Anna Rybak, University of Bialystok, Poland Istv á n L é n á rt, ELTE University, Hungary

1. Play computer game ... and learn spherical geometry Anna Rybak, University of Bialystok, PolandIstván Lénárt, ELTE University, Hungary

2. Children like computer games. Is it possible to teach topics from beyond the curriculum using computer game? The main question

3. Five-in-a-line

4. Experiment • Main idea: From game on the plane (circle and cross) through game on the sphere (five-in-a-line) to models • Main aim: to investigate and learn some properties of figures on the sphere • Groups that took part in the experiment:- students of Math Institute (math teachers-to-be)- kids 11 and 12 years old (members of computer science circle from middle school in Bialystok)- practising math teachers (participants of seminar for teachers in the Institute of Computer Science, University of Bialystok) • Time: 2 lessons (1,5 hour) for each group

5. First step • Playing circle and cross on the plane 3x3, then 4x4. • First mathematical question:What should we know about geometry in order to play circle and cross? • Answer: We should know what shape is called “straight line”.

6. Next step • Presentation of the game five-in-a-line on the sphere. • Second mathematical question:What should we know about geometry in order to play this game? • Answer: We should know what shape is called “straight line” on the sphere. • So: What is a straight line on the sphere? • Answer: Equator!!! Remark:Students had big troubles with answering this question. Kids immediately answered.Teachers were in the middle between these two groups.

7. Very important question Why is the equator a spherical straight line? One boy from 5th grade of middle school (11 years old) answered immediately: Because it divides a sphere into two identical parts, just like planar straight line divides a plane into two identical parts. Students did not find this property of spherical straight line, teachers needed a hint.

8. Next steps • Playing the game in pairs • Question from a leader:Which properties of spherical straight line are different from properties of planar straight line? • Answer:Straight line on a plane is infinite, on a sphere is finite.

9. Next steps • Remark from a leader:So maybe properties of other figures on a sphere are different from properties of figures on a plane. • Discussion about “spherical” figures: segments, triangles, other polygons. • Looking for different figures on the sphere from game.

10. Investigation of the sum of internal angles in the spherical triangle It is possible to rotate the sphere, so participants can easily estimate measures of angles in the marked triangle. They discover that the sum is not equal 180°! Then it is possible to mark other triangles and to discover that sum of angles is not constant.

11. Very important moment: from game to models Remark from a leader: You could only estimate angles in some special triangles using a model of the sphere on the screen of computer. It is possible to construct any triangle on 3D-model and measure angles of constructed triangle.

12. Presentation of models and tools

13. Problems for investigations with use of models and tools • What is the value of the sum of interior angles in a spherical triangle? • Does “spherical π” exist i.e. Is the ratio of circle's circumference to its diameter is constant on a sphere? • Can we use a spherical square as a unit of area on a sphere? Justify all your answers.

15. Survey (for kids) • What do you think about the game that you learnt about today? • Would you prefer the game four-in-a-line? • What do you think about geometry that you learnt about today? • Did the game help you to learn geometry on a sphere? • What activity you prefer: playing game or working with models? Why? • Would you like to learn more about geometry on a sphere? • How do you think: is this geometry useful for anything?

16. The most interesting answers • This geometry is a little bit strange because of figures. • This geometry is much more difficult and interesting in comparison with geometry on a plane. • Game is interesting, requires a lot of thinking, it is much more difficult to put points on a straight line on a sphere than on a plane. • This game is not easy. It is necessary to think a lot if one wants to win. • This game is strange, but interesting. • It requires a lot of logic.

17. The most interesting answers • I prefer game. • Game is always more interesting, but we can get more knowledge from models, so I prefer to work with models. • I prefer to work with models because it is possible to find much more properties. • I prefer playing football.

18. Survey (for teachers) • What do you think about the game – from didactical point of view? • What do you think about such methodology of introducing spherical geometry? • Can spherical geometry be taught at school? Why? • Does the game help in learning spherical geometry? Why? • What do you prefer: to play game or to work with models? Why?

19. The most interesting answers • Game helps to develop imagination, concentration, looking for strategy. • Method is good: from playing to learning. • These ideas could be taught at mathematical circles, for interested students. • Work with models lets you touch the tools, models are 3-D, picture on the screen is flat.

20. Conclusions • It is possible to use computer game (not educational computer game) for introducing quite new topic (from beyond curriculum) to middle school, make this topic interesting and understable for kids. • It is possible to use two kinds of media: electronic and manipulative in the same teaching process and to raise kids' interest in using manipulative media as more useful in investigations.

21. Conclusions • The results of joining different kinds of media and learning new topics are better when learners are younger. • Computer game may raise kids' interest for 3-D geometry and non-Euclidean geometries. • At the same time, it helps concept formation in Euclidean plane geometry too.

22. Thank you for your attention