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