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

Theoretical aspects Practical 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 GEONE X T 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 Geometric model Conjecture Theorem Formalize Construct Investigate Test Deductive proof Analytical solution Generalization

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

Synergy 1 IYOUWE Consider a problem + Formalize problem  Construct Geometric Model + Test Geometric Model +   Investigate + Make a conjecture +  Test the conjecture    Formulate final result = Theorem  Deliver a deductive proof or analytical solution +   Try to generalize   - discussion between 2 pupils  check each other  - discussion with the whole class PROBLEM-SOLVING STRATEGY

Synergy 2 IYOUWE (Real-life) Problem ConsiderDiscussFormalize Geometric Model (GM) Construct GM Test GM Investigate GM Conjecture Make it Test conjecture Discuss. Formulate final result Theorem Deliver a deductive proof or analytical solution Test Conclusion. Try to Generalize PROBLEM-SOLVING STRATEGY

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 pedagogical- educational 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 mathematics applets (GEONExT) collection 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 IYouWe Step-by- Step problem solving strategy first lesson situation 1situation 2situation 3 second lesson situation 4situation 5situation 6

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

Lesson 1 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 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 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

Lesson 2 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 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.TesselationTesselation

GEONExT Polygons.Tessalation

Real-life problem. BoxBox

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