ABSTRACT:Â This paper presents new trends in technology and learning through Geogebra, which could be especially important for the future development of e-learning for College mathematics. Also, contribution of this paper is presentation of methodological frames on several specifics examples for teaching mathematics at the college level on interactive and creative way

Palve, Madhukar M.

English

International Research Journal of Multidisciplinary Studies (IRJMS)

  
VOLUME-II, ISSUE-I                                                                                              ISSN (Online): 2454-8499             
 INTERNATIONAL RESEARCH JOURNAL OF MULTIDISCIPLINARY STUDIES  
1st January, 2016   Page 1 
Website: www.irjms.in                                       Email: irjms2015@gmail.com, irjms.in@gmail.com 
Learning Concepts of Mathematics using GeoGebra at the College Level 
 
Dr. Madhukar M. Palve, 
Department of Mathematics, 
Prof. Ramkrishna More ACS College, Akurdi, Pune, 
India – 411044 
 
ABSTRACT: 
 This paper presents new trends in technology and learning through Geogebra, which could be 
especially important for the future development of e-learning for College mathematics. Also, 
contribution of this paper is presentation of methodological frames on several specifics examples 
for teaching mathematics at the college level on interactive and creative way. 
KEYWORDS: GeoGebra, college mathematics, creative environment. 
 
GeoGebra:  
At the moment, GeoGebra (from Geometry 
and Algebra) is one of the most innovative, 
open -code math software (GNU General 
Public License) which can be freely 
downloaded from www.geogebra.org. 
GeoGebra works on a wide spectrum of 
operating system platforms which have Java 
virtual machine installed on. Markus 
Hohenwarter created free, open-source 
dynamic mathematics software   GeoGebra,   
which   is   used   for   both   teaching   and   
learning mathematics from middle school 
through college to the University level (see 
Hohenwarter & Preiner, 2007). GeoGebra 
offers geometry, algebra and calculus 
features in a fully connected, compacted and 
easy-to-use software environment. In other 
words, this tool extends the concepts of 
dynamic geometry to the fields of algebra 
and mathematical analysis. Designed  
specifically  for  educational  purposes,  
GeoGebra  can  help students grasp 
experimental, problem-oriented and 
research-oriented learning of  mathematics,  
both  in  the   classroom  and  at  home.  
Students can simultaneously use a computer 
algebra system and an interactive geometric 
system; by doing this, they can increase 
their cognitive abilities in the best way. 
   GeoGebra is dynamic geometry software 
that supports constructions with points, 
lines and all conic sections. It also 
provides typical features for a Computer 
Algebra System such as finding important 
points of functions (roots, local extrema and 
inflection points of functions), direct input 
of equations and coordinates, finding 
derivates and integrals of the entered 
functions. That is the reason why GeoGebra 
is a good choice for multiple presentations 
of mathematical objects. 
   The basic idea of GeoGebra’s interface is 
to provide two presentations of each 
mathematical object in its algebra and 
graphics windows. If you change an object 
in one of these windows, its presentation in 
the other one will be immediately updated. 
Computer algebra systems (such as 
Mathematica, Maple) and dynamic 
geometry software (such as Geometer’s 
Sketchpad, Cabri Geometry, and so on, e.g.) 
are powerful technological tools for teaching 
mathematics. Numerous research results 
suggest that these software packages can be 
used to encourage discovery, 
experimentation and visualization in 
traditional teaching of mathematics. 
However, researches suggest that, for the 
majority of teachers, the main problem is 
how to provide the technology necessary for 
the successful integration of technology into 
teaching (Ruthven & Hennessy, 2004). 
     So, the suggested solution for applying 
technology in the college math teaching and 
learning is the software packet GeoGebra..  
  
VOLUME-II, ISSUE-I                                                                                              ISSN (Online): 2454-8499             
 INTERNATIONAL RESEARCH JOURNAL OF MULTIDISCIPLINARY STUDIES  
1st January, 2016   Page 2 
Website: www.irjms.in                                       Email: irjms2015@gmail.com, irjms.in@gmail.com 
 
