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Geometric construction in real-life problem solving

Geometric construction in real-life problem solving. Valentyna Pikalova. Ukraine. Manfred J. Bauch. Germany. Theoretical aspects Practical realization. Theoretical aspects. Synergy of the two educational strategies Content and structure of a dynamic learning environment

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Geometric construction in real-life problem solving

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  1. Geometric construction in real-life problem solving Valentyna Pikalova Ukraine Manfred J. Bauch Germany

  2. Theoretical aspects • Practical realization

  3. Theoretical aspects • Synergy of the two educational strategies • Content and structure of a dynamic learning environment • Different teaching and learning traditions • Interdisciplinary aspects • Dynamic mathematics software

  4. Ukrainian side • German side • Joint work

  5. Ukrainian side • Students' worksheets for secondary school geometry course • Dynamic learning environments with DG • Implementation at Ukrainian schools • Intel “Teach to the Future”

  6. German side • I –You – We concept • Dynamic learning environments with GEONEXT • Implementation at German schools • Evaluation and feedback

  7. Joint work • Synergy of two educational models • Dynamic learning environments • Joint publications

  8. Step-by-step (real-life) problem-solving tasks strategy (Real-life) problem Formalize Construct Generalization Geometric model Theorem Investigate Test Deductive proof Analytical solution Conjecture

  9. I – YOU – WE • I – individual work of the single student • You – cooperation with a partner • We – communication in the whole class

  10. - discussion between 2 pupils check each other  - discussion with the whole class Synergy 1 PROBLEM-SOLVING STRATEGY

  11. Practical realization • The comparative study of the curricula in Ukraine and Germany • Selection of topics for explorative learning environments based on a combination ofthe two pedagogical-educational models • Collect the set of tasks for each topic

  12. Practical realization • Consider different types of explorative learning environments • Design a learning environment • Implementation in German and Ukrainian schools

  13. Dynamic learning environments • sequence of HTML pages including • text • graphics • dynamic mathematics applets (GEONExT) • collection of the dynamic models in DG

  14. Types of explorative learning environments • Getting practical skills • for working in dynamic geometry packages • in constructing geometrical models • Gaining research skills through problem solving • Gaining new knowledge through investigation

  15. Example1 . Vectors Lesson1 Addition of Vectors. The Parallelogram Rule Lesson 2 Solving Strategies with Vectors

  16. Pedagogical Model

  17. Lesson 1Addition of Vectors. The Parallelogram Rule • Situation 1 • Construct the sum of 2 vectors using the parallelogram rule.

  18. Lesson 1Addition of Vectors. The Parallelogram Rule • Situation 2.1 • Investigate the sum of 2 vectors • Make a conjecture about it properties. • *Situation 2.2 • Repeat the same steps for 3 vectors.

  19. Lesson 1Addition of Vectors. The Parallelogram Rule • Situation 3 • Conclusions • *Problem discussion – more general problem • construct and investigate the sum of 4, 5, … vectors; • create and save new tools the Sum of 2, 3, … vectors by using macroconstructions.

  20. Lesson 2 Problem Solving Strategies with Vectors Problem: Investigate the position of point O in any given triangle ABC for which the expression is true • Situation 4 • Construct the given geometric model • Construct the sum of 3 vectors • Test it

  21. Lesson 2Problem Solving Strategies with Vectors • Situation 5.1 • Investigate the geometric model • Investigate the position of the point O • Make a conjecture • Check it in many cases • *Situation 5.2 • Deliver deductive proof

  22. Lesson 2 Problem Solving Strategies with Vectors • Situation 6 • Final conclusions • *Related problems • 4 vectors • 6 vectors

  23. DGGeometrical Place of points • Problem • Construct two segments AB and CD on the plane. Point E and F are points on the segments AB and CD respectively. Conjecture about the set of midpoints of the segment EF when dragging points E and F along AB and CD respectively

  24. GEONExTGeometrical Place of points

  25. DGPolygons.Tesselation

  26. GEONExTPolygons.Tessalation

  27. Real-life problem. Box

  28. Thank you! ObDiMat Lehren und Lernen mit dynamischer Mathematik Обучение с динамической математикой Teaching and Learning with dynamic mathematics

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