on students’ procedures and understandings. In addition, DGS’s interface provides students with the opportunity to use visual reasoning in mathematics and helps them, through the dragging facilities, to generalize problems and relationships (Christou, Mousoulides, Pittalis, & Pitta-Pantazi, 2005, p. 127). [/justify][/*][*][justify]DGS can be used and transformed by students in visually confirming and negating conjectures and in developing a new perspective on solving and posing original problems (Meira, 1998; Sinclair, 2004).[/justify][/*][/list]We used some methods of instructions in dynamic GeoGebra environment, which has been created, based on the concept of problem-posing situations (E. N. Stoyanova, 1997). The methods of instruction in mathematics include the following main points:[br][br][table][tr][td]1. Usage of geometrical problems for:[br][list][*]extracurricular or class work in Geometry for acquiring new knowledge related to learning objects such as arbelos and Archimedean circles;[/*][*]forming and developing skills for creating new problems;[/*][*]forming and developing skills for solving problems in the field of work;[/*][*]stimulating and maintaining the interest of students towards the mathematical field studied;[/*][*]stimulating mathematical creativity in the interests of the trainees;[/*][/list][/td][/tr][tr][td]2. Usage of three kinds of problem-posing situations (P-P situations) for forming and developing of problem-posing skills:[br][list][*]Free P-P situations;[/*][*]Semi-structured P-P situations; [br][/*][*]Structured P-P situations;[/*][/list][/td][/tr][tr][td]3. Usage of  P-P situations for:[list][*]solving problems with mathematical tools in GeoGebra environment; [/*][*]solving problems via constructing or proving statements in GeoGebra environment;[/*][*]posing new problems with GeoGebra; [br][/*][*]creating interesting problems from the perspective of the learner and in accordance with the scope of work; [/*][/list][/td][/tr][tr][td]4. Usage of GeoGebra for:[br][list][*]individual work;[/*][*]mathematical investigations;[/*][*]experimentation;[/*][*]a logical evidence;[/*][*]analysis of geometric structures;[/*][*]creating new structures and tasks;[/*][*]problem formulating and problem solving;[/*][*]different approaches, such as exploring students' posed problems and confirming their new approaches, analysing, exchanging, and evaluating posed problems (Fukuda & Kakihana, 2009);[/*][*]mastering students’ new knowledge and developing the students’ skills for working with GeoGebra.[br][/*][/list][/td][/tr][/table]

[i][size=100][justify]Create mathematical problems in a dynamic GeoGebra environment, which include the geometric figure arbelos.[/justify][/size][/i]

[justify][i]Calculate the length [/i][i][math]L[/math][/i][i] of[/i][i] the external arc of arbelos, if the length of the inner arcs  [math]L_1[/math][/i] [i] and [math]L_2[/math][/i] [i] are known.[/i][/justify]

[justify][i]A.      [/i][i]Create a formula to calculate the sum of the areas  [math]F_1, F_2, F_3[/math][/i]  of[i] the [/i][i]arbeloses, which are constructed externally on the sides of the equilateral triangle, if the tangential points of the two small semicircles of each arbelos are the medium points of the triangle sides. Post GeoGebra construction.[/i][/justify][justify][i]B.      [/i][i]Post GeoGebra construction, which is a generalisation of the problem in the condition A. Bring a new formula for calculating the sum of the areas of arbeloses constructed externally on the sides of a triangle.[/i][/justify]

[justify][i]If on the sides of the equilateral triangle are built arbeloses and the tangential points of the two small semicircles of each arbelos are the medium points of the triangle sides, then the formula for calculating the sum of areas of the arbeloses is[br] [/i][i][math]F=F_1+F_2+F_3=\frac{3\pi}{2}\left(R^2-2r^2\right),[/math][br]where [math]R[/math] is the radius of the biggest semicircle and [math]r[/math]is the radius of the small semicircles of each arbelos.[/i][/justify]

[justify][color=#666666][i]In each semicircle of the arbelos are inscribed triangles [math]\Delta,\Delta_1,\Delta_2[/math] with an acute angle of [math]30^\circ[/math][/i][i] two of the vertices of which are at the end of the diameters of the respective semicircles[/i][/color][color=#666666][i].[br]Prove that[/i][br][/color][center][math]a:a_1:a_2=b:b_1:b_2=c:c_1:c_2=n,[/math][/center][color=#666666]where [math]a,b,c; a_1,b_1,c_1;a_2,b_2,c_2[/math] are the sides of the triangles [math]\Delta,\Delta_1,\Delta_2[/math], respectively. [/color][/justify]

