Kyoto Workshop 2012: Research Community ofComputer Algebra System, July4-6, 2012,RIMS, Japan

The Pedagogical Implementation of GeoGebra as Technology Based Mathematics Learning

MBAITIGA Zacharie

Okinawa National College ofTechnology,Department ofMedia Information Engineering, Japan

E-Mail: zacharie@okinawa-ct.acjp

Abstract

Mathematical proof, the art to understandthe mathematics process, or the way eachargument, definition and theorem are used isabout to switch off its light among manystudents. In order to keep mathematical proofalive in the heart of students, we conducted asurvey to find out what does mathematicalproof really mean for college students.

Keywords: Mathematics, proof college students,

Mathematicalproof

I. Introduction

No one can doubt that the world we are livingnow compare to the world that our ancestorslived hundred and thousands years ago becomesthe world that Technology is playing animportant role in everyday aspects of ourlifestyle. As an example, (1) telephone andfacsimile is more used than airmail as a fast wayto pass the information, (2) we can fmd the placeor location we want to go easily using GPS, (3)

in real time we can exchange ideas with $\theta$iendsusing chat mail or we can from house transfermoney to bank without to going to the bank viainternet money transfer system and so forth tomention only a few. All those systems would notbe possible without the support of mathematics.Mathematics is the most important field that allsciences need to prosper. For example theapplication of statistics in agronomy andmeteorology. In agronomy experimentation, tocompare a number of fertihzers, it was thoughtnecessary to devote only a single plot to eachtreatment and determine yields in order to arriveat valid conclusions conceming the relativevalues of the treatments. However, theagronomist found soon that the yields of a seriesof plots treated alike differed greatly amongthemselves even when soil conditions appeareduniform and experimental conditions werecarefully designed to reduce errors in harvesting.For this reason it becomes necessary to findsome means for determining whether differencesin yields were due to differences in treatment orto uncontrollable factors that also contribute tothe variability of the yields. Statistical methodswere applied, and their value in the scientific

investigation of agronomic practices was soonproved.Next, the modern science of meteorology is to agreat degree dependent on statistical methods forits existence. For example, the methods that giveweather forecasting the accuracy it has todayhave been developed using modern samplesurvey techniques. The mathematics educationneeds some kind of support. Thus, Technology.In this study the pedagogical implementation ofgeoGebra as technology based mathematicsleaming is used. Because geogebra is a greatdynamic mathematics software for teaching andlearning mathematics as it covers all ofthe basicsof dynamic mathematics software. It offersinteractive geometry, algebra, tables, graphicsand calculus including statistics tools in oneenvironment. It is a great support indeed forlinking mathematics concept.But there is one question that we should make

it clear is that:

Is geogebra easy to use? Response: Yes and No

Yes, for mathematics instructors or teachersbecause they know the mathematics and cantherefore easily understand the ideas andlogic behind the tools.

Yes, for students who have been instructedor under training on how to use the toolsand understand the mathematics and logicbehind it. They can use it in solvingproblems and for investigatingmathematical relationships.

No, for students, especially those who havenot leamed the basic of graphing, equations,and geometric relationships, the use ofgeogebra is limited to manipulating ready-made geogebra applets. Since, geogebraapplets are easy to use but most of the timeif they do not know the mathematics behindthe construction or can’t construct it bythemselves, then the learning of themathematics may be superficial. And it iswhat we should avoid.

数理解析研究所講究録

第 1930巻 2015年 88-92 88

Kyoto Workshop 2012: Research Community of Computer Algebra System, July4-6, 2012,RIMS, Japan

II. What GeoGebra is Capable of

There are uncountable numbers ofmathematics learning based software on themarket. Each of this mathematics software hastheir strong points as well as weak points and theevaluation of each of this software dependent onthe users. What makes geogebra different fromthese software is the flexibility of its userinterface that can be adapted to the need of anystudents’ level. Starting from elementary pupil touniversity graduated students. The geogebra userinterface consists of graphics view and algebraview, where graphs constructions can be doneusing points, vectors, segments, lines and conicsections as well as function while changing themdynamically afterwards. On the top of that,equations and coordinates can be entered directlyas well as finding the derivatives and integrals offunctions. It offers also commands such as rootor vertex respectively. One of the most featuresofthe geogebra is that it provides geometry toolswith the mouse to create geometric constructionsand the graphical representation of all objects isdisplayed in the graphics view respectively.