The advantages of using GeoGebra are 
1) In  comparison  to  a  graph  
calculator,  GeoGebra  is  more  
user-friendly. GeoGebra offers easy-
to-  use interface, multilingual 
menus, commands and  help. 
2) GeoGebra   encourages   students`   
projects   in   mathematics,   
multiple presentations and   
            experimental and guided discovery 
learning. 
3) Students can personalize their own 
creations through the adaptation of 
interface (e.g. font size, language, 
quality of graphics, color, 
coordinates, line thickness, line style 
and other features). 
4) GeoGebra was created to help 
students gain a better 
understanding of mathematics. 
Students can manipulate variables 
easily by simply dragging “free” 
objects around the plane of 
drawing, or by using sliders. 
Students can generate changes using 
a technique of manipulating free 
objects, and then they can learn how 
the dependent objects will be 
affected. In this way, students   have   
the opportunity   to   solve   
problems   by investigating 
mathematical relations dynamically. 
5)    Cooperative  learning  is  the  right  
context  for  a  mathematics  course 
Lecturing should be   
     replaced by a task-  oriented  
interactive  classroom. The  primary  
role  of  teaching   is  not  to 
     lecture, explain, or otherwise attempt 
to "transfer" mathematical knowledge, 
but  to  create   
     situations  for  students  that  will  
foster  their  making  the necessary 
mental constructions. In  
    that sense, GeoGebra provides a 
good opportunity for  cooperative 
learning, i.e. cooperative  
   problem solving in small  groups,  
or whole class interactive  teaching,  or  
individual/group  
     student presentations. 
6) The algebra input allows the user to 
generate new objects or to modify 
those already existing, by the  
command line. The worksheet files 
can easily be published as Web pages. 
7) GeoGebra   stimulates   teachers   to   
use   and   assess   technology   in: 
visualization  of  mathematics;  
investigations  in  mathematics;  
interactive mathematics  classes  on  
site  or  at  a  distance;  mathematics 
and  its applications, etc. 
 
The deficiencies of using GeoGebra are: 
• Students without previous programming 
experience   will hardly enter algebraic 
commands in the 
   input box. Although the basic commands 
are not difficult to learn, students may feel 
embarrassed or 
    quite at a loss of what to do. 
• Some   methodological   approaches   
(e.g.   Independent   exploring   and 
experimenting) cannot be  
    appropriate for many students. 
• In  a  technical  sense,  GeoGebra  does  
not  have  an  in-built  support  for 
animation. So, including  
  the modules for animating in GeoGebra 
should become an important technical 
element for future  
   versions. 
• Future extensions of the software 
GeoGebra will surely include 
moresymbolic features of  
   computer algebra systems which will 
further increase possible  complex  
applications  in  the 
   mathematical ana lyses,  and  3D 
extensions. 
  
VOLUME-II, ISSUE-I                                                                                              ISSN (Online): 2454-8499             
 INTERNATIONAL RESEARCH JOURNAL OF MULTIDISCIPLINARY STUDIES  
1st January, 2016   Page 3 
Website: www.irjms.in                                       Email: irjms2015@gmail.com, irjms.in@gmail.com 
• Limited research on the impact of 
GeoGebra on teaching and learning of 
mathematics. 
GeoGebra has been rapidly gaining 
popularity among teachers and researchers 
around the world, because it is easy-to-use 
dynamic mathematics software that 
combines many aspects of different 
mathematical packages. In addition, because 
of its open-source nature, an extensive user 
community has developed around it. 
Math Visualization and Exploration by 
Using GeoGebra 
    The visualization that is possible with 
today’s dynamic software enables the 
student to see and explore mathematical 
relations and concepts that were difficult  to  
“show”  in past  prior  to  technology.  The  
most  meta-analysis  of research  in  the  
area  of  instructional  technology  in  
education  show  that students who use 
technology in their learning had positive 
gains in learning outcomes over students 
who learned without technology  
      Although my teaching and learning  
methods are still traditional (textbook, 
lecture notes, lectures, assignments, final 
exam) my plan is to try to make some 
changes in the way of work in order to 
make my teaching courses  more 
effect ive to improve the learning 
outcomes of my students. The goal is to use 
GeoGebra to provide an environment for 
active exploration of mathematical 
structures through multiple representations, 
or to show students some aspects of the 
mathematics that are not possible with pen 
and paper. 
Interactive Examples in Teaching and 
Learning Linear Algebra with  
      GeoGebra:  
Using examples and questions given below, 
the teacher will guide students to fully 
understanding of the meaning and nature of 
solutions to linear systems of equations. 
These examples illustrate low-order systems 
to assist students in personalizing important 
concepts about systems of linear equations 
by "discovery method". The teacher should 
define the terms "consistent, inconsistent, 
homogeneous", "one solution, no solution, 
infinite solutions", and precise definitions of 
"linear equation, linear system, nonlinear 
system," and "solution". Also, the teacher 
should strongly encourage geometric 
visualizations through the lesson plan 
(Dikovic, 2007). 
 
