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Geometric construction in real-life problem solving Valentyna Pikalova Ukraine Manfred J. Bauch Germany Theoretical Practical aspects realization 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 Ukrainian side German side Joint work Ukrainian side Students' worksheets for secondary school geometry course Dynamic learning environments with DG Implementation at Ukrainian schools Intel “Teach to the Future” German side I –You – We concept Dynamic learning environments with GEONEXT Implementation at German schools Evaluation and feedback Joint work Synergy of two educational models Dynamic learning environments Joint publications Step-by-step (real-life) problem-solving tasks strategy (Real-life) problem Theorem Geometric model Conjecture I – YOU – WE I – individual work of the single student You – cooperation with a partner We – communication in the whole class - discussion between 2 pupils Synergy 1 check each other - discussion with the whole class PROBLEM-SOLVING STRATEGY I Consider a problem YOU + Formalize problem Construct Geometric Model WE Investigate + + + Make a conjecture + + Test Geometric Model Test the conjecture Formulate final result = Theorem Deliver a deductive proof or analytical solution Try to generalize Practical realization The comparative study of the curricula in Ukraine and Germany Selection of topics for explorative learning environments based on a combination of the two pedagogicaleducational models Collect the set of tasks for each topic Practical realization Consider different types of explorative learning environments Design a learning environment Implementation in German and Ukrainian schools Dynamic learning environments sequence of HTML pages including text graphics dynamic collection mathematics applets (GEONExT) of the dynamic models in DG 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 Example1 . Vectors Lesson1 Addition of Vectors. The Parallelogram Rule Lesson 2 Solving Strategies with Vectors Pedagogical Model I – You – We Step-byStep problem solving strategy I You We first lesson situation 1 situation 2 situation 3 second lesson situation 4 situation 5 situation 6 Lesson 1 Addition of Vectors. The Parallelogram Rule Situation 1 Construct the sum of 2 vectors using the parallelogram rule. Lesson 1 Addition 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. Lesson 1 Addition 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. Lesson 2 Problem Solving Strategies with Vectors Problem: Investigate the position of point O in any given triangle ABC for which the expression OA OB OC 0 is true Situation 4 Construct the given geometric model Construct the sum of 3 vectors Test it Lesson 2 Problem 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 Lesson 2 Problem Solving Strategies with Vectors Situation 6 Final conclusions *Related problems 4 vectors 6 vectors DG Geometrical 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 GEONExT Geometrical Place of points DG Polygons.Tesselation GEONExT Polygons.Tessalation Real-life problem. Box Thank you! ObDiMat Lehren und Lernen mit dynamischer Mathematik Обучение с динамической математикой Teaching and Learning with dynamic mathematics