1. English, L. & Sriraman, B. (2010). Problem solving for the 21 th century. In B. Sriraman & L. English (Eds.), Theories of Mathematics Education: Seeking new frontier. Berlin: Springer, 263-290. Mousoulides, M. Sriraman, B., Pittalis, M., & Christou, C. (2007). Tracing students' modeling processes in elementary and secondary school. Paper presented at 5th Congress of the European Society for Research in Mathematics Education (CERME – 5), Cyprus.

2. בעייה מתמטיקה

3. בעייה מתמטיקה The best way to teach mathematical ideas is to start with interesting problems whose solution requires the use of ideas. Use problem solving as a powerful means to develop substantive mathematical concept. Real word problem Integration within all topic areas Outside the class a. complex systems. b. plan, monitor and assess progress c. adapt quickly to technologies.

4. בעייה מתמטיקה

5. Modeling Mathematical models = "systems of elements, operations, relationships and rules that can be used to describe, explain, or predict the behavior of some other familiar system” Modeling problems are realistically complex situations where the problem solver engages in mathematical thinking beyond the usual school experience and where the products to be generated often include complex artefacts or conceptual tools that are needed for some purpose, or to accomplish some goal.

6. Modeling Mathematical models = "systems of elements, operations, relationships and rules that can be used to describe, explain, or predict the behavior of some other familiar system” Elicit my own mathematics meaningful Make sense identifying selecting Interpreting making representation Cyclic process sorting organizing selecting quantifying Transforming large data

7. University’s Cafeteria Construct models for selecting the best among a number of employees

8. Elicit my own mathematics identifying selecting Interpreting making representation Cyclic process sorting organizing selecting quantifying Transforming large data Real world Outside the class מתמטיקה  בעייה סטטיסטיקה Reformulating the problem Not just one right answer Complex system Hours, money, seasons

9. Leikin, R. (2004). Towards high quality geometrical tasks: reformulation of a proof problem. In M. J. Hoines & A. B. Fuglestad (Eds.) Proceedings of the 28 th International Conference for the Psychology of Mathematics Education, Vol. 3, 209-216.

10. בעייה The person has to be motivated to find a solution No readily available procedures The person has to make an attempt and persists to reach a solution Several solving approaches 4 קריטריונים לקביעת איכות הבעייה :

11. בעיית חקר The person has to be motivated to find a solution No readily available procedures The person has to make an attempt and persists to reach a solution Several solving approaches 4 קריטריונים לקביעת איכות הבעייה : challenging Cognitively demanding High motivated

12. עד כמה לפתוח את הבעייה ? 1. הוכח שגובה הטרפז שווה לקטע האמצעים 2. השווה בין הגובה לקטע האמצעים. 6. מה תוכל לומר על טרפז זה ? מסעיף 2 שימוש בתוכנת גיאומטריה דינמית. 3. מצא יחסים אפשריים בין הקטעים היוצאים מאמצעי הצלעות הנגדיות. 4. אותה שאלה ללא הנחייה לבניית עזר. ( לא רבלנטי לבעייה לעיל ). 5. מצא גדלים שווים בטרפז זה.

13. מה ההבדל ?