Systems of computer algebra
and dynamic geometry
as tools of mathematical investigation
Roman Hašek
University of South Bohemia, Czech Republic
Faculty of Education
University of South Bohemia
České Budějovice, Czech Republic
http://www.pf.jcu.cz
Objective
To share experience in the use of computers in mathematics
teacher training courses.
To present selected examples that we practice with students of
mathematics teaching and with in-service mathematics teachers.
Outline
- Main idea
- Utilization of CAS and DGS in mathematics teacher-training
courses at the Faculty of Education USB.
- Selected good practice examples
Computer programs
as tools of mathematical investigation
The use of computer programs in such a way that reveals
the mathematical essence of a problem or phenomena
compared to its mere computation which reveals only
a result.
- Exploration of the given phenomena
- Formulation of a hypothesis
- Verification or proving of the hypothesis
Example: Length of the graph of a function y=f(x)
between two points
wxMaxima <http://andrejv.github.com/wxmaxima>
GeoGebra 4.2 <http://www.geogebra.org>
Promotion of the use of computers
in mathematics education at FE USB
A regular biennial national “Conference on the use of computers in
mathematics education”, since 2003.
Czech localization of Derive 6 in 2004. Organizing of courses for
math teachers on the use of Derive.
Establishment of the first Czech GeoGebra Institute in 2011.
Czech localization of new versions of GeoGebra.
Utilization of CAS and DGS in math teacher
training
Two aims
Technological support of the university mathematical courses.
Preparation of students for practice, i.e. for the effective use of
computers in mathematics teaching at basic and secondary
schools.
Combination of both commercial and free software:
Maple, wxMaxima, CoCoa, Sage, GeoGebra, Cabri II+, Cabri 3D
Good practice examples
Maple
wxMaxima
The joint use of CAS (wxMaxima) and DGS (GeoGebra)
Maple
http://www.maplesoft.com
Commercial CAS
Long tradition at the USB.
Used in regular higher-level mathematics courses
Unsuitable for the use in basic and secondary schools
because of its high price and no Czech localization.
Maple: “Introduction to Quadrics”, university textbook
[Hašek, R., Pech, P., Kvadratické plochy a jejich reprezentace v programu Maple,
České Budějovice: Pedagogická fakulta JU v Č. B., 2010. ISBN 978-80-7394-271-7.]
Maple: Smart documents for financial education
[Petrášková, V., Hašek, R., Multimediální pomůcka pro výuku předmětu Úvod do financí,
WWW stránka pro podporu výuky, http://www.pf.jcu.cz/stru/katedry/m/uf, 2009.]
wxMaxima
http://andrejv.github.com/wxmaxima
Free downloadable open source CAS
Czech localization
Used in regular freshmen courses of linear algebra and in
courses of the use of computers in mathematics education
Seems to be the most perspective CAS for utilization in
schools (i.e. the most convenient substitute for Derive)
wxMaxima: Freshmen course of Linear algebra
Joint use of CAS and DGS
The joint use of spreadsheet, dynamic geometry and symbolic
algebra forms a strong tool for the investigation of various
mathematical phenomena
wxMaxima plus GeoGebra: Logarithmic spiral
Photo by Edward Weston
wxMaxima plus GeoGebra: Tractrix
wxMaxima plus Cabri 3D: Rotational hyperboloid
wxMaxima plus GeoGebra: Nephroid
GeoGebra 4.2 (CAS)
GeoGebra 4.2 (CAS)
Example
An envelope of a one-parameter family of lines
L ( x, y , t ) = 0
∂L( x, y, t )
=0
∂t
Astroid
Astroid
∆L( x, y, t ) i L( x, y, t i ) − L( x, y, t i −1 )
=
=0
∆t
∆t
Astroid
∆L( x, y, t ) i ∂L( x, y, t )
=
=0
lim
∆t
∂t
∆t →0
Thank you for your attention
[email protected]