geogebra in the secondary mathematics classroom: a literature review

GeoGebra in the Secondary Mathematics

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GeoGebra in the Secondary Mathematics Classroom:

A Literature Review

Dan Schellenberg

February, 2009


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GeoGebra in the Secondary Mathematics Classroom

GeoGebra is open source software that combines dynamic geometry software,

computer algebra system and spreadsheet functionality. This paper explores the current

literature on each of these topics in an attempt to determine the advantages and barriers to

using GeoGebra to enhance instruction in a secondary mathematics classroom.

The paper begins with an explanation of why technology is seen as an important part

of mathematics teaching and learning. This is followed by a description of the types of

technologies often used in mathematics teaching and learning. Specific attention is given to

dynamic geometry software, computer algebra systems and spreadsheets. Opportunities and

precautions are identified for each category of software. Finally, the important characteristics

of GeoGebra itself are examined.

Technology in Mathematics Education – Why Bother?

Technology plays an important part in the learning of mathematics. Students must

become familiar with the technological tools utilized in mathematics, whether that be an

abacus or a graphing calculator. Modern technology allows for easier exploration of

mathematics than was previously possible. “The speed of computers and calculators enables

students to produce many examples when exploring mathematical problems. This supports

the observation of patterns, and the making and justification of generalizations” (British

Education Communication Technology Agency, 2004, p. 1).

According to the National Council of Mathematics Teachers position statement

regarding technology, appropriate use of technology allows more students access to

mathematical concepts (National Council of Teachers of Mathematics, 2008). A motivating

factor for increasing the accessibility of mathematics is that mathematics knowledge has

become as an important part of critical citizenship (Adler, Ball, Krainer, Lin, & Novotna,

2005, p. 360). To help students gain the skills that will be useful as citizens, students must


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have the opportunity to use the same technology that is available outside the walls of their

classrooms. (Haapasalo, 2007, p. 9). Using the same technology that is available outside the

classroom allows students to transfer their knowledge into the world as they move beyond

formal education.

Some teachers and school systems remain wary of integrating technology into

mathematics education. The three most common reasons are curriculum scope (convincing

teachers the benefits are worth the change), availability of the technology (open computer

labs, for example) and accessibility of the programs (technology that is easy enough to learn

that the focus remains on the math) (Little, 2008, p. 49). Equipment failure can also be a

major roadblock to the adoption of technology, as teachers will not commit to using

something they cannot rely on in their daily teaching (Cuban, Kirkpatrick, & Peck, 2001, p.

829).

The views of the mathematics teacher greatly influence whether and how technology

will be incorporated into the classroom. According to a recent study, middle-aged and more

experienced teachers were more likely to integrate technology than their younger

counterparts, despite having a more negative attitude regarding technology (Hung & Hsu,

2007, p. 233). This suggests that familiarity with technology might not correlate to increased

technology use in the classroom. A base level of technical skill is required, however, as a

previous study notes that “effective teachers who use ICT [information and communications

technology] are teachers who are confident with ICT” (Bramald, Miller, & Higgins, 2000, p.

5).

Types of Technology Used in Mathematics Education

The technology used in mathematics teaching and learning can be categorized into

two major types, virtual manipulatives and general software tools (Preiner, 2008, p. 26). A

virtual manipulative can be defined as “an interactive, Web-based visual representation of a


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dynamic object that presents opportunities for constructing mathematical knowledge”

(Moyer, Bolyard, & Spikell, 2002). Virtual manipulatives allow a student to interact with the

mathematical situation without any additional skills or training required, though the student’s

exploration is limited by the design of the virtual manipulative. By contrast, general software

tools allow the student to explore any number of mathematical concepts, but require some

training to use.

A variety of general software tools are used in mathematics, including dynamic

geometry software, computer algebra systems and spreadsheets. Barzel

defines general software tools as “tools [that] can be used for a wide set of tasks and

be considered to be general purpose tools that are not useful for only a limited number of

specific tasks – that is the character and as well the most important benefit of general tools”

(2007, p. 81).

The remainder of this literature review will be spent on examining the research on

dynamic geometry software, computer algebra software and spreadsheets. These are the

types of software that GeoGebra seeks to integrate into one coherent tool.

Dynamic Geometry Software

Dynamic Geometry Software (DGS) is the most easily adopted form of general

software tools, as it was explicitly designed for classroom use (Ruthven, 2008, p. 1). DGS is

controlled primarily with the mouse, allowing the basic functionality to be easily learned.