$m$. Basic Implementation of GeoGebra inMathematics Class

I have been teaching mathematics for more thanten years now starting from algebra, geometry,probability statistics, Laplace transforms, Fourierseries and discrete mathematics to mention a few.During these ten years of teaching experience inmathematics, I always advised my students not touse software for studying mathematics, as it willmake them become lazy and will slow theirability of thinking. But during the $15^{th}$ AsianTechmology Conference in Mathematics that washosted by the University of Malaya, KualaLumpur, Malaysia in 2012 that I first heard aboutgeogebra. Since then I started to leam and now itbecomes one of my tools of teachingmathematics to my students. Below are someexamples of problems that my students wereasked to solve using geogebra.

$\bullet$ Problem-l:Consider a connected graph G. The distancebetween vertices and vertex $v$ in $G$, written$G(u, v)$ is the length of the shortest pathbetween $u$ and $v$. The diameter of $G$, written$\dim(G)$ is the maximum distance betweenany two points in G.

$\bullet$ Question-l:Draw the graph $G$ and find the distancebetween $u$ and $v$ as well as the diameter ofthe graph $G$ using geogebra.

$\bullet$ Answer-l:To solve this problem the class was divided into

two groups and each group was asked to proposedifferent solution to solve the problem and theiranswers if correct should be the same. To thisinstruction two options were given by thestudents.

$i$. Option-l:First group proposed the use of grid,where seven points $(A,$ $B,$ $C,$ $D,$ $E,$ $F$, andG) on the graphics view were placed. Allseven points have the same distance fromone another and all connected exceptpoints $D$ and G. From the coordinates ofthese points on the algebra view(dependent objects) the graph distancebetween vertices $B$ and vertex $F$ , written $d$

$(B, F)$ is calculated and found to be 2.Thatis to say $d(B, F)=2$ . Then the diameter ofthe graph, written diam (G) is alsocalculated using the same coordinateswhich is 3. That is to say diam (G) $=3$

respectively. Fig.1 and Fig.2 are theoutputs ofthe d(B,F) and diam (G).

Fig. 1 Proposed distance output for groupl.

$\prime\alpha$

Diam$(G)-=3$.

$\aleph^{P_{*}}*$$K\cdot/!\backslash$

Fig.2 Proposed diameter output for groupl.

89

Kyoto Workshop 2012: Research Community of Computer Algebra System, July4-6, 2012,RIMS, Japan

$u$ Option-2:Second group proposed the use of circles,where seven circles $(A,$ $B,$ $C,$ $D,$ $E,$ $F$, andG) on the graphics view were placed. Allseven circles have the same distance fromone another and two as radius. As for thegrpoupl all seven circles are connectedexcept circles $D$ and G. Next from theequations of the circles on the algebraview (dependent objects) the distancebetween vertices $B$ and vertex $E$ written $d$

$(B, E)$ is calculated and has found to be 2.Hence $d(B, E)=2$ . The graph diameter isalso calculated using the circle equationsdisplayed on the algebra view which wasfound to be 3. That is to say diam $(G)=3$

Fig.3 and Fig.4 are the outputs of thed(B,F) and diam (G) respectively

Fig.3 Proposed distance output for group2.

Fig.4 Proposed diameter output for group2.

$\bullet$ Problem-2:The following function. $f$ : $x arrow\frac{x3}{(x-1)^{2}}$

Has as:

$D=\}-\infty,$ $1[\cup]1,$ $+\infty[as$ domain

While its sign and variation table is

$\bullet$ Question-2:Using the above information draw the graphof this function using both geogebra andusually graph draw method. Here again theclass is divided into two groups and eachgroup should report the time they spent todraw the graph. The reason of doing this kindof exercise is that some students no familiarin using software for mathematics learningare still hesitating to use geogebra. To letthem know that using geogebra they can savea lot of time

$\bullet$ Answer-2:The following is the graph drawn by students of$\Psi^{opu1}$ using geogebra following with the timespent for the both of the groups. For spaceconstrains the graph drawn by group 2 is notreported.

