1. Numerics with Geogebra in High School dr Dragoslav Herceg dr Đorđe Herceg Faculty of Science and Mathematics Novi Sad, Serbia {hercegd | herceg}@im.ns.ac.yu

2. 2 of 18 Faculty of Science and Mathematics  Department of Mathematics and Informatics offers undergraduate and graduate studies in mathematics, mathematics of finance and computer science  Many of our faculty teach in the "Jovan Jovanović Zmaj" high school

3. 3 of 18 The Software that we use  Mathematica 5  Matlab 6  GeoGebra 2  C# (MSDN AA)  Java  We develop course materials in Mathematica and GeoGebra

4. 4 of 18 Our textbooks  Many books on Numerical Mathematics

5. 5 of 18 Zmajtematika

6. 6 of 18 How we use GeoGebra  In a high school course in Numerical Mathematics, held for fourth grade students during five years (2002-2007)  As a teaching tool in the class  In the lab – for student assignments  How does using a computer in the class help the students learn more than before? ... and what will they really learn?

7. 7 of 18 Course outline  A course in numerical mathematics, covering standard topics  The textbook written by us  A CD with interactive lessons in Mathematica and programs to perform numerical methods step by step  GeoGebra drawings with geometrical interpretation of numerical methods  Let the students use the computer to solve mathematical problems, not the programming ones!

8. 8 of 18 Our goals  To let students perform some tasks on their own, to experiment and vary the solutions  To teach each student something (at least)  Strict criteria – the solutions must match!  Relaxed testing conditions – students may use whatever they have on hand, but not communicate among themselves

9. 9 of 18 Testing  All students have/can use a computer  But not too enthusiastic about learning another programming language  All exams based on textbook examples, slightly modified  Same grading criteria during 5 years, so we were able to compare five generations of students

10. 10 of 18 Grade average  Same types of tests during 5 years  The generation of 2004/05 consisted of seven students who were eager to learn, while the other six were uninterested. Year 2002/ 03 2003/ 04 2004/ 05 2005/ 06 2006/ 07 No. of students2422133028 Grade average3.143.553.254.044.17

11. 11 of 18 2005/06 and 2006/07  A slightly different testing procedure  Students split into two groups  Two additional tests were given:  Graphical localization of roots  Localization of solutions of algebraic equations Year2005/062006/07 GroupIIII No. of students171314 Grade average4.203.804.384.00

12. 12 of 18 Two different approaches to teaching have been tested  In the first test, the first group was encouraged during lectures to experiment and try, using some of the programs from the CD, to find the solutions of equations. No particular method was favored and the choice was completely up to the students.  The second group was taught in the more "classical" way to first analyze the function theoretically, and then graph the function and show where the solutions are.  The roles were reversed during the preparations for the second test.

13. 13 of 18 Results  Not surprisingly, the results were:  Obviously, letting a student chose her/his own way of solving a problem gives better results The first testThe second test GroupIIII No. of students14 Grade average4.182.622.503.57

14. 14 of 18 Conclusion  Computers free students of tedious manual work  However, they shouldn't free them of having to reason, draw conclusions and act on them  The possibility to build the problem in reverse, starting from the desired solution  Personalized tests  Careful preparation of tests ensures that only the real knowledge is valued  Trial and error – the good way to do math?

15. 15 of 18 The role of GeoGebra?  GeoGebra provides essential support to the teacher  A number of drawings were developed and students were encouraged to try to duplicate them ..\primeri\Njutnova metoda.ggb..\primeri\Njutnova metoda.ggb..\primeri\Njutnova metoda.ggb ..\primeri\Opsti iterativni postupak.ggb..\primeri\Opsti iterativni postupak.ggb..\primeri\Opsti iterativni postupak.ggb

16. 16 of 18 What we would like  What we would like to see in GeoGebra:  Lists  Nested functions  Calculated properties:  AA: If[boolexpr == 1, A, (1000, 0)]  More flexible external control  BIG FONTS AND DOTS

17. 17 of 18 Work in progress...  Formula Editor, for entering and displaying formulas  Accompanying software for the book "Zmajtematika"

18. Numerics with Geogebra in High School dr Dragoslav Herceg dr Đorđe Herceg Faculty of Science and Mathematics Novi Sad, Serbia {hercegd | herceg}@im.ns.ac.yu Thank you!