Example 1. Consider families of equations which take the form 
 
                                                                            ax + by = c. ….                                 (1) 
 
For different values of a, b, c , the graphs of these equations are lines in a two dimensional 
coordinate system. Let the students choose any real numbers  a1 , b1 , c1 , a2 , b2 , c2  to define a 
system of two equations in two unknowns: 
 
                                                                                                                   a1 x + b1 y = c1 ,                             (2) 
                                                                              a2 x + b2 y = c2 .    
These graphs can easily be generated using 
GeoGebra. GeoGebra allows modifying 
these equations dynamically by moving the 
points or the sliders. In this way, students 
are able to investigate all possible situations 
including the nteresting special cases of no 
solution and infinitely many solutions                            
 
 
  
VOLUME-II, ISSUE-I                                                                                              ISSN (Online): 2454-8499             
 INTERNATIONAL RESEARCH JOURNAL OF MULTIDISCIPLINARY STUDIES  
1st January, 2016   Page 4 
Website: www.irjms.in                                       Email: irjms2015@gmail.com, irjms.in@gmail.com 
Q1. Can you change the numbers of the second equation so that the resulting system has 
• No solution? (see figure 2) 
• Exactly one solution? (see figure 1) 
• Infinitely many solutions? 
• If you do get one (infinite) solution for your final answer, is this system consistent   or   
inconsistent?   Would   the   equations   be   dependent   or independent? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Q2. Student can set a1 , b1 , a2 , b2 to values such that 
                                                     a1 ⋅ b2 − a2 ⋅ b1 = 0      ………..               (3) 
 
 
and  consider  the  next  questions:  How  are  the  two  lines  positioned  with respect to each 
other? How many solutions the system has? What is the relationship between the slopes of the 
two lines? 
Q3. Student can set all 6 constants to values such that 
 
a1    =  b1   =  c1)              (4) 
a2       b2      c2                                              
 
and explain how many solutions the system has in that case. 
Q4. Ask the students to randomly select 
numbers of a third equation and to graph it 
along with the first two. Normally such a 
random selection should generate an 
inconsistent system with a large probability, 
whose lines bound a triangular region. Does 
this new system have a solution and 
whether they have a solution or not? 
      Also, GeoGebra brings to students and 
teachers the opportunity to individualize 
learning - to generate illustrative examples, 
to follow interesting topics to the desired 
  
VOLUME-II, ISSUE-I                                                                                              ISSN (Online): 2454-8499             
 INTERNATIONAL RESEARCH JOURNAL OF MULTIDISCIPLINARY STUDIES  
1st January, 2016   Page 5 
Website: www.irjms.in                                       Email: irjms2015@gmail.com, irjms.in@gmail.com 
depth, to choose their own problems and 
appropriate tools for solving them. The 
students can concentrate on ideas instead of 
trying to  get  the  arithmetic  right  in  the  
solution  of  the  some  linear  system. 
Additionally in examples of solving linear 
system, GeoGebra can stimulate the 
students' geometric intuition through 
interesting visualizations in 2D. 
  Conclusion:
In this paper, I highlighted some 
opportunities and examples on how 
GeoGebra can be used in classrooms to 
explore some basic concepts in linear 
algebra and calculus It was showed that 
GeoGebra has many possibilities to help 
students to get an intuitive feeling and to 
visualize adequate math process. The use of 
this software’s tools allows students to 
explore a wider range of function types, and 
provides students to make the connections 
between symbolic and visual 
representations. 
 
 
 
References: 
 van Joolingen, Wouter (1999): Cognitive tools for discovery learning. International 
 Journal of Artificial Intelligence in Education, 10, p. 385-397. 
 www.bsrim.org.uk@theauthor-49 
 NCTM   (2000).  Principles     and     Standards     for     School     Mathematics 
 http://standards.nctm.org 
 Herceg  Dj.,  Herceg  D.(2007),  Numerics  with  GeoGebra  in  high  school,  First Central- and 
Eastern European Conference on Computer Algebra- and Dynamic Geometry Systems in     
Mathematics Education, University of Pécs, Pollack Mihály Faculty of Engineering, Hungary. 
 Dikovich Lj.(2007), „An Interactive Learning and Teaching of Linear Algebra by Web     
Technologies: Some Examples“, Journal the Teaching of mathematics, Publisher: The 
Mathematical Society of Serbia, Beograd, ISSN: 1451-4966, Issue: X_2, Pages: 109 – 116. 
 Wurnig O. (2008), Some Problem Solving examples of Multiple solutions using cas and dgs,    
Proceedings of the Discussing Group 9 : Promoting Creativity for All Students   in   
Mathematics Education,   The   11th  International Congress on  Mathematical Education, 
Monterrey, Mexico..  
 http://www.geogebra.org 
 http://www.learning-theories.com/discovery-learning-bruner.html 
 
 


Learning Concepts of Mathematics using GeoGebra at the College Level

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Learning Concepts of Mathematics using GeoGebra at the College Level

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International Research Journal of Multidisciplinary Studies (IRJMS)