geometric construction in real-life problem solving valentyna pikalova manfred j. bauch ukraine...
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Geometric construction in real-life problem solving
Valentyna Pikalova
Manfred J. Bauch
Ukraine
Germany
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Theoretical aspects
Practical realization
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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
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Ukrainian side
German side
Joint work
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Ukrainian side
Students' worksheets for secondary school geometry course
Dynamic learning environments with DG
Implementation at Ukrainian schools Intel “Teach to the Future”
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German side
I –You – We concept
Dynamic learning environments with GEONEXT
Implementation at German schools
Evaluation and feedback
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Joint work
Synergy of two educational models
Dynamic learning environments
Joint publications
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Step-by-step (real-life) problem-solving tasks strategy
(Real-life) problem
Geometric model
Conjecture
Theorem
FormalizeConstruct
Investigate
Test
Deductive proof
Analytical solution
Generalizati
on
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I – YOU – WE
I – individual work of the single student
You – cooperation with a partner
We – communication in the whole class
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Synergy 1I YOU WE
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
PR
OB
LEM
-SO
LVIN
G S
TR
AT
EG
Y
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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
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Practical realization
Consider different types of explorative learning environments
Design a learning environment
Implementation in German and Ukrainian schools
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Dynamic learning environments
sequence of HTML pages including textgraphics dynamic mathematics applets (GEONExT)
collection of the dynamic models in DG
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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
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Example1 . Vectors
Lesson1 Addition of Vectors. The Parallelogram Rule
Lesson 2
Solving Strategies with Vectors
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Pedagogical Model
I – You – We
I You We
Step-by-Step
problem solving strategy
first lesson
situation 1 situation 2 situation 3
second lesson
situation 4 situation 5 situation 6
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Lesson 1Addition of Vectors. The Parallelogram Rule
Situation 1 Construct the
sum of 2 vectors using the parallelogram rule.
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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.
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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.
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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 4Construct the given geometric model
Construct the sum of 3 vectors Test it
0 OCOBOA
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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
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Lesson 2 Problem Solving Strategies with Vectors
Situation 6 Final conclusions *Related problems
4 vectors 6 vectors
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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
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GEONExTGeometrical Place of points
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DGPolygons.Tesselation
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GEONExTPolygons.Tessalation
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Real-life problem. Box
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Thank you!
ObDiMat
Lehren und Lernen mit dynamischer Mathematik
Обучение с динамической математикой
Teaching and Learning with dynamic mathematics