Using DGS, teachers and students are able to quickly and accurately explore geometrical

figures, changing their dimensions while maintaining the mathematical relationships in the

figure. For example, a figure could be drawn showing a perpendicular bisector of a line

segment. As the line segment is dragged, changing its position and length, the perpendicular

bisector automatically moves as well. The important features of DGS are listed by Kokol-

Voljic as:


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- a dynamic modeling of the traditional paper and pencil (blackboard and chalk)

teaching environment through the drag mode

- an option to condense a sequence of commands to form a "new command", a

macro

- an option to visualize the paths of the movements of geometrical objects, a locus

(2007, p. 56)

Figure 1. Screenshot of dynamic geometry software Geometer’s Sketchpad.

DGS has the ability to profoundly change the way we teach proof, one of the most

crucial ideas in mathematics. DGS allows students to instantly create and test their

conjectures, allowing them the freedom to explore geometry and discover patterns. Although

students can easily find patterns using DGS, researchers are advising users of DGS to use

exploration merely as the foundation for deductive proof, since some teachers have begun to

use exploration as a replacement for proof (Hanna, 2000, p. 14). Teachers’ tendency to

replace formal proof with dynamic exploration is seen as a reaction to improper use of formal

proof, such as only proving things students are already convinced is true (Hoyles & Jones,

1998, p. 122).


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Although the power and flexibility of DGS is enticing, we must understand and

acknowledge that changing the medium of teaching geometry will cause important changes in

the way students construct meaning about geometry. Jones lists a number of specific areas in

which DGS has a mediational impact, including:

- The students’ understanding that the order in which objects were created leads to a

hierarchy of functional dependency within a figure.

- The constraint of robustness of a figure under drag becoming linked with using

points of intersection to try to hold the figure together.

- The ‘dynamic’ nature of the software influencing the form of explanation given by

the students. (Jones, 2000, p. 80)

The role of the teacher is shifted when DGS is utilized in the classroom, but the

teacher’s role remains critically important; the teacher’s guidance is crucial as the student

tries to construct meaning from the explorations they are involved in.

The artefact [DGS] is exploited by a double use, with respect to which it functions as

semiotic mediator. On the one hand, meanings emerge from the activity – the learner

uses the artefact in actions aimed at accomplishing a certain task; on the other hand,

the teacher uses the artefact to direct the development of meanings that are

mathematically consistent. (Mariotti, 2000, p. 37)

There are pitfalls inherent with free exploration in DGS, such as students

inadvertently creating a special case by dragging a generic drawing (Sinclair, 2003, p. 291).

This could lead students to incorrect assumptions about mathematical figures, such as

thinking that the Pythagorean theorem holds for all triangles, when in fact it is only true with

right triangles. While these pitfalls should not stop us from using DGS, we must be aware of

them as we begin to incorporate the use of DGS in our classrooms.


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Computer Algebra System

Another type of general software being used in mathematics education is a Computer

Algebra System (CAS). A CAS can be defined as “a piece of software which is capable of

working symbolically as well as numerically. In principle it is a program which does on a

computer the manipulation that has traditionally been done with pencil and paper” (Lawson,

1997, p. 228). CAS are primarily controlled by the keyboard through textual and numerical

input. It is important to note that CAS was created for use by practicing mathematicians, not

for mathematics education (Ruthven, 2008, p. 1). This has caused slower adoption of CAS

into the classroom, and teachers and researchers are still attempting to come to terms with the

effects of using CAS in the classroom. Much of the discussion on CAS in the classroom

revolves around what portions of the curriculum students need to know how to do by hand,

and what portions they can off-load to a computer. The answers to these questions greatly

influence what is taught, and how it is assessed.

Figure 2. Screenshot of factoring with the computer algebra system Mathematica.

Supporters of CAS in education emphasize the ability of students to access higher

level concepts, without having to drudge through tedious algebraic manipulations (Atiyah,

Monaghan, & Pierce, 2004, p. 157). Access to these higher-level concepts allows students to

leave contrived problems behind, giving them a chance to explore real world situations


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instead (Heid & M. T. Edwards, 2001, p. 128). Leigh-Lancaster as gives a fairly

comprehensive list of possible benefits resulting from the use of CAS, including

- the possibility for improved teaching of traditional mathematical topics

- opportunities for new selection and organization of mathematical topics

- access to important mathematical ideas that have previously been to difficult to

teach effectively

- a vehicle for mathematical discovery

- long and complex calculations can be carried out by the technology, enabling

students to concentrate on the conceptual aspects of mathematics

- the technology provides immediate feedback so that students can independently

monitor and verify their ideas

- the need to express mathematical ideas in a form understood by the technology

helps students to clarify their mathematical thinking

- situations and problems can be modeled in more complex and realistic ways

(2003, p. 5)