Fig.5 Graph drawn by groupl using geogebra

$\bullet$ Time spent by group-l: 90 seconds$\bullet$ Time spent by group-2: 20 minutes

90

Kyoto Workshop 2012: Research Community of Computer Algebra System, July 4-6, 2012,RIMS, Japan

$\bullet$ Problem-3:A graph is said to be complete if everyvertex in $G$ is connected to every othervertex in G. Thus a complete graph mustbe connected. The complete graph with $n$

vertices is denoted by $K_{n}.$

$\bullet$ Question-3:Draw $K_{5}$ and $K_{6}$ using geogebra. Which oneof the two graphs is Eulerian graph and tellwhy?

$\bullet$ Answer-3:To draw the graph of $K_{5}$ and $K_{6}$ all studentsused the grid and placed on it five pints $(A,$ $B,$

$C,$ $D$, E$)$ for graph $K_{5}$ and six points $(A,$ $B,$ $C,$

$D,$ $E$, F) for graph K6. Then connected allvertices are connected to each other. Beloware the output

Fig.6 Graph $K_{5}$

Fig.7 Graph $K_{6}$

$K5is$ the Eulerian graph as each vertex has evendegree. That is to say:

$\deg(A)=\deg(B)=\deg(c)=\deg(D)=\deg(E)=4$

While K\’o is not due to its odd vertex degree.

$\bullet$ Problem-4:From the following two signs andvariation tables using Geogebra

$\bullet$ Question-4:Find the appropriate functions and drawtheir graphs.

$\bullet$ Answer-4:The two functions that were found bystudents are as follows:

$i$ . Functionl: $g(x)=\ln x$

ii. Function2: $f(x)=ex$

And their graphs are plotted in Fig.8

Fig.8 shows the graphs of the two functionsabove.

91

Kyoto Workshop 2012: Research Community ofComputer Algebra System, July 4-6, 2012,RIMS, Japan

Next students changed the graphs’s size from theobject’s properties and added some colors tomake the graphs beautiful (Fig.9).

Fig.9 dynamic change of functions fromGeogebra its properties window

$\bullet$ Problem-5:Evaluate and demonstrate the Riemannsun for

$f(x)=2x^{3}+8x^{2}+4x-2$

Taking the sample points to be under thecurve where

$a=-3,$

$b=-1,$

$n=2$

And upper the curve where

$a=1,$

$b=3,$

$n=4$

$\bullet$ Question-5:Sketch a graph of the function and theRiemann rectangle and use geogebra todetermine the areas.

$\bullet$ Answer-3:The lower sum of the function $f(x)$ isbetween-3 and-l with 8 rectangles. Theupper sum of the function $f$ is between 1and 3 with 8 rectangles as well. On the other

hand, the Area for LowerSum:8.06 and Area forUpper Sum: 135.25 respectively.

Fig. 10 Riemann rectangle and its areas.

IV Conclusion

The outcomes of this discussion are as follows:The Dynamism of Geogebra can be used tocustom training to use mathematical conceptsand methods in addressing everyday situations orto solve practical problems. Training motivationto study mathematics as an area relevant tosocial and professional life. Expressing thegeometric representation of the concepts relatedto geometry, function and so on. Development ofteaching strategies based on specific skills inschool curriculum.

References

htto://www.theffeedictionaIv.com/mathematical+

proof

http://en.wikioedia.org/wiki/Mathematical proof

http://www.math.uconn.edu/\sim hurley/math315/pr

oofgoldberger.pdf

MBAITIGA $Z$ (2008). Proof of BernhardRiemann‘s Functional Equation using GammaFunction, Joumal of Mathematics and Statistics4(3):181-185, 2009, Science Publications, NewYork.MBAITIGA Z (2009). Why College orUniversity Students hate Proofs in Mathematics.Journal of Mathematics and Statistics 5(1):31-41,2009, Science Publications, New York.MBAITIGA $Z$ (2010). Proof of AnalyticExtension Theorem for Zeta Function using Abel

Transformation and Euler Product, Journal ofMathematics and Statistics 6(3):294-299, 2010,Science Publications, New York.

92