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 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

Geometric model

Conjecture

Theorem

FormalizeConstruct

Investigate

Test

Deductive proof

Analytical solution

Generalizati

on

I – YOU – WE

I – individual work of the single student

You – cooperation with a partner

We – communication in the whole class

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

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 textgraphics 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

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

Lesson 1Addition 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 is true

Situation 4Construct the given geometric model

Construct the sum of 3 vectors Test it

0 OCOBOA

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

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

GEONExTGeometrical Place of points

DGPolygons.Tesselation

GEONExTPolygons.Tessalation

Real-life problem. Box

Thank you!

ObDiMat

Lehren und Lernen mit dynamischer Mathematik

Обучение с динамической математикой

Teaching and Learning with dynamic mathematics