[justify][i][color=#666666]The triangles set in this way are rectangular because two of their vertices are parts of the diameters of the semicircles. Therefore, they are similar. Hence: [br][/color][/i][/justify][center][math]a:a_1:a_2=b:b_1:b_2=c:c_1:c_2.[/math][/center]

[justify][i][color=#666666]In each semicircle of the arbelos are inscribed triangles [math]\Delta,\Delta_1,\Delta_2[/math] with equal angles, two of the vertices of which are at the end of the diameters of the respective semicircles. For the elements of the triangles [math]\Delta,\Delta_1,\Delta_2[/math] prove that if [br][/color][/i][/justify][center][i][math]a:a_1;a_2=b:b_1:b_2=c:c_1:c_2=n,[/math][/i][/center][i][color=#666666]then[br][/color][/i][center][i][color=#666666][math]r:r_1:r_2=n,[/math][br][br][math]R:R_1:R_2=n[/math],[br][br][math]s:s_1:s_2=n[/math],[br][br][math]h_{\left(c\right)}:h_{\left(c_1\right)}:h_{\left(c_2\right)}=n[/math],[br][br][math]F:F_1:F_2=n^2,[/math][/color][/i][/center][i][color=#666666]where [math]r,r_1,r_2[/math] are the radii of the inscribed circles, [br][math]R,R_1,R_2[/math] – radii of the circumscribed circles,[br][math]s,s_1,s_2[/math] – semiperimeters, [br][math]h_{\left(c\right)},h_{\left(c_1\right)},h_{\left(c_2\right)}[/math] – heights to the hypotenuses,[br][math]F,F_1,F_2[/math] - areas of the triangles.[/color][/i]

[justify][color=#666666]The observed results from the creation and the use of GeoGebra environment have shown that:[br][/color][/justify][list][*][color=#666666][justify]Problem-posing situations foster the development of strong interest in the use of GeoGebra environment for creating and solving mathematical problems. They change strongly the role of the trainees - from passive consumers of knowledge to active creators of knowledge.[/justify][/color][/*][*][color=#666666][justify]Students understood the problems and the methods for creating or solving problems more deeply by using technology.[/justify][/color][/*][/list][color=#666666][justify]The use of the GeoGebra environment stimulates the mathematical creativity among the students and the problems, which they created, have creative character.[/justify][/color]

[justify][/justify][list=1][*][font=Arial][color=#666666][justify]Abu-Elwan, R. (1999). The development of mathematical problem posing skills for prospective middle school teachers. [i]Proceedings of the International conference on Mathematical Education into the 21st Century: Social challenges, Issues and approaches[/i] (Vol. 2, pp. 1–8).[/justify] [/color][/font][/*][*][justify]Bilchev, S. & Velikova, E. A. (2007). Transformation Methods. In G. Makrides (Ed.), [i]MATHEU, Identification, Motivation and Support of Mathematical Talents in European Schools[/i] (Vol. 3, pp. 129–146). Intercollege, Cyprus: MATHEU Project.[/justify][/*][*][font=Arial][color=#666666][justify]Brown, S. I. & Walter, M. I. (2005). The Art of Problem Posing. [i]Lawrence Erlbaum Associates (Bks)[/i]. ERIC.[/justify][/color][/font][/*][*][font=Arial][color=#666666][justify]Christou, C., Mousoulides, N., Pittalis, M. & Pitta-Pantazi, D. (2005). Problem solving and problem posing in a dynamic geometry environment. 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[color=#666666]Member of the GeoGebra Institute of Ruse, http://ggir.uni-ruse.bg/ [br][br][br]Department of Mathematics[br]Faculty of Natural Sciences and Education[br]University of Ruse, [/color][url=http://www.uni-ruse.bg/][color=#666666]www.uni-ruse.bg[br][/color][/url][color=#666666]8 Studentska Str., 7017 Ruse[br]BULGARIA [br][br][i]Cell phone:[/i] (+359) 886 162 384[br][i]E-mails:[/i][br][/color][url=mailto:mpetkova@uni-ruse.bg][color=#666666]mpetkova@uni-ruse.bg[br][/color][/url][url=mailto:magipetkova@ymail.com][color=#666666]magipetkova@ymail.com[br][/color][/url][br][i]Google Scholar Profile Page:[/i][br][url]https://scholar.google.bg/citations?user=1v6y8KcAAAAJ&hl=bg[/url]