Despite the perceived benefits of CAS, some researchers find fault with the

underlying assumption that concepts and skills can be separated. While this does not

necessarily lead them to reject the notion of using CAS in the classroom, it does change the

way in which CAS is used. The French researcher Lagrange is among the leading voices in

this camp. For Lagrange, manual skills (or more generally, techniques) are required for the

student to construct meaning. In my own experiences as a classroom teacher, I have found

that if students are shown a technological solution before having a chance to practice a

technique by hand, they may not ever truly understand the nature of what the technology is

doing. For example, if a student is taught to multiply matrices using a graphing calculator,

they may be very proficient at typing the numbers into the calculator, but may have no idea


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about how to interpret the elements of the resulting matrix. I have found it to be much more

effective to introduce matrix multiplication by guiding the students through a word problem

and having the students define matrix multiplication themselves. In the words of Lagrange,

At certain moments a technique can take the form of a skill. This is particularly

the case when a certain ‘routinisation’ is necessary… It is certain that the availability

of new instruments reduces the urgency of this routinisation… But techniques must

not be considered only in their routinised form. The work of constituting techniques in

response to tasks, and of theoretical elaboration on the problems posed by these

techniques remains fundamental to learning. (Lagrange, as cited in Ruthven, 2002, p.

16)

A more pragmatic concern with the use of CAS in the classroom is that students may

be confused by the results given by the CAS (Artigue, 2002, p. 265). For example, when a

secondary mathematics student is taught to factor a difference of cubes, they are taught a

rigid algorithm, which will result in all students achieving the same answer. The CAS may or

may not represent the factored form of the expression in the same manner the student is used

to seeing. A student working by hand would factor as follows:

€

8x 3 − 27( )

= 2x − 3( ) 4x 2 +12x + 9( )

When the CAS feature of GeoGebra performs this factorization, the result is:

€

8x 3 − 27( )

= 12x + 8x 2 +18( ) x − 3/2( )

Although these expressions are in fact equivalent, recognizing that fact may not be trivial for

a student without the ability to perform such tasks mentally or by hand. This sort of situation

can lead to students being unable to determine if the answers given by the CAS are

reasonable (Waits & Demana, 1998).


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The power of CAS will fundamentally change mathematics learning and assessment.

“Whereas graphics calculators, for many teachers, slotted easily into the curriculum and

enhanced their teaching with little threat, CAS demands a more thorough response” (Kendal,

Stacey, & Pierce, 2005, p. 105). Paper and pencil algorithms (techniques, in the terminology

of Lagrange) must be examined individually to determine if they contribute understanding for

the student. Algorithms that do not contribute to a student’s understanding should be

performed with technology (Waits & Demana, 1998).

Spreadsheets

Another type of general mathematics software is the spreadsheet. A spreadsheet is

simply an array of rows and columns that allow calculations to be quickly performed. More

recently, “the basic paradigm of an array of rows-and-columns with automatic update and

display of results has been extended with libraries of mathematical and statistical functions

[and] versatile graphing and charting facilities” (Baker & Sugden, 2003, p. 19). This

extension of the functionality of spreadsheets has allowed spreadsheets to become useful

when teaching a variety of mathematical topics.

Figure 3. Screenshot of spreadsheet software Microsoft Excel.

Spreadsheets are useful tools for exploring a large range of mathematical topics. One

of the simplest uses of the spreadsheet at a secondary level occurs when teaching statistics.


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Although spreadsheets may not be suited to deal with in-depth mathematical statistics, they

can be very useful for introductory level statistics, as would be seen in a secondary

mathematics curriculum (Nash, 2008, p. 4603). Performing simple calculations on statistical

data becomes a trivial with a spreadsheet. Spreadsheets can also be used in teaching such

diverse mathematical topics as inequalities (Abramovich, 2005), limits in calculus

(Abramovich & Levin, 1994) and the concept of infinity (Abramovich & Norton, 2000).

Studies have shown increased student understanding of statistical graphs as a result of

using spreadsheet explorations in statistics (Wu & YoongWong, 2007). As the use of

spreadsheets in the classroom becomes more complex, however, new issues arise. Unless

they are taught otherwise, students tend to create spreadsheets that are not reliable when cell

values are changed. Students must be taught how to create spreadsheets that can solve

general problems, instead of only being useful for only one specific case (Niess, 2006, p.

199).

GeoGebra’s Defining Features

GeoGebra is software that attempts to combine DGS, CAS and spreadsheets into one

application. “On the one hand, GeoGebra is a dynamic geometry system in which you work

with points, vectors, segments, lines, and conic sections. On the other hand, equations and

coordinates can be entered directly” (Sangwin, 2007, p. 36). Every object in GeoGebra has a

representation in both the algebra window, as well as the geometry window. The user can

adjust the value of the object through either representation, allowing them to either drag the

geometric figure using the mouse, or change the symbolic representation using the keyboard.

While GeoGebra attempts to combine aspects of DGS, CAS and spreadsheets, small

annoyances reveal that this combination is done imperfectly. One such annoyance is the need

to learn arcane syntax in order to show dynamic text or calculations when using the DGS

feature of the software. For example, to create dynamic text that updates as the location of a


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point changes, one would have to enter something like ”a = ” + a + ”cm”. By

contrast, the ease of use of Geometer’s Sketchpad (another popular DGS) when doing the

same task allows students to show dynamic calculations without having to learn any syntax at

all. Some of these annoyances may be a result of GeoGebra still being relatively new

software, having been initially created by Markus Hohenwarter in 2001 as part of his

Master’s thesis in mathematics education (Preiner, 2008, p. 36).

Figure 4. Screenshot of finding the area of a triangle with GeoGebra.

The CAS abilities of GeoGebra are currently quite limited, though in the pre-release

version of GeoGebra the CAS aspects of the software have been dramatically enhanced. In a

recent post to the GeoGebra CAS mailing list, Markus Hohenwarter explained that

development version (pre-release version) of GeoGebra now includes a full featured CAS

system by incorporating an open source CAS into GeoGebra (M. Hohenwarter, 2008).


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The development version of GeoGebra also incorporates spreadsheet functionality,

though certain limitations exist. Currently, only 100 rows of data can be viewed in the

spreadsheet mode. Spreadsheet cell ranges must be typed when performing calculations, not

selected with the mouse. For example, one can type Mean[A1:A9], but you cannot type

Mean[] and then highlight which cells you want to calculate the median of. Many advanced

features of popular spreadsheet applications such as Excel are not available in GeoGebra,

though most functions that would be used at a high school level are already available.

GeoGebra is an open source application, which gives GeoGebra both moral and

pragmatic benefits over proprietary software. GeoGebra is freely available to schools and

students, eliminating the cost factor for schools with limited budgets. Students are able to use

the application at home on their private computers with no site licensing concerns (M.

Hohenwarter, J. Hohenwarter, Kreis, & Lavicza, 2008, p. 2). Markus Hohenwarter, the

creator of GeoGebra, has stated the reason GeoGebra is released as a free, open source

application is that he believes education should be free (Edwards & Jones, 2006). A side

benefit of being open source is that a development community has grown around the project,

which has allowed GeoGebra to be translated into many languages (39 different languages as

of GeoGebra 3.0 in March 2008), making it accessible to many more students and educators.

GeoGebra is written in Java, which allows it to run on virtually any platform (Mac,

Windows, Linux). Being written in Java also allows GeoGebra to easily export files as

dynamic webpages. This allows for simple creation of online math explorations, often called

math applets. The ease with which GeoGebra sketches can be shared online has lead to a

community of teachers using GeoGebra freely sharing their resources with one another online

at http://www.geogebra.org/en/wiki.

While an online wiki of resources is useful for early adopters of GeoGebra, the

majority of teachers will not begin using GeoGebra on the basis of resources being available


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online. Simply providing technology to teachers does not lead to successful integration of

that technology in their teaching (Cuban et al., 2001). To address this concern, an

International GeoGebra Institute has been created, with the purpose of providing structured

training and support to teachers interested in using GeoGebra (M. Hohenwarter & Lavicza,

2007). Various chapters have been set up across the world, allowing those interested in

learning GeoGebra to have local support.

Conclusion

Incorporating technology into mathematics teaching and learning allows greater

access to mathematical concepts. General mathematics software allows students to explore

any number of mathematical situations, but require students to learn the software first.

Dynamic Geometry Software is quite easy to use, allowing students and teachers to test

conjectures by exploring geometrical figures. The manner in which proof is taught in

mathematics has been greatly affected by the introduction of DGS to the classroom.

Computer Algebra Systems are able to perform much of the symbolic manipulation that

students do by hand. Educators must determine which algorithms can be delegated to a CAS

and which must be done by hand. Spreadsheets are particularly useful when teaching

statistics, but can also be used to teach a wider variety of mathematical topics.

GeoGebra combines DGS and CAS into one application (development/future versions

also include a spreadsheet). GeoGebra is open source software, which allows anyone to

download the software and use it for free. As a Java application, it can run on any platform.

To help teachers learn how to incorporate GeoGebra into their classrooms, an International

GeoGebra Institute has been created to provide structured training and support.


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geogebra in the secondary mathematics classroom: a literature review

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