1. Solving of reverse constructions with GeoGebra in mathematics teacher training RNDr. Kitti Vidermanová, PhD. PaedDr. Janka Melušová, PhD. Department of Mathematics Constantine the Philosopher University in Nitra

2. Idea  while checking homework  Students sent their solutions in GeoGebra and Cabri II by e-mail  Mistake of one student – she used wrong given data in construction task – time consuming and demanding

3. The Constructional Tasks  Belong to most difficult tasks in school geometry, because they cannot be solved/taught by any algorithm, they have to be solved by deduction …should be taught because  they offer space for independent creative activity of students,  they help to development of creativity and visualization,  Logical thinking,  Sence for aesthetics.

4. The Reverse Constructions  Teach how to read from figure,  While the solving process, students do not construct but inquire and explain separate steps of the construction,  Transformation of wording to symbolic language,  Deeper understanding.

5. Study – brief description  3 tasks (2 planar, 1 spatial) – constructions done in GeoGebra,  8 student working independently with the computer and wrote their reasoning on the paper,  Tasks: „Based on given construction, try to find out, what shape should be constructed and which objects were given.“

6. Task 1 Given length of base AB, size of angle by A, height of the trapezoid. Task Draw the isosceles trapezoid ABCD.

7. Task 1 - results Number of students Given dataDrawn objectNote 3 students AB, v,  Isosceles trapezoid Correct solution 2 students AB, v,  + BC TrapezoidUnawareness of isosceles 2 studentsNot writtenIsosceles trapezoid Instead of formulating the task, students wrote how to draw the object 1 studentNot writtenTrapezoidInstead of formulating the task, student wrote how to draw the object, Student used the equality of legs BC and AD, but he did not write that the final object is isosceles trapezoid

8. Task 2 Given lengths of medians. Task Draw the triangle ABC.

9. Task 2 -results Number of Students Given dataDrawn objectNote 2 studentsIsosceles triangle ABC Rectangular triangle ACG´ They did not use the dynamicity of GeoGebra 2 studentstriangle ABCIsosceles triangle ACG´ Wrong interpretation of symmetry according to point Dynamicity not used 2 studentstriangle ABCIsosceles triangle ACG´ 2 studentstriangle ABCTriangle with area 1/3 of given Triangle We consider it as the correct solution

10. Task 3 Given data length of side of the base altitude. Task Draw the regular pyramid with triangle base in sloping parallel projection

11. Task 3 - results Number of Students Given dataDrawn objectNote 5 studentsNot writenRegular 3-faced pyramid in sloping parallel Projection According to the reasoning of students the task was solved correctly, but given data is missing 1 studentSide of the base and the altitude of the pyramid Regular 3-faced pyramid in sloping parallel Projection Only complete solution 1 studentNot writenPyramidThe object specification is missing 1 studentNot writenRegular tetrahedralThe given object was not regular

12. Students’ opinions (future teachers) ICT in geometry  It is interesting and motivating, but it should be used more for revising, it is important to get the skills in drawing on the paper,  Students spend too much time sitting by a computer, I prefer the standard (pen and paper) way to teach. Reverse constructions  Interesting, new and demanding,  Like a puzzle, so they can be motivating  I would use this kind of task because students usually do not like classic drawing with pencil, it may be useful to alternate this task with classic ones  I would implement them at upper secondary and tertiary education, students should develop the drawing skills.

13. Survey of in-service teachers Survey in 2007 (129 respondents, average length of practice 17, 8 years)  66% of teacher have the possibility to use the computers in mathematics lessons  Only 40 % of these really use it. Reasons:  Lack of time for innovative methods (teach students to work with program),  Lot of teachers is not able to use the software.

14. Additional information Since 2008/09 – curricular reform  Less content, less time. Project MVP - design of teachers book aimed to use of ICT in school mathematics (including the training of teachers)

15. Thank you